Rationalizability and Learning in Games with Strategic Heterogeneity
Anne-Christine Barthela, Eric Homanna,∗
aWest Texas A&M University, 2501 4th Ave, Canyon, TX 79016, USA
Abstract
It is shown that in games of strategic heterogenity (GSH), where both strategic complements and substitutes are present, there exist upper and lower serial undominated strategies which provide a bound for all other rationalizable strategies. We establish a connection between learning in a repeated setting and the iterated deletion of strictly dominated strategies which provides necessary and su cient conditions for dominance solvability and stability of equilibria. These results not only extend monotonicity analysis to a wider class of games, but generalize many results in the literature on games of strategic complements and substitutes. Lastly, we provide conditions under which games that do not exhibit monotone best responses can be analyzed as a GSH. Multiple examples are given. Keywords: JEL
1. Introduction
Games in which players exhibit monotone best response correspondences have been the focus of extensive study, and describe two basic strategic interactions. Under one scenario, such as Bertrand price competition, players nd it optimal to best respond to their opponents' decision to take a higher action by also taking a higher action, describing games of strategic complements (GSC). Under the opposite scenario, including Cournot quantity competition, players nd it optimal to best respond to their opponents' decision to take a higher action by taking a lower action, describing games of strategic substitutes (GSS). One advantage of
∗Corresponding author Email addresses: [email protected] (Anne-Christine Barthel), [email protected] (Eric Homann)
Preprint submitted to Elsevier January 5, 2017 describing games in terms of the monotonicty of their best response correspondences is that by doing so, we can guarantee that the set of rationalizable strategies as well as the set of Nash equilibria possess a certain order structure, which greatly aids in their analysis. Among other properties, in both GSC and GSS, best response dynamics starting at the highest and lowest actions in the strategy set lead to highest and lowest serially undominated (SU) strategies. Thus, convergence of such dynamics is su cient to guarantee the existence of globally stable, dominance solvable Nash equilibria. This paper shows that both GSC and GSS are special cases of a class of games whose set of rationalizable and equilibrium strategies possess nice order properties such as the ones described above, called games of strategic heterogeneity (GSH). GSH describe situations where some players may have monotone increasing best responses while allowing for others to have monotone decreasing best response responses, and hence encompass a wide range of games, including matching pennies type scenarios, Bertrand-Cournot competition, games on networks, as well as all GSC and GSS. Our main result shows that in GSH, a surprising connection between the two distinct solution concepts of learning in games and rationalizabil- ity is established: As long as players learn to play adaptively (including Cournot dynamics and ctituous play, among others) in a repeated setting, then resulting play will eventually be contained within an interval [a, a] de ned by upper and lower SU strategies, a and a, re- spectively.1 As a consequence, new results concerning equilibrium existence, uniqueness, and stability are established. In particular, if a GSH is dominance solvable, so that a = a, then all adaptive learning necessarily converges to a unique Nash equilibrium. We also provide conditions under which an arbitrary game, which may not exhibit either montone increasing or decreasing best responses, may be embedded into a corresponding GSH. Most impor- tantly, this embedding preserves the order structure of the set of rationalizable strategies, allowing us to draw analogous conclusions in the original game. Monotonicity analysis in games goes back to Topkis [10] and Vives [12], which led to a general formulation of GSC by Milgrom and Roberts [4] and Milgrom and Shannon [5],
1Recall that a serially undominated strategy is one that survives the iterated deletion of strictly domi- nanted strategies.
