Characterizing the Interplay Between Information and Strength in Blotto Games

Characterizing the interplay between information and strength in Blotto games

Keith Paarporn, Rahul Chandan, Mahnoosh Alizadeh, and Jason R. Marden

Abstract— In this paper, we investigate informational asym- introduced by Borel in 1921 [7], researchers have incre- metries in the Colonel Blotto game, a game-theoretic model mentally contributed to this body of work over the last of competitive resource allocation between two players over one hundred years [8], [9]. In 1950, Gross and Wagner a set of battlefields. The battlefield valuations are subject to randomness. One of the two players knows the valuations with [8] provided an equilibrium solution of the two battlefield certainty. The other knows only a distribution on the battlefield case with asymmetric values and resources, as well as for n realizations. However, the informed player has fewer resources homogeneous battlefields and symmetric resources. In 2006, to allocate. We characterize unique equilibrium payoffs in a the work of Roberson [9] generalized the solution to n two battlefield setup of the Colonel Blotto game. We then battlefields and asymmetric forces by leveraging the solutions focus on a three battlefield setup in the General Lotto game, a popular variant of the Colonel Blotto game. We characterize the of all-pay auctions and the theory of copulas [10]. unique equilibrium payoffs and mixed equilibrium strategies. There recently has been renewed interest in the Colonel We quantify the value of information - the difference in Blotto game, along with its many variants and extensions equilibrium payoff between the asymmetric information game [2], [11]–[15]. Notably, results for heterogeneous battlefield and complete information game. We find information strictly valuations have appeared in Colonel Blotto as well as the improves the informed player’s performance guarantee. How- ever, the magnitude of improvement varies with the informed General Lotto variant, which admits a more tractable analysis player’s strength as well as the game parameters. Our analysis [13], [14]. Blotto games have also received attention in highlights the interplay between strength and information in engineering application areas such as the security of cyber- adversarial environments. physical systems [3] and the formation of large-scale networks [16], [17]. The vast majority of these studies assume I.INTRODUCTION the players have complete information about the opposing In adversarial interactions, informational asymmetries may player’s resource budget and the values of each battlefield. provide an advantage to one competitor over the other. Few works have considered the Blotto or Lotto game with Adversarial contests appear in a wide range of applications, incomplete information. In [18], the authors study a Blotto such as political campaigns [1], [2], security of cyber- game in which players have incomplete information about physical systems [3], and competitive advertising [4]. Rigor- the other’s resource budgets. In [19], the players are subject ous studies on the interplay between asymmetric information to incomplete information about the battlefield valuations. In and strategic decision-making in adversarial settings has both of these works, all players are equally uninformed about received attention in the recent literature [5], [6]. Though the parameters, and hence symmetric Bayes-Nash equilibria these works analyze general zero-sum game settings, another are provided. To the best of our knowledge, one-shot Blotto popular framework for analyzing such contests is the Colonel games where the players possess asymmetric information has Blotto game. Two players strategically allocate their limited not received attention in the literature. resources over a finite set of battlefields. A player secures In this paper, we formulate a Bayesian game framework a battlefield if it has allocated more resources to it than the in which one player is completely informed and the other opponent. Each player aims to secure as many battlefields does not receive any side information about the battlefield arXiv:1909.03382v2 [cs.GT] 13 Sep 2019 as possible. The goal of the present paper is to quantify the valuations (Section III). Our main contributions characterize performance improvements one player in the Blotto game unique equilibrium payoffs and strategies in representative may experience when it possesses system-level information scenarios of this framework. We first analyze the Colonel while its opponent does not. Blotto game with two battlefields (Section IV). With three Though simply posed, the Colonel Blotto game features battlefields, we are able to solve for unique mixed-strategy highly complex strategies. The established literature on the equilibria in a representative General Lotto game under Colonel Blotto game and its variants primarily focuses on the same informational asymmetry (Section V). We do this characterizing mixed-strategy Nash equilibria, due to non- by leveraging established results on all-pay auctions with existence of pure equilibria in most cases of interest. First asymmetric information [20] (Section VI). We can then quantify the difference between the equilibrium payoff in this K. Paarporn ([email protected]), R. Chandan ([email protected]), M. Alizadeh ([email protected]), game and the scenario where both players are uninformed. and J. R. Marden ([email protected]) are with the These characterizations allow us to determine conditions Department of Electrical and Computer Engineering at the University of under which the informed player has an advantage in the California, Santa Barbara, CA. This work is supported by UCOP Grant LFR-18-548175, ONR grant Lotto game, despite having fewer resources to allocate. #N00014-17-1-2060, and NSF grant #ECCS-1638214. As an illustrative analysis, we consider a scenario where both players are uninformed and the option of