Approximate Equilibria in Generalized Colonel Blotto and Generalized

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Both firms’ objective is to make allocation decisions to maximize their aggregate shares from all markets. The Colonel Blotto game can also be used to model problems in military logistics, see e.g., Gross (1950); Gross & Wagner (1950); in politics (where political parties distribute their budgets to compete over voters), see e.g., Kovenock & Roberson (2012a); Laslier & Picard (2002); Myerson (1993); Roberson (2006); in cybersecurity (where attack/defense resources are distributed over sensitive targets), see e.g., Chia (2012); Schwartz et al. (2014); in online advertising (where marketing campaigns allocate ads broadcasting time over web users), see e.g., Masucci & Silva (2014, 2015); in telecommunication (where network service providers distribute and lease their spectrum to users), see e.g., Hajimirsaadeghi & Mandayam (2017). In many of these applications, it is often the case that the number of battlefields under consideration is very large.1 The main focus of the literature on the Colonel Blotto game is its Nash equilibria (henceforth, simply referred to as equilibria). Completely characterizing and computing an equilibrium of the Colonel Blotto game, however, is a notoriously difficult problem. A standard approach used in the literature is to first find candidate equilibrium marginal distributions corresponding to players’ allocations toward battlefields—they are called the optimal univariate distributions of the game—; and then construct an n-variate joint distribution (of these univariate distributions) whose realizations satisfy the budget constraints. Constructing this n-variate distribution, with this particular coupling constraint, is the main challenge in studying equilibria of the Colonel Blotto game. So far, there are results available only under restricting assumptions on the game’s parameters such as assuming players symmetry or battlefields homogeneity, and assuming that the battlefields evaluations are identical for both players—see our detailed discussion on the related work below. In several applications of the Colonel Blotto game (e.g., in the example of advertising competitions), such assumptions are not satisfied in practice; studying more general variants of the Colonel Blotto game is therefore of prominent importance. In this paper, we consider the most general version of the Colonel Blotto game; where the evaluations of the battlefields’ values can be heterogeneous across battlefields and different between the two players, and the players’ budgets can be asymmetric—we refer to it as the Generalized Colonel Blotto game (hereinafter, GCB game). The GCB game was first formulated and studied by Kovenock & Roberson (2020).2 Yet, the characterization of equilibria in the GCB game remains an open question—even the existence of an equilibrium has not been proved or disproved in general cases; and the key question remains open: how to play strategically in the GCB game to obtain good guarantees on payoffs? In this work, we also study the Nash equilibrium but we take a different angle: instead of looking for an exact equilibrium, we focus on approximate equilibria, i.e., strategies such that the extra payoff a player can gain by unilaterally deviating from them is bounded by a small term (relative to the scale of the players’ total payoffs). Our first contribution is to identify a class of approximate equilibria of the GCB game, called the independently uniform strategies, henceforth denoted as IU strategies.3 These strategies ensure that the budget constraints are satisfied. Moreover, we construct the IU strategies such that, although their corresponding marginals do not form a set of optimal univariate distributions of the game, they approximate well-known ones. Based on this, we characterize the approximation error of this solution according to the games’ parameters and show that it is negligible when the number of battlefields is sufficiently large (it quickly decreases as the number of battlefields increases). The IU strategies are also simple and efficiently computable even in large-scale problems; which provides desired scalability in practice.

1For example, in advertising competitions between pharmacy companies, they deploy their medical representatives’ effort to present/advertise new products to persuade (a large number of) doctors to prescribe their drugs (see Fu & Iyer (2019) for more details). There, the doctors correspond to the markets over which the firms compete. 2Note that we consider a tie-breaking rule that is more general than that of Kovenock & Roberson (2020) (see Section 2.1). 3We explain the name IU in Section 4.1 and give the formal definition of approximate equilibria in Section 4. 2 Going beyond the Colonel Blotto game, we then consider the Lottery Blotto game, which relaxes the winner-takes-all assumption. In the Lottery Blotto game, each player only gains a fraction of their value in each battlefield; alternatively, we can interpret it as each player winning a battlefields’ value with a certain probability depending on the players’ allocations on that battlefield (which can be non-zero even for the player with smaller allocation). For instance, in some advertising competitions, the sales in each market may be shared between the two firms based on their advertising expenditures, but with no firm having total dominance; these situations are discussed in Duffy & Matros (2015a); Friedman (1958); Kovenock & Arjona (2019) as motivating examples for different variants of the Lottery Blotto game. The Lottery Blotto game model may also prove useful in other areas such as political contests for voters’ attention, research and development activities, or radio-wave transmission with noises. In this paper, we specifically consider the Lottery Blotto game that results from replacing the winner- determination-rule in the GCB game by a (generic) contest success function—we refer to it as the Generalized Lottery Blotto game (henceforth, GLB game). Contest success functions, studied profoundly in the rent- seeking literature (see e.g., Corchón (2007); Skaperdas (1996)), are functions that take the players’ allocations as inputs and output the probability of winning a battlefield. The definition of a contest success functions that we adopt (see Section 2.2) includes the winner-takes-all rule as a special case, so that the GCB game is a particular case of the GLB game. Similar to the GCB game, the equilibrium characterization is an open question for the GLB game in its general setting, with exceptions limited to a few special cases—see our detailed discussion of the related work below. Our second main contribution is to prove that the IU strategy is also an approximate equilibrium of the GLB game with an approximation error that decreases quickly as the number of battlefields increases and the corresponding contest success functions converge pointwise to that of the GCB game. As an illustration, we consider the family of GLB games with ratio-form contest success functions—an important class that is often studied in the literature. We analyze the IU strategies with two of the most well-known cases of ratio-form contest success functions: the power form and the logit form; in these cases, we obtain more precise results on the convergence of the approximation error.

1.1. Related Work

As stated above, the GCB game was introduced by Kovenock & Roberson (2020), who provide a set of distributions that are optimal univariate distributions of the GCB game. They then indicate a sufficient condition4 for these optimal univariate distributions to constitute an equilibrium that only covers a restricted range of games; and they also show a necessary condition without which there is no equilibrium satisfying such a set of univariate distributions. On the other hand, most works in the literature focus on the more restricted constant-sum Colonel Blotto game, where both players assign the same value to each battlefield.5 Even in this simpler version, complete equilibrium characterization results are available only under restrictive assumptions. When players have symmetric budgets, equilibria are constructed by Borel & Ville (1938) in the constant-sum Colonel Blotto game involving three battlefields and by Gross (1950); Gross & Wagner (1950) in the constant-sum Colonel Blotto game containing any number of battlefields (see also Laslier (2002); Laslier & Picard (2002); Thomas (2017) for a modern presentation of this solution). For the constant-sum Colonel Blotto game with asymmetric budgets, equilibria characterization remains an open question in general; the exceptions are the following restricted cases: the games with only two battlefields (Macdonell & Mastronardi (2015)), the games with any number of battlefields but homoge- neous values (Roberson (2006)), and the games where there exists a sufficient number of battlefields of each

4The set of battlefields are partitioned such that two battlefields are in the same partition if they have the same (normalized) values; the sufficient condition on the attainability of equilibria requires a sufficient number of battlefields in each partition. 5It is trivial to see that in this case, the summation of players’ payoffs always equals the summation of battlefields’ values. We sometimes refer to the GCB game (without further assumptions) as the non-constant-sum Colonel Blotto game to distinguish it from the constant-sum version. 3 possible value (Schwartz et al. (2014)). Note that our results for the GCB game can be trivially adapted to the constant-sum version, i.e., the IU strategy is also an approximate equilibrium in this version. Moreover, for the constant-sum Colonel Blotto game, we show the additional result that the IU strategy is also an approximate max-min strategy (with the same approximation error). To the best of our knowledge, only very few works in the Colonel Blotto game literature mention approximate results, and in very different settings compared to the GCB game we consider. Weinstein (2005) briefly discusses approximate equilibria (with a fixed approximation error) of the game with only 3 battlefields in a variant of the constant-sum Colonel Blotto game with the majority objective.6 Also for the majority objective, Behnezhad et al. (2019) propose a polynomial time approximation scheme for (u,p)-maximin strategies.7 A strategy construction similar to the IU strategies can be found in Vu et al. (2018) for the (constant-sum) discrete Colonel Blotto game (i.e., where the budgets and every allocation are required to be integers) with asymmetric budgets and heterogeneous battlefields. Due to the discrete condition, their analysis has essential differences to our work.8 A few works have considered other extensions and related versions of the Colonel Blotto game, though with a significantly different flavor than ours. In particular, Boix-Adseràet al. (2020) study a Colonel Blotto game between more than two players with symmetric budgets; Kovenock & Roberson (2020); Myerson (1993) completely find equilibria of the General Lotto game—a relaxed version of the Colonel Blotto game where budget constraints are only required to hold in expectation; Ahmadinejad et al. (2016); Behnezhad et al. (2018, 2019, 2017); Hart (2008); Hortala-Vallve & Llorente-Saguer (2012); Vu et al. (2018) focus on the discrete Colonel Blotto game; Powell (2009); Rinott et al. (2012) consider sequential Colonel Blotto games and Adamo & Matros (2009); Kovenock & Roberson (2011); Paarporn et al. (2019) study Colonel Blotto games with incomplete information. Note that the GCB game can also be considered as an instance of larger classes such as multiple-battlefields conflicts (see e.g. Kovenock & Roberson (2012b)) and multi-item contests (see e.g., Fu & Wu (2019); Kvasov (2007); Robson (2005)) where battlefields’ outcomes and players’ payoffs are defined by more general functions. The idea of a game formulation similar to the GLB game (with generally defined contest success functions) appeared previously in Kovenock & Roberson (2012b), where it was not, however, explicitly defined. More importantly, Kovenock & Roberson (2012b) only study with details a particular case where players’ gains in each battlefield follow the Tullock contest success function (termed after Tullock (1980) and also called the lottery contest success function).9 This Lottery Blotto game with the Tullock function is also the instance that attracted the most attention in the literature: Friedman (1958) investigates the pure equilibrium of its constant-sum version, Robson (2005) generalizes these results in the case where the Tullock function is modified with a multiplicative constant representing advantages of a certain player, Kovenock & Arjona (2019) characterize a best-response mechanism of the non-constant-sum version, Duffy & Matros (2015b); Kim & Kim (2019) prove the existence and partially characterize the equilibria in the majority version and Kim et al. (2018) provide similar results in a version with an infinite number of players. Note that the terms “lottery Blotto game” or “lottery Colonel Blotto game” are sometimes used in the literature (e.g., by Kovenock & Arjona (2019); Kovenock & Roberson (2012b)) to refer to this particular game variant that is essentially simpler than the GLB that we study. Coincidentally, the term “Generalized Lottery

6In the game with majority objective, a player wins the whole game and gains a positive payoff only if the aggregate values (or the number) of battlefields won by her exceed a given threshold, e.g. 50% (see also Kvasov (2007); Kovenock & Roberson (2012a); Laslier (2005)). 7Strategies guaranteeing that a player gains a payoff at least u with a probability at least p. 8Particularly, their asymptotic results involve a double limits of the number of battlefields and the ratio of players’ budgets; moreover, the convergence of the players’ payoffs in their work does not have the difficulties of continuous allocation encountered in our work. 9 That is the GLB game where two players, called A and B, commonly evaluate each battlefield i with a value wi; if players A B A A B B A B allocates xi ,xi to battlefield i then Player A gains xi wi/(xi + xi ) and Player B gains xi wi/(xi + xi ) from this battlefield. 4 Blotto” is also used by Osório (2013) to refer to a game version with a slight generalization of the Tullock function (also studied by Shubik & Weber (1981)); however, only numerically computed approximate-results of the equilibrium are proposed and no tractable close-form solution is provided in general cases where battlefields’ values are asymmetric across players. Note that Lottery Blotto games with Tullock function and its generalizations are included in the class of Lottery Blotto games with ratio-form contest success functions that we study in this work. Finally, a sequential Blotto-type game with generic contest success functions also appears in Klumpp et al. (2019), but it concerns the majority rule; moreover, results are only obtained under a sufficiently-concave assumption on the contest success functions.

1.2. Roadmap and Notation

The remainder of this paper is organized as follows. Section 2 gives the formal definitions of the GCB and GLB game. Although the GLB game model is essentially more general, we first focus on the GCB game as it is a more classical game. Section 3 provides preliminary results needed in the rest of the analysis. In Section 4, we define the IU strategy and show that any IU strategy is an approximate equilibrium of the GCB game. In Section 5, we study how the IU strategy is also an approximate equilibrium of the GLB game. We conclude in Section 6. Finally, detailed proofs of all lemmas and theorems are given in Appendix. Throughout the paper, we use bold symbols (e.g., x) to denote vectors and subscript indices to denote its elements (e.g., x = (x1, x2,...,xn)). The notation [n] denotes the set {1, 2,...,n}, for any n ∈ N\{0}. We n often use the letter P to denote a player and use −P to indicate her opponent in the games. R≥0 denotes 1 the set of all n-tuples whose elements are non-negative (R≥0 := R≥0). We denote the Euler’s number by e. For any random variable X, we use FX and EX to denote its corresponding cumulative density function (abbreviated by CDF) and its expectation respectively; for an event E, we denote the probability that it happens by P(E). We use the asymptotic notation O with its standard definition and O˜ as a variant of O where logarithmic terms are ignored (see Appendix A for formal definitions). We sum up the notations used in this work in Table A.2 (Appendix A).

2. Games Formulation

In this section, we deﬁne the two games that are our main focus: in Section 2.1, we introduce the Generalized Colonel Blotto game (henceforth, GCB game) and its restricted version—the constant-sum Colonel Blotto game; in Section 2.2, we present the Generalized Lottery Blotto game (henceforth, GLB game), as an extension of the GCB game.

2.1. The Generalized Colonel Blotto Game

We consider the following one-shot, complete information game between two players A and B. Each player has a fixed amount of resources (called the budgets), denoted XA and XB, respectively. Without loss of generality, we assume that 0 0, corresponding to the values at which Player A and Player B respectively assess this battlefield. A pure strategy of player xP P Rn n P P ∈ {A, B} is a vector = xi i∈[n] ∈ ≥0 that satisfies the budget constraint i=1 xi ≤ XP . In each battlefield i, when player P allocates strictly more than her opponent, she gains completely her embedded P A B P A values wi while the opponent gains 0. In case of a tie, i.e., if xi = xi , then Player A receives αwi and B Player B receives (1 − α)wi , where α ∈ [0, 1] is a fixed parameter. Each player’s payoff is the summation of values she gains from all battlefields; formally, for any pure strategy profile (xA, xB), the payoffs of players xA xB n A A B xA xB n B A B A and B are ΠA( , )= i=1 wi · βA xi , xi and ΠB ( , )= i=1 wi · βB xi , xi respectively; P P 5 here, βA and βB (henceforth, we called them the Blotto functions) are functions defined as follows:

1 , if x>y 1 , if y > x β (x, y)= α , if x = y and β (x, y)= 1 − α , if y = x , for all x, y ∈ R . (2.1) A  B  ≥0  0 , if x

To lighten the notation, we only include the subscript n (the number of battlefields) and XA, XB (the values XA,XB of players’ budgets) in the notation GCBn and omit the other parameters; in particular, the values α and A B wi , wi for i ∈ [n]. In this game, a mixed strategy is a joint distribution on the allocations of all battlefields, such that any drawn pure strategy of a player is an n-tuple that satisfies her budget constraint. We reuse the A B A B notations ΠA s ,s and ΠB s ,s to denote the payoffs of players A and B when they play the mixed strategies sA and sB, respectively. Note that the definition of GCBXA,XB above allows asymmetry in players’ n budgets and heterogeneity in battlefields values; moreover, it allows battlefield values to differ between the two players. Furthermore, the defined payoff functions can be understood as if we randomly break the tie (if it happens) such that Player A wins battlefield i with probability α while Player B wins it with probability (1 − α). This includes all the classical tie-breaking rules considered in the literature; for instance, the rule of giving the whole value to Player B used by Roberson (2006); Schwartz et al. (2014) corresponds to α = 0; the 50-50 rule used by Ahmadinejad et al. (2016); Behnezhad et al. (2017); Kovenock & Roberson (2020) corresponds to α =1/2. A A In this paper, we also often work with the normalized values of the battlefields defined as vi := wi /WA B B n A n B and vi := wi /WB , where WA := j=1 wj and WB := j=1 wj for i ∈ [n]. We trivially observe that vP ∈ [0, 1] for all i and that n vP = 1. Most of our analysis relies on an additional assumption that i j=1Pj P the battlefields’ values are bounded away from zero and infinity; particularly, we work with the following P condition—namely, Assumption (A0)—:

∃w, w>¯ 0 : w ≤ wP ≤ w,¯ ∀i ∈ [n], ∀P ∈{A, B}. (Assumption (A0)) ¯ ¯ i Assumption (A0) is a fairly mild assumption that is satisfied in most of (if not all) practical applications. As a direct consequence, the normalized values satisfy w w¯ ¯ ≤ vP ≤ , ∀i ∈ [n], ∀P ∈{A, B}. (2.2) nw¯ i nw ¯ Finally, we note that most works in the literature (the only exception, to our knowledge, being the work of Kovenock & Roberson (2020)) focus only on the constant-sum Colonel Blotto game where players have XA,XB the same evaluations on battlefields’ values. The game GCBn given in Definition 2.1 is more general; XA,XB hence all our results for GCBn can be straightforwardly applied to this constant-sum version as well. However, for the purpose of comparing with the literature and because we can show stronger results in this special case, it is useful to also formally define the constant-sum game variant as follows.

XA,XB Deﬁnition 2.2. A constant-sum Colonel Blotto game, denoted by Const-CBn , is a game that has XA,XB A B the same formulation as the game GCBn but with the additional condition that wi = wi , ∀i ∈ [n].

XA,XB As a trivial corollary of this additional condition, in Const-CBn , players also have common normalized A B valuation on battleﬁelds, i.e., vi =vi for all i ∈ [n] and the players’ maximum payoﬀs are equal, i.e., WA = WB. 6 2.2. Contest Success Functions and the Generalized Lottery Blotto Game

In this section, we present the Generalized Lottery Blotto game (GLB game) that extends the model of the GCB game. This new game is based on the notion of contest success functions (henceforth, CSFs), that we introduce below before defining the game model. Contest success functions are functions that quantify the winning probability in contests (also called rent-seeking competitions) where several players compete for a single prize by exerting resources/efforts. CSFs can be defined for any number of players (see e.g., a general definition by Skaperdas (1996)), but in this work, we focus only on the case of two players. R2 R R2 R Definition 2.3. ζA : ≥0 → and ζB : ≥0 → is a pair of contest success functions (CSFs) if and only if the following two conditions are satisfied:

(C1) ζA(x, y), ζB (x, y) ≥ 0 and ζA(x, y)+ ζB(x, y)=1, ∀x, y ≥ 0.

(C2) ζA(x, y) (resp. ζB(x, y)) is non-decreasing in x (resp. in y) and non-increasing in y (resp. in x).

Intuitively, the function ζA (resp. ζB) maps any pair of players’ invested resources to the probability that Player A (resp. Player B) wins the prize. Condition (C1) indicates that the outputs of any pair of the CSFs always satisfy the condition of a probability distribution. On the other hand, Condition (C2) states that a player’s winning probability increases (or at least stays the same) when she increases her effort and decreases (or at least stays the same) when her opponent increases her effort. We note that Definition 2.3 allows a more general definitions of the CSFs (in two-player cases) compared to the definition given by Clark & Riis (1998); Hirshleifer (1989); Skaperdas (1996) that contains other assumptions.10 While many of the CSFs considered in the literature are continuous functions, we do not include continuity in Definition 2.3 to keep XA,XB the generality. Importantly, the Blotto functions βA,βB of the game GCBn (i.e., the winner-takes-all rule) satisfy Conditions (C1) and (C2), hence βA,βB are CSFs. Besides these functions, some examples of other CSFs considered in the literature are:

(a) ζA(x, y)= x/(x + y) and ζB (x, y)= y/(x + y), proposed by Tullock (1980);

1 (b) ζA(x, y) = max min 2 +C(x−y), 1 , 0 and ζB(x, y)=1 − ζA(x, y), proposed by Che & Gale (2000), where C > 0 is a ﬁxed parameter; 1 y−x 1 x−y (c) ζA(x, y)= 2 − 2y if x ≤ y and ζA(x, y)= 2 + 2x if x ≥ y; and ζB (x, y)=1 − ζA(x, y), proposed by Alcalde & Dahm (2007).

Building on the notion of CSFs and the GCB game, we now define a new game model based on the XA,XB following idea: in a game GCBn , we view each battlefield as a contest between players where the prize is the battlefield’s value and players’ effort correspond to their allocations; by doing this, each pair of CSFs defines an instance of a game where the probability of winning a battlefield follows them accordingly.

Definition 2.4. Let ζ = (ζA, ζB) be a pair of CSFs, a Generalized Lottery Blotto game with n bat- XA,XB tlefields and budgets XA,XB, denoted GLBn (ζ), is the game with the same players A and B and the XA,XB xA xB same strategy sets as in GCBn ; but where payoffs are given, for any pure strategy profile ( , ), by n n ζ xA xB A A B ζ xA xB B A B Π ( , )= wi · ζA xi , xi and Π ( , )= wi · ζB xi , xi . A i=1 B i=1 X X 10 For example, Skaperdas (1996) defines ζA, ζB with an axiom of anonymity; they also require that any player who puts a strictly positive amount of resources has a strictly positive probability of winning the prize; Clark & Riis (1998) considers the CSFs additionally satisfying the Choice Axiom. These are technical conditions needed for proving their results and we omit them here lest they unnecessarily limit our scope of study.