2 henceforth referred to as MS. By assuming that each player's strategy set is a complete lattice, MS show that if utility functions satisfy certain ordinal properties (supermodularity and the single crossing property) which guarantee that the bene t of choosing a higher action against a lower action is increasing in opponents' actions, then best responses are non-decreasing in opponents' strategies, capturing the notion of strategic complementarities. In this setting, MS show that dynamics starting from the highest and lowest available strategies converge respectively to upper and lower SU strategies a and a, which are also Nash equilibria, and hence an interval [a, a] containing all SU and rationalizable strategies can be constructed. This observation has found a number of applications. For example, in a global games setting, Frankel, Morris, and Pauzner [2] show that in a GSC, as the noise of private signals about a fundamental approaches zero, the issue of multiplicity is resolved as the upper and lower SU Bayesian strategies converge to a unique global games prediciton. Mathevet [3] studies supermodular mechanism design, and gives conditions under which a social choice function is optimally supermodular implementable, giving the smallest possible interval of rationalizable strategies around the truthful equilibrium. Using similar methods, Roy and Sabarwal [9] show that dynamics also lead to extremal SU strategies in GSS, although they need not be Nash equilibria. The dynamics of GSH and how they relate to the set of solutions remains an open question. Monaco and Sabarwal [6] show that the set of equilibria in a GSH is completely unordered, and also give su cient conditions for comparative statics. However, these results assume the existence of equilibria, which need not be true in general GSH. It is therefore of interest to understand when equilibria can be guaranteed to exist, when they are stable, and under what conditions players can learn to play such strategies. Consider the following example:
Example 1. Cournot-Bertrand Duopoly Consider a particular case of the Cournot-Bertrand duopoly studied in Naimzada and
Tramontana (NT)[7], where rm 1 competes by choosing quantity, and rm 2 competes by
3 choosing price.2 Suppose that each rm i = 1, 2 faces a constant marginal cost of 3, so that
pro ts can be written as πi = (pi − 3)qi, and the demand system is given by the following linear equations:
1 p = 81 − q − q 1 1 2 2 1 q = 81 − p − q 2 2 2 1
Solving for best response functions then gives
1 q∗(p ) = 25 + p 1 2 3 2
1 p∗(q ) = 42 − q 2 1 4 1
Because player 1's best response is increasing in p2, and player 2's best response is decreasing
in q1, this is a GSH. We will be interested in deriving conditions which guarantee the existence and stability of equilibria in this setting. Suppose WLOG that the maximium output for
rm 1 and price for rm 2 are given by q1 = 168 and p2 = 429, respectively, so that the largest element in the strategy space is given by z0 = (168, 429). Obviously, the smallest strategy in the strategy space is given by y0 = (0, 0). Starting from these points, consider the following dynamic process:
• z0 = (168, 429), y0 = (0, 0).
1 ∗ 0 ∗ 0 , 1 ∗ 0 ∗ 0 . • z = (q1(z2), p2(y1)) y = (q1(y2), p2(z1))
For general , k ∗ k−1 ∗ k−1 , k ∗ k−1 ∗ k−1 . • k ≥ 1 z = (q1(z2 ), p2(y1 )) y = (q1(y2 ), p2(z1 ))
The rst three iterations of the above dynamics are shown in the graph below. The path in
the northeast corner of the graph represents the evolution of the sequence zk, while the path in the southwest corner of the graph represents the evolution of the sequence yk. Notice that by construction, for each n ≥ 0, zk ≥ yk, and that if both sequences converge to the same
2For example, consider the market for produce, where a farmer competes in quantity, and a store competes in prices.
4 point ∗ ∗ , then that point is necessarily a Nash equilibrium. Indeed, we see that both (q1, p2) sequences are converging to the unique Nash equilibrium, given by ∗ ∗ . (q1, p2) = (36, 33)
NT show that under Cournot best response dynamics, where each period's play is a best response to the play in the previous period, such equilibria are stable. However, by exploiting the monotonicity of the best reponse functions, the convergence of our dynamics allow us to not only conclude that a unique equilibrium exists, but also that it is the unique rationalizable strategy, as well as being stable under a much wider range of learning processes. To see why this is true, consider y0 and z0. By allowing player 1, the strategic complements player, to best respond to the highest strategy possible by her opponent, 0, then 1 constitutes the z2 z1 largest best response possible by player 1 regardless of player 2's initial strategy. Likewise, by allowing player 2, the strategic substitutes player, to best respond to the lowest strategy possible by her opponent, 0, then 1 constitutes the largest best response possible by player y1 z2 2 regardless of player 1's initial strategy. Hence, z1 is the largest possible joint best response after one round of iterated play starting from any initial strategy, and likewise y1 is the smallest. By the same logic, z2 and y2 are, respectively, the largest and smallest possible joint best responses after 2 rounds of iterated play starting from any initial strategy. By continuing in this manner, we see that play starting at any initial strategy and updated
5 according to Cournot dynamics must be contained within the interval de ned by [yn, zn] after n rounds of play. Theorem shows that this same logic holds true as long as players choose strategies adaptively, including Cournot learning, ctitious play, and a wide range of other updating rules. Hence, the convergence of k and k to ∗ ∗ not only implies y z (q1, p2) the stability of ∗ ∗ , but also that any adaptive dynamic starting at any initial strategy (q1, p2) will converge to ∗ ∗ , which is called global stability. (q1, p2) (q1, p2) Seeing that the equilibrium ∗ ∗ is the unique rationalizable strategy is a (q1, p2) = (36, 33) much more subtle point, and shown in Theorem 1. Example 2 expands on this analysis and shows that under general conditions, such equilibria are are dominance solvable and globally stable. This constitutes a new result in the understanding of Cournot-Bertrand duopolies.