purchasing budget to be met only in expectation with respect to their information with a fraction of its budget is available only to marginal distributions. That is, the weaker player. We quantify a measure of informational ( n ) X j value determined by the largest proportion of resources the L(X ) := F : E j [x ] = X (LC) z z Fz z z player is willing to give up in exchange for information j=1 (Section V). is the set of such feasible mixed strategies. Note that both the budget constraint (LC) and expected payoff (2) only depend II.MODEL on the univariate marginal distributions and not the joint n- Two players, I and U, allocate their non-atomic forces variate distribution Fz. Hence, a player’s choice amounts across n battlefields simultaneously. In each battlefield j ∈ to selecting n independent univariate marginals that satisfy {1, . . . , n}, the player that sends more forces wins the (LC). We specify General Lotto game with GL(XI ,XU , v). battlefield and receives payoff vj, while the losing player receives −vj. In the case of a tie on a battlefield, both players III.BLOTTOAND LOTTOGAMESWITHASYMMETRIC receive zero payoff. The utility to each player is the sum INFORMATION of payoffs across all battlefields. The players have a budget Using a standard Bayesian games framework, we wish on their forces, XI ,XU > 0. We assume XI < XU . The to model a situation in which the battlefield valuations j n are subject to randomness. Suppose there are possible budgets and battlefield valuations v := {v }j=1 are common m knowledge. For z ∈ {I,U}, a pure strategy is a non-negative sets of battlefield valuations, labeled by the states Ω = 1 n n z { } vector xz = (xz, . . . , xz ) ∈ R+. We call x an allocation. ω1, . . . , ωm . The state ωi is realized with probability The payoff to player z is pi > 0, and the probability vector p ∈ ∆(Ω) satisfying Pm n i=1 pi = 1 is common knowledge to the players. Given j X j j j state ωi is realized, battlefield j is worth v > 0, and we uz(xz, x−z; v) := v sgn(x − x ), (1) i z −z × j j=1 denote V as the m n matrix with elements vi . We consider the setting where player I observes the true where we follow the convention sgn(0) = 0. This defines a state realization, and player U does not receive any side zero-sum game since the payoff to player −z is the negative information about the realization1. We call U the “unin- of the above. A mixed strategy for player z is an n-variate formed” player. Specifically, player I has m distinct types n distribution function Fz on R+. That is, an allocation xz TI := {t1, . . . , tm}, receiving type ti whenever state ωi is is drawn from the distribution Fz. A distribution Fz has realized. That is, I knows the realization with certainty. We j n n univariate marginal (cumulative) distributions {Fz }j=1, refer to TI as its type space. Player U has a single type, and specifying the allocation distributions to each battlefield. hence infers the realization (and I’s type) according to the Extending the definition of (1) to mixed strategies, the common prior p. The information structure is known to both expected payoff for each player can be expressed as players. Z Z n A strategy for player I consists of n-variate distributions X j j j m uz(Fz,F−z; v) = v sgn(xz − x−z)dFzdF−z {FI (ti)}i=1, one for each type ti, i = 1, . . . , m. We refer Rn Rn m + + j=1 to the collection FI = {FI (ti)}i=1 as I’s strategy. Player n Z ∞ U’s strategy FU is a single n-variate distribution. For the X j j j = v (2F−z − 1)dFz . Lotto game, FI (ti) ∈ L(XI ) for each t1, . . . , tm, and 0 j=1 FU ∈ L(XU ). For the Blotto game, they belong to B(XI ) (2) j n and B(XU ), respectively. We denote {FI (tI )}j=1 as the j A. The Colonel Blotto game univariate marginals of FI (tI ) for each tI ∈ TI , and FU the univariate marginals of FU . All of these components specify In the Colonel Blotto game, a feasible allocation xz for a Lotto or Blotto game with incomplete information, which player z ∈ {I,U} lies in the set we denote as GL(XI ,XU ,V, p) and CB(XI ,XU ,V, p), re- ( n ) spectively. The expected utility of player U is n X j B(Xz) := xz ∈ R : x = Xz . (BC) m + z X j=1 πU (FU ,FI ) := piuU (FU ,FI (ti), ωi) (3) i=1 Furthermore, the support of a mixed strategy Fz is contained in B(Xz). Thus, an allocation xz drawn from Fz belongs where with some abuse of notation, we use ωi to refer to the to B(Xz) with probability one. We denote the set of all battlefield valuation set in state i. The ex-interim utility for such feasible mixed strategies with B(Xz). We refer to the player I, given type ti is the expected payoff Colonel Blotto game with CB(XI ,XU , v). πI (FI (ti),FU |ti) := uI (FI (ti),FU , ωi). (4)

B. The General Lotto game 1A more general framework of asymmetric information may be formulated with Bayesian games. That is, arbitrary partial information structures The General Lotto game relaxes the support constraint in may be assigned to the players. Since our focus is on one informed and one the Colonel Blotto game, requiring each player’s allocation uninformed player, we leave such generalizations to future work. Player I 1 1

v1 1 2 v2 xI xI 0.8 -0.1 0.8 0.4 Pr= 1 0.3 v¯ 2 v 0.6 -0.2 0.6 0.4 -0.3 0.4 0.2 v 1 v¯ Pr= 2 0.2 -0.4 0.2 0.1 1 2 xU xU -0.5 0 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 Player U

(b) (c) (a) Fig. 1: (a) A diagram of the Colonel Blotto game with two battlefields and asymmetric budgets and information. There are two possible 1 battlefield valuation sets, each realized with probability 2 . Player I has fewer resources than player U, but knows the true realization. (b) 1 The equilibrium payoff (8) to the informed player in the Blotto game CB(XI ,XU ,V, 2 ), where v¯ = 1 and v = α ∈ (0, 1). It is negative 1 ∗ GW when γ ∈ ( 2 , 1) and α ∈ (0, 1). (c) The value of information, πI − πI , is non-negative. Information offers the most improvement for 1 low budget (γ near 2 ) and when there is higher priority in the diagonal battlefields (α low).

A Bayes-Nash equilibrium (BNE) is a strategy profile For n even, consider player U’s deterministic strategy that ∗ ∗ (FI ,FU ) satisfying places 2XI /n to the first n/2 battlefields, and the remaining ∗ ∗ resources allocated arbitrarily. This guarantees U attains at FI (ti) ∈ arg max πI (FI (ti),FU |ti), ∀i = 1, . . . , m FI (ti) least half of the total available value. That is, U’s security ∗ ∗ (5) value for this strategy is at least zero. Similar reasoning and FU ∈ arg max πU (FU ,FI ). FU applies to the case of n odd. ∗ ∗ An equivalent BNE condition is that (FI ,FU ) satisfies the For two battlefields, the weaker resource player I never best-response correspondences in ex-ante payoffs, given that wins the game, despite having better information. The ∗ ∗ each pi > 0 [21]. That is, FU ∈ arg max πU (FU ,FI ) and following analysis focuses on the two battlefield case of FU 1 ∗ ∗ CB(XI ,XU ,V, p) under the uniform prior p = 2 12. Al- FI ∈ arg max πI (FI ,FU ), where I’s ex-ante payoff is m though I never wins, we quantify how information improves {FI (ti)}i=1 m its payoff relative to when both players are uninformed. A X solution for a general prior p = [p, 1−p] remains a challenge πI (FI ,FU ) := piπI (FI (ti),FU |ti). (6) in this analysis, due to the complexity of finding suitable i=1 equilibrium distributions. Hence for the rest of this section, We note that since the underlying complete information game we assume p = 1 . is zero-sum, the games with asymmetric information are 2 also zero-sum with respect to ex-ante utilities (3), (6). This A. Two battlefields case allows us to speak of a unique equilibrium ex-ante payoff If XI < 1 , player U can secure both battlefields re- ∗ ∗ XU 2 πI = −πU of the Blotto and Lotto games with asymmetric gardless of what I does. We thus restrict our attention to information. the regime XI ∈ ( 1 , 1). We state the main result of this XU 2 IV. COLONEL BLOTTO RESULTS section, which characterizes the equilibrium payoff to I. Here, we assume a symmetric structure on the battlefields In the following structural result, we characterize sufficient v¯ v XI 1 conditions on the budget ratio γ := such that the V = v¯+v for v¯ > v > 0. Figure 1a illustrates the XU v v¯ stronger player U is guaranteed a positive equilibrium payoff setup of this game. regardless of the information asymmetry. XI 1 1 Theorem 1. Let X ∈ ( , 1), and assume p = . Define XI U 2 2 Proposition 1. Assume < 1. Consider the game XU XU q = b c. Then the ex-ante equilibrium payoff to the XU −XI CB(XI ,XU ,V, p) where V ∈ m × n, i.e. there are n informed player I is battlefields and m possible state realizations. A sufficient  −1 ∗ q−1 k condition for which πU ≥ 0 is − P 2 − if odd ∗  2 k=0 (¯v/v) 1 q ( πI = q −1 (8) XI 2 v −1 k X < n , if n is even − P 2 U (7)  v¯+v k=0 (¯v/v) if q even XI < 2 , if n is odd XU n+1 We provide the proof in the Appendix, which details a Pm j Proof. Denote Ej = i=1 pivi as the prior expected value set of equilibrium mixed strategies. When both players are of battlefield j. Without loss of generality, assume the n uninformed, i.e. both have a single type, we have a complete battlefields are ordered according to E1 ≥ E2 ≥ · · · ≥ En. information game where the two battlefield valuations are 1 their expected values, 2 . The equilibrium payoff is then all-pay auctions. An iterative algorithm is formulated in given by an application of Gross and Wagner’s solution [8], [20] to construct mixed equilibrium strategies. To verify GW 1 which yields πI := − q . The following result verifies the that the set of constructed strategies indeed is a BNE of 1 equilibrium payoff improves when I obtains information. GL(XI ,XU ,Vαβ, 3 13), the constraint (LC) must be met. Corollary 1. We have π∗ > πGW. That is, information strictly I I B. The value of information in the Lotto game improves the equilibrium payoff for I. In the following analysis, we consider a scenario in which Proof. v 1 v¯ Consider the q even case. Since v¯+v < 2 and v > 1 both players are uninformed, and the option to purchase q − −1 v P 2 1 k 1 information with a fraction of its budget is available to player for v¯ > v, from (8) we get − v v k=0 (¯v/v) > − q . ¯+ I. We quantify a value of information as the equilibrium Similar arguments apply in the q odd case. payoff gain or loss in purchasing information, and specify We may quantify a value of information in this setting the maximal cost I is willing to pay before it experiences a ∗ GW as the payoff difference πI − πI . Setting v¯ = 1 and v = payoff loss. In the case both players are uninformed, we may ∗ α ∈ (0, 1), we plot the equilibrium values πI in Figure 1 , as use the algorithm of [20] to solve the Lotto game. In this well as the value of information. In the two battlefield Blotto setting, we arrive at the equilibrium payoff γ − 1 for I. We game, an informed player still cannot gain an advantage, as first highlight some immediate consequences of Theorem 2. ∗ for all parameters ∈ × . Information πI < 0 (α, γ) (0, 1) (0, 1) Corollary 2. We have that π∗(α, β, γ) > γ−1 for (α, β, γ) ∈ offers the most payoff improvement when ’s budget is low I I (0, 1)3. That is, information strictly improves the equilibrium and the minor battlefield is worth less (Fig. 1c). α payoff for I. Additionally, π∗ is strictly decreasing in α and In the next section, we consider the three battlefield case in I β, and strictly increasing in γ. the General Lotto game. In general, equilibrium solutions for n ≥ 3 heterogeneous battlefields have not been characterized Proof. The first claim follows from comparing the expres- in the Colonel Blotto game, due to the complexity of finding sions (9) and (10) to γ − 1. The second claim follows by suitable copulas for the marginal distributions [2]. The Lotto showing the respective signs of partial derivatives hold. constraint relaxation allows for analytic tractability in cases of more than two heterogeneous battlefields [14]. We also characterize the parameter region in which an informed I wins the game for the special case α = β. V. RESULTS ON GENERAL LOTTO ∗ Corollary 3. Fix a budget ratio γ. Then πI (α, α, γ) > 0 if 1 −γ We restrict our attention to a representative three- and only if 3 . α < 3γ2−4γ+ 1 battlefield Lotto game GL(X ,X ,V , p), where p = 3 I U αβ 1 α β Proof. Player cannot win in the first region ∈ 1 , 1 1 1 1 I γ (0, 3 ] ( , , ), and Vαβ = β 1 α. where 1 > α ≥ ∗ 3 3 3 1+α+β as πI ≤ 0. We find that the second expression of (9) is α β 1 equivalent to (10) when β = α. We then solve the equation β > 0. In each set of battlefields (rows), the total valuation π∗(α, α, γ) = 0 for α, from which we obtain the result. is normalized to one. Though this formulation may be quite I specific, the representative game serves as an illustrative and We now provide a general quantity of describing the tractable scenario highlighting the informational asymmetry value of information. In particular, we consider a scenario between players. Intuitively, for small α, β values, the valu- where both players are uninformed. Player I, with the able battlefield is worth 1 but sits at a different location budget ratio γ, has an opportunity to purchase information in each realization. An informed player would be able to with a fraction cI of its budget. The value of information take the most advantage by focusing its resources on this quantifies the equilibrium payoff gain or loss in purchasing battlefield without wasting resources on the other battlefields. the information at the budget fraction cost cI . Formally, the value of information is the quantity A. Main result: characterization of equilibrium payoff X ∗ Theorem 2. Let γ = I ≤ 1. Then player I’s equilibrium VoI(α, γ) := π (α, α, (1 − cI )γ) − (1 − γ). (11) XU I payoff, π∗(α, β, γ), for α, β, γ ∈ (0, 1) takes the values I An instance of the value of information is plotted in Figure 3γ − if ∈ 1 1 1+α+β 1, γ (0, 3 ] 2b, when the cost is cI = 5 . We note there is a regime in 1 h 1 i 1 2 which information is not worth the cost, i.e. VoI(α, γ) < 0. 1+α+β 1− 3γ (3γα + (1−α)) + 1 −1, if γ ∈ ( 3 , 3 ] (9) We also seek to find the highest cost on information that I and is willing to pay before it experiences an equilibrium payoff loss. To quantify this cost, let γe be defined as the value that 1 h 1 1 2 i ∗ 1+α+β 2 − 3γ + α 2 − γ + 3βγ 1 − 3γ −1 (10) satisfies πI (α, α, γe) = γ − 1. Such a value is unique and ∗ well-defined for any α, since πI (α, α, γ) > γ − 1 and is ∈ 2 if γ ( 3 , 1]. strictly increasing in γ, by Corollary 2. Then CI (α, γ) := ∗ γ−γe A plot of πI is shown in Figure 2a for the special case α = γ is the largest fraction of resources I can give up. That ∗ β. We provide the details of the proof in Section VI, where is, for all cI ∈ [0,CI (α, γ)], we have πI (α, α, (1 − cI )γ) ≥ we draw upon known results in asymmetric information γ − 1, with equality if and only if cI = CI (α, γ). Then, 1 1 1 0.5 0.6 0.6 0.8 0.8 0.8 0.5 0.4 0.6 0 0.6 0.6 0.4

0.2 0.3 0.4 0.4 0.4 -0.5 0.2 0.2 0.2 0 0.2 0.1

0 -1 0 0 0 0.5 1 0 0.5 1 0.2 0.4 0.6 0.8 1

(a) (b) (c)

∗ 1 Fig. 2: (a) The equilibrium payoff πI to the informed player in the Lotto game GL(XI ,XU ,Vαβ , 13), under the special case α = β. ∗ 3 The dashed line are the points (γ, α) for which πI = 0. That is, for parameters below the line, the informed player “wins” the game and “loses” for parameters above the line (see Corollary 3). (b) The value of information, or payoff gain or loss to player I from purchasing 1 information at the cost cI = 5 . The dashed black line in Figure 2b indicates the parameters for which VoI(α, γ) = 0. (c) The maximal cost (12), or the largest fraction of its resource budget γ that player I is willing to give up in exchange for information before it experiences a payoff loss. Information is more valuable in lower α ranges - when the battleﬁeld rewards are concentrated at diverse locations.

Corollary 4. Fix a budget ratio γ. Then νI (ti) and player U’s valuation is νU (ti). Assume the types in TI are ordered such that ( 1 γ (γ − γe(α, γ)) if (α, γ) ∈ B CI (α, γ) = 1 (12) 1 − 3c if (α, γ) ∈/ B α νI (ti) ≥ νI (tk) (13) √ 2 2 −(2(1−α)cα−γ)+ (2(1−α)cα−γ) +4cαα(1−α) where γe(α, γ):= , 6αcα 1 2 1 whenever i < k, i, k ∈ {1, . . . , m}. In the two player all-pay cα := , and B := {(α, γ) ∈ (0, 1) : γe(α, γ) ≥ }. 1+2α 3 auction with asymmetric information and valuations, player ∗ Proof. We solve the equation πI (α, α, γe) = γ − 1 for γe. I selects the distributions of bids FI : TI × R+ → [0, 1] Recall the second expression of (9) is equivalent to (10) when contingent on its type. We write FI (ti) to refer to the 1 distribution of bids in type i, FI (ti, x) to refer to its value β = α. The resulting value is valid only if γe > 3 . If not, we solve for γe using the ﬁrst entry of (9). at x ≥ 0, and fI to refer to its density function. As player U has a single type, it selects a single distribution of bids, Figure 2c plots CI (α, γ). As an example, when γ = 1 FU : R+ → [0, 1]. The best-response problems for each 2 and α = 0, player I can exchange up to 3 of its budget for player are information and still obtain a payoff greater than γ − 1 = 0. When α is high, player I cannot afford to give up resources, m X Z ∞ as information is less valuable in this regime. max pi [νI (ti)FU − x] dFI (ti) FI (ti) i=1 0 i=1,...,m (14) VI.PROOFOF THEOREM 2 m X Z ∞ We show that the equilibria in the Lotto game max pi [νU (ti)FI (ti) − x] dFU . FU 0 GL(XI ,XU ,V, p) coincides with equilibria of n independent i=1 all-pay auctions with asymmetric information and valuations. The work of Siegel [20] provides an iterative algorithm for B. Algorithm of [20] which to construct equilibrium mixed strategies in the all-pay auction. We leverage this procedure to construct proposed An iterative procedure is formulated in [20] to construct equilibrium distributions in the Lotto game, and verify that equilibrium mixed strategies satisfying the correspondences they satisfy the constraint (LC). (14). We note that this procedure handles general player information structures in the two-player all-pay auction. In A. Two-player all-pay auctions with asymmetric information this paper, we restrict our attention to its application in our and valuations informed-uninformed setting. In particular, the constructed Here, we specialize the setup of [20] to our setting of an marginals are proven to be piecewise constant functions informed player I and an uninformed player U. Assume the with ﬁnite support, with the possibility of having point two players compete in an all-pay auction over an indivisible masses placed at zero. We do not give further details of this good, where the players I and U have the same type spaces algorithm due to space limitations and for ease of exposition. as before. When type ti ∈ TI is realized (i ∈ {1, . . . , m}) We refer the reader to [20] for a general proof that the output with probability pi > 0, player I’s valuation of the good is distributions satisfy (14). C. Connection of General Lotto to all-pay auctions j is an independent all-pay auction, whose problem is

In the game GL(XI ,XU ,V, p), player U’s Lagrangian at " m Z ∞ j ! # X 2vi j j both ex-ante and interim levels may be written as max pi FI (ti) − x dFU (19) F j λU U i=1 0 n " m j ! # Z ∞ " m j ! # X X 2vi j j Z ∞ 2v p max pi F (ti) − x dF (15) X i i j j j I U max pi F − x dF (ti) (20) F λU j U I j=1 U i=1 0 F (t ) λI I i i=1 0 i=1,...,m where λU > 0 is the Lagrange multiplier on U’s expected which coincides with (14) with auction valuations νU (ti) = budget constraint (LC), and we have removed constant j j 2vi 2vi pi additive and multiplicative terms in the expression that do λ and νI (ti) = λ . Here, we have multiplied each max- j n U I not depend on the decision variables {FU }j=1. In a similar imization problem of by pi > 0, which does not change the fashion, player I’s Lagrangian maximization at the interim optimal solutions. In this setting, mixed-strategy equilibria level may be written of the Lotto game are equivalent to that of n independent two-player all-pay auctions with asymmetric information and n Z ∞ j ! i X 2vi j j valuations. max λI i FU − x dFI (ti) (16) F j (t ) λ We may then apply the algorithm of [20] to construct equi- I i j=1 0 I j=1,...,n j j m librium distributions FU and {FI (ti)}i=1 for each battlefield j. The constructed distributions are functions of the known where the multiplier i corresponds to the budget λI > 0 parameters as well as the Lagrange multipliers. If there exists constraint (LC) in type t . When two sets of battlefields i unique multipliers λ∗, λ∗ > 0 such that the Lotto constraints contain the same valuations, we can deduce equivalence I U (LC) is met for all types {ti }m for I and for type t for U, between the corresponding Lagrange multipliers. I i=1 U then it is clear the n constructed strategy profiles constitute Lemma 1. Consider the game GL(XI ,XU ,V, p). Suppose a BNE for GL(XI ,XU ,V, p). Indeed, we have applied the the rows i and k of V have the same elements, each algorithm to obtain a set of distributions, and verified there is ∗ ∗ with identical multiplicities. Then the equilibrium ex-interim a unique λI , λU > 0 such that (LC) is satisfied. The details payoff to player I for type ti is equivalent to that of type tk. of this calculation are outlined as follows. i k Furthermore, λI = λI . 1 Proof of Theorem 2. In GL(XI ,XU ,Vαβ, 3 13), we deduce Proof. An equivalent formulation of (16) in type i is i from Lemma 1 that λI = λI for all i = 1, 2, 3. Hence we need only apply the algorithm of [20] to a single max πI (FI ,FU |ti) (17) “column” of the game to obtain all marginals of the BNE FI (ti)∈L(XI ) since the valuations are identical in each battlefield, i.e. the The corresponding problem for type tk is identical to the correspondences (19), (20) are the same for all battlefields above, because the valuations in both rows are the same j, with a permutation of indices i ∈ {1, 2, 3}. (possibly with some permutation of indices j), and the Denote the constructed marginal distributions as FU for optimization for player U remains (15). This shows the d α β player U and {FI ,FI ,FI } for player I, where “d” is for equivalence of interim equilibrium payoffs. the diagonal battlefield value 1. We find that depending on To show i k, let ∗ solve (17) for both types and λI λI 1 λI 1 1 λI = λI FI ti whether λ ≥ 1, λ ∈ ( 2 , 1), or λ ∈ ( 2 , 3 ), the algorithm I U U U tk. Any allocation xj ∈ R+ to battlefield j in the support of determines three distinct sets of marginal distributions for vj j∗ 2 i j∗ I and U. In each case, there are unique λI , λU > 0 such FI solves the one-dimensional problem maxx i FU −x as λI j that the constraint (LC) is met. For brevity, we illustrate the 2vi j∗ well as maxx k FU −x. Therefore, the first-order necessary calculation for one such case, as the other two follow similar λI i j j∗ I k methods. If λI ≥ 1, the algorithm of [20] gives condition for optimality that holds is λI = 2vi fU (xj ) = λI . λU 3λI 2c FU (x) = x, x ∈ 0, (21) To solve for a BNE, we write the best-response corre- 2c 3λI spondences at the ex-ante level. Player U’s ex-ante best- h i response problem is given by (15), while Player I’s ex-ante F d(x) = 3λU x + 1 − λU , x ∈ 0, 2c I 2c λI 3λI optimization problem may be written α (22) FI (x) = 1, x ≥ 0 β n " m Z ∞ j ! # F (x) = 1, x ≥ 0 X X i 2vi pi j j I max λ F − x dF (ti) (18) j I i U I F (ti) 0 λI j=1 I i=1 The constraint (LC) requires that EF d + EF α + E β = XI i=1,...,m I I FI and 3EFU = XU , from which we obtain the unique solutions 1 i λI = and λU = 3γλI . This solution implies the after removing additive constants. Under the conditions λI = XU (1+α+β) 1 λI > 0 for all i = 1, . . . , m (by Lemma 1), each battlefield budget ratio satisfies γ ∈ (0, 3 ). Using these marginals, we calculate the equilibrium ex-ante payoff (6) as investigate General Lotto games where the players hold Z ∞ Z ∞ Z ∞ arbitrary information structures. 2 d α β FU dFI +α FU dFI +β FU dFI −1. 1+α+β 0 0 0 APPENDIX (23) XU λ Proof of Theorem 1. Let d := XU −XI and q := b c From this, we obtain (9). The other cases I ∈ ( 1 , 1) and XU −XI λU 2 so that XU = qd + r, where 0 ≤ r < d. Denote a delta λI ∈ ( 1 , 1 ) correspond to the budget ranges γ ∈ [ 1 , 2 ) and λU 2 3 3 3 mass function centered at y ∈ R by δy. Define c :=v/v ¯ . 2 γ ∈ [ 3 , 1), respectively. We prove the Theorem by proposing a set of mixed strategy For completeness, we present the marginals for the other ∗ ∗ ∗ ∗ distributions FI = {FI (t1),FI (t2)}, and FU each satisfying two cases. When γ ∈ [ 1 , 2 ), the unique solution of the ∗ ∗ 3 3 (BC), and showing the strategy Fz is a best-response to F−z, 1 h 1−α i multipliers are λI = 2 + α and λU = z = I,U. For brevity, we prove the case when q is odd, as XU (1+α+β) 9γ h 2c α α i the even case provides similar mixed strategies and follows 3γλI . Denote the intervals I1 = 0, 3 ( λ − λ ) and I2 = h i I U similar arguments. Let e ∈ (r, d), and consider the strategies 2c ( α − α ), 2c ( α + 1−α ) . Then the equilibrium marginals 3 λI λU 3 λI λU q−1 are 2 1 X q+1 ( 3λ ∗ 2 −k I x x ∈ I FU = c δe+(k−1)d + δ q−1 2αc 1 e+ 2 d FU (x) = (24) sA 3λI λI k=1 x+(1− )(1−α) x ∈ I2 (28) 2c λU q ! X k− q+1 + c 2 δ F d(x) = 3λU x − α λU − 1 x ∈ I e+(k−1)d I 2c λI 2 q+1 k=q−1 +1 λ λ 2 q−1 ! F α(x) = 3 U x + 2 − U x ∈ I (25) 1 vc 2 X I 2αc λI 1 ∗ q−1−k β FI (t1) = δ q−1 d + c δkd F (x) = 1, x ≥ 0 sB v¯ + v 2 I k= q−1 +1 2 (29) 2 q−1 When γ ∈ [ , 1), the unique solution of the 2 −1 q−1 ! 3 2 c h 1 i ∗ 1 X k vc q−1 multipliers are λI = X β + 9γ2 (1 + 3α − 4β) and FI (t2) = c δkd + δ d U sB v¯ + v 2 h 2c β 2β i k=0 λU = 3γλI . Denote the intervals I1 = 0, ( − ) , 3 λI λU h β 2β β α−2β i where q−1 q−1 2c 2c q−1 −1 I2 = 3 λ − λ , 3 λ + λ , and I3 = 2 2 h I U I i U X k vc 2 X k 2c β + α−2β , 2c β + α−2β+1 . Then the equilibrium sA := 1 + 2 c , sB := + c (30) 3 λI λU 3 λI λU v¯ + v marginals are calculated to be k=1 k=0 are normalizing factors. Before proceeding, we make a  3λI  x x ∈ I1 few remarks. These strategies are similar in nature to the  2βc 3λI λI β strategies provided in Gross & Wagner [8]. There, the authors FU (x) = 2αc x + 1−2 λ 1− α x ∈ I2  U show that a strategy composed of equally spaced delta  3λI λI  x + (1 − β) + (2β − α − 1) x ∈ I3 2c λU functions with geometrically decreasing weights equalizes (26) the payoff of the other player, and vice versa. In a similar d 3λU λU F (x) = x − β − 2 − α x ∈ I3 I 2c λI fashion, the strategies (28) and (29) equalize the ex-ante F α(x) = 3λU x − β λU − 2 x ∈ I (27) payoffs in certain intervals of the players’ allocation space. I 2αc α λI 2 β 3λU λU Any allocation in these intervals give a best-response to the F (x) = x + 3 − x ∈ I1 I 2βc λI other player’s equilibrium strategy. Let xU ∈ [0,XU ] be an allocation to battlefield 1, leaving ∗ XU −xU to battlefield 2. The payoff uU (xU ,FI (t1), ω1) (2) VII.CONCLUSION ∗ of any allocation xU against FI (t1) in battlefield set 1 is In this paper, we extended the Colonel Blotto and General   q−1 Lotto games to a setting where players have asymmetric 1 X q−1−k  c (¯vsgn(xU −kd)+vsgn((k+1)d−xU )) information about the valuations of the battlefields. We sB   k= q−1+1 focused on the case when one player is completely informed 2 " q−1 # about the valuations but has fewer resources to allocate, 1 vc 2 q−1 q+1 + v¯sgn xU − d +vsgn d−xU and the other player is uninformed. Our analysis on the sB v¯ + v 2 2 two battlefield case in the Colonel Blotto game shows (31) an informed player still cannot defeat its opponent in a For notational purposes, let HU (¯v, v, xU ) denote the above ∗ ∗ mixed-strategy Nash equilibrium. We find a three battlefield quantity. Then against FI (t2), uU (xU ,FI (t2), ω2) = 1 scenario presents enough complexity such that the informed HU (v, v,¯ xU ) in battlefield set 2. Given p = 2 , the ∗ 1 ∗ player in the General Lotto game can attain the advantage ex-ante utility is πU (xU ,FI ) = 2 uU (xU ,FI (t1), ω1) + 1 ∗ for certain parameters. 2 uU (xU ,FI (t2), ω2). After some algebra, we arrive at A direction of future research involves generalizing the ( v¯ if xU ∈ (0, qd ] connection of the all-pay auctions with asymmetric infor- ∗ sB πU (xU ,FI ) = (32) mation to the General Lotto game. This will allow us to 0 if xU ∈ (qd, XU ] Any mixed strategy FU with support on the interval (0, qd) [13] G. Schwartz, P. Loiseau, and S. S. Sastry, “The heterogeneous colonel ∗ ∗ ∗ blotto game,” in 2014 7th International Conf. on NETwork Games, is a best-response to FI , and hence FU ∈ BRU (FI ). Now, 1 ∗ COntrol and OPtimization (NetGCoop), Oct 2014, pp. 232–238. any pure allocation xI ∈ [0,XI ] of player I against FU in [14] D. Kovenock and B. Roberson, “Generalizations of the general lotto 1 ∗ type t1 gives the ex-interim utility πI (xI ,FU |t1) (4), as and colonel blotto games,” 2015. [15] A. Ferdowsi, A. Sanjab, W. Saad, and T. Basar, “Generalized colonel  q−1 blotto game,” in 2018 Annual American Control Conference (ACC). 2 −1 X q+1−k 1 1 IEEE, 2018, pp. 5744–5749.  2 c v¯sgn(e+(k−1)d−xI )−vsgn(e+(k−2)d−xI ) [16] E. M. Shahrivar and S. Sundaram, “Multi-layer network formation via sA  k=1 a colonel blotto game,” in 2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE, 2014, pp. 838–841. q − 1 1 q−1 1 [17] S. Guan, J. Wang, H. Yao, C. Jiang, Z. Han, and Y. Ren, “Colonel + v¯sgn e+ d−xI − vsgn e+ −1 d−xI 2 2 blotto games in network systems: Models, strategies, and applications,”  IEEE Transactions on Network Science and Engineering, 2019. q q+1 [18] T. Adamo and A. Matros, “A blotto game with incomplete informa- X k− 1 1 2  + c v¯sgn(e+(k−1)d−xI )−vsgn(e+(k−2)d−xI )  tion,” Economics Letters, vol. 105, no. 1, pp. 100–102, 2009. q+1 [19] D. Kovenock and B. Roberson, “A blotto game with multi-dimensional k= 2 +1 (33) incomplete information,” Economics Letters, vol. 113, no. 3, pp. 273– 275, 2011. 1 For notational purposes, let HI (¯v, v, xI ) denote the above [20] R. Siegel, “Asymmetric all-pay auctions with interdependent valua- quantity. One can list all possible values this takes as a tions,” Journal of Economic Theory, vol. 153, pp. 684 – 702, 2014. function of 1 (not shown due to space constraint). This [21] F. Vega-Redondo, Economics and the Theory of Games. Cambridge xI University Press, 2003. 1 is increasing in xI , and attains the maximum value when 1 q−1 1 ∗ x ∈ (e + − 1 d, XI ), giving max 1 πI (x ,F |t ) = I 2 xI I U 1 − v (1 + c). sA 2 Any pure allocation xI ∈ [0,XI ] of player I against ∗ 2 ∗ FU in type t2 gives ex-interim utility πI (xI ,FU |t2) = 2 HI (v, v,¯ xI ). Through a similar analysis, we find the max- v imal value − (1 + c) is attained for x2 ∈ (0, e + q−1 d). sA I 2 Hence, any mixed strategy FI (t1) with support on (e + q−1 − q−1 2 1 d, XI ) and FI (t2) with support on (0, e+ 2 d) is ∗ ∗ ∗ a best-response to FU . Consequently, FI ∈ BRI (FU ). The quantities v and v (1 + c) coincide, giving the value of sB sA the game.

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