7 The GLB game model is more flexible than that of the GCB game, as it allows choosing the CSFs that define the players’ payoffs for each specific practical situation. Throughout the paper, to refer to a GCB XA,XB A B XA,XB game GCBn that has the same parameters n,XA,XB, wi , wi , ∀i ∈ [n] as a GLB game GLBn , XA,XB XA,XB we call GCBn the corresponding game of GLBn and vice versa. Intuitively, the players’ payoffs in XA,XB the GLBn game can be seen as the expected payoffs in the corresponding GCB game with respect to the following random process determining the winner in any battlefield i: Player A wins with probability A B A B A B ζA(xi , xi ) and Player B wins with probability ζB(xi , xi ) if they allocate xi and xi respectively. Similar XA,XB XA,XB to the game GCBn , players’ payoffs in the GLBn game are also monotonic with respect to the XA,XB allocations in a battlefield (due to Condition (C2)). Note that, to derive our results for GLBn , we will also use Assumption (A0) introduced above. Besides the GLB game with generally defined CSFs, we additionally consider the games with CSFs that belong to a special class called the ratio-form CSFs. These are the CSFs that are studied the most profoundly in the literature. We will use the games with these ratio-form CSFs to illustrate the results obtained in the GLB game. R2 R Definition 2.5. Functions ζA, ζB : ≥0 → ≥0 are called ratio-form CSFs if they have the form:

η(x) κ(y) ζ (x, y)= and ζ (x, y)= , A η(x)+ κ(y) B η(x)+ κ(y) where η,κ : R≥0 → R are non-negative functions such that ζA and ζB satisfy Conditions (C1) and (C2) of Deﬁnition 2.3.

Two classical ratio-form CSFs in the literature (see e.g., Corchón & Dahm (2010); Hillman & Riley (1989)) are the power form where η(z)= κ(z)= zR, ∀z ≥ 0 and the logit form where η(z)= κ(z)= eRz, ∀z ≥ 0, where R > 0 is a parameter chosen a priori. These functions yield the sharing 50-50 tie-breaking rule, i.e., ζA(x, y) = ζB (x, y)=1/2 if x = y. We define in Table 1 the generalized versions of these ratio-form CSFs—namely, µR and νR—using the parameter α ∈ (0, 1) that leads to the general tie-breaking rule as XA,XB 11 R R R R in the GCB game GCBn . It is trivial to verify that both pairs (µA,µB) and (νA ,νB ) satisfy the Conditions (C1) and (C2). Henceforth, we use the terms power and logit form to indicate the CSFs µR and νR respectively and use the term Lottery Blotto game with ratio-form CSF to commonly address the games XA,XB R XA,XB R GLBn (µ ) and GLBn (ν ) (i.e., the games formulated by replacing ζ in Definition 2.4 by either µR or νR). An important remark is that both the power and logit form CSFs converge pointwise toward the Blotto functions βA,βB as R tends to infinity (see Section 5.2 for more details). This convergence can be observed in Figure 1 that illustrates several instances of the ratio-form CSFs in comparison with the Blotto functions.

Table 1: Power and logit form CSFs with generalized tie-breaking rule (α ∈ (0, 1)).

Name Notation If x2 + y2 > 0 If x = y =0

R R R R R R αxR R (1−α)y µA(x, y)= α Power form µ :=(µA,µB) µA(x, y)= αxR+(1−α)yR ; µB(x, y)= αxR+(1−α)yR R µB(x, y)=1 − α yR R R R R R αexR R (1−α)e νA (x, y)= α Logit form ν :=(νA ,νB ) νA (x, y)= αexR+(1−α)eyR ; νB (x, y)= αexR+(1−α)eyR R νB (x, y)=1 − α

11When α = 1/2, the CSFs µR and νR match the classical power form and logit form CSFs. Note that we exclude the cases where α =0 or α = 1 since these are the trivial cases: in the corresponding GLB game, a player, say P ∈ {A, B}, always has the payoﬀ WP while player −P ’s payoﬀ is always zero regardless how they allocate their resources.

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hneot,OUDs) (henceforth, ) n X rit seeg,Kvnc Roberson & Kovenock e.g., (see traints ( ,By ta.(96;Hlmn&Riley & Hillman (1996); al. et Baye ., snwt h ltofunctions. Blotto the with ison γ ( , A lstigof setting al v ) v ntanst ehl nexpectation in hold be to onstraints i D uhta t elztosalways realizations its that such UDs ,X A i v B sn oainsmlrt htof that to similar notation a using ) i A sn(00) neulbimi to is equilibrium an (2020)), rson B 2 taeueu o u nlssof analyses our for useful are at sapoc ocntuta ap- an construct to approach is . ,frany for ), noe usin However, question. open an X h qiiracharacteriza- equilibria the ions. A dprmtr,bsdon based parameters, ed 10 = GCB γ i α , ∈ ) X n 12 F A (0 0 = n i A ,X fpaesi well- is players of i , F , i l-a auctions all-pay from ∞ B . from 6. i oeta if that Note . B ,w deﬁne we ), aesupports have F F i P i − , P ∀ i . (3.1) ∈ [ n ], (3.1) Let us denote by Sn the set containing all positive solutions of Equation (3.1) corresponding to the XA,XB XA,XB 13 game GCBn (or GLBn ). Based on the intermediate value theorem, the following lemma is proved by Kovenock & Roberson (2020).

XA,XB XA,XB Lemma 3.1. For any game GCBn (or GLBn ), Equation (3.1) has at least one positive solution; (3.1) i.e., Sn 6= ∅.

Equation (3.1) may have more than one solution and solving it (i.e., ﬁnding algebraic expressions of its 14 ∗ (3.1) solutions) can be done in O(n ln(n)) time. Now, corresponding to each positive solution γ ∈ Sn , we 15 ∗ ∗ deﬁne two constants, namely λA and λB as follows:

2 (γ∗)2 vB 1 λ∗ := i + vA, (3.2) A 2X vA 2X i B ∗ i B ∗ i∈ΩXA(γ ) i/∈ΩXA(γ ) 2 1 1 vA λ∗ := vB + i . (3.3) B 2X i 2(γ∗)2X vB A ∗ A ∗ i i∈ΩXA(γ ) i/∈ΩXA(γ ) ∗ ∗ ∗ Note importantly that we have γ = λA/λB (see Lemma A1 in Appendix A for a proof). We now use these ∗ ∗ constants λA and λB to deﬁne several important distributions.

XA,XB XA,XB ∗ (3.1) Deﬁnition 3.2. Given a game GCBn (or GLBn ), for any γ ∈ Sn and the corresponding ∗ ∗ 16 constants λA, λB , we deﬁne the following random variables and distributions, for each i ∈ [n]:

A B ∗ vi vi S W (a) If i ∈ ΩA(γ ) (i.e., ∗ > ∗ ), we deﬁne Aγ∗,i and Bγ∗,i as the random variables whose distributions are λA λB

∗ B xλB vi FAS (x) := , ∀x ∈ 0, , (3.4) γ∗,i vB λ∗ i B A B vi vi ∗ − ∗ ∗ B λA λB xλA vi W FB ∗ (x) := A + A , ∀x ∈ 0, ∗ . (3.5) γ ,i vi ∗ vi λB λA

A B ∗ vi vi W S (b) If i∈ / ΩA(γ ) (i.e., ∗ ≤ ∗ ), we deﬁne Aγ∗,i and Bγ∗,i as the random variables whose distributions are λA λB

B A vi vi ∗ − ∗ ∗ A λB λA xλB vi W FA ∗ (x) := B + B , ∀x ∈ 0, ∗ , (3.6) γ ,i vi ∗ vi λA λB ∗ A xλA vi S FB ∗ (x) := , ∀x ∈ 0, . (3.7) γ ,i vA λ∗ i A

13 (3.1) XA,XB Note that (3.1) and Sn also depend on other parameters of the game GCBn but we use the notation with only the subscript n and omit other parameters to lighten the notation. 14 A B To solve this equation, we first sort out all ratios vi /vi in a non-decreasing order (which can be done in O(n ln(n))), then ∗ A B ∗ A B ∗ A B A B there are three possible cases: γ < min{vi /vi ,i ∈ [n]} or γ ≥ max{vi /vi ,i ∈ [n]} or ∃j : γ ∈ hvj /vj , vj+1/vj+1. In all of these cases, Equation (3.1) becomes a cubic equation; therefore, it can be solved algebraically. 15These constants are the Lagrange multipliers corresponding to the budget constraints in finding players’ best-response; see Kovenock & Roberson (2020) for more details. 16Here, the superscripts S and W , standing for strong and weak, are used to emphasize the intuition on players’ incentive ∗ A ∗ B ∗ to play according to these distributions in the Colonel Blotto games: if i ∈ ΩA(γ ) := i : vi /λA > vi /λB , Player A has a ∗ “stronger” incentive to win battlefield i and Player B has a “weaker” incentive; if i∈ / ΩA(γ ), the roles of players are exchanged.

10 To lighten the notation, hereinafter, we often commonly denote these random variables as follows (the ∗ ∗ corresponding distributions are denoted by FAi and FBi ):

S ∗ S ∗ ∗ Aγ∗,i if i ∈ ΩA(γ ) ∗ Bγ∗,i if i∈ / ΩA(γ ) Ai := W ∗ and Bi := W ∗ . (3.8) A ∗ if i∈ / Ω (γ ) B ∗ if i ∈ Ω (γ ) γ ,i A γ ,i A

We term these distributions the uniform-type distributions: F S (x) is the continuous uniform distri- Aγ∗,i A B A B ∗ vi vi vi W ∗ ∗ ∗ bution on 0, v /λ and FB ∗ (x) is the distribution placing a positive mass − at 0 and i B γ ,i λA λB λA B ∗ uniformly distributing the remaining mass on 0, v /λ ; similarly, F S is the uniform. distribution on i B Bγ∗,i A ∗ A ∗ [0, v /λ ] and F W is uniform on (0, v /λ ] with a positive mass at 0. i A Bγ∗,i i A If Player A can construct and plays a mixed strategy such that her sampled allocation to any battleﬁeld

∗ ∗ i ∈ [n] follows the distribution FAi , it is optimal for Player B to play such that her allocation to i follows FBi (if it is possible) and vice versa. We will revisit this result (with more details) in Section 4 and in Lemma B3 in Appendix B. Importantly, under the condition that Player A and Player B can respectively construct

∗ ∗ joint distributions of FAi , ∀i ∈ [n] and FBi , ∀i ∈ [n] such that their sampled allocations satisfy the budget XA,XB constraint, these mixed strategies yield an equilibrium of the game GCBn . However, in general, that ∗ ∗ 17 condition does not always hold. For instance, although Ai and Bi have finite upper-bounds, we note that among these random variables, some may (with strictly positive probability) exceed the budgets XA,XB ∗ ∗ for certain parameters’ configuration of the game; therefore, allocating according to FAi , FBi may violate ∗ ∗ the budget constraints and it is then trivial that there exists no equilibrium yielding FAi , FBi , ∀i ∈ [n] as ∗ ∗ marginals. On the other hand, given fixed XA,XB, if n is large enough, we can guarantee that Ai ,Bi do not exceed the budgets for each i; however, even in this case, we still do not have guarantees on the summation ∗ ∗ of allocations sampled from all Ai ,Bi ,i ∈ [n], i.e., it is still unknown if there exists an equilibrium yielding ∗ ∗ ∗ ∗ FAi , FBi ,i ∈ [n] as marginals. Note importantly that the budget-constraints violation of Ai ,Bi does not affect our work and our results hold for any parameters’ configuration of the games. Finally, under Assumption (A0), we obtain a novel result, presented below as Proposition 3.3, stating ∗ ∗ ∗ that the parameters γ , λA and λB are all bounded. The proof of this proposition is given in Appendix A. ∗ ∗ ∗ From this proof, we observe that the bounds of γ , λA and λB depend only on the ratiosw/ ¯ w and XB/XA; ∗ ∗ ∗ ¯ when these ratios increase, the ranges in which γ and λA, λB belong to also become larger (i.e., the ratios γ/¯ γ and λ/¯ λ also increase). The main results of this work are based on asymptotic analyses in terms of the ¯ number¯ of battlefields of the games; therefore, it is noteworthy that the bounds of these parameters do not depend on n. Proposition 3.3. Given w, w,X¯ ,X > 0 (w ≤ w¯, X ≤ X ), there exist constants γ, γ,¯ λ, λ>¯ 0 such that ¯ A B ¯ A B ¯ XA,XB XA,XB ∗ (3.1) ¯ for any game GCBn (or GLBn ) satisfying Assumption (A0), any γ ∈Sn and its corresponding ∗ ∗ ∗ ∗ ∗ ¯ λA, λB , we have γ ≤γ ≤γ¯ and λ≤λA, λB ≤λ. ¯ ¯

4. Approximate Equilibria of the Generalized Colonel Blotto Game

XA,XB In this section, we propose a class of strategies in the GCB game GCBn , called the independently uniform strategies, and we show that it is an approximate equilibrium (and an approximate min-max strategy in the constant-sum case). Note that the independently uniform strategies are also approximate equilibria XA,XB of the GLB game GLBn , we analyze that in Section 5.

17Trivially from Proposition 3.3, the random variables A∗, B∗, ∀n, ∀i ∈ [n] are upper-bounded byw/ ¯ (wnλ). In the remainders i i ¯ ¯ of the paper, we sometimes need an upper-bound of these random variables that does not depend on n: we can prove that they are bounded by 2XB (see Lemma A1 in Appendix A).

11 We begin by recalling the concept of approximate equilibria (see e.g., Myerson (1991); Nisan et al. XA,XB (2007)) in the context of our games: for any ε ≥ 0, an ε-equilibrium of the game GCBn is any strategy ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ proﬁle (s ,t ) such that ΠA(s,t )≤ΠA(s ,t )+ε and ΠB (s ,t)≤ΠB (s ,t )+ε for any strategy s and t of ζ ζ players A and B. Replacing ΠA and ΠB by ΠA and ΠB , we have the deﬁnition of ε-equilibria of the game XA,XB GLBn (ζ). Hereinafter, we use the generic term approximate equilibrium whenever the approximation error ε need not be emphasized.

4.1. The Independently Uniform Strategies

XA,XB XA,XB (3.1) Given a game GCBn (or GLBn ), let us consider the corresponding Equation (3.1) and set Sn . ∗ (3.1) For any γ ∈ Sn , we define in Definition 4.1 a mixed strategy via an algorithm, called Algorithm 1. We ∗ term this strategy as the independently uniform strategy (or IUγ strategy), parameterized by γ∗. Intuitively, this strategy is constructed by a simple procedure: players start by independently drawing n numbers from the uniform-type distributions defined in Definition 3.2, then they re-scale these numbers to guarantee the budget constraints.

Deﬁnition 4.1 (The independently uniform strategy). In any game GCBXA,XB (or GLBXA,XB ), for ∗ n n ∗ (3.1) γ xP any γ ∈ Sn and any player P ∈{A, B}, IUP is the mixed strategy of player P where her allocation is randomly generated from Algorithm 1.

∗ Algorithm 1: IUγ strategy-generation algorithm. N A B ∗ (3.1) Input: n ∈ , wi , wi ∈ [w, w¯], ∀i ∈ [n], budgets XA,XB, γ ∈ Sn xA xB Rn ¯ Output: , ∈ ≥0 1 ∗ ∗ Draw ai ∼ FAi ,bi ∼ FBi , ∀i ∈ [n] independently 2 if j∈[n] aj =0 then 3 xA := 0, ∀i ∈ [n] Pi 4 else 5 xA := ai X , ∀i ∈ [n] i aj A Pj∈[n] 6 if j∈[n] bj =0 then 7 xB := 0, ∀i ∈ [n] Pi 8 else 9 xB := bi X , ∀i ∈ [n] i bj B Pj∈[n]

∗ ∗ ∗ Henceforth, we use the term IUγ strategy to denote the strategy profile (IUγ , IUγ ). We also simply ∗ ∗ A B γ∗ γ γ use the notation IU in some places to commonly address either IUA or IUB strategy in case the name of the player need not be specified. Observe that for any player P ∈{A, B}, any output xP from Algorithm 1 ∗ is an n-tuple that satisfies her budget constraint. In other words, IUγ is a mixed strategy that is implicitly P ∗ defined by Algorithm 1 and each run of this algorithm outputs a feasible pure strategy sampled from IUγ . ∗ P γ ∗ ∗ Note that the marginals of the IU strategy are not the uniform-type distributions FAi , FBi ,i ∈ [n] defined in Section 3. In terms of computational complexity, Algorithm 1 terminates in O(n) time. Below we discuss the specificity of the outputs of Algorithm 1 in the cases where j∈[n] aj =0 or j∈[n] bj = 0.

γ∗ P P Remark 4.2. If j∈[n] aj = 0 or j∈[n] bj = 0, the IUP strategy allocates zero resource to all battlefields for the corresponding player (line 3 and line 7 of Algorithm 1). It may seem more natural that, if P P A j∈[n] aj =0, Player A allocates equally on all battlefields, i.e., set xi := XA/n, ∀i ∈ [n] in line 3 of Algo- rithm 1 (and similarly for Player B). In reality though, these assignments can be chosen to be any arbitrary P 12 xP Rn P n-tuple in ≥0 as long as i∈[n] xi ≤ XP without affecting the results in our work. This comes from the fact that in most cases, the conditions in line 2 and 6 hold with probability zero. They can happen with a positive probability only when oneP player is the “weak player” and the other is the “strong player” on all of the ∗ ∗ XA,XB battlefields (i.e., either ΩA(γ )= ∅ or ΩA(γ ) = [n]), e.g., in a constant-sum game Const-CBn . Yet, even in this case, this probability decreases exponentially as the number of battlefields increases (see (B.18) in Appendix B). The asymptotic order of the approximation error in all of our results is larger than this probability; therefore, it does not matter which assignment we choose in lines 3 and 7 of Algorithm 1. Here, A B we choose to assign xi =0, ∀i and xi =0, ∀i to ease the notation in the proofs of the results in the following sections; in particular, it avoids creating a discontinuity outside 0 in the CDF of the effective allocation in each battlefield (see also Lemma B1 in Appendix B).

XA,XB 4.2. Approximate Equilibria of the Generalized Colonel Blotto Game GCBn

∗ We now present our main result stating that the IUγ strategy is an approximate equilibrium with an error that is only a negligible fraction of the maximum payoﬀs that the players can achieve, quickly decreasing as n increases. In the following results, note that since we focus on the setting of games with a large number of battleﬁelds, we now focus on characterizing the approximation error according to n and XA,XB treat other parameters of the GCBn games, including XA,XB, w, w¯ and α, as constants (but not the P P ¯ ˜ values wi , vi , ∀i ∈ [n], P ∈ {A, B}). Note also that we often use the notation O that is a variant of the big-O notation where logarithmic factors are ignored.18 Theorem 4.3. Consider GCB games satisfying Assumption (A0), we have the following results:

XA,XB ˜ −1/2 ∗ (3.1) (i) In any game GCBn , there exists a positive number ε = O(n ) such that for any γ ∈ Sn , the following inequalities hold for any pure strategy xA and xB of players A and B:

∗ ∗ ∗ xA γ γ γ ΠA( , IUB ) ≤ ΠA(IUA , IUB )+ εWA, (4.1) ∗ ∗ ∗ γ xB γ γ ΠB (IUA , ) ≤ ΠB (IUA , IUB )+ εWB. (4.2)

∗ XA,XB (ii) Given XA,XB > 0 (XA ≤ XB), there exists C > 0 such that for any ε∈ (0, 1], in any game GCBn ∗ −2 −1 ∗ (3.1) A B with n ≥ C ε max 1, ln(ε ) , (4.1) and (4.2) hold for any γ ∈ Sn , any pure strategy x , x of players A and B respectively. A proof of this theorem is presented in Appendix B. First, note that although the results in Theorem 4.3 are stated under Assumption (A0), it does not restrict their scope of application since this, as discussed, is a very mild assumption. Moreover, the two results of Theorem 4.3 are two equivalent statements. In the remainders of this section, we give some interpretations of these two results. Result (i) of Theorem 4.3 states that, given a game GCBXA,XB , there exists no unilateral deviation ∗ n from an IUγ strategy that can proﬁt any player P ∈ {A, B} more than a small portion of her maximum ∗ γ XA,XB payoﬀ WP . As a direct corollary, the IU strategy is an approximate equilibrium of the game GCBn with a bounded approximation error (depending on n); this is formally stated as follows:

XA,XB XA,XB Corollary 4.4. (Approximate equilibrium of the GCBn game) In any game GCBn satisfying ∗ −1/2 ∗ (3.1) γ Assumption (A0), there exists a positive number ε = O˜ n such that for any γ ∈ Sn , the IU strategy is an εW -equilibrium where W := max{W , W }. A B We now discuss the interpretation of this corollary and Result (i) of Theorem 4.3 in a sequence of games XA,XB GCBn (satisfying Assumption (A0)) that have larger and larger numbers of battleﬁelds (i.e., n increases)

18See Appendix A for formal deﬁnitions of these notations.

13 ∗ to see the relation between the games’ parameters and the approximation error of the IUγ strategies. First, observe that as n increases, the error εW of this approximate equilibrium might not decrease to 0. This is due to the fact that although ε decreases as n increases, W does not. Indeed, recall that WA and WB are the total values that players A and B assign on all battlefields; as the number of battlefield increases, by Assumption (A0), WA, WB and W = max{WA, WB} inevitably increase (at a rate O(n)). This, however, ∗ does not question the significance of the IUγ strategies as a good approximate equilibrium (with n is ∗ large enough). In fact, it is not meaningful to compare the magnitude of approximation errors of IUγ XA,XB strategies in two GCBn games having different sizes. Instead, we should compare the ratio between the approximation error and W —an upper-bound on players’ payoffs—which is relative to the scale of the considered games. This ratio is exactly ε (we call this the level of error). Importantly, as we consider XA,XB ˜ −1/2 GCBn games with larger and larger n, ε indeed quickly tends to 0; the bound O(n ) indicates the order of this relation showing how fast ε decreases as n increases. ∗ Said differently, the result of Corollary 4.4 states that, by playing the IUγ strategies, a player guarantees to not lose more than a small fraction ε of her maximum total payoff W , which is indeed the meaningful XA,XB statement in our setting. An alternative formulation would be to consider a sequence of GCBn games (where n increases) whose the battlefields values are re-scaled such that W = 1 in all games. In this ∗ γ XA,XB case, Corollary 4.4 would indicate that the IU strategy is an ε-equilibrium of GCBn with ε → 0 when n → ∞. We prefer our formulation without re-scaling as it better emphasizes the relationship to the battlefield values. XA,XB We also note that the upper-bound on ε also depends on other parameters of the game GCBn , including X ,X , w andw ¯ (given by Assumption (A0)).19 We can extract from the proof of Theorem 4.3 A B ¯ (see Appendix B) that asw/ ¯ w and/or XB/XA increases, ε also increases, i.e., for games with higher het- ¯ ∗ erogeneity of the battlefields values and/or higher asymmetry in players’ budgets, the IUγ strategy yields higher errors. Additionally, to keep the generality, Result (i) of Theorem 4.3 is presented such that the ∗ γ ∗ (3.1) approximation error ε is commonly addressed for any IU strategy with any γ ∈ Sn . For each specific ∗ ∗ ∗ ∗ γ solution γ of Equation (3.1) (implying λA and λB), the corresponding IU strategy is an approximate XA,XB equilibrium of GCBn with an approximation error that is at most (and it might be strictly smaller than) ε. On the other hand, Result (ii) of Theorem 4.3 indicates the number of battlefields that a GCB game needs to contain—relative to players’ budgets and bounds on battlefields’ values—in order to guarantee a ∗ desired level of the approximation error by using the IUγ strategy as an approximate equilibrium. Hence, ∗ in practical situations involving large instances of the Colonel Blotto game, the IUγ strategy (simply and efficiently constructed by Algorithm 1) can be used as a safe replacement for any Nash equilibrium whose construction may be unknown or too complicated. Now, let us introduce an important notation:

n Deﬁnition 4.5. Corresponding to the players’ allocations toward each battleﬁeld i ∈ [n], let FAi and γ∗ γ∗ n FBi denote the univariate marginal distributions of the IUA and IUB strategies (see (B.1) and (B.2) n n in Appendix B for a more explicit formulation of FAi and FBi ). Intuitively, Result (ii) can be proved by showing the following two results: (a) when Player B’s allocation

∗ to the battlefield i ∈ [n] follows FBi , the best response of Player A is to play such that her allocation to i ∗ n n follows the distribution FAi (and vice versa); (b) as n—the number of battlefields—increases, FAi and FBi ∗ ∗ ∗ γ uniformly converge toward the distributions FAi and FBi , i.e., the marginal distributions of the IU strategy ∗ ∗ approximate the distributions FAi and FBi . This convergence can be proved by applying concentration ∗ ∗ inequalities on the random variables j∈[n] Aj and j∈[n] Bj (see Lemma B4 in Appendix B); moreover, P P 19This dependency is implicitly presented in the asymptotic notation O˜ in Result (i) and the constant C∗ in Result (ii) of Theorem 4.3. 14 the relation between ε and n in the results of Theorem 4.3 depends directly on the rate of this convergence. In this work, we use the Hoeffding’s inequality (Hoeffding (1963)) that yields a better convergence rate than working with other types of concentration inequalities (e.g., Chebyshev’s inequality). To complete the γ∗ proof of Result (ii), we finally show that as n increases, when player −P ∈{A, B} plays the IU−P strategy, γ∗ the IUP ’s payoff of player P converges toward her best-response payoff. Note that these payoffs can be written as expectations with respect to different measures (see (B.3), (B.4) and Lemma B2 in Appendix B). To prove the convergence of payoffs, we use a variant of the portmanteau theorem (see Lemma B5 in Appendix B) regarding the equivalent definitions of the weak convergence of a sequence of measures. Note importantly that a direct application of the portmanteau theorem leads to a slow convergence rate (notably, (4.1) and (4.2) only hold when n = Ω(ε−4)). This is due to the fact that the players’ payoffs involve the

∗ ∗ bounded Lipschitz functions FAi and FBi and that these functions depend on n, particularly, their Lipschitz ∗ A ∗ B constants (that are either λA/vi or λB /vi ) increase as n increases. In order to obtain the convergence rate n ∗ n as indicated in Theorem 4.3, we exploit the special relation between FAi and FAi , and between FBi and ∗ FBi ; then we apply a telescoping-sum trick allowing us to avoid the need of using the Lipschitz properties (for more details, see the proof of Lemma B5 in Appendix B.5).

4.3. Approximate Equilibria of the Constant-sum Colonel Blotto Game

XA,XB In this section, we discuss the constant-sum game variant: the game Const-CBn defined in Def- inition 2.2. As an instance of the GCB game, the game Const-CBXA,XB satisfies all results presented in ∗ n Sections 4.1 and 4.2. Additionally, we show that any IUγ strategy is an approximate max-min strategy of XA,XB the game Const-CBn . XA,XB ∗ In any game Const-CBn , Equation (3.1) has a unique solution γ =XB/XA ≥1 which, in turn, ∗ ∗ 2 XA,XB uniquely induces λA =1/(2XB) and λB = XA/(2XB ). Moreover, in the game Const-CBn , observe A B ∗ ∗ ∗ that vi /vi =1 ≤ XB/XA = γA/γB, ∀i ∈ [n]; therefore, we have ΩA(γ )= ∅. Intuitively, Player A is the XA,XB “weak player” (and B the “strong player”) in all battlefields. Now, note that in the game Const-CBn , W := max{WA, WB} = WA = WB, we apply Theorem 4.3 and obtain the following result.

XA,XB Corollary 4.6. In any game Const-CBn satisfying Assumption (A0), there exists a positive number ∗ −1/2 γ ∗ (3.1) ε ≤ O˜(n ) such that the IU strategy is an εW -equilibrium with γ ∈ Sn = {XB/XA}.

XA,XB Note that if an equilibrium exists in Const-CBn (note that its existence still remains an open question), then the set of OUDs is unique (see e.g., Corollary 1 of Kovenock & Roberson (2020)) and they correspond ∗ ∗ to the distributions F W and F S for any i ∈ [n] as deﬁned in (3.6) and (3.7) (where λ and λ are Aγ∗,i Bγ∗,i A B 2 γ∗ respectively replaced by the values 1/(2XB) and XA/(2XB )). The marginals of the IU strategy with ∗ γ = XB/XA converge toward these unique OUDs. ∗ γ XA,XB Finally, we deduce that the IU strategy is an approximate max-min strategy of the game Const-CBn ; this is formally stated as follows.

XA,XB Corollary 4.7. In any game Const-CBn satisfying Assumption (A0), there exists a positive number −1/2 ∗ (3.1) ε ≤ O˜(n ) such that the following inequalities hold for γ ∈ Sn = {XB/XA} and any strategy s˜ and t˜ of players A and B: γ∗ min ΠA(˜s,t) ≤ min ΠA(IU ,t)+ εW, (4.3) t t A γ∗ min ΠB(s, t˜) ≤ min ΠB (s, IU )+ εW. (4.4) s s B γ∗ Intuitively, if player P ∈ {A, B} plays the IUP strategy, she guarantees a near-optimal payoff even in the worst-case scenario when her opponent −P plays strategies that minimize P ’s payoff (no matters how it affects −P ’s payoff). The proofs of Corollary 4.6 and Corollary 4.7 can be trivially deduced by applying specifically Theorem 4.3 to the constant-sum Colonel Blotto games and thus are omitted in this work. 15 5. Approximate Equilibria of the Generalized Lottery Blotto Game

∗ In this section, we present the results regarding the IUγ strategy in the GLB game. In Section 5.1, we ∗ XA,XB γ analyze the game GLBn (ζ) with an arbitrary pair of CSFs ζ = (ζA, ζB) and show that the IU strategy XA,XB is an approximate equilibrium of GLBn (ζ) with an error depending on the number of battleﬁelds as well as the dissimilarity between ζA and βA (and between ζB and βB). In Section 5.2, we illustrate this result in two particular instances, the games GLBXA,XB (µR) and GLBXA,XB (νR), belonging to the class of n n ∗ Lottery Blotto games with ratio-form CSFs. We characterize the approximation error of the IUγ strategy in these games according to n and the parameter R of these CSFs.

XA,XB 5.1. Approximate Equilibria of Generalized Lottery Blotto games GLBn (ζ)

We start by defining a parameter that expresses the dissimilarity between a given pair of CSFs ζ = (ζA, ζB) and the Blotto functions βA,βB (defined in (2.1)). First, note that for any n and i ∈ [n], the random vari- ∗ ∗ n n ables Ai ,Bi (corresponding to the distributions defined in Definition 3.2) and Ai ,Bi (corresponding to the 20 distributions defined in Definition 4.5) are all upper-bounded by 2XB. Now, given any ε> 0, for any pair ∗ ∗ of CSFs ζ = (ζA, ζB ) and any x ∈ [0, 2XB] and y ∈ [0, 2XB] (i.e., any number that can be sampled from

∗ ∗ n n FAi , FBi , FAi or FBi ), we introduce the following sets:

ζ ∗ ∗ ∗ X (y ,ε) := {x ∈ [0, 2XB]: |ζA(x, y ) − βA(x, y )|≥ ε} , (5.1) ζ ∗ ∗ ∗ Y (x ,ε) := {y ∈ [0, 2XB]: |ζB (x ,y) − βB(x ,y)|≥ ε} . (5.2)

∗ (3.1) 21 Deﬁnition 5.1. For any pair of CSFs ζ = (ζA, ζB), ε> 0 and γ ∈ Sn , we deﬁne the following set

∆ ∗ (ζ,ε) := δ ∈ [0, 1] : max max dF ∗ (x) ≤ δ, and max max dF ∗ (y) ≤ δ . γ ∗ Ai ∗ Bi i∈[n] y ∈[0,2X ] ζ ∗ i∈[n] x ∈[0,2X ] ζ ∗ ( B ZX (y ,ε) B ZY (x ,ε) ) ∗ Intuitively, the set ∆γ∗ (ζ,ε) contains all numbers δ ∈ [0, 1] such that for any allocation y of Player B toward ∗ an arbitrary battlefield i, if Player A draws an allocation x from the distribution FAi , it only happens with ∗ probability at most δ that the value of the CSF ζA at (x, y ) is significantly different (i.e., ε-away) from that ∗ of the Blotto function βA; and we have a similar statement for the distribution FBi of Player B and any ∗ ∗ ∗ ∗ ∗ allocation x of Player A. Note that the set ∆γ (ζ,ε) depends on FAi and FBi , thus it depends on γ . We can trivially see that ∆γ∗ (ζ,ε) is an interval with the form [δ0, 1] since if δ0 ∈ ∆γ∗ (ζ,ε) then δ ∈ ∆γ∗ (ζ,ε) for any δ ≥ δ0.

n n ∗ ∗ Based on the convergence of FAi and FBi toward FAi and FBi (see the details in Lemma B4 in Appendix B), we can prove the following lemma (a formal proof is given in Appendix C.1):

Lemma 5.2. Given w, w,X¯ A,XB > 0 (w ≤ w¯, XA ≤ XB), there exists a constant L0 > 0 such that for ¯ −2 ¯ −1 XA,XB any ε ∈ (0, 1] and n ≥ L0ε max 1, ln(ε ) , for any game GLBn (ζ) satisfying Assumption (A0), ∗ (3.1) any γ ∈ Sn , δ ∈ ∆γ∗ (ζ,ε) and i∈ [n], we have:

n n max sup dFAi (x), sup dFBi (y) ≤ δ + ε. (5.3) ∗ ζ ∗ ∗ ζ ∗ (y ∈[0,2XB ] ZX (y ,ε) x ∈[0,2XB ] ZY (x ,ε) )

20 n n ∗ ∗ By deﬁnition, Ai , Bi are bounded by XA, XB and thus by 2XB ; for a proof that Ai , Bi admit the same upper-bound, see Lemma A1 in Appendix A. 21Note that F ∗ ,F ∗ are continuous, bounded functions on [0, 2X ]; therefore, they attain a maximum on this interval. Ai Bi B

16 Intuitively, this lemma provides an upper-bound for the probability of the value of the CSFs ζ being ε-away from the Blotto functions when Player A (resp. Player B) plays such that her allocation to battleﬁelds i γ∗ n n follows FAi (resp. FBi ), i.e., when she plays the IU strategy. Based on the deﬁnition of ∆γ∗ (ζ,ε) and Lemma 5.2, we can now show the following result regarding the ∗ IUγ strategy in the GLB game. Theorem 5.3. Consider GLB games satisfying Assumption (A0), we have the following results:

XA,XB ˜ −1/2 ∗ (3.1) (i) In any game GLBn (ζ), there exists a positive number ε ≤ O(n ) such that for any γ ∈ Sn A B and δ ∈ ∆γ∗ (ζ,ε), the following inequalities hold for any pure strategy x and x of players A and B:

∗ ∗ ∗ ζ xA γ ζ γ γ ΠA( , IUB ) ≤ ΠA(IUA , IUB )+(8δ + 13ε) WA, (5.4) ∗ ∗ ∗ ζ γ xB ζ γ γ ΠB(IUA , ) ≤ ΠB(IUA , IUB )+(8δ + 13ε) WB. (5.5)

∗ (ii) Given XA,XB > 0 (XA ≤ XB), there exists L > 0 such that for any ε ∈ (0, 1], in any game XA,XB ∗ −2 −1 ∗ (3.1) GLBn (ζ) where n ≥ L ε max 1, ln(ε ) , (5.4) and (5.5) hold for any γ ∈ Sn , δ ∈ A B ∆ ∗ (ζ,ε) and any pure strategy x , x of players A and B. γ The proof of this theorem is given in Appendix C.2. The main idea to prove these results is that we ∗ XA,XB γ can approximate the players’ payoffs in the game GLBn (ζ) when they play the IU strategies by that XA,XB in the corresponding game GCBn (the difference between these payoffs is controlled by the parameter XA,XB δ ∈ ∆γ∗ (ζ,ε)); and then use the results from Section 4 for the game GCBn (involving the error ε) to prove (5.4) and (5.5). The coefficients (8 and 13) in front of the parameters δ and ε come from the application of several triangle inequalities to connect these approximate results. Note that if the CSFs ζA and ζB are Lipschitz continuous on [0, 2XB] × [0, 2XB], we can avoid the need to approximate several terms ∗ γ XA,XB involved in the analysis of using the IU strategy in the game GLBn (ζ) via the corresponding terms in XA,XB the game GCBn ; thus, we can improve the results in Theorem 5.3 to obtain an approximation error of 2δ +5ε instead of 8δ +13ε (see Remark C3 in Appendix C.5 for more details). Here, to keep the generality, we do not include the continuity assumption of the CSFs in Theorem 5.3 (recall that our definition of a CSF allows for discontinuity). ∗ Intuitively, Result (i) of Theorem 5.3 determines the order of the approximation error while using IUγ in ∗ XA,XB γ any given game GLBn (ζ). Straightforwardly, we can deduce that the IU strategy is an approximate XA,XB equilibrium of the game GLBn (ζ), formally stated as follows.

XA,XB XA,XB Corollary 5.4. (Approximate equilibria of the GLBn game) In any game GLBn (ζ) sat- −1/2 ∗ (3.1) isfying Assumption (A0), there exists a positive number ε ≤ O˜(n ) such that for any γ ∈ Sn and γ∗ δ ∈ ∆γ∗ (ζ,ε), the IU strategy is an (8δ + 13ε) W -equilibrium where W := max{WA, WB }.

We observe that the error bound in Theorem 5.3 (and in Corollary 5.4) is valid for any δ of the set ∆γ∗ (ζ,ε). Naturally, it is the tightest for δ0 = min{δ : δ ∈ ∆γ∗ (ζ,ε)}; but this quantity is not always easy to compute; XA,XB R for example, in the Lottery Blotto games with the power and logit form CSFs (i.e., GLBn (µ ) and XA,XB R R R GLBn (ν )). Still, in Section 5.2, we show that there exists an element of ∆γ∗ (µ ,ε) and ∆γ∗ (ν ,ε) that is negligibly small, given appropriate parameter’s conﬁgurations of the games; in other words, we can still ob- ∗ tain a good error’s upper-bound for the IUγ strategy in these games. Note that, on the other hand, the GCB XA,XB XA,XB game GCBn can be considered as an instance of the game GLBn (ζ) where the CSFs are ζA = βA and XA,XB ζ ∗ ζ ∗ ζB = βB; therefore, it also satisﬁes Theorem 5.3. In GCBn , we trivially have X (y ,ε)= Y (x ,ε)= ∅ ∗ ∗ 22 for any x ,y ; thus ∆γ∗ (ζ,ε) = [0, 1] for any ε> 0 and min{δ : δ ∈ ∆γ∗ (ζ,ε)} = 0. This is consistent with results obtained in Theorem 4.3 in Section 4.

22 XA,XB Note also that for the case of the game GCBn , the left-hand side in (5.3) equals zero for any n and i ∈ [n]. 17 In Theorem 5.3, Result (ii) is an equivalent statement of Result (i). It indicates the number of battlefields ∗ needed to guarantee a certain level of approximation error when using the IUγ strategy in the game XA,XB XA,XB GLBn (ζ). For instance, to obtain an approximate equilibrium of the game GLBn (ζ) where the level of error is less than a certain numberε ¯, one needs ε ≤ ε¯ such that we can find a δ ∈ ∆γ∗ (ζ,ε) satisfying 8δ + 13ε ≤ ε¯; from these parameters, by Result (ii), one can deduce the sufficient number of battlefields XA,XB needed for an GLBn game to yield that desired level of error. Finally, in the constant-sum variant of the GLB game, denoted by Const-LBXA,XB (ζ) (i.e., when ∗ n A B γ wi = wi , ∀i ∈ [n]), we can easily deduce from Theorem 5.3 that the IU strategy is also an approximate max-min strategy:

XA,XB ˜ −1/2 Corollary 5.5. In any game Const-LBn (ζ) satisfying Assumption (A0), there exists ε ≤ O(n ) ∗ (3.1) such that for any γ ∈ Sn = {XB/XA} and δ ∈ ∆γ∗ (ζ,ε), the following inequalities hold for any strategy s˜ and t˜ of players A and B:23

∗ min Πζ (˜s,t) ≤ min Πζ (IUγ ,t)+(8δ + 13ε)W, t A t A A ∗ min Πζ (s, t˜) ≤ min Πζ (s, IUγ )+(8δ + 13ε)W. s B s B B 5.2. Approximate Equilibria of the Lottery Blotto Games with Ratio-form CSFs

XA,XB R XA,XB R R R We now consider the games GLBn (µ ) and GLBn (ν ) where µ and ν are pairs of CSFs that are defined in Table 1. Note again that for those CSFs, we do not consider the degenerate cases where α =0 XA,XB R or α = 1 in which trivial equilibria exist in the corresponding GLB games. The games GLBn (µ ) and GLBXA,XB (νR) are instances of the game GLBXA,XB (ζ) studied in Section 5.1; therefore, by Theorem 5.3 n ∗ n (and Corollary 5.4), the IUγ strategy is also an approximate equilibrium of them. In this section, we focus ∗ on characterizing the approximation error of the IUγ strategy in these games according to n (the number of battlefields) and R (the corresponding parameter of the CSFs). We will show that this error quickly tends to zero as n and R increase under appropriate conditions. To do this, we first notice that although it is R R non-trivial to analyze the closed form of the sets ∆γ∗ (µ ,ε) and ∆γ∗ (ν ,ε) and find their minimum, we can find small elements of theses sets. Lemma 5.6. Fix n ≥ 2, R> 0 and α ∈ (0, 1), for any ε< min{α, 1 − α}, we have:24

XA,XB R (i) In any game GLBn (µ ) satisfying Assumption (A0) and having α as the tie-breaking parameter, −1/R R ∗ (3.1) there exists δµ =min{1, O n(ε −1) } such that δµ ∈ ∆γ∗ (µ ,ε) for any γ ∈ Sn .

XA,XB R (ii) In any game GLBn (ν ) satisfying Assumption (A0) and having α as the tie-breaking parameter, −1 −1 R ∗ (3.1) there exists δν =min{1, O nR ln(ε ) } such that δν ∈ ∆γ∗ (ν ,ε) for any γ ∈ Sn .

The proof of Lemma 5.6 is given in Appendix D.1. Note that for the sake of generality, the parameters ∗ ∗ (3.1) δµ and δν are indicated in this lemma in such a way that they do not depend on γ , but for each γ ∈ Sn , R R we can find smaller elements of the corresponding sets ∆γ∗ (µ ,ε) and ∆γ∗ (ν ,ε). More importantly, for a fixed n, the numbers δµ and δν decrease as R increases; but δµ and δν increase as ε decreases. While R R the lemma is valid for any parameter values, since 1 is a trivial element of ∆γ∗ (µ ,ε) and ∆γ∗ (ν ,ε), it −1 is useful only if δµ,δν < 1; this is guaranteed whenever R ≥ O n ln(ε ) . Note finally that the condition ε< min{α, 1−α} in the statement of Lemma 5.6 does not limit its use since our goal is to obtain asymptotic ∗ γ XA,XB R XA,XB R results on the IU strategy when ε tends to 0. Moreover, in the games GLBn (µ ) and GLBn (ν )

23 Recall that in the constant-sum variant, W := max{WA,WB} = WA = WB. 24The asymptotic notations are taken w.r.t. when ε → 0.

18 where α is either very close to 0 or 1, one player has a very high advantage and always obtains large gains from all battlefields (where her allocation is strictly positive) while her opponent gains very little regardless of her allocations; therefore, there exists (many) trivial approximate equilibria with small errors. Combining the results of Corollary 5.4 and Lemma 5.6, we can deduce directly that in any game XA,XB R XA,XB R ˜ −1/2 ∗ (3.1) GLBn (µ ) (resp. GLBn (ν )), there exists ε ≤ O(n ) such that for any γ ∈ Sn , the γ∗ IU strategy is an (8ε + 13δµ)W -equilibrium (resp. (8ε + 13δν )W -equilibrium). Next, we look for the asymptotic relation between these error terms and the parameters n, R of the games. First, as n increases, the error level ε decreases; on the other hand, from Lemma 5.6, the number δµ (and δν ) decreases if R increases with a faster rate than O˜(n). However, there is a trade-off between ε and δµ (or δν ): as ε decreases, δµ (and δν ) increases and vice versa. To handle this trade-off between δµ and ε (resp. δν and ε), we can first find a condition on n that generates a small error ε, and then find a condition on R (with respect to n) such that the error δµ (resp. δν ) is of the same order as ε. Formally, we state the result that the ∗ γ XA,XB R XA,XB R IU strategy yields an approximate equilibrium of the games GLBn (µ ) and GLBn (ν ) with any arbitrary small error in the next theorem. Theorem 5.7. (Approximate equilibria of Lottery Blotto games with ratio-form CSFs) Given w, w,X¯ ,X > 0 (w ≤w¯, X ≤X ) and α ∈ (0, 1), there exists L>˜ 0 such that for any ε¯ ∈ (0, min{α, 1−α}), ¯ A B ¯ A B in any game GLBXA,XB (µR) and GLBXA,XB (νR)—satisfying Assumption (A0) and having α as the tie- n n ∗ ˜ −2 −1 n 1 γ breaking-rule parameter—where n ≥ Lε¯ max 1, ln(¯ε ) , R ≥ O ε¯ ln ε¯ , the IU strategy is an εW¯ - ∗ (3.1) equilibrium for any γ ∈ Sn . The proof of this theorem is based on Theorem 5.3 and Lemma 5.6 (see Appendix D.2 for more details). Theorem 5.7 involves a double limit in R and n. Intuitively, if n and R increase but R increases with a slower rate, then ε (in Theorem 5.3) decreases but the corresponding δµ and δν (in Lemma 5.6) do not decrease; thus, the total error is not guaranteed to decrease. Therefore, in order to obtain an approximate equilibrium XA,XB R XA,XB R of the games GLBn (µ ) or GLBn (ν ) with a small level of error (i.e.,ε ˜ in Theorem 5.7), one needs that these games have a large number of battlefields and that the parameter R is large enough such R R R R that the CSFs µA,µB (and νA ,νB ) approximate well the Blotto functions βA,βB. This is consistent with R R R R the observation that as R increases, µA,µB (and νA ,νB ) converge pointwise towards βA,βB respectively.

6. Conclusion

In this work, we consider the Generalized Colonel Blotto game—the most general version of the Colonel Blotto game with heterogeneous battlefields and asymmetric players. While most of (if not all) works in the literature attempt (but do not completely succeed) to construct an exact equilibrium of the Colonel Blotto game by looking for a joint distribution with the uniform-type marginals that satisfies the budget ∗ constraints, we take a different angle. We propose a class of strategies called the IUγ strategies that is ∗ simply constructed by an efficient algorithm; the IUγ strategies guarantee the budget constraints but their ∗ marginals are not the optimal univariate distributions. Yet, we prove the IUγ strategy to be an approximate equilibrium of the GCB games. We also study an extended game called the Generalized Lottery Blotto game and obtain similar results. We characterize the approximate error in our results in terms of the number of battlefields of the games. Our work extends the scope of application of the GCB games and its variants. Throughout the paper, we emphasized the dependence of the approximation error on the number of battlefields n. Yet, although the dependence on other parameters of the games is not explicitly emphasized, it can be extracted from our analysis and the proofs of the stated results. Note that most results in this paper are obtained under an assumption on the games’ parameters, but this assumption holds for all relevant practical applications. It is also interesting to note that although the notion of approximate equilibrium ∗ is defined in terms of payoffs (the payoffs when players play the IUγ strategy are close to optimal), the

19 ∗ IUγ strategy also approximates the equilibrium marginals (if an equilibrium exists)—that is, it is also an approximate equilibrium in terms of strategies. Our approximation results are valid even in the case where no equilibrium exists (and we do not include the assumption that requires the existence of the equilibrium). Particularly in the cases of the Colonel Blotto game where it is known that there exists no equilibrium yielding the uniform-type marginals, the ∗ IUγ strategy is still an approximate equilibrium, yet we suspect that in those cases the approximation error might be large. On the other hand, our work does not solve the question of the existence of an exact Nash equilibrium. In particular, we leave as future work the investigation of possible conditions under which a Nash equilibrium exists, for instance for a large-enough number of battlefields. We also finally note that, in the GCB game, the existence of multiple solutions γ∗ of Equation (3.1) leads to problems of equilibrium ∗ selection (in practical contexts involving a social welfare measurement) among the IUγ strategies with ∗ (3.1) different γ ∈ Sn , which we also leave as future work.

Acknowledgment

This work was supported by the French National Research Agency through the “Investissements d’avenir” program (ANR-15-IDEX-02) and through grant ANR-16-TERC0012; and by the Alexander von Hum- boldt Foundation.

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23 Appendix A. Nomenclatures and Preliminaries

Table A.2: Table of Notation

Abbreviation XA,XB , GCB (or GCBn ) Generalized Colonel Blotto game (with n battlefields) XA,XB , GLB (or GLBn ) Generalized Lottery Blotto game (with n battlefields) XA,XB , XA,XB Const-CBn the constant-sum versions of GCBn XA,XB , XA,XB Const-LBn the constant-sum versions of GLBn CSF , contest success function OUD , optimal univariate distribution ∗ ∗ ∗ γ γ γ , ∗ IU (= (IUA , IUB )) independent uniform strategy (corresponding to γ ) Games’ Parameters XA,XB , budgets of Player A and B respectively n , number of battlefields A B , wi , wi values of battlefield i assessed by Player A and B respectively w, w¯ , lower and upper bounds of battlefields’ values ¯ , n A n B WA, WB sums of battlefields’ values, WA := i=1 wi , WB := i=1 wi W , max{WA, WB} A B , P P vi , vi normalized values of battlefield i assessed by Player A and B A B , xi , xi the allocation to battlefield i of Player A and B respectively ΠA(s,t), ΠB (s,t) , players’ payoffs inG CB games when playing the strategies s and t α , the tie-breaking parameter βA,βB , Blotto functions (see (2.1)) ζ , (ζA, ζB)—the generic CSFs XA,XB , GLBn (ζ) the GLB game with CSFs ζA, ζB R , R R µ (µA,µB)—the power form CSFs with parameter R (see Table 1) R , R R ν (νA ,νB )—the logit form CSFs with parameter R (see Table 1) ζ ζ , XA,XB ΠA(s,t), ΠB (s,t) players’ payoffs in GLBn (ζ) games when playing the strategies s, t ζ ∗ ζ ∗ X (y ,ε), Y (x ,ε) , the sets characterizing the dissimilarity between (βA,βB) and (ζA, ζB ) (see (5.1), (5.2)) ∆γ∗ (ζ,ε) , see Definition 5.1 ∗ IUγ Strategies γ∗ , a positive solution of Equation (3.1) (3.1) , XA,XB Sn the set of positive solutions of Equation (3.1) w.r.t. GCBn ∗ (3.1) γ, γ¯ , lower and upper bounds of any γ ∈ Sn (see Proposition 3.3) ¯ ∗ , A B ∗ ΩA(γ ) i ∈ [n]: vi /vi >γ λ∗ , λ∗ , Lagrange multipliers corresponding to γ∗ (see (3.2), (3.3)) A B λ, λ¯ , lower and upper bounds of λ∗ , λ∗ (see Proposition 3.3) ¯ A B ∗ ∗ , FAi , FBi uniform-type distributions (see (3.4)-(3.8) S W n , Aγ∗,i, Aγ∗,i, Ai random variables defined in (3.4)-(3.8) S W n Bγ∗,i,Bγ∗,i,Bi γ∗ n n , FAi , FBi the marginals corresponding to battlefield i of the IU strategy , ∗ ∗ A=0, A>0 the events { i∈[n] Ai =0} and { i∈[n] Ai > 0}, respectively B ,B , the events { B∗ =0} and { B∗ > 0}, respectively =0 >0 Pi∈[n] i Pi∈[n] i P P Throughout the paper, we use the asymptotic notation (Bachmann–Landau notations) O by its standard 24 25 Rn z definition, i.e., for any (real-valued) functions f, g defined on an unbounded subset S ⊂ >0, and g( ) > 0 for any z ∈ S, we write f(z)= O (g(z)) if ∃M,C > 0 : |f(z)|≤ Cg(z), ∀z ∈ S : zi ≥ M, ∀i ∈ [n]. Moreover, we also use another variant of O that is O˜ where the logarithmic terms (in zi) are ignored. We also write h(z) ≤ O(g(z)) if there exists a term f(z) = O(g(z)) such that h(z) ≤ f(z), ∀z ∈ S (similar notation for O˜ can be deduced). In the remainders of this section, we introduce and prove several preliminary lemmas that are useful for our analysis.

XA,XB XA,XB ∗ (3.1) Lemma A1. Given a game GCBn (or GLBn ), for any γ ∈ Sn , we have: ∗ ∗ ∗ ∗ ∗ (i) λA, λB > 0 and γ = λA/λB.

B A 2 ∗ A B 2 ∗ E S 1 vi E W vi λB E S 1 vi E W vi λA (ii) For any i ∈ [n], [Aγ∗,i]= ∗ , [Aγ∗,i]= ∗ B , [Bγ∗,i]= ∗ and [Bγ∗,i]= ∗ A . 2 λB λA 2vi 2 λA λB 2vi E ∗ E ∗ (iii) XA = i∈[n] [Ai ] and XB = i∈[n] [Bi ].

∗ ∗ (iv) For anyPi ∈ [n], Ai and Bi haveP a constant upper-bound; particularly, P ∗ P ∗ (Ai ≤ 2XB)= (Bi ≤ 2XB)=1.

Proof.

∗ ∗ ∗ ∗ ∗ (i) The positivity of λA and λB follows from the positivity of γ and the deﬁnitions of λA and λB in (3.2) ∗ ∗ ∗ and (3.3). By dividing (3.2) by (3.3) and combining with (3.1), we trivially have that γ = λA/λB.

(ii) These results come directly from the deﬁnitions of the distributions F S , F W F S and F W . Aγ∗,i Aγ∗,i Bγ∗,i Bγ∗,i

∗ ∗ (iii) We multiply both sides of (3.3) by XA/λB and both sides of (3.2) by XB/λA then using the fact that ∗ ∗ ∗ γ = λA/λB to obtain the following:

2 B A ∗ 1 vj vj λB XA = ∗ + ∗ B , (A.1) ∗ 2 λB ∗ λA ! 2vj j∈ΩXA(γ ) j∈ /ΩXA(γ ) 2 B ∗ A vj λA 1 vj XB = ∗ A + ∗ . (A.2) ∗ λB ! 2vj ∗ 2 λA j∈ΩXA(γ ) j∈ /ΩXA(γ ) E ∗ E ∗ Combining with (ii), we deduce that XA = i∈[n] [Ai ] and XB = i∈[n] [Bi ]. ∗ ∗ S P ∗ W P (iv) If i ∈ ΩA(γ ), we have Ai = Aγ∗,i and Bi = Bγ∗,i. Recalling Deﬁnition 3.2, we have that P S B ∗ P W B ∗ Ai ≤ vi /λB = 1 and Bi ≤ vi /λB = 1. On the other hand, from (A.1), we deduce

B B vj vi XB ≥ XA ≥ ∗ ≥ ∗ . ∗ 2λB 2λB j∈ΩXA(γ ) P S P S B ∗ P W P W B ∗ Therefore, (Ai ≤ 2XB) ≥ Ai ≤vi /λB = 1 and (Bi ≤ 2XB) ≥ (Bi ≤ vi /λB) = 1. We conclude that for any i ∈ Ω (γ∗), A∗,B∗ are bounded by 2X . A i i B

25This deﬁnition is used by textbooks e.g., Brassard & Bratley (1996); Cormen et al. (2009).

25 ∗ ∗ W ∗ S If i∈ / ΩA(γ ), we have Ai = Aγ∗,i and Bi = Bγ∗,i. Recalling Deﬁnition 3.2, we have that P W A ∗ P S A ∗ Ai ≤ vi /λA = 1 and Bi ≤ vi /λA = 1. On the other hand, from (A.2), we deduce

A A vj vi XB ≥ ∗ ≥ ∗ . ∗ 2λA 2λA j∈ /ΩXA(γ ) P W P W A ∗ P S P S A ∗ Therefore, (Ai ≤ 2XB) ≥ Ai ≤ vi /λA = 1 and (Bi ≤ 2XB) ≥ (Bi ≤ vi /λA)= 1. We conclude that for i∈ / Ω (γ∗), A∗,B∗ are also bounded by 2X . A i i B Proposition 3.3. Given w, w,X¯ ,X > 0 (w ≤ w¯, X ≤ X ), there exist constants γ, γ,¯ λ, λ>¯ 0 such that ¯ A B ¯ A B ¯ XA,XB XA,XB ∗ (3.1) ¯ for any game GCBn (or GLBn ) satisfying Assumption (A0), any γ ∈Sn and its corresponding ∗ ∗ ∗ ∗ ∗ ¯ λA, λB , we have γ ≤γ ≤γ¯ and λ≤λA, λB ≤λ. ¯ ¯ ∗ (3.1) Proof. Let γ ∈ Sn , we consider the following cases:

A ∗ vi ∗ ∗ Case 1: If 0 <γ < min B . In this case, ΩA(γ ) = [n], and since γ is a solution of (3.1), we deduce: i∈[n] vi n o n B w 4 ∗ XB i=1 vi XB n n¯w¯ XB w γ = B ≥ 2 = ¯ . (v )2 w¯ XA n i XA XA w¯ i=1 A nw P vi n w¯ ¯ nw¯ P Here, the inequality comes directly from (2.2).

A ∗ vi ∗ ∗ Case 2: If γ ≥ max B . In this case, ΩA(γ )= ∅, and since γ is a solution of (3.1), we deduce: i∈[n] vi n o A 2 n (vi ) 4 X i=1 vB X w¯ γ∗ = B i ≤ B . X n vA X w A P i=1 i A ¯ P

A A 2 vi ∗ vj ∗ w 2 w¯ ¯ Case 3: If ∃i, j : vB ≤ γ < vB . In this case, trivially from (2.2), we have γ ∈ w¯ , w . i j ¯ 4 2 4 X w 4 w 2 X w¯ w¯ X w¯ In conclusion, by denoting γ := min B ¯ , ¯ andγ ¯ := max B , = B , XA w¯ w¯ XA w w XA w ¯ ¯ ¯ ¯ we have the conclusion on the bounds ofnγ∗. o ∗ On the other hand, from the deﬁnition of λA in (3.2), we deduce

∗ 2 2 ∗ (γ ) w 1 1 w λA ≥ ¯ + ¯ i∈Ω (γ∗) w¯ i/∈Ω (γ∗) 2XB A nw¯ nw 2XB A nw¯ X ¯ X (γ∗)2 1 1 w 3 ≥ min , · ¯ 2XB 2XB i∈[n] n w¯ (γ∗)2 1 Xw 3 ≥ min , · ¯ . 2XB 2XB w¯ Similarly, we have the upper-bound

∗ 2 3 3 ∗ 2 3 ∗ (γ ) 1 1 w¯ 1 w¯ (γ ) 1 w¯ λA ≤ max , · + = max , · . 2XB 2XB  ∗ n w ∗ n w  2XB 2XB w i∈ΩXA(γ ) ¯ i/∈ΩXA(γ ) ¯ ¯   26 3 1 1 w 3 ∗ 1 1 w¯ Similarly, we can prove that min , ∗2 ¯ ≤λ ≤ max , ∗2 ; therefore, 2XA 2γ XA w¯ B 2XA 2γ XA w ¯ n o n o ∗2 3 ∗2 3 γ 1 1 1 w ∗ ∗ γ 1 1 1 w¯ min , , , ∗ 2 ¯ ≤ λA, λB ≤ max , , , ∗ 2 . 2XB 2XB 2XA 2(γ ) XA w¯ 2XB 2XB 2XA 2(γ ) XA w ¯ ∗ ∗ ∗ ¯ Since γ ∈ [γ, γ¯], λA and λB are bounded in [λ, λ], where ¯ ¯ γ2 1 1 1 w 3 λ := min ¯ , , , 2 ¯ , ¯ (2XB 2XB 2XA 2¯γ XA ) w¯ 2 3 ¯ γ¯ 1 1 1 w¯ λ := max , , , 2 . 2XB 2XB 2XA 2γ XA w ¯ ¯

Finally, we prove a trivial result that will be used quite often in the remainder of this work.

ˆ ˆ 1 Cˆ Lemma A2. For any ε>ˆ 0 and C ≥ 1, we have that (ln(C) + 1) ln min{ε,ˆ 1/e} ≥ ln εˆ . Proof.

Case 1: Ifε< ˆ 1/e. In this case, we have ln(1/εˆ) > 1; therefore,

1 1 1 1 (ln(Cˆ) + 1) ln = (ln(Cˆ) + 1) ln = ln(Cˆ) ln + ln min{ε,ˆ 1/e} εˆ εˆ εˆ 1 > ln(Cˆ) + ln εˆ Cˆ = ln . εˆ !

Case 2: Ifε ˆ ≥ 1/e. We have ln(1/εˆ) ≤ 1; therefore,

1 1 1 Cˆ (ln(Cˆ) + 1) ln = (ln(Cˆ) + 1) ln = ln(Cˆ)+1 ≥ ln(Cˆ) + ln = ln . min{ε,ˆ 1/e} 1/e εˆ εˆ !

Appendix B. Proof of Theorem 4.3

First note that in the remainders of the paper, for any bounded, non-negative random variable Z P R ∞ (i.e., ∃C > 0 : (Z ∈ [0, C]) = 1), any measurable function g on , we write 0 g(x)dFZ (x) instead of C 0 g(x)dFZ (x) if there is no need to emphasize the bounds of Z. For the sake ofR notation, we also denote ∗ ∗ Rby A=0 the event j∈[n] Aj =0 and by A>0 its complement event, that is j∈[n] Aj > 0 . Similarly, n o ∗ n∗ o we denote by B=0 theP event j∈[n] Bj =0 and by B>0 the event j∈[n] BjP> 0 . n n Recall the notation FAi andn FBi as theo univariate marginal distributionsn correspondingo to battleﬁeld ∗ ∗ P P i ∈ [n] of the IUγ and IUγ strategies (the corresponding random variables are denoted An and Bn). Due A B∗ i i to the deﬁnition of the IUγ strategy (via Algorithm 1), for any x ≥ 0 and i ∈ [n], we have:

n n n P P FAi (x)= {Ai ≤ x} A=0 + {Ai ≤ x} A>0 \ 27 \ ∗ P P Ai · XA = (A=0)+ ∗ ≤x A>0 . (B.1) ( j∈[n] Aj ) ! \ ∗ P n Here, we have used the fact that if j∈[n] Aj = 0 (i.e., when A=0 happens), then Ai = 0 by deﬁnition and thus, P(An ≤ x) = 1 and P ({An ≤ x} A )= P(A ). Similarly to (B.1), for any x ≥ 0 and i ∈ [n], i i P =0 =0 T ∗ Bi · XB F n (x)= P (B )+P ≤x B . (B.2) Bi =0 ∗ >0 ( j∈[n] Bj ) ! \ n n P Regarding the random variables Ai and Bi (i ∈ [n]), we prepare a lemma stating several useful results as follows (its proof is given in Appendix B.1). Lemma B1. For any n and i ∈ [n], we have P n P ∗ P n P ∗ (i) (Ai =0)= (Ai = 0) and (Bi =0)= (Bi = 0). P n P n (ii) (Ai = x)= (Bi = y)=0 for any x ∈ (0, ∞)\{XA} and y ∈ (0, ∞)\{XB}.

2 n−1 2 n−1 n λ w n λ w P ¯ ¯ P ¯ ¯ (iii) (Ai = XA) ≤ 1 − λ¯ w¯2 and (Bi = XB) ≤ 1 − λ¯ w¯2 . n n Intuitively, Result (ii) states that the function FAi (resp. FBi ) is continuous on (0,XA) (resp. (0,XB)). n n The discontinuity of FAi (resp. FBi ) at XA (resp. at XB) is due to the normalization step involved in ∗ γ n n the deﬁnition of the IU strategy; note that the probability that Ai = XA (resp. Bi = XB) quickly tends to zero when n increases as has been shown in Result (iii). Finally, Result (i) shows that in some

n n ∗ ∗ cases, FAi and FBi may be discontinuous at 0. This is due to the fact that the functions FAi and FBi may be discontinuous at 0. Moreover, recall that we chose the assignments of the outputs in line 3 and

n n 7 of Algorithm 1 to be allocating zero to every battleﬁeld, i.e., the mass at 0 of FAi and FBi is added by a (negligibly small) positive probability. While other assignments do not aﬀect our results, they make

n n FAi (resp. FBi ) be discontinuous at some points diﬀering from 0 and XA (resp. XB), e.g., if in line A n 3 of Algorithm 1, we assign xi = XA/n, the distribution FAi would also be discontinuous at the point XA/n. Our choice of assignments provides more convenience in our analysis since we have to consider their discontinuity at 0 in any case. Finally, with all the preparation steps mentioned above, we are ready to prove Theorem 4.3. Theorem 4.3. Consider GCB games satisfying Assumption (A0), we have the following results:

∗ ∗ ∗ xA γ γ γ ΠA( , IUB ) ≤ ΠA(IUA , IUB )+ εWA, (4.1) ∗ ∗ ∗ γ xB γ γ ΠB (IUA , ) ≤ ΠB (IUA , IUB )+ εWB. (4.2)

∗ XA,XB (ii) Given XA,XB > 0 (XA ≤ XB), there exists C > 0 such that for any ε∈ (0, 1], in any game GCBn ∗ −2 −1 ∗ (3.1) A B with n ≥ C ε max 1, ln(ε ) , (4.1) and (4.2) hold for any γ ∈ Sn , any pure strategy x , x of players A and B respectively. Proof. In this section, we ﬁrst give a proof of Result (ii) of Theorem 4.3. Based on this, we then deduce Result (i) of Theorem 4.3. We ﬁrst look for the condition on n such that (4.1) holds for any pure strategy xA of Player A. The proof that (4.2) holds for any pure strategy of Player B under the same condition can be done similarly and thus is omitted.

28 ∗ First, we write explicitly the payoffs of Player A when Player B plays the IUγ strategy and Player A ∗ B xA γ plays either the pure strategy or the IUA strategy: n n ∗ xA γ AP n A AP n A ΠA( , IUB )= α wi (Bi = xi )+ wi (Bi < xi ), (B.3) i=1 i=1 Xn Xn ∗ ∗ γ γ AP n n AP n n ΠA(IUA , IUB )= α wi (Bi = Ai )+ wi (Bi < Ai ) i=1 i=1 X X n ∞ n ∞ A n A n P n P n = α wi (Bi = x)dFAi (x)+ wi (Bi < x)dFAi (x). (B.4) i=1 0 i=1 0 X Z X Z We then prepare a useful lemma, its proof is given in Appendix B.2. Intuitively, this lemma shows that as n is large enough, we can prove (4.1) without the need of analyzing separately the case where players get tie allocations (that is our results hold regardless of the tie-breaking-rule parameter α). Lemma B2. Given w, w,X¯ ,X > 0 (w ≤ w¯, X ≤ X ), there exists a constant C∗ > 0 such that for ¯ A B ¯ A B 0 ∗ −1 XA,XB ∗ (3.1) any ε ∈ (0, 1] and n ≥ C0 max 1, ln(ε ) , for any game GCBn and γ ∈ Sn the following inequality is a sufficient condition of (4.1): n n ∞ A A A ε v FBn x ≤ v FBn (x)dFAn (x)+ . (B.5) i i i i i i 2 i=1 i=1 Z0 X X In the remainders of the proof, we focus on (B.5) and look for the condition of n such that it holds; this will be done in the following five steps. After that, from Lemma B2, we can conclude that (4.1) also holds with the corresponding condition on n.

∗ ∗ Step 1: Prove that {FAi }i is optimal against {FBi }i.

XA,XB xA ∗ (3.1) Lemma B3. In any game GCBn , for any pure strategy of Player A and γ ∈ Sn , we have

n n ∞ A ∗ A A ∗ ∗ vi FBi xi ≤ vi FBi (x)dFAi (x). (B.6) i=1 i=1 Z0 X X The proof of Lemma B3 is given in Appendix B.3. This lemma can be interpreted as follows: if the allocation ∗ of Player B to battlefield i follows the distribution FBi , then it is optimal for Player A to play such that ∗ her allocation at this battlefield follows FAi (we do not know if it is possible to construct a mixed strategy ∗ such that Player A’s allocation at battlefield i follows FAi for all i ∈ [n]; however, this does not affect our results in this work). Using this lemma, we will analyze the validity of (B.5) by proving that, as n → ∞, the terms in (B.5) respectively converge toward the terms in (B.6). To do this, we consider the next step.

n n ∗ ∗ Step 2: Prove that FAi and FBi uniformly converge toward FAi and FBi as n increases.

Lemma B4. Given w, w,X¯ A,XB > 0 (w ≤ w¯, XA ≤ XB), there exists C1 > 0 such that for any ε1 ∈ (0, 1], −2 ¯ −1 ¯ n ≥ C1ε1 max 1, ln(ε1 ) and i ∈ [n],

n ∗ n ∗ sup FAi (x) − FAi (x) ≤ ε1 and sup FBi (x) − FBi (x) ≤ ε1. (B.7) x∈[0,∞) x∈[0,∞)

A proof of this lemma is given in Appendix B.4. The main intuition of this res ult comes from the fact n n ∗ that Ai (resp. Bi ) is the normalization of Ai ,i ∈ [n] (except for the special cases of the events A=0 and ∗ ∗ B=0) and the use of concentration inequalities on the random variables j∈[n] Aj (and j∈[n] Bj ). In this work, we apply the Hoeffding’s inequality (Theorem 2, Hoeffding (1963)) to obtain the rate of convergence P P indicated here in Lemma B4. 29 Step 3: Prove that the left-hand-side of (B.5) converges toward the left-hand-side of (B.6). Take C1 as indi- ∗ 2 ∗ C1 −1 4 1 26 cated in Lemma B4, we define C1 :=16C1(ln(4)+1) and deduce that ε2 max 1, ln(ε ) ≥C1 ε ln ε 1 . min{ 4 , e } ∗ −2 −1 −2 −1 Therefore, take ε1 := ε/4, for any n ≥ C1 ε max 1, ln(ε ) , we have n ≥ C1 ε1 max 1, ln(ε1 ) ; apply Lemma B4, for any pure strategy xA of Player A, we have n n n

A n A A ∗ A A n ∗ vi FBi xi − vi FBi xi ≤ vi sup FBi (x) − FBi (x) x∈[0,∞) i=1 i=1 i=1 X X Xn ε ε ≤ vA = . (B.8) i 4 4 i=1 X

Step 4: Prove that the right-hand-side of (B.5) converges toward the right-hand-side of (B.6). We consider the diﬀerence of the involved terms as follows.

n ∞ n ∞ A A n n ∗ ∗ vi FBi (x)dFAi (x) − vi FBi (x)dFAi (x) i=1 0 i=1 0 X Z X Z n ∞ n ∞ ∞

A n ∗ n A ∗ n ∗ ∗ ≤ vi FBi (x) − FBi (x) dFAi (x)+ vi FB i (x)dFAi (x) − FBi (x)dFAi (x) . (B.9) i=1 Z0 i=1 Z0 Z0 X X ∗ Let us deﬁne C2 := C1 · 64(ln(8) + 1) (again, C1 is the constant indicated in Lemma B4), we have that ∗ −2 −1 8 2 1 27 ∗ −2 −1 C2 ε max 1, ln(ε ) ≥ C1 ε ln ε 1 . Therefore, take ε1 := ε/8, for any n ≥ C2 ε max 1, ln(ε ) , min{ 8 , e } −2 −1 we have n ≥ ε1 max 1, ln(ε1 ) and by Lemma B4, we have

n ∞ n ∞ n A A ε A ε v FBn (x) − FB∗ (x) dFAn (x) ≤ v dFAn (x)= v . (B.10) i i i i i 8 i i 8 i=1 Z0 i=1 Z0 i=1 X X X

Now, we need to ﬁnd an upper-bound of the second term in the right-hand-side of(B.9). To do this, we present a lemma, called Lemma B5 (stated below), that is based on the portmanteau lemma (see, e.g., Van der Vaart (2000)) regarding the weak convergence of a sequence of measures. Note importantly that by ∗ n a direct application of the portmanteau lemma (since FBi is Lipschitz continuous and from Lemma B4, FAi ∞ ∞ uniformly converges to F ∗ ), we can prove that F ∗ (x)dF n (x) converges toward F ∗ (x)dF ∗ (x) Ai 0 Bi Ai 0 Bi Ai as n → ∞; however, note that the convergence rate obtained by doing this is large due to the fact that ∗ ∗ A R R the Lipschitz constant of FBi (that is λA/vi ) increases as n increases. To obtain a better convergence rate as indicated in Lemma B5, we exploit the properties of the involved functions that allow us to use the telescoping sum trick (see Appendix B.5 for more details).

Lemma B5. Given w, w,X¯ A,XB > 0 (w ≤ w¯, XA ≤ XB), there exists a constant C2 > 0 such that for ¯ −2 −¯1 any ε2 ∈ (0, 1], n ≥ C2 · ε2 max 1, ln(ε2 ) and i ∈ [n], we have ∞ ∞ ∗ n ∗ ∗ FBi (x)dFAi (x) − FBi (x)dFAi (x) ≤ ε2. (B.11) Z0 Z0

26 ∗ −2 −1 4 2 −1 4 2 4 This is due to C · ε max 1, ln(ε ) = C1 · (ln(4) + 1) max 1, ln(ε ) ≥ C1 · ln ; here, we have applied 1 ε ε ε Lemma A2 withε ˆ := ε and Cˆ := 4; moreover, ε = min{ ε , 1 } since ε ≤ 1; thus, we can rewrite ln 4 = ln 1 . 4 4 e ε min{ε/4,1/e} 2 2 27This is due to C∗ · ε−2 max 1, ln(ε−1) = C 8 · (ln(8) + 1) max 1, ln(ε−1) ≥ C · 8 ln 8 ; here, we have applied 2 1 ε 1 ε ε Lemma A2 withε ˆ := ε and Cˆ := 8; moreover, ε = min{ ε , 1 } since ε ≤ 1; thus, we can rewrite ln 8 = ln 1 . 8 8 e ε min{ε/8,1/e}

30 ∗ 2 The proof of Lemma B5 is given in Appendix B.5. Based on this constant C2, we deﬁne C3 := 8 C2(ln 8+1). ∗ −2 −1 −2 −1 28 ∗ −2 −1 Now, take ε2 := ε/8, C3 ε max 1, ln(ε ) ≥ C2ε2 max 1, ln(ε2 ) ; thus, for n ≥ C3 ε max 1, ln(ε ) , we have n ≥ C ε−2 max 1, ln(ε−1) and by Lemma B5, we deduce 2 2 2 ∞ ∞ ∗ n ∗ ∗ FBi (x)dFAi (x) − FBi (x)dFAi (x) ≤ ε/8. Z0 Z0

∗ ∗ −2 −1 Combine this with (B.9) and (B.10), for any n = max{C2 , C3 }ε max 1, ln(ε ) , we have

n ∞ n ∞ n n A A A A ε n n ∗ ∗ vi FBi (x)dFAi (x) − vi FBi (x)dFAi (x) ≤ vi ε/8+ vi ε/8= . (B.12) 0 0 4 i=1 Z i=1 Z i=1 i=1 X X X X

∗ ∗ ∗ −2 −1 xA Step 5: Conclusion. For any n ≥ max{C1 , C2 , C3 }ε max 1, ln(ε ) and any pure strategy of Player A, we conclude that n n A A A A ε v FBn x ≤ v FB∗ x + (from (B.8)) i i i i i i 4 i=1 i=1 X X n ∞ A ε ≤ v FB∗ (x)dFA∗ (x)+ (from (B.6)) i i i 4 i=1 0 X Z n ∞ A ε ε ≤ v FBn (x)dFAn (x)+ + (from (B.12)) i i i 4 4 i=1 0 X Z n ∞ A ε = v FBn (x)dFAn (x)+ . i i i 2 i=1 0 X Z ∗ ∗ ∗ ∗ ∗ This is exactly (B.5); therefore, apply Lemma B2 and denote C(4.1) :=max{C0 , C1 , C2 , C3 }, we have proved ∗ −2 −1 that (4.1) holds for any n ≥ C(4.1)ε max 1, ln(ε ) . Similarly, we can prove that there exists a constant ∗ ∗ −2 −1 ∗ ∗ ∗ C(4.2) such that (4.2) holds for any n ≥ C(4.2) ε max 1, ln(ε ) . Finally, deﬁne C := max{C(4.1), C(4.2)}, we conclude the proof for Result (ii) of Theorem 4.3. Now, to obtain Result (i) of Theorem 4.3, we prove that it is implied by Result (ii) of Theorem 4.3. Note that the constant C∗ found in the Result (ii) of Theorem 4.3 does not depend on neither n nor ε. Moreover, the function

ξ : (0, ∞) → (0, ∞) 1 ε˜ 7→ C∗ε˜−2 ln . min{ε,˜ 1/e} is continuous and increases to inﬁnity when ε tends to zero. Therefore, for any n ≥ 1, there exists an ε> 0 ∗ −2 −1 XA,XB such that n = C ε max 1, ln(ε ) . Now, apply Result (ii), (4.1) and (4.2) hold in the game GCBn ∗ (3.1) xA xB for any γ ∈ Sn and pure strategies , . We conclude the proof by notice that if ε ≥ 1/e, we have ∗ −2 ∗ −1/2 1 1 n = C ε and thus ε = n/C = O(n ); on the other hand, if ε< e , we have ln ε > 1 that induces ∗ −2 1 ∗ −2 C∗ 1 n C∗ 1 C∗ n ˜ −1/2 n = C ε ln ε ≥ C ε p≥ ε , thus, ε ≤ C∗ . We deduce that ε = n ln ε ≤ n ln C∗ = O(n ). q q

28 ∗ −2 −1 8 2 −1 8 2 8 Once again, apply Lemma A2, C ε max 1, ln(ε ) = C2 (ln(8)+1) max 1, ln(ε ) ≥ C2 ln ; moreover, 3 ε ε ε we have ε := ε =min ε , 1 . 2 8 8 e

31 Appendix B.1. Proof of Lemma B1

∗ ∗ n ∗ (i) Assuming Ai = 0, if j6=i Aj = 0 then Ai = 0 (due to line 3 of Algorithm 1) and if j6=i Aj > 0 then An = A∗ A∗ = 0. Reversely, assuming An = 0, then regardless whether A∗ = 0 i i j∈[nP] j i Pj∈[n] j or A∗ > 0, we have A∗ = 0. Therefore, An =0 ⇔ A∗ = 0 for any n and i ∈ [n]. Similarly, we j∈[n] j P i i i P can prove that Bn =0 ⇔ B∗ = 0. P i i n n (ii) The results are trivial in cases where x>XA and y>XB due to the deﬁnition of Ai and Bi (that n n guarantees that with probability 1, Ai ≤ XA and Bi ≤ XB). In the following, we consider the case ∗ where x ∈ (0,XA). For any n, i ∈ [n], we denote Zi := j6=i Aj and obtain: P n (Ai = x) P P n = {Ai = x} A>0 (since x> 0) \ P ∗ x ∗ = Ai = Aj A>0 XA j∈[n] X \ P ∗ x x ∗ = Ai 1 − = Aj A>0 X X j6=i A A Z · x X \ =P A∗ = i A (note that X − x> 0) i X − x >0 A A \ z · x ≤P({A∗ = Z =0} ∩ A )+ P A∗ = dF (z) i i >0 i X − x Zi Zz>0 A P ≤ (A=0 ∩ A>0)+ 0 dFZi (z) Zz>0 =0. Here, the second-to-last inequality comes from the fact that zx > 0, ∀z > 0, ∀x ∈ (0,X ) and XA−x A P ∗ P n (Ai = a) =0 for any a> 0. Similarly, we can prove that (Bi = y)=0 for any y ∈ (0,XB). (iii) We have

P(An = X )= P A∗ = A∗ A i A  i j  >0  jX∈[n]  \     ≤ P A∗ =0  j  Xj6=i P ∗  ∗ = Aj =0 (since Aj , j ∈ [n] are non-negative and independent). j6=i Y ∗ P ∗ ∗ S Now, if there exists j 6= i such that j ∈ ΩA(γ ), then (Aj = 0) = 0 due to the fact that Aj = Aγ∗,j S P ∗ and the deﬁnition of Aγ∗,j (see (3.4)). In this case, j6=i Aj =0 = 0. On the other hand, if ∗ ∗ W j∈ / Ω (γ ) for any j 6= i, then A = A ∗ for j 6= i; therefore, A j γ ,j Q n−1 B A B A w n−1 v v v v λ∗ λ ¯ λ w2 P A∗ =0 = j − j j = 1 − j B ≤ 1 − ¯ nw¯ = 1 − ¯ ¯ . j λ∗ λ∗ λ∗ vB λ∗ λ¯ w¯ λ¯ w¯2 j6=i j6=i " B A !, B # j6=i j A ! nw ! Y Y Y ¯ A B Here, to obtain the last equality, we use (2.2) for the bounds of vj , vj and Proposition 3.3 for the ∗ ∗ bounds of λA, λB. 2 n−1 n λ w P ¯ ¯ Similarly, we can obtain (Bi = XB) ≤ 1 − λ¯ w¯2 . 32 Appendix B.2. Proof of Lemma B2

Fix ε ∈ (0, 1] and assume that (B.5) is satisﬁed, we prove that (4.1) also holds by comparing the terms in (B.5) with the terms in (4.1). First, due to the fact that α ≤ 1, we can ﬁnd a lower bound of the left-hand side of (B.5) as follows:

n n n

A n A AP n A AP n A vi FBi xi = vi (Bi = xi )+ vi (Bi < xi ) i=1 i=1 i=1 X Xn Xn AP n A AP n A ≥ α vi (Bi = xi )+ vi (Bi < xi ) i=1 i=1 X ∗ X xA γ = ΠA( , IUB )/WA. (B.13)

Now, we turn our focus to the right-hand-side of (B.5), we can rewrite the involved term as follows.

n ∞ n ∞ n ∞ A A n A n n n P n P n vi FBi (x)dFAi (x)= vi (Bi = x)dFAi (x)+ vi (Bi < x)dFAi (x). i=1 0 i=1 0 i=1 0 X Z X Z X Z n ∗ ∞ A A γ We observe that v F n (x)dF n (x) is very similar to the expression of Π (x , IU ) stated in (B.4). 0 i Bi Ai A B i=1 The main differenceP R lies at the coefficient of the term related to the tie cases that is the tie-breaking parameter α. Therefore, we consider the following two cases of α: A A Case 1: α = 1. Forany n, divide two sides of (B.4) (with α = 1)by WA and recall that vi := wi /WA, ∀i, n ∗ ∗ ∞ A γ γ we trivially have v F n (x)dF n (x) = Π (IU , IU )/W . 0 i Bi Ai A A B A i=1 P n n−1 Case 2: α< 1.P DueR to Results (ii) and (iii) of Lemma B1, for any x> 0, we have (Bi = x) ≤ D λ w2 ¯ ¯ where we define D := 1 − λ¯ w¯2 < 1. We consider two cases of α as follows. • ˆ 1 29 ˆ −1 If 2(1−α) ≤ 1, define C1 := ln(1/D) +1 > 0, we have that C1 max 1, ln(ε ) ≥ logD ε + 1; therefore, ˆ −1 for any n ≥ C1 max 1, ln(ε ) , we obtain n − 1 ≥ logD ε and ε Dn−1 ≤ DlogD ε = ε ≤ (note that D< 1 and in this case 2(1 − α) ≤ 1). 2(1 − α)

• ˆ 1 ln(2−2α) ˆ −1 ε 30 If 2(1−α)>1, deﬁne C2 := ln(1/D) + ln(1/D) +1>0; we have C2 max 1, ln(ε ) ≥ logD 2(1−α) + 1. ˆ −1 ε We conclude that for any n ≥ C2 max 1, ln(ε ) , we obtain n − 1 ≥ logD 2(1 −α) and log ε ε Dn−1 ≤ D D 2(1−α) = . 2(1 − α)

∗ ˆ ˆ ∗ −1 Let us deﬁne C0 = max{C1, C2} > 0, we conclude that for any α< 1, n ≥ C0 max 1, ln(ε ) , i ∈ [n] and x> 0, we have ε P(Bn = x) ≤ Dn−1 ≤ . (B.14) i 2(1 − α)

29If ε < 1/e, then ln(1/ε) > 1 and Cˆ max 1, ln(ε−1) = ln(1/ε) + ln(1/ε) > log ε + 1; otherwise, if ε ≥ 1/e, we have 1 ln(1/D) D ln(1/ε) ≤ 1 and Cˆ max 1, ln(ε−1) = 1 + 1 ≥ ln(1/ε) + 1 = log ε + 1 (note that ln(1/D) > 0 since D < 1). 1 ln(1/D) ln(1/D) D 30 −1 1 If ε< 1/e, we have Cˆ2 max 1, ln(ε ) = log ε + log (2 − 2α) + 1 ln > log ε−log (2 − 2α)+1; otherwise, if D 1/D ε D D ε≥1/e, we have Cˆ max 1, ln(ε−1) = Cˆ ≥ ln(1/ε) + ln(2−2α) +1 ≥ log ε−log (2−2α)+1 (since 1 ≥ ln 1 ). 2 2 ln(1/D) ln(1/D) D D ε 33 P n n P n P n P ∗ P ∗ 31 Note also that (Ai = Bi =0)= (Ai = 0) (Bi =0)= (Ai = 0) (Bi =0)=0, ∀i, we conclude ∗ −1 that when α< 1, for any n ≥ C0 max 1, ln(ε ) , we have n ∞ A n n vi FBi (x)dFAi (x) i=1 0 X Z n ∞ n ∞ A n A n P n P n = vi (Bi < x)dFAi (x)+ α vi (Bi = x)dFAi (x) "i=1 0 i=1 0 # X Z X Z n ∞ AP n n + (1 − α) vi (Bi = x)dFAi (x) i=1 Z0 n X n ∗ ∗ γ γ A ε A n n =ΠA(IU , IU )/WA + (1 − α) v dFAn (x)+(1 − α) v P(A = B = 0) A B i 2(1 − α) i i i i i=1 Z(0,∞) i=1 Xn X ∗ ∗ γ γ A ε =ΠA(IU , IU )/WA + (1 − α) v dFAn (x)+0 A B i 2(1 − α) i i=1 Z(0,∞) ∗ ∗ Xε ≤Π (IUγ , IUγ )/W + (1 − α) A A B A 2(1 − α) γ∗ γ∗ =ΠA(IUA , IUB )/WA + ε/2. ∗ −1 In conclusion, regardless of the value of α, for any n ≥ C0 max 1, ln(ε ) , we have

n ∞ ∗ ∗ A γ γ n n vi FBi (x)dFAi (x) ≤ ΠA(IUA , IUB )/WA + ε/2. (B.15) i=1 0 X Z ∗ −1 Combine (B.13), (B.15) and the assumption that (B.5) holds, for any n ≥ C0 max 1, ln(ε ) , we have ∗ ∗ ∗ xA γ (B.13) n (B.5) n ∞ (B.15) γ γ ΠA( , IUB ) A A A ΠA(IUA , IUB ) ≤ v FBn x ≤ v FBn (x)dFAn (x)+ε/2 ≤ +ε. W i i i i i i W A i=1 i=1 Z0 A X X By multiplying both sides of this inequality by WA, we obtain (4.1).

Appendix B.3. Proof of Lemma B3

∗ ∗ We compute the right-hand-side of (B.6) based on the deﬁnition of FAi and FBi (see Deﬁnition 3.2). n ∞ ∞ ∞ A A A v FB∗ (x)dFA∗ (x)= v F W (x)dF S (x)+ v F S (x)dF W (x) i i i i Bγ∗,i Aγ∗,i i Bγ∗,i Aγ∗,i i=1 0 ∗ 0 ∗ 0 X Z i∈ΩXA(γ ) Z i/∈ΩXA(γ ) Z vB A B vA i vi vi i λ∗ ∗ − ∗ λ∗ B A λA λB x 1 A A x 1 = vi A + A B dx + vi A B dx vi vi vi vi vi ∗ 0  ∗ ∗  ∗ ∗ 0 ∗ ∗ i∈ΩA(γ ) Z λ λ λ i/∈ΩA(γ ) Z λ λ X A A B X A B B ∗  A vi γ A 2 1 = vi 1 − A + (vi ) ∗ B . (B.16) ∗ 2vi ∗ 2γ vi i∈ΩXA(γ ) i/∈ΩXA(γ ) On the other hand, for any pure strategy xA of Player A, we have: n A A A A A A v FB∗ x = v F W x + v F S x i i i i Bγ∗,i i i Bγ∗,i i i=1 i∈Ω (γ∗) i/∈Ω (γ∗) X XA XA

31 ∗ P ∗ ∗ P ∗ Note that if i ∈ ΩA(γ ) then (Ai =0)=0, if i∈ / ΩA(γ ) then (Bi = 0) = 0 (see (3.4)-(3.8)); therefore, P ∗ P ∗ (Ai =0) (Ai =0)=0, ∀i. 34 A B vi vi ∗ − ∗ A ∗ A ∗ A λA λB xi λA A xi λA ≤ vi A + A + vi A vi ∗  ∗ vi  ∗ vi i∈ΩA(γ ) λ i/∈ΩA(γ ) X A X A B  n vi vi ∗ ∗ A ≤ − λA + λAXA (since xi ≤ XA) ∗ ∗ i=1 ∗ λA λB i∈ΩXA(γ ) X B ∗ A vi γ A 2 1 = vi 1 − A + (vi ) ∗ B . (B.17) ∗ 2vi ∗ 2γ vi i∈ΩXA(γ ) i/∈ΩXA(γ )

Here, to obtain the last equality, we use (A.1) to rewrite XA. Finally, from (B.16) and (B.17), we conclude that (B.6) holds for any xA and γ∗.

Appendix B.4. Proof of Lemma B4

n P P Since the definition of FAi involves (A=0) (see (B.1)), we first look for an upper-bound of (A=0). For ∗ (3.1) ∗ ∗ S P ∗ any n and γ ∈ Sn , if ΩA(γ ) 6= ∅, i.e., there exists i such that Ai = Aγ∗,i, then (Ai = 0) = 0 due S P P ∗ to the definition of Aγ∗,i (see (3.4)); in this case, (A=0) = j∈[n] Aj =0 = 0. On the other hand, if ∗ ∗ W Ω (γ )= ∅, then A = A ∗ for any j ∈ [n]; therefore, A j γ ,j Q B A B A ∗ 2 n ∗ vj vj vj vj λB λ w P(A=0)= P A =0 = − = 1 − ≤ 1 − ¯ ¯ . (B.18) j λ∗ λ∗ λ∗ vB λ∗ λ¯ w¯2 j∈[n] j∈[n] " B A !, B # j∈[n] j A ! Y Y Y λ w2 ¯ ¯ Here, the last inequality comes directly from (2.2) and Proposition 3.3. Recall the notation D := 1 − λ¯ w¯2 ˜ ln(4)+1 ˜ −1 ε1 32 ˜ −1 and define C0 := ln(1/D) >0, we have C0 max 1, ln(ε1 ) ≥logD 4 . Therefore, for any n≥C0 max 1, ln(ε1 ) we have n ≥ log (ε /4) and since D< 1 we have: D 1

n log (ε1/4) P(A=0) ≤ D ≤ D D = ε1/4. (B.19)

ε w P 1 ¯ Now, we look for an upper-bound of (A>0). For any n, deﬁne the constants ǫn := 4 nw¯λ¯ and ¯ 2 τ := 1 w¯ 1 +1 = 1 4λ w¯ +1 , we consider the following term for any i ∈ [n]: XA nwλ ǫn XA ε1λ w ¯ ¯ ¯ ¯ ∗ P ∗ Ai Ai − ∗ XA >ǫn A>0 ( j∈[n] Aj ) ! \ P ∗ P ∗ Ai ≤ Ai − ∗ XA >ǫn A>0 ( j∈[n] Aj ) !

\ ∗ P ∗ ∗ ≤ P A A − XA >ǫn A i j∈[n] j j∈[n] j X X P ∗ ∗ ∗ = Ai Aj − XA >ǫnXA −ǫn XA − Aj j∈[n] j∈[n] X X ∗ ∗ ∗ ≤ P A A − XA >ǫnXA −ǫn A − XA i j∈[n] j j∈[n] j X X P ∗ ǫnXA = Aj − XA > j∈[n] A∗ +ǫ i n X

32 −1 1 −1 ln(4/ε1) ε1 This is due to the fact that C˜0 · max 1, ln(ε ) = (ln(4) + 1) max 1, ln(ε ) ≥ = log ; here, we n 1 o ln(1/D) n 1 o ln(1/D) D 4 have applied Lemma A2 (see Appendix A) forε ˆ := ε1 and Cˆ := 4. 35 ∗ ǫnXA ≤ P A − XA > j∈[n] j w¯ + ǫ nwλ n ! X ¯ ¯ ∗ 1 = P A − XA > . (B.20) j∈[n] j τ X ∗ Here, the second-to-last inequality comes from the fact that for any i ∈ [n], Ai is upper-bounded by either vA/λ∗ or vB /λ∗ (see (3.4) and (3.6)), thus, it is bounded byw/ ¯ (nwλ) (due to (2.2) and Proposition 3.3). i A i B ¯ ¯ n E ∗ Recall that XA = Aj (see Lemma A1-(iii)), we use the Hoeﬀding’s inequality (see e.g., Theorem "j=1 # 2, Hoeﬀding (1963)) on theP random variables A∗,i ∈ [n] (bounded in [0, w/¯ (nwλ)]) to obtain i ¯ ¯

−2 1 ∗ 1  τ 2  P Aj − XA > ≤2exp 2  τ  w¯ j∈[n] X  nwλ   j∈[n] ¯ ¯       P 2  −2n λw =2 exp ¯ ¯ . (B.21) τ 2 w¯ " #

2 2 ˜ 1 4 λ¯ w¯2 1 w¯ 33 Now, we deﬁne C := 2 + (ln 8 + 1) ; due to the deﬁnition of τ, we have that 1 2 XA λ w XA wλ ¯ ¯ 2 ¯ ¯ ˜ 1 −1 τ 2 8 w¯ ˜ −2 −1 C1 · 2 max 1, ln(ε1 ) ≥ ln ; therefore, for any n ≥ C1ε1 max 1, ln(ε1 ) , we can deduce ε1 2 ε1 wλ 2 ¯ ¯ 2n wλ 8 that 2 ¯ ¯ ≥ln and thus, τ w¯ ε1 −2n λw 2 8 ε 2exp ¯ ¯ ≤ 2exp − ln = 1 . (B.22) τ 2 w¯ ε 4 " # 1 Combining (B.20), (B.21) and (B.22), we deduce that

∗ P ∗ Ai ε1 ˜ −2 −1 Ai − ∗ XA >ǫn A>0 ≤ , ∀n ≥ C1ε1 max 1, ln(ε1 ) . (B.23) ( j∈[n] Aj ) ! 4 \ Finally, note that for anyP n, i ∈ [n] and x ≥ 0, we also have

∗ P Ai · XA ∗ ≤ x A>0 ( j∈[n] Aj ) ! \ P ∗ ∗ P Ai XA ∗ Ai XA =  ∗ ≤x Ai − ∗ ≤ǫn A>0 (B.24) Aj Aj  j∈[n]   j∈[n]   \ \   P P      ∗ ∗ P Ai ·XA ∗ Ai XA +  ∗ ≤ x Ai − ∗ >ǫn A>0 Aj Aj  j∈[n]   j∈[n]   \ \   P P      2 2 2 33 ˜ 1 −1 1 1 4 λ¯ w¯2 1 −1 w¯ τ 2 8 w¯ This is due to C1 · 2 max 1, ln(ε ) = 2 + (ln(8)+1) max 1, ln(ε ) ≥ ln ; ε n 1 o 2 h XA ε1 λ w ε1 i n 1 o wλ 2 ε1 wλ 1 ¯ ¯ ¯ ¯ ¯ ¯ here, we have used Lemma A2 withε ˆ := ε1 and Cˆ := 8 and the fact that 1/ε ≥ 1.

36 ∗ P ∗ P ∗ Ai XA ≤ ({Ai ≤ x+ǫn})+ Ai − ∗ >ǫn A>0 . (B.25) Aj  j∈[n]   \   P   ˜ ˜  −2 −1 Therefore, deﬁne C1 := max{C0, C1}, for any n ≥ C1ε1 max 1, ln(ε1 ) , from (B.1), we have

n ∗ FAi (x) − FAi (x) ∗ A · XA P P i ∗ = (A=0)+ ∗ ≤ x A>0 − FAi (x) ( j∈[n] Aj ) ! \ P ∗ ε1 ∗ ∗ Ai XA ≤ + P ({A ≤ x+ǫ })+ P A − >ǫ A − F ∗ (x) (due to (B.19) and (B.25)) i n  i ∗ n >0 Ai 4 Aj  j∈[n]   \  ε1 ε1  P  ≤ + FA∗ (x + ǫn)+ − FA∗(x)  (due to (B.23)). (B.26) 4 i 4 i 

∗ ∗ The ﬁnal step is to bound the term FAi (x + ǫn) − FAi (x); we present this as the following lemma.

ǫλ∗ ∗ ∗ B Lemma B6. For any ǫ> 0, n> 0, i ∈ [n] and x ∈ [0, ∞), we have FA (x + ǫ) − FA (x) ≤ B . i i vi

Proof. ∗ ∗ S If i ∈ ΩA(γ ), then Ai = Aγ∗,i and

∗ ∗ ∗ B (x+ǫ)λB xλB ǫλB vi B − B = B , if 0 ≤ x< ∗ − ǫ vi vi vi λB ∗ ∗ B B xλB ǫλB vi vi S S FA ∗ (x + ǫ) − FA ∗ (x)=  1 − B ≤ B , if ∗ − ǫ ≤ x ≤ ∗ . (B.27) γ ,i γ ,i vi vi λB λB  B B  ǫvi vi  1 − 1 ≤ ∗ , if x> ∗ λB λB  ∗ ∗  W On the other hand, if i∈ / ΩA(γ ), then Ai = Aγ∗,i and we have

∗ ∗ ∗ A (x+ǫ)λB xλB ǫλB vi B − B = B , if 0 ≤ x< ∗ − ǫ vi vi vi λA vB vA i i ∗ ∗  λ∗ − λ∗ A A B A xλB ǫλB vi vi W W ∗ ∗ FA ∗ (x + ǫ) − FA ∗ (x)=  1 − vB − B ≤ B , if λ − ǫ ≤ x ≤ λ . (B.28) γ ,i γ ,i  i vi vi A A  λ∗  B  B A ǫvi vi 1 − 1 ≤ λ∗ , if x> λ∗  B A  

∗ B Combine this lemma with (B.26) and recall the deﬁnition of ǫn (which induces that ǫnλB/vi ≤ ε1/2), we −2 −1 −2 −1 n ∗ conclude that FAi (x)−FAi (x) ≤ ε1 for any n ≥ C1ε1 max 1, ln(ε1 ) . Similarly, for n ≥ C1ε1 max 1, ln(ε1 ) n ∗ and i ∈ [n], we can deduce that FAi (x)−FAi (x)≥−ε1 for any x ∈ [0, ∞). We conclude that for any −2 −1 n ∗ n ∗ n ≥ C1ε1 max 1, ln(ε1 ) , sup FAi (x) − FAi (x) ≤ ε1. The inequality sup FBi (x)−FBi (x) ≤ ε1 x∈[0,∞) x∈[0,∞) can be proved in a similar way.

Appendix B.5. Proof of Lemma B5 E ∞ E In this proof, we will use the notation f(X) := 0 f(z)dFZ (x) and I f(X) := I f(z)dFZ (x) for any λ¯ w¯2 function f, random variable Z and interval I. To simplify the notation, let us define M := λ w2 and we R R ¯ ¯ B ∗ ε2 denote by Ii the interval 0, vi /λB . For any ε2 ∈ (0, 1], we define δ2 := 6+2M . We first consider the case 37 ∗ ∗ W ∗ n ∗ where i ∈ ΩA(γ ), i.e., Bi = Bγ ,i. Note that FAi (x)= FAi (x) =1 for any x ≥ 2XB (see Lemma A1-(iv)); the left-hand-side of (B.11) can be rewritten as follows.

E ∗ n E ∗ ∗ FBi (Ai ) − FBi (Ai )

∗ n ∗ ∗ = FBi (x)dFAi (x) − FBi (x)dFAi (x) [0,2XB ] [0,2XB ] Z Z

n ∗ E B ∗ W E B ∗ W n ∗ ≤ [0,v /λ ]FB ∗ (Ai ) − [0,v /λ ]FB ∗ (Ai ) + dFA (x)− dFA (x) i B γ ,i i B γ ,i i i

vB /λZ∗ ,2X vB /λZ∗ ,2X ( i B B ] ( i B B ]

n ∗ B ∗ B ∗ E W E W n n ∗ ∗ = Ii FB ∗ (Ai ) − Ii FB ∗ (Ai ) + FA (2XB) − FA (vi /λB) − FA (2XB)+ FA (vi /λB) γ ,i γ ,i i i i i n ∗ E W E W n ∗ ≤ Ii FB ∗ (Ai ) − Ii FB ∗ (Ai ) + 2 sup FAi (x) − FAi (x) . (B.29) γ ,i γ ,i x∈[0,∞)

M We now focus on bounding the first term in (B.29). Let us define K := and K + 1 points xj such δ2 B vi K that x0 := 0 and xj := xj−1 + ∗ , ∀j ∈ [K]. In other words, we have the partitions Ii = Pj λB K j=1 where we denote by P the interval [x , x ] and by P the interval (x , x ] for j = 2,...,K. For any 1 0 1 j j−1 j S ′ W x, x ∈ Pj , ∀j ∈ [K], from the definition of Bγ∗,i, we have ∗ ∗ B ¯ ′ λA ′ λA vi 1 λnw¯ w¯ 1 M |FBW (x) − FBW (x )| = |x − x |≤ · · ≤ · · = ≤ δ2. (B.30) γ∗,i γ∗,i vA vA λ∗ K w nwλ K K i i B ¯ ¯ ¯ K W Now, we define the function g(x) := FB ∗ (xj )1Pj (x). Here, 1Pj is the indicator function of the set Pj . j=1 γ ,i From this definition and Inequality (B.30),P we trivially have |F W (x) − g(x)|≤ δ2, ∀x ∈ Ii. Therefore, Bγ∗,i

n n E E n n Ii F W (A ) − Ii g(A ) ≤ F W (x) − g(x) dFA (x) ≤ δ2dFA (x) ≤ δ2, (B.31) Bγ∗,i i i Bγ∗,i i i I I Z i Z i

E ∗ E ∗ ∗ ∗ Ii F W (A ) − Ii g(A ) ≤ F W (x) − g(x) dFA (x) ≤ δ2dFA (x) ≤ δ2. (B.32) Bγ∗,i i i Bγ∗,i i i I I Z i Z i

j W W W Now, we note that for any j ∈ [K], FB ∗ (xj ) = FB ∗ (xm) − FB ∗ (xm−1) ; here, for the sake of γ ,i m=0 γ ,i γ ,i notation, we denote by x−1 an arbitrary negative numberh (that is F W (x−1) = 0).i Using this, we have: P Bγ∗,i E n E ∗ | Ii g (Ai ) − Ii g (Ai )| K n ∗ W E E = FB ∗ (xj ) Ii 1Pj (Ai ) − Ii 1Pj (Ai ) γ ,i j=1 X

K n ∗ W P P = FB ∗ (xj )[ (Ai ∈ Pj )− (Ai ∈ Pj )] γ ,i j=1 X

K j n ∗ W W P P = FB ∗ (xm) − FB ∗ (xm−1) [ (Ai ∈ Pj )− (Ai ∈ Pj )] γ ,i γ ,i j=1 m=0 ! X X h i

K n ∗ W W P P ≤ FB ∗ (x0) − FB ∗ (x−1) [ (Ai ∈ Pj )− (Ai ∈ Pj )] γ ,i γ ,i j=1 h i X

K K n ∗ W W P P + FB ∗ (xm) − FB ∗ (xm−1) [ (Ai ∈ Pj )− (Ai ∈ Pj )] . (B.33)  γ ,i γ ,i  m=1 j=m X h i X n ∗ 34  Note that P (A ∈ Pj ) −P (A ∈ Pj )=FAn (xj )−FAn (xj−1)−FA∗ (xj )+FA∗ (xj−1). Moreover, due to the i i i i i i fact that x0 = 0 and F W (x−1) = 0, we can rewrite the ﬁrst term in (B.33) as follows: Bγ∗,i

K n ∗ W W P P FB ∗ (x0) − FB ∗ (x−1) [ (Ai ∈ Pj )− (Ai ∈ Pj )] γ ,i γ ,i j=1 h i X

= FBW (0) · FAn (xj ) − FAn (xj−1) − FA∗ (xj )+ FA∗ (xj−1) γ∗,i  i i i i  j=1 X   = FBW (0) · FAn (xK ) − FAn (x0) − FA∗ (xK )+ FA∗ (x0) γ∗,i i i i i

W n ∗ ≤F B ∗ (0) · 2 sup FAi (x) − FAi (x) γ ,i x∈[0,∞)

n ∗ ≤2 sup FAi (x) − FAi (x) . (B.34) x∈[0,∞)

B ∗ Now, recall that xm = xm−1 + v /(λ · K), ∀m ∈ [K], by the deﬁnition of F W , we deduce that i B Bγ∗,i B ∗ 2 vi λA λ¯ w¯ 1 M W W ∗ FB ∗ (xm) − FB ∗ (xm−1)= λ K vA ≤ λ w2 K = K , ∀m ∈ [K]. Therefore, the second term in (B.33) is γ ,i γ ,i B i ¯ ¯ K K n ∗ W W P P FB ∗ (xm) − FB ∗ (xm−1) [ (Ai ∈ Pj )− (Ai ∈ Pj )]  γ ,i γ ,i  m=1 j=m X h i X   K K ∗ ∗ = FBW (xm) − FBW (xm−1) FAn (xj ) − FAn (xj−1) − FA (xj )+ FA (xj−1)  γ∗,i γ∗,i i i i i  m=1 j=m X h i X K  

W W n n ∗ ∗ = FB ∗ (xm) − FB ∗ (xm−1) FA (xK ) − FA (xm−1) − FA (xK )+ FA (xm−1) γ ,i γ ,i i i i i m=1 h i X K ≤ F W (xm) − F W (xm−1) · 2 sup FAn (x) − FA∗ (x) Bγ∗,i Bγ∗,i i i m=1 x∈[0,∞) X K M ≤ · 2 sup FAn (x) − FA∗ (x) K i i m=1 x∈[0,∞) X n ∗ =2M sup FAi (x) − FAi (x) . (B.35) x∈[0,∞)

Inject (B.34) and (B.35) into (B.33), we obtain that

n ∗ E E n ∗ | Ii g (Ai ) − Ii g (Ai )|≤ (2+2M) sup FAi (x) − FAi (x) . (B.36) x∈[0,∞)

Apply the triangle inequality and combine (B.31), (B.32), (B.36), we have that:

n ∗ E W E W n ∗ Ii FB ∗ (Ai ) − Ii FB ∗ (Ai ) ≤ 2δ2 +(2+2M) sup FAi (x) − FAi (x) . γ ,i γ ,i x∈[0,∞)

34 For any j ≥ 2, this is trivially since Pj := (xj−1,xj ]. For P1 = [0,x1], we have that P n P ∗ P n P ∗ P n P ∗ Ai ∈ P1 − Ai ∈ P1 = (Ai ∈ (0,x1])− (Ai ∈ (0,x1])+ (Ai = 0)− (Ai = 0); moreover, due to Lemma B1-(i), we P n P ∗ also note that (Ai =0)= (Ai =0). 39 From this and (B.29), we obtain that

E ∗ n E ∗ ∗ n ∗ | FBi (Ai ) − FBi (Ai )|≤ 2δ2 +(4+2M) sup FAi (x) − FAi (x) . (B.37) x∈[0,∞)

2 Recall the constant C1 indicated in Lemma B4, we deﬁne C2 := C1 · (6+2M) [ln(6 + 2M) + 1] (note that −2 −1 −2 1 35 C2 does not depend on n nor ε2) and deduce that C2ε max 1, ln(ε ) ≥ C1δ ln . 2 2 2 min{δ2,1/e} −2 −1 −2 −1 Take ε1 := δ2, for any n ≥ C2ε2 max 1, ln(ε2 ) , we have n≥C1ε1 max 1, ln(ε1 ) and by applying n ∗ Lemma B4, we obtain sup FAi (x) − FAi (x) ≤ ε1 = δ2 and thus by (B.37), we have: x∈[0,∞)

E ∗ n E ∗ ∗ | FBi (Ai ) − FBi (Ai )|≤ 2δ2 +(4+2M) δ2 =(6+2M)δ2 = ε2.

∗ This is exactly (B.11). We can have a similar result in the case where i∈ / ΩA(γ ) (its proof is omitted here) and we conclude the proof of this lemma.

Appendix C. Proof of Results in Section 5

Appendix C.1. Proof of Lemma 5.2

n n max sup dFAi (x), sup dFBi (y) ≤ δ + ε. (5.3) ∗ ζ ∗ ∗ ζ ∗ (y ∈[0,2XB ] ZX (y ,ε) x ∈[0,2XB ] ZY (x ,ε) ) ∗ Fix y ∈ [0, 2X ], we look for the condition on n such that ∗ dF n (x) ≤ δ+ε holds. The condition B X ζ (y ,ε) Ai n ∗ corresponding to the inequality ζ ∗ dFB (x) ≤ ε + δ with x ∈ [0, 2XB] can be proved similarly and Y (x ,ε) i R thus is omitted in this section. ζ ∗ R First, we note that if X (y ,ε) is empty, ∗ dF n (x) = 0 and the result trivially holds. Now, let X ζ(y ,ε) Ai ζ ∗ ζ ∗ 36 us assume that X (y ,ε) 6= ∅, we can write XR (y ,ε)= I1 I2 I3 with ∗ ∗ I1 := {x ∈ [0, 2XB]: x = y , |ζSA(x,S y ) − α|≥ ε}, ∗ ∗ I2 := {x ∈ [0, 2XB]: xy , 1 − ζA(x, y ) ≥ ε}.

It is trivial that I1 is either an empty set or a singleton; on the other hand, due to the monotonicity of the CSF ζA (see (C2), Deﬁnition 2.3), I2 and I3 are either empty sets or half intervals. Moreover, for any arbitrary distribution F , we have that

0 , if I′ = ∅, dF (x)= F (a) , if I′ = {a}, i.e., I′ is a singleton, x∈I′  ′ ′ Z  F (b) − F (a) , if I = (a,b], i.e., I is a half interval.  2 2 35 −2 −1 6+2M −1 6+2M 6+2M Apply Lemma A2, C2ε max 1, ln(ε ) =C1 [ln(6+2M)+1] max 1, ln(ε ) ≥C1 ln . 2 n 2 o ε2 n 2 o ε2 ε2 ε2 ε2 1 1 Moreover, since = min , = min δ2, (due to the fact that δ2 = ε2/(6 + 2M) < 1/e). Therefore, we 6+2M n 6+2M e o e −2 −1 −2 1 have C2ε max 1, ln(ε ) ≥ C1δ ln . 2 n 2 o 2 min{δ2,1/e} 36 ∗ ∗ ∗ ∗ ∗ ∗ Recall that by deﬁnition, βA(x,y )= α if x = y , βA(x,y )=0 if xy 40 Therefore, we can deduce that

n ∗ n ∗ n ∗ dFAi (x) − dFAi (x)= dFAi (x) − dFAi (x) ≤ 5 sup |FAi (x) − FAi (x)| ζ ∗ ζ ∗ X (y ,ε) X (y ,ε) j=1 Ij Ij ! x∈[0,∞) Z Z X Z Z 2 Recall the constant C1 indicated in Lemma B4, we deﬁne L0 := C15 (ln(5)+1). Note that L0 does not depend ∗ −2 −1 2 −1 37 on the choice of y . Take ε1 := ε/5, we can deduce that L0ε max 1, ln(ε ) ≥ C1ε1 max 1, ln(ε1 ) . Therefore, for any n ≥ L ε−2 max 1, ln(ε−1) , we have n ≥ C ε2 max 1, ln(ε−1) and by Lemma B4, 0 1 1 1 n ∗ −2 −1 ∗ sup |FAi (x) − FAi (x)|≤ ε1 = ε/5. Hence, for any n ≥ L0ε max 1, ln(ε ) and δ ∈ ∆γ (ζ,ε), x∈[0,∞)

n ∗ dFAi (x) ≤ dFAi (x)+5 · ε/5 ≤ δ + ε. ζ ∗ ζ ∗ ZX (y ,ε) ZX (y ,ε)

Appendix C.2. Proof of Theorem 5.3 Theorem 5.3. Consider GLB games satisfying Assumption (A0), we have the following results:

∗ ∗ ∗ ζ xA γ ζ γ γ ΠA( , IUB ) ≤ ΠA(IUA , IUB )+(8δ + 13ε) WA, (5.4) ∗ ∗ ∗ ζ γ xB ζ γ γ ΠB(IUA , ) ≤ ΠB(IUA , IUB )+(8δ + 13ε) WB. (5.5)

∗ (ii) Given XA,XB > 0 (XA ≤ XB), there exists L > 0 such that for any ε ∈ (0, 1], in any game XA,XB ∗ −2 −1 ∗ (3.1) GLBn (ζ) where n ≥ L ε max 1, ln(ε ) , (5.4) and (5.5) hold for any γ ∈ Sn , δ ∈ A B ∆ ∗ (ζ,ε) and any pure strategy x , x of players A and B. γ Proof. We first give the proof of Result (ii). For the sake of brevity, we only focus on (5.4). The proof that (5.5) holds under the same condition can be done similarly and thus is omitted. Note that in this proof, we often use the Fubini’s Theorem to exchange the order of the double integrals. xA A XA,XB Recall that = (xi )i∈[n], by the definition of the payoff functions in GLBn (ζ), (5.4) can be rewritten as

n ∞ n ∞ ∞ A A A v ζA x ,y dFBn (y) − v ζA (x, y) dFAn (x) dFBn (y) ≤ 8δ + 13ε. (C.1)  i i i   i i i  i=1 Z i=1 Z Z X 0 X 0 0     We now prove that (C.1) holds under appropriate parameters values. To do this, we prepare two useful lemmas as follows.

∗ Lemma C1. For any pair of CSFs ζ = (ζA, ζB), any ε ∈ (0, 1] and x ∈ [0, 2XB], the following results hold:

(i) For any n, i ∈ [n] and δ ∈ ∆γ∗ (ζ,ε),

∞ ∞ ∗ ∗ ζA (x ,y) dFB∗ (y) − βA (x ,y) dFB∗ (y) ≤ δ + ε. (C.2) i i Z Z 0 0

37 ε −2 −1 5 2 −1 5 2 5 Note that ε1 = and apply Lemma A2, L0 ·ε max 1, ln(ε ) = C1 ·(ln(5)+1) max 1, ln(ε ) ≥ C1 · ln ; 5 ε ε ε moreover, ε = min{ ε , 1 } since ε ≤ 1; thus, we can rewrite ln 5 = ln 1 = ln 1 . 5 5 e ε min{ε/5,1/e} min{ε1,1/e} 41 −2 −1 (ii) There exists a constant L1 > 0 such that for any n ≥ L1ε max 1, ln(ε ) , i ∈ [n] and δ ∈ ∆γ∗ (ζ,ε),

∞ ∞ ∗ ∗ ζA (x ,y) dFBn (y) − βA (x ,y) dFBn (y) ≤ δ +2ε. (C.3) i i Z Z 0 0

Lemma C2. Given w, w,X¯ A,XB > 0 (w ≤ w¯, XA ≤ XB), there exists a constant L2 > 0 such that for any ¯ −2 −1 ¯ XA,XB ε ∈ (0, 1] and n ≥ L2ε max 1, ln(ε ) , in any game GLBn (ζ) with any δ ∈ ∆γ∗ (ζ,ε) and i ∈ [n], we have: ∞ ∞

ζA (x, y) dFBn (y) − ζA (x, y) dFB∗ (y) ≤ 2δ +4ε, ∀x ≥ 0, (C.4) i i Z Z 0 0 ∞ ∞ ∞ ∞

ζA (x, y) dFA∗ (x)dFB∗ (y) − ζA (x, y) dFA∗ (x)dFBn (y) ≤ 2δ +3ε, (C.5) i i i i Z Z Z Z 0 0 0 0 ∞ ∞ ∞ ∞

ζA (x, y) dFBn (y)dFA∗ (x) − ζA (x, y) dFBn (y)dFAn (x) ≤ 2δ +4ε. (C.6) i i i i Z Z Z Z 0 0 0 0

Lemma C1 states the relation between the ﬁrst term appearing in the left-hand-side of (C.1) and the n ∗ corresponding terms when we replace the CSF ζ by the Blotto functions β and replace FBi by FBi . These relations are useful to connect the statement we want to prove and the results obtained in Section 4. A proof of Lemma C1 is given in Appendix C.3. On the other hand, Lemma C2 indicates several useful ∗ XA,XB γ inequalities involving the players’ payoﬀs in the game GLBn (when they play according to the IU ∗ ∗ strategy or playing such that the marginals are FAi , FBi ). Its proof is given in Appendix C.4 that is based n n ∗ ∗ on Lemma C1 and the convergence of the distributions FAi , FBi toward FAi , FBi (i.e., Lemma B4). We have another remark: for any n and i ∈ [n],

P ∗ ∗ (Ai = Bi = x)=0, ∀x ≥ 0. (C.7) P ∗ ∗ P ∗ P ∗ This can be trivially proved as follows: ﬁrst, (Ai = Bi = x) = (Ai = x) (Bi = x) since they are ∗ ∗ P ∗ P ∗ independent; now, if x > 0, FAi and FBi are continuous at x and thus (Ai =x)= (Bi =x)=0; on the ∗ ∗ S P ∗ other hand, if x = 0, in the case where i ∈ ΩA(γ ), since Ai = Aγ∗,i, we have (Ai = x) = 0, in the case ∗ ∗ S P ∗ where i∈ / ΩA(γ ), since Bi = Bγ∗,i, we have (Bi = x) = 0. ∗ ∗ −2 −1 Finally, use Lemma C1 and C2 and take L = max{L1,L2}, for any n ≥ L ε max 1, ln(ε ) , A δ ∈ ∆ ∗ (ζ,ε) and any pure strategy x of Player A, we have: γ

n ∞ A A v ζA x ,y dFBn (y)  i i i  i=1 Z X 0 n  ∞  n A A A ≤ v ζA x ,y dFB∗ (y) + v (2δ +4ε) (due to (C.4))  i i i  i i=1 Z i=1 X 0 X n  ∞  n A A A = v ζA x ,y dFB∗ (y) +2δ +4ε (note that v = 1)  i i i  i=1 i i=1 Z X 0 X n  ∞  A A ≤ v βA x ,y dFB∗ (y) +3δ +5ε (due to (C.2))  i i i  i=1 Z X 0   42 n A P ∗ A P ∗ A = vi α (Bi = xi )+ (Bi < xi ) +3δ +5ε i=1 Xn

A ∗ A ≤ vi FBi xi +3δ +5ε (since α ≤ 1) i=1 X n ∞ A ≤ v FB∗ (x)dFA∗ (x) +3δ +5ε (due to Lemma B3)  i i i  i=1 X Z0 n  ∞  A ∗ = v P(B < x)dFA∗ (x) +3δ +5ε (due to (C.7))  i i i  i=1 X Z0 n  ∞ ∞  A ≤ v βA (x, y) dFB∗ (y) dFA∗ (x) +3δ +5ε  i i i  i=1 X Z0 Z0 n  ∞ ∞  A ≤ v ζA (x, y) dFB∗ (y) dFA∗ (x) +4δ +6ε (due to (C.2))  i i i  i=1 X Z0 Z0 n  ∞ ∞  A ≤ v ζA (x, y) dFA∗ (x)dFBn (y) +6δ +9ε (due to (C.5))  i i i  i=1 X Z0 Z0 n  ∞ ∞  A ≤ v ζA (x, y) dFBn (y)dFAn (x) +8δ + 13ε (due to (C.6)).  i i i  i=1 X Z0 Z0   Hence, we conclude that for n ≥ L∗ε−2 max 1, ln(ε−1) (C.1) holds and thus, (5.4) also holds. To prove that Result (ii) implies Result (i), we can proceed similarly to the proof that Theorem 4.3-(ii) implies Theorem 4.3-(i) (see Appendix B). We conclude this proof.

Appendix C.3. Proof of Lemma C1

∗ First, we prove (C.2). Note that FBi (y)=1, ∀y > 2XB (see Lemma A1-(iv)), for any n, i ∈ [n] and δ ∈ ∆γ∗ (ζ,ε), we have

∞ ∞ ∗ ∗ ζA (x ,y) dFB∗ (y) − βA (x ,y) dFB∗ (y) i i Z Z 0 0

∗ ∗ ∗ ∗ ≤ |ζ (x ,y) − β (x ,y)| dF ∗ (y)+ |ζ (x ,y) − β (x ,y)| dF ∗ (y) A A Bi A A Bi ζ ∗ ζ ∗ ZY (x ,ε) Z[0,∞)\Y (x ,ε)

∗ ∗ ∗ ∗ ∗ ∗ = |1−ζB(x ,y)−1+βB(x ,y)| dFBi (y)+ |1−ζB(x ,y)−1+βB(x ,y)| dFBi (y) ζ ∗ ζ ∗ ZY (x ,ε) Z[0,2XB ]\Y (x ,ε)

∗ ∗ ∗ ∗ ∗ ∗ = |ζB (x ,y) − βB(x ,y)| dFBi (y)+ |ζB(x ,y) − βB(x ,y)| dFBi (y) ζ ∗ ζ ∗ ZY (x ,ε) Z[0,2XB ]\Y (x ,ε)

∗ ∗ ≤ dFBi (y)+ εdFBi (y) ζ ∗ ζ ∗ ZY (x ,ε) Z[0,2XB ]\Y (x ,ε) ≤δ + ε. (C.8)

Here, the second-to-last inequality comes from the fact that 0 ≤ ζB(x, y),βB (x, y) ≤ 1 for any x, y and the ζ ∗ deﬁnition of Y (x ,ε) while the last inequality is due to the deﬁnition of ∆γ∗ (ζ,ε).

43 Now, in order to prove (C.3), we proceed similarly as in (C.8) to show that

∞ ∞ ∗ ∗ ζA (x ,y) dFBn (y) − βA (x ,y) dFBn (y) i i Z Z 0 0

≤ dF n (y)+ εdF n (y) Bi Bi ζ ∗ ζ ∗ ZY (x ,ε) Z[0,2XB ]\Y (x ,ε)

n ≤ dFBi (y)+ ε. (C.9) ζ ∗ ZY (x ,ε) −2 −1 Finally, by Lemma 5.2, for any n ≥ L ε max 1, ln(ε ) and δ ∈∆ ∗ (ζ,ε), we have ∗ dF n (y)≤ε+δ. 0 γ Yζ (x ,ε) Bi −2 −1 Combine this with (C.9), we conclude that (C.3) holds for any n ≥ L0ε max 1, ln(ε R ) and δ ∈ ∆γ∗ (ζ,ε). Take L := L , we conclude the proof. 1 0

Appendix C.4. Proof of Lemma C2 E ∞ E ∞ In this proof, we use the notation h(X,y) := 0 h(x, y)dFX (x) and h(x, Y ) := 0 h(x, y)dFY (y) where X, Y are arbitrary non-negative random variables and h is any function. R R Proof of (C.4): For any i ∈ [n] and x ≥ 0, we have

∞ ∞

ζA (x, y) dFBn (y) − ζA (x, y) dFB∗ (y) i i Z Z 0 0 n n n ∗ ∗ ∗ ≤ |EζA(x, B )−EβA(x, B )|+|EβA(x, B )−E βA(x, B )|+|EβA(x, B )−EζA(x, B )| . (C.10) i i i i i i We notice that upper-bounds of the ﬁrst and third terms in the right-hand-side of (C.10) are given by (C.3) and (C.2) from Lemma C1. We focus on ﬁnding an upper-bound of the second term of (C.10); to do this, we rewrite this term as follows.

E n n P n n P n βA(x, Bi )= dFBi (y)+ α (Bi = x)= FBi (x) − (1 − α) (Bi = x), (C.11) Zy

E ∗ ∗ P ∗ ∗ P ∗ and βA(x, Bi )= dFBi (y)+ α (Bi = x)= FBi (x) − (1 − α) (Bi = x). (C.12) Zy

E n E ∗ n ∗ If α = 1, we trivially have | βA(x, Bi ) − βA(x, Bi )| = FBi (x) − FBi (x) . In the following, we assume that α< 1 and consider three cases:

P n P ∗ Case 1: If x = 0. From Lemma B1-(i), we have (Bi =0)= (Bi = 0) and thus

n ∗ n ∗ E E n ∗ P P | βA(0,Bi ) − βA(0,Bi )| = dFBi (y) − dFBi (y)+ α (Bi = 0) − α (Bi = 0) =0. Zy<0 Zy<0

P ∗ Case 2: If x > 0, (Bi = x) = 0 by definition. On the other hand, from Results (ii) and (iii) of 2 n n−1 λ w P ¯ ¯ Lemma B1, we have (Bi = x) ≤ D where we define D := 1 − λ¯ w¯2 . Following (B.14), for any −1 n−1 ε n ≥ C0 max 1, ln(ε ) (here, C0 is defined as in Appendix B.2), we have D ≤ 2(1−α) . Therefore, for −1 any n ≥ C0 max 1, ln(ε ) , we have E n E ∗ | βA(x, Bi ) − βA(x, Bi )|

n ∗ P n ≤|FBi (x) − FBi (x)| + (1 − α) | (Bi = x)| (due to (C.11) − (C.12)) 44 ε n ∗ ≤ sup |FBi (x) − FBi (x)| + (1 − α) x∈[0,∞) 2(1 − α) ε n ∗ = sup |FBi (x) − FBi (x)| + . x∈[0,∞) 2

E n E ∗ n ∗ In conclusion, | βA(x, Bi ) − βA(x, Bi )|≤ sup |FBi (x) − FBi (x)| + ε/2 for any x ≥ 0, α ∈ [0, 1] and x∈[0,∞) −1 ′ n ≥ C0 max 1, ln(ε ) . Now, let us deﬁne C1 = C1 ·4(ln(2)+1) (where C1 is indicated in Lemma B4); take ′ −2 −1 −2 −1 ′ −2 −1 ε1 := ε/2, we have C1ε max 1, ln(ε ) ≥ C1ε1 max 1, ln(ε1 ) . Thus, for any n ≥ C1ε max 1, ln(ε ) , −2 −1 n ∗ we have n ≥ C1ε1 max 1, ln(ε1 ) and apply Lemma B4, we have sup |FBi (x) − FBi (x)|≤ ε1 = ε/2. x∈[0,∞) ′ −2 −1 We deduce that for any x ≥ 0, for any n ≥ max{C0, C1}ε max 1, ln(ε ) and i ∈ [n], we have: E n E ∗ | βA(x, Bi ) − βA(x, Bi )|≤ ε/2+ ε/2= ε. (C.13)

Finally, apply Lemma C1 to (C.10) to bounds the ﬁrst and third term of its right-hand-side, use (C.13) to ′ −2 −1 bound its second-term and take L(C.4) = max{L1, C0, C1}, we deduce that for any n ≥ L(C.4)ε max 1, ln(ε ) and δ ∈ ∆ ∗ (ζ,ε), γ ∞ ∞

ζA (x, y) dFBn (y) − ζA (x, y) dFB∗ (y) ≤ (δ +2ε)+ ε + (δ + ε)=2δ +4ε. i i Z Z 0 0

Proof of (C.5): To prove this inequality, we note that similar to the proof of (C.2) in Lemma C1 (by replacing ∗ ∗ ∗ ∗ ∗ ∗ FBi by FAi and replacing ζA(x ,y),βA(x ,y) by ζA(x, y ),βA(x, y )), we can prove that for any n, i ∈ [n], ∗ δ ∈ ∆γ∗ (ζ,ε) and y ∈ [0, 2XB], the following inequality holds

∞ ∞ ∗ ∗ ζA (x, y ) dFA∗ (x) − βA (x, y ) dFA∗ (x) ≤ δ + ε. (C.14) i i Z Z 0 0

Using this, we have ∞ ∞ ∞ ∞

ζA (x, y) dFA∗ (x)dFB∗ (y) − ζA (x, y) dFA∗ (x)dFBn (y) i i i i Z Z Z Z 0 0 0 0 ∞ ∞ ∞

≤ ζA(x, y)dFA∗ (x)− βA(x, y)dFA∗ (x) dFB∗ (y) i i i Z Z Z 0 0 0 ∞ ∞ ∞ ∞

+ βA(x, y)dFA∗ (x)dFB∗ (y)− βA(x, y)dFA∗ (x)dFBn (y) i i i i Z Z Z Z 0 0 0 0 ∞ ∞ ∞

+ βA(x, y)dFA∗ (x) − ζA(x, y)dFA∗ (x) dFBn (y) i i i Z Z Z 0 0 0 ∞ ∞ ∞ ∞ ∗ E ∗ ∗ E n ∗ n ≤ (δ+ε)dFBi (y)+ βA(x, Bi )dFAi (x)− βA(x, Bi )d FAi (x) + (δ+ε)dFBi (y) Z0 Z0 Z0 Z0 ∞ n ∗ ≤2δ+2ε+ |Eβ (x, B ) − Eβ (x, B )| dF ∗ (x). A i A i Ai Z0 ′ −2 −1 Finally, take L(C.5) = max{C0, C1} and apply (C.13), we deduce that for any n ≥ L(C.5)ε max 1, ln(ε ) , (C.5) holds. 45 Proof of (C.6) To prove this inequality, we note that similar to the proof of (C.3) in Lemma C1 (by

n n ∗ ∗ ∗ ∗ replacing FBi by FAi and replacing ζA(x ,y),βA(x ,y) by ζA(x, y ),βA(x, y )), we can prove that for −2 −1 n ≥ L1ε max 1, ln(ε ) , i ∈ [n] and δ ∈ ∆γ∗ (ζ,ε),

∞ ∞ ∗ ∗ ζA (x, y ) dFAn (x) − βA (x, y ) dFAn (x) ≤ δ +2ε. (C.15) i i Z Z 0 0

Now, similar to the proof leading to (C.13), we can prove that E ∗ E n | βA(Ai ,y) − βA(Ai ,y)|≤ ε, (C.16)

′ −2 −1 for any n ≥ max{C0, C1}ε max 1, ln(ε ) , i ∈ [n] and y ≥ 0. ′ −2 −1 Finally, take L = max{L , C , C }, for any n ≥ L ε max 1, ln(ε ) , i ∈ [n] and δ ∈ ∆ ∗ (ζ,ε), (C.6) 1 0 1 (C.6) γ we have ∞ ∞ ∞ ∞

ζA (x, y) dFBn (y)dFA∗ (x) − ζA (x, y) dFBn (y)dFAn (x) i i i i Z Z Z Z 0 0 0 0 ∞ ∞ ∞

≤ ζA(x, y)dFA∗ (x)− βA(x, y)dFA∗ (x) dFBn (y) i i i Z Z Z 0 0 0 ∞ ∞ ∞ ∞

+ βA(x, y)dFA∗ (x)dFBn (y)− βA(x, y)dFAn (x)dFBn (y) i i i i Z Z Z Z 0 0 0 0 ∞ ∞ ∞

+ βA(x, y)dFAn (x) − ζA(x, y)dFAn (x) dFBn (y) i i i Z Z Z 0 0 0 ∞ ∞ ∞

∗ n ∗ E E n n ≤ (δ+ε)dFAi (x)+ | βA(Ai ,y)− βA(Ai ,y)| dFBi (y)+ (δ+2ε)dFAi (x) (due to (C.14), (C.15)) Z0 Z0 Z0 ≤2δ +4ε (due to (C.16)). In conclusion, take L2 := max{L(C.4),L(C.5),L(C.6)}, we conclude the proof of this lemma.

Appendix C.5. Remark on the Generalized Lottery Blotto games with continuous CSFs

In this section, we present and prove the remark stating that under the additional assumption that the CSFs ζA and ζB are Lipschitz continuous on [0, 2XB] × [0, 2XB], the statements in Theorem 5.3 also hold with (C.17) and (C.18) (see below) in places of (5.4) and (5.5). For the sake of completeness, we formally state this result as follows.

Remark C3. For any CSF ζA and ζB that are Lipschitz continuous on [0, 2XB] × [0, 2XB], the following results hold (here, we denote ζ := (ζA, ζB)):

∗ ∗ ∗ ζ xA γ ζ γ γ ΠA( , IUB ) ≤ ΠA(IUA , IUB )+(2δ +5ε) WA, (C.17) ∗ ∗ ∗ ζ γ xB ζ γ γ ΠB(IUA , ) ≤ ΠB(IUA , IUB )+(2δ +5ε) WB. (C.18) 46 (ii) For any ε ∈ (0, 1], there exists a constant Lζ > 0 (that depends on ζ but does not depend on ε) such XA,XB −2 −1 that in any game GLBn (ζ) where n ≥ Lζ ε max 1, ln(ε ) , (C.17) and (C.18) hold for any (3.1) ∗ ∗ xA xB γ ∈ Sn , δ ∈ ∆γ (ζ,ε) and any pure strategy , of players A and B. Proof. We deﬁne the Lipschitz constant of ζA, ζB respectively by LζA , LζB and let Lζ := max{LζA , LζB }. We focus on proving Result (ii) of this Remark; Result (i) can be deduced from Result (ii) and thus is omitted. ∗ ∗ Step 1 : We prove that for any x ,y ∈ [0, 2XB], there exists a constant Cζ (that does not depend on ε ∗ ∗ −2 −1 nor x ,y ) such that for any n ≥ Cζ ε max 1, ln(ε ) , the following inequalities hold: ∞ ∞ ∗ ∗ ζA (x, y ) dFAn (x) − ζA (x, y ) dFA∗ (x) ≤ ε, (C.19) i i Z Z 0 0 ∞ ∞

∗ ∗ ζA (x ,y) dFBn (y) − ζA (x ,y) dFB∗ (y) ≤ ε. (C.20) i i Z Z 0 0

The proof of this statement is quite similar to the proof of Lemma B5 (see Appendix B.5). We present here the proof of (C.19); the proof of (C.20) can be done similarly. Fix y∗ ∈ [0, 2X ]; to simplify the notation, we deﬁne f(x) := ζ (x, y∗)andε ˜ := ε . From Lemma A1, B A 1 4+4XB Lζ n ∗ FAi (x)= FAi (x)=1, ∀x> 2XB; therefore, the left-hand-side of (C.19) can be rewritten as follows.

∞ ∞ 2XB 2XB ∗ n ∗ ∗ n ∗ ζA (x, y ) dFAi (x) − ζA (x, y ) dFAi (x) = f(x)dFAi (x) − f(x)dFAi (x) . (C.21) 0 0 Z0 Z0 Z Z

Let us deﬁne K := 2XB Lζ and K +1 points x such that x := 0 and x := x + 2XB , ∀j ∈ [K ]. In other ε˜1 j 0 j j−1 K K words, we have the partitions [0, 2XB]= j=1 Pj where we denote by P1 the interval [x0, x1] and by Pj the ′ interval (xj−1, xj ] for j =2,...,K. For any x, x ∈ Pj , ∀j ∈ [K], since f is Lipschitz continuous, we have S 2X |f(x) − f(x′)| ≤ L |x − x′| ≤ L B ≤ ε˜ . (C.22) ζ ζ K 1 K

Now, we deﬁne the function g(x):= f(xj )1Pj (x). Here, 1Pj is the indicator function of the set Pj . From j=1 this deﬁnition and Inequality (C.22),P we have |f(x) − g(x)|≤ ε˜1, ∀x ∈ [0, 2XB]. Therefore,

2XB 2XB 2XB n n n f(x)dFAi (x) − g(x)dFAi (x) ≤ ε˜1dFAi (x) ≤ ε˜1, (C.23) 0 0 0 Z Z Z

2XB 2XB 2XB ∗ ∗ ∗ f(x)dFAi (x) − g(x)dFAi (x) ≤ ε˜1dFAi (x) ≤ ε˜1. (C.24) 0 0 0 Z Z Z

Now, we note that for any j ∈ [K], f(xj )= [f(xm) − f(xm−1)]; here, by convention, we denote by x−1 m=0 an arbitrary negative number and set f(x−1)P = 0. Using this, we have:

2XB 2XB n ∗ g(x)dFAi (x) − g(x)dFAi (x) 0 0 Z Z K 2XB 2XB

= f(xj ) 1P (x)dFAn (x) − 1P (x) dFA∗ (x) j i j i j=1 "Z0 Z0 # X

K = f(x )[P (An ∈ P )−P (A∗ ∈ P )] j i j i j j=1 X

K j = [f(x ) − f(x )] [P (An ∈ P )−P (A∗ ∈ P )] m m−1 i j i j j=1 m=0 ! X X

K ≤ [f(x ) − f(x )] [P (An ∈ P )−P (A∗ ∈ P )] 0 −1 i j i j j=1 X

K K + [f(x ) − f(x )] [P (An ∈ P )−P (A∗ ∈ P )] . (C.25)  m m−1 i j i j  m=1 j=m X X n ∗  38  Note that P (A ∈ Pj )−P (A ∈ Pj )=FAn (xj )−FAn (xj−1)−FA∗ (xj )+FA∗ (xj−1). Now, we can rewrite i i i i i i the ﬁrst term in (C.25) as follows.

K [f(x ) − f(x )] [P (An ∈ P )−P (A∗ ∈ P )] 0 −1 i j i j j=1 X

= f(0) · FAn (xj ) − FAn (xj−1) − FA∗ (xj )+ FA∗ (xj−1)  i i i i  j=1 X  n n ∗ ∗  = f(0) · FAi (xK ) − FAi (x0) − FAi (xK )+ FAi (x0)

n ∗ ≤2 sup FAi (x) − FAi (x) . (C.26) x∈[0,∞)

Here, the last inequality comes from the fact that f(x) ≤ 1, ∀x ∈ [0, 2XB] (since it is a CSF). 2XB Lζ Now, we recall that for any m ∈ [K], f(xm) − f(xm−1) ≤ K . Therefore, the second term in (C.25) is

K K [f(x ) − f(x )] [P (An ∈ P )−P (A∗ ∈ P )]  m m−1 i j i j  m=1 j=m X X   K K

= [f(xm) − f(xm−1)] FAn (xj ) − FAn (xj−1) − FA∗ (xj )+ FA∗ (xj−1)  i i i i  m=1 j=m X X K  

n n ∗ ∗ = [f(xm) − f(xm−1)] FAi (xK ) − FAi (xm−1) − FAi (xK )+ FAi (xm−1) m=1 X K 2XBLζ ≤ · 2 sup FAn (x) − FA∗ (x) K i i m=1 x∈[0,∞) X n ∗ =4XBLζ sup FAi (x) − FAi (x) . (C.27) x∈[0,∞)

Inject (C.26) and (C.27) into (C.25), we obtain that

2XB 2XB n ∗ n ∗ g(x)dFAi (x) − g(x)dFAi (x) ≤ (2+4XBLζ ) sup FAi (x) − FAi (x) . (C.28) 0 0 x∈[0,∞) Z Z

38 For any j ≥ 2, this is trivially since Pj := (xj−1,xj ]. For P1 = [0,x1], we have that P n P ∗ P n P ∗ P n P ∗ Ai ∈ P1 − Ai ∈ P1 = (Ai ∈ (0,x1])− (Ai ∈ (0,x1])+ (Ai = 0)− (Ai = 0); moreover, due to Lemma B1-(i), we P n P ∗ also note that (Ai =0)= (Ai =0). 48 Apply the triangle inequality and combine (C.23), (C.24), (C.28), we have that:

2XB 2XB n ∗ n ∗ f(x)dFAi (x) − f(x)dFAi (x) ≤ 2˜ε1 +(2+4XBLζ ) sup FAi (x) − FAi (x) . (C.29) 0 0 x∈[0,∞) Z Z

2 Recall the constant C1 indicated in Lemma B4, we deﬁne Cζ :=C1 (4+4XBLζ ) [ln(4+4XBLζ )+1] (note 39 that Cζ does not depend on n nor ε) and deduce that

1 C ε−2 max 1, ln(ε−1) ≥ C ε˜−2 ln . ζ 1 1 min{ε˜ , 1/e} 1 −2 −1 −2 −1 Take ε1 :=ε ˜1, for any n ≥ Cζ ε max 1, ln(ε ) , we have n ≥ C1ε1 max 1, ln(ε1 ) and by applying n ∗ Lemma B4, we obtain that sup FAi (x) − FAi (x) ≤ ε1 =ε ˜1 and thus by (C.21) and (C.29), we have: x∈[0,∞)

∞ ∞ ∗ ∗ ζA (x, y ) dFAn (x) − ζA (x, y ) dFA∗ (x) ≤ 2˜ε1 +(2+4XBLζ )˜ε1 =(4+4XBLζ )˜ε1 = ε. i i Z Z 0 0

This is exactly (C.19). Step 2 : Based on (C.19) and (C.20), we can trivially deduce that the following inequalities hold for any −2 −1 n ≥ Cζ ε max 1, ln(ε ) and i ∈ [n]:

∞ ∞

ζA (x, y) dFBn (y) − ζA (x, y) dFB∗ (y) ≤ ε, ∀x ≥ 0, (C.30) i i Z Z 0 0 ∞ ∞ ∞ ∞

ζA (x, y) dFB∗ (y) dFA∗ (x) − ζA (x, y) dFBn (y) dFA∗ (x) ≤ ε, (C.31) i i i i Z Z Z Z 0 0 0 0 ∞ ∞ ∞ ∞

ζA (x, y) dFA∗ (x) dFBn (y) − ζA (x, y) dFAn (x) dFBn (y) ≤ ε. (C.32) i i i i Z Z Z Z 0 0 0 0

We notice that the left-hand-sides of these inequalities are exactly the terms considered in Lemma C2; moreover, the upper-bounds given in(C.30), (C.31) and (C.32) are smaller than that in (C.4), (C.5) and (C.6) of Lemma C2. Step 3 : To complete the proof of Remark C3, we follow the proof of Theorem 5.3 where we use (C.30), (C.31) and (C.32) instead of (C.4), (C.5) and (C.6). By doing this, we obtain (C.17) and (C.18).

Appendix D. Proof of Lemma 5.6 and Theorem 5.7

Appendix D.1. Proof of Lemma 5.6

XA,XB R (i) We ﬁrst consider the games GLBn (µ ).

2 39 −2 −1 4+4XB Lζ −1 Apply Lemma A2, C ε max 1, ln(ε ) =C1 ln(4+4X L )+1 max 1, ln(ε ) ζ ε B ζ 2 4+4XB Lζ 4+4XB Lζ ε ε 1 1 ≥C1 ln . Moreover, since = min , = min ε˜1, (due to the fact ε ε 4+4XB Lζ n 4+4XB Lζ e o e ε 1 thatε ˜1 = < ). 4+4XB Lζ e

49 −1/R µR ∗ ∗ ∗ Step 1: We want to prove that there exists δ0 = O(ε − 1) such that X (y ,ε) ⊂ [y − δ0,y + δ0] ∗ µR ∗ for any y ∈ [0, 2XB]. Note that this is trivial if X (y ,ε) = ∅. In the following, we consider the case µR ∗ where X (y ,ε) 6= ∅. We denote by f : [0, 2XB] × [0, 2XB] → [0, 1] the function:

αxR ∗ αxR+(1−α)(y∗)R , if xy

R R Trivially, y∗ ∈/ X µ (y∗,ε). Take an arbitrary x ∈ X µ (y∗,ε). If xy∗, we have: αxR x 1 − ε 1 − α 1/R f(x, y∗) ≥ ε ⇒ 1 − ≥ ε ⇒ ≤ . αxR + (1 − α)(y∗)R y∗ ε α ∗ ∗ 1−ε 1−α 1/R Therefore we have 0 < x − y ≤ y ε α − 1 . Here the right-hand side is positive (due to the −1/R condition α + ε< 1) and is upper-boundedh by O (ε i− 1). −1/R µR ∗ ∗ ∗ In conclusion, for any ε< min{α, 1−α}, there exists δ0 =O(ε −1) such that X (y ,ε) ⊂ [y −δ0,y +δ0]. −1/R ∗ Note that a similar proof can be done to prove that there exists δˆ0 = O(ε − 1) such that for any x ∈ [0, 2XB], µR ∗ ∗ ∗ Y (x ,ε) ⊂ [x − δˆ0, x + δˆ0].

∗ ∗ ∗ ∗ Step 2: For any y ∈ [0, 2XB] and δ0 ≥ 0, let us deﬁne the set I0(y ):=[y −δ0,y +δ0] [0, 2XB]; we want 2nλδ¯ 0w¯ to show that ∗ dFA∗ (x) ≤ , ∀i ∈ [n]. x∈I0(y ) i w T ∗ ∗¯ S Case 1: ForR i ∈ ΩA(γ ), then Ai Aγ∗,i, we have that

∗ ∗ ∗ S S dFAi (x) ≤FA ∗ (y + δ0) − FA ∗ (y − δ0) ∗ γ ,i γ ,i x∈I0(y ) Z ∗ ∗ ∗ (y +δ0)λB 2δ0λB ∗ B ≤ B , if 0 ≤ y ≤ δ0 vi vi ∗ ∗ ∗ ∗ ∗ B (y +δ0)λB (y −δ0)λB 2δ0λB ∗ vi  B − B = B , if δ0 ≤ y < ∗ − δ0 vi vi vi λB =  ∗ ∗ B ∗ ∗ ∗ ∗ B B  (y −δ0)λB vi −y λB +δ0λB 2δ0λB vi ∗ vi  1 − B = B ≤ B , if ∗ − δ0 ≤ y ≤ ∗ + δ0  vi vi vi λB λB 1 − 1=0, otherwise  2nλδ¯ w¯ ≤  0 . w ¯ ∗ ∗ W Case 2: For i∈ / ΩA(γ ), then Ai = Aγ∗,i. We have

∗ ∗ ∗ W W dFAi (x) ≤FA ∗ (y + δ0) − FA ∗ (y − δ0) ∗ γ ,i γ ,i x∈I0(y ) Z ∗ ∗ (y +δ0)λB 2nλδ¯ 0w¯ ∗ B ≤ , if 0 ≤ y ≤ δ0 vi w ∗ ∗ ∗ ¯ ∗ ∗ A (y +δ0)λB (y −δ0)λB 2δ0λB ∗ vi  B − B = B , if δ0

∗ ∗ ∗ ∗ dFAi (x)= dFAi (x) and dFBi (y)= dFBi (x). ∗ ∗ ∗ ∗ ∗ ∗ Zx∈[y −δ,y +δ0] Zx∈I0(y ) Zy∈[x −δ,x +δ0] Zy∈I0(x )

2nλδ¯ 0w¯ −1/R Let us deﬁne δµ := min{1, w } = O n(ε − 1) and we conclude that: ¯ max max dF ∗ (x), max dF ∗ (y) ≤ δ . ∗ Ai ∗ Bi µ y ∈[0,2X ] µR ∗ x ∈[0,2X ] µR ∗ ( B ZX (y ,ε) B ZY (x ,ε) )

R This implies that δµ ∈ ∆γ∗ (µ ,ε).

XA,XB R (ii) We now turn our focus on the games GLBn (ν ). We ﬁrst prove the existence of δ1 > 0 such that νR ∗ ∗ ∗ ∗ X (y ,ε) ⊂ [y − δ1,y + δ1] for any y ∈ [0, 2XB]. Similar to step 1 in the above analysis for the game XA,XB R GLBn (µ ), we denote by g : [0, 2XB] × [0, 2XB] → [0, 1] the function:

αexR ∗ ∗ αexR+(1−α)ey R , if xy .

R R Trivially, y∗ ∈/ X ν (y∗,ε). Take an arbitrary x ∈ X µ (y∗,ε). If x

xR ∗ αe g(x, y ) ≥ ε ⇒ ∗ ≥ ε. αexR + (1 − α)ey R

∗ 1 1−ε α Therefore, 0 < y − x ≤ R ln ε 1−α . Here, we note that the right-hand side is positive (due to the condition ε < α). On the other hand, if x>y∗, we have:

xR ∗ αe g(x, y ) ≥ ε ⇒ 1 − ∗ ≥ ε. αexR + (1 − α)ey R

∗ 1 1−ε 1−α Therefore, 0 < x−y ≤ R ln ε α . Here, the right-hand side is positive (due to the condition α+ε< 1). −1 −1 νR ∗ ∗ ∗ In conclusion, let us denote δ1 = O(R ln(ε )), we have proved that X (y ,ε) ⊂ [y − δ1,y + δ1] ∗ ∗ ∗ ∗ for any y ∈ [0, 2XB]. Now, we deﬁne I1(y ) := [y − δ1,y + δ1] [0, 2XB]. Similar to step 2 of the XA,XB R ∗ ¯ above analysis regarding the game GLBn (µ ), we can prove that ∗ dFA (x) ≤ 2nλδ1w/¯ w for any TI1(y ) i ¯ y∗ ∈ [0, 2X ]. Therefore, B R

max max dF ∗ (x), max dF ∗ (y) ≤ δ , ∗ Ai ∗ Bi ν y ∈[0,2X ] νR ∗ x ∈[0,2X ] νR ∗ ( B ZX (y ,ε) B ZY (x ,ε) ) ¯ 2nλδ1w¯ −1 −1 ∗ R where δν := min{1, w } = O nR ln(ε ) and δν ∈ ∆γ (ν ,ε). ¯ 51 Appendix D.2. Proof of Theorem 5.7

Theorem 5.7. (Approximate equilibria of Lottery Blotto games with ratio-form CSFs) Given w, w,X¯ ,X > 0 (w ≤w¯, X ≤X ) and α ∈ (0, 1), there exists L>˜ 0 such that for any ε¯ ∈ (0, min{α, 1−α}), ¯ A B ¯ A B in any game GLBXA,XB (µR) and GLBXA,XB (νR)—satisfying Assumption (A0) and having α as the tie- n n ∗ ˜ −2 −1 n 1 γ breaking-rule parameter—where n ≥ Lε¯ max 1, ln(¯ε ) , R ≥ O ε¯ ln ε¯ , the IU strategy is an εW¯ - ∗ (3.1) equilibrium for any γ ∈ Sn . Proof. Take ε =ε/ ¯ 21 and L˜ = L∗212(ln(21) + 1) (where L∗ is indicated in Theorem 5.3). We note that L˜ε¯−2 max 1, ln(¯ε−1) ≥ L∗ε max 1, ln(ε−1) ;40 therefore, for any n ≥ L˜ε¯−2 max 1, ln(¯ε−1) , we ∗ ∗ −1 γ have n ≥ L ε max 1, ln(ε ) and thus, apply Theorem 5.3-(ii), for any R > 0, the IU strategy is an XA,XB R (8δµ + 13ε)W -equilibrium of the game GLBn (µ ) (recall that W := max{WA, WB}). Similarly, the ∗ γ XA,XB R IU strategy is an (8δν + 13ε)W -equilibrium of the game GLBn (ν )). XA,XB R ∗ (3.1) We ﬁrst consider the game GLBn (µ ). Now, by applying Lemma 5.6, for any γ ∈ Sn and 1 ε 1 n any R ≥ O ln ln +1 = O ln , we have δµ ≤ ε. Therefore, we deduce that for any ε n ε ε ∗ n ≥ L˜ε¯−2 max 1, ln(¯ε−1) , R ≥ O n ln 1 , the IUγ strategy is an 21εW -equilibrium (i.e.,εW ¯ -equilibrium) ε¯ ε¯ of the game LB(µR). ∗ (3.1) n 1 Similarly, apply Lemma 5.6, for any γ ∈ Sn and R ≥ O ln , we have δν ≤ ε. Therefore, ∗ ε¯ ε¯ for any n ≥ L˜ε¯−2 max 1, ln(¯ε−1) , R ≥ O n ln 1 , the IUγ strategy is an 21εW -equilibrium (i.e., ε¯ ε¯ εW¯ -equilibrium) of the game LB(νR).

2 40Note that ε =ε/ ¯ 21 and apply Lemma A2 to have that L˜ε¯−2 max 1, ln(¯ε−1) ≥ L∗ 21 ln 21 ; moreover, we recall that ε¯ ε¯ ε¯ < 1 ; therefore, ln 21 = ln 1 = max 1, ln(ε−1) . 21 e ε¯ min{ε/¯ 21,1/e} 52