2. Theoretical Framework
This paper will use standard lattice de nitions (See Topkis [11] for a complete discus-
sion.). Recall that a partially ordered set3 (X, ) is a lattice if for each x, y ∈ X, x ∨ y ∈ X and x ∧ y ∈ X, where x ∨ y and x ∧ y are the supremum (least upper bound) and in mum (greatest lower bound) of x, y, respectively. If, in addition, we have that for each S ⊂ X, ∨S, ∧S ∈ X, where ∨S and ∧S are the supremum and in mum of S, then (X, ) is a complete lattice.
Let I be a non-empty set of nitely many players. For each player i ∈ I, we will let
Ai denote player i's action space, which has an associated partial ordering i. Without
mention, we will write as the the product order of the i when referring to the product Q Q space A = Ai as well as the product space A−i = Aj. Each player has a utility function i∈I j6=i given by . A game can then be described by a tuple , πi : Ai × A−i → R Γ = {I, (Ai, πi)i∈I } bringing us to the following de nition:
De nition 1. A game Γ = {I, (Ai, πi)i∈I } is a game of strategic heterogeneity (GSH) if for each i ∈ I, the following hold:
1. (Ai, i) is a complete lattice.
3(X, ) is a partially ordered set if is re exive, transitive, and anti-symmetric.
6 4 2. πi is continuous in a (in the order interval topology), and quasisupermodular in ai.
5 6 3. πi satis es either the single crossing or decreasing single crossing property in (ai; a−i).
Notice that this de nition is very general and allows for games with strategic substitutes, games with strategic complements, or any mixture of the two. That is, by Milgrom and Shannon [5], under Conditions 1 and 2 above, the best response correspondence
BRi(a−i) = argmax πi(ai, a−i) ai∈Ai is guaranteed to be a non-empty, complete lattice. By Roy and Sabarwal (2010) and Milgrom and Shannon [5], if πi satis es the (decreasing) single crossing property in (ai; a−i), then
7 Ai is increasing (decreasing) in the strong set order. If best responses are BRi : A−i 2 singletons, this is equilivalent to BRi(a−i) being monotone non-decreasing (non-increasing).
Notice that the quasisupermodularity of πi in ai will always be satis ed in the case when is linearly ordered, such as when it is a subset of . The single crossing and decreasing Ai R single crossing properties are weaker ordinal versions of increasing dierences and decreasing dierences, respectively, which can easily be veri ed in the case when πi is dierentiable. That is, as long as the cross partials between own strategy and opponents' strategy are non-negative (non-positive), then πi satis es increasing (decreasing) dierences (See Topkis
∂π2 1 ∂π1 1 (1998)). In Example 1, we had = and = − so that π2 and π1 satis y the ∂p1∂q2 3 ∂q2∂p1 4 single crossing and decreasing single crossing property, respectively.
It will be useful to partition the set of players I into IS and IC , where each i ∈ IS
is such that πi exhibits the decreasing single crossing property in (ai; a−i), and each player
i ∈ IC is such that πi exhibits the single crossing property in (ai; a−i). A typical element can thus be described as S C , where S and C . Thus, for a ∈ A (a , a ) a = {ai}i∈IS a = {ai}i∈IC
4 is quasisupermodular in if for each 0 , and , 0 πi ai ai, ai ∈ Ai a−i ∈ A−i πi(ai, a−i) ≥ πi(ai ∧ ai, a−i) ⇒ 0 0 and 0 0 0 . πi(ai ∨ ai, a−i) ≥ πi(ai, a−i) πi(ai, a−i) > πi(ai ∧ ai, a−i) ⇒ πi(ai ∨ ai, a−i) > πi(ai, a−i) 5 satis es the single crossing property in if for every 0 and , 0 πi (ai; a−i) ai i ai a−i −i a−i πi(ai, a−i) ≥ 0 0 0 and 0 0 0 0 . πi(ai, a−i) ⇒ πi(ai, a−i) ≥ πi(ai, a−i) πi(ai, a−i) > πi(ai, a−i) ⇒ πi(ai, a−i) > πi(ai, a−i) 6 satis es the decreasing single crossing property in if for every 0 and , πi (ai; a−i) ai i ai a−i −i a−i 0 0 0 0 and 0 0 0 0 . πi(ai, a−i) ≥ πi(ai, a−i) ⇒ πi(ai, a−i) ≥ πi(ai, a−i) πi(ai, a−i) > πi(ai, a−i) ⇒ πi(ai, a−i) > πi(ai, a−i) 7 The strong set order is de ned as follows: For non-empty subsets A, B of Ai, A v B if for every a ∈ A, and for every b ∈ B a ∧ b ∈ A and a ∨ b ∈ B.
7 a ∈ A, the largest and smallest joint best responses ∨BR(a) and ∧BR(a) can be described
as S C , where S and C ∨BR(a) = (∨BR (a), ∨BR (a)) ∨BR (a) = {∨BRi(a−i)}i∈IS ∨BR (a) = , and likewise for . {∨BRi(a−i)}i∈IC ∧BR(a)
3. Main Results
In this section we develop the main results of the paper. First, we draw a connec- tion between learning in a GSH and the iterative deletion of strictly dominated strategies, which establishes a relationship between the set of Nash equilibria and serially undominated strategies. Second, we show that a wide range of games which do not satisfy De nition 1 can nevertheless be analyzed as a GSH by considering alternative orderings on the strategy space.
3.1. Characterizing Solution Sets
We begin with the following de nitions:
De nition 2. Suppose that Γ is a GSH, and let i ∈ I.
1. We say that is strictly dominated if there exists 0 such that for all ai ∈ Ai ai ∈ Ai
a−i ∈ A−i,
0 πi(ai, a−i) > πi(ai, a−i).
2. For S ⊂ A−i, we de ne player i's undominated responses as
0 0 URi(S) = {ai ∈ Ai | ∀ai ∈ Ai, ∃a−i ∈ S, πi(ai, a−i) ≥ πi(ai, a−i)}.
That is, the set of undominated responses to a subset of opponents' actions S are those actions that are not strictly dominanted by any of player i's actions. Thus, given any X ⊂ A, we can de ne the set of joint undominated responses as
UR(S) = (URi(S−i))i∈I
8 where S−i is the projection of S on A−i. We can then de ne the following iterative pro- cess: Let S0 = A, and for all k ≥ 1, de ne Sk = UR(Sk−1). A strategy a ∈ A is then serially undominated if a ∈ ∩ Sk = SU. It is straightforward to con rm that all ratio- k≥0 nalizable strategies as well as Nash equilibria are serially undominated. The rst Lemma shows that in a GSH, we have a nice characterization of the interval containing all seri- ally undominated responses to an interval of strategies [a, b]. To this end, we will de ne
URd(S) = [∧UR(S), ∨UR(S)] to be the smallest order interval containing UR(S).
Lemma 1. For each b a, URd([a, b]) = [(∧BRS(b), ∧BRC (a)), (∨BRS(a), ∨BRC (b))].
Proof. We will rst show that UR([a, b]) ⊆ [(∧BRS(b), ∧BRC (a)), (∨BRS(a), ∨BRC (b))]. Suppose this isn't the case, and that for some y ∈ A, y∈ / [(∧BRS(b), ∧BRC (a)), (∨BRS(a), ∨BRC (b))].
Then either y (∧BRS(b), ∧BRC (a)) or y (∨BRS(a), ∨BRC (b)). In particular, consider an such that . Then strictly dominates against i ∈ IS yi ∨BRi(a−i) (yi) ∧ (∨BRi(a−i)) yi
every x−i ∈ [a−i, b−i]. To see this, note that
πi((yi) ∨ (∨BRi(a−i), a−i) − πi(∨BRi(a−i), a−i) < 0
⇒ πi(yi, a−i) − πi((yi) ∧ (∨BRi(a−i), a−i)) < 0
⇒ πi(yi, x−i) − πi((yi) ∧ (∨BRi(a−i)), x−i) < 0,
where the rst implication follows from quasi-supermodularity and the second implication
follows from the decreasing single crossing property. Thus yi ∈/ URi([a, b]), and because a
similar argument holds for all i ∈ IC , we have y∈ / UR([a, b]). Thus UR([a, b]) is contained in [(∧BRS(b), ∧BRC (a)), (∨BRS(a), ∨BRC (b))], from which it follow that URd([a, b]) is as well.
Alternatively, because ∧BRS(b), ∨BRS(a), ∧BRC (a), and ∨BRC (b) all consist of best re- sponses to elements in [a, b], we have that both (∧BRS(b), ∧BRC (a)) and (∨BRS(a), ∨BRC (b)) are contained in UR([a, b]) ⊂ URd([a, b]) and hence [(∧BRS(b), ∧BRC (a)), (∨BRS(a), ∨BRC (b))] ⊂ URd([a, b]) is as well.
We will now de ne an adaptive dynamic (following Milgrom and Roberts [4]). First,
9 given a sequence of repeated play k ∞ , let us de ne (a )k=0
P (K, k) = {at | K ≤ t < k} as those elements that were played between time periods and . Then k ∞ is an K k (a )k=0 adaptive dynamic if
∀K ≥ 0, ∃K0 ≥ 0, ∀k ≥ K0, ak ∈ URd([∧P (K, k), ∨P (K, k)]).
That is, a sequence of play is adaptive as long as there exists a point beyond which all play falls within the interval de ned by the highest and lowest undominated responses to the past play. This de nition is quite broad, and includes such learning processes as Cournot dynamics and ctitious play, among others. We will now proceed to draw a connection between the evolution of repeated play, specif- ically an adaptive dynamic, and the set of serially undominated strategies. This will be done in a series of Lemmas, which bring us to our rst main result, Theorem 1. In order to do so, we will study the evolution of upper and lower dynamics, which are de ned below: De ne the following sequences:
1. z0 = ∨A, y0 = ∧A.
2. z1 = (∨BRS(∧A), ∨BRC (∨A)) = (∨BRS(y0), ∨BRC (z0)). y1 = (∧BRS(∨A), ∧BRC (∧A)) = (∧BRS(z0), ∧BRC (y0)).
3. In general, zk = (∨BRS(yk−1), ∨BRC (zk−1)), yk = (∧BRS(zk−1), ∧BRC (yk−1)).
Then the sequence k ∞ is the upper dynamic starting at and k ∞ is the lower (z )k=0 ∨A (y )k=0 dynamic starting at ∧A. Notice that these sequences are similar to those de ned in Milgrom and Roberts [4] and Roy and Sabarwal [? ]. In the former case, the upper dynamic and lower dynamic are de ned, respectively, as best response dynamics starting from the highest and lowest strateiges, which, by strategic complementarities, de ne decreasing and increasing sequences. In the latter case, the lower dynamic was de ned as the best response to the highest possible strategy, and then the best response to the best response of the lowest
10 possible strategy, etc., which by strategic substitutes, resulted in an increasing sequence, and
likewise a decreasing sequence for the upper dynamic. Our dynamics allow for players in IS to follow the same procedure as in Roy and Sabarwal, and for those players in IC to follow the same procedure as in Milgrom and Roberts. The next Lemma then shows that such a procedure converges to upper and lower strategies z and y, respectively, and that when they converge to the same strategy, or z = y, a Nash equilibrium exists.
Lemma 2. If Γ is a GSH, then the following are true:
1. k ∞ and k ∞ are decreasing and increasing sequences, respectively. (z )k=0 (y )k=0
2. zk → z and yk → y for some z, y ∈ A.
3. For i ∈ IS, yi ∈ BRi(z−i) and zi ∈ BRi(y−i). For i ∈ IC , yi ∈ BRi(y−i) and
zi ∈ BRi(z−i). Hence, if y = z, this strategy is a Nash equilibrium.
Proof. First note that
z1 = (∨BRS(∧A), ∨BRC (∨A)) ∨A = z0
and
y1 = (∧BRS(∨A), ∧BRC (∧A)) ∧A = y0.
n−1 n n n−1 Suppose that ∀k = 1, 2, ..., n, z z and y y . Then, since for all i ∈ IC (respec-
tively i ∈ IS), best responses are increasing (respectively decreasing), we have that
zn+1 = (∨BRS(yn), ∨BRC (zn)) (∨BRS(yn−1), ∨BRC (zn−1)) = zn,
and
yn+1 = (∧BRS(zn), ∧BRC (yn)) (∧BRS(zn−1), ∧BRC (yn−1)) = yn.
n n Therefore, (z )n≥0 and (y )n≥0 are decreasing and increasing sequences, respectively, estab- lishing Claim 1. Because monotone sequences in complete lattices converge in the order
interval topology, establishing Claim 2. For Claim 3, suppose that for i ∈ IS, yi ∈/ BRi(z−i).
11 Then there exists some xi ∈ Ai such that ui(xi, z−i) > ui(yi, z−i), and hence by the continuity of , there exists an such that n n+1 n . However, this contradicts ui n ≥ 0 ui(xi, z−i) > ui(yi , z−i) the optimality of n+1 against n , hence . The other cases follow similarly. yi z−i yi ∈ BRi(z−i)
Claim 3 above extends known results in the GSC and GSS literature to more general GSH. That is, when a GSH is a pure GSC, the existence of a Nash equilibrium is guaranteed
regardless of whether y = z, as y ∈ BR(y) and z ∈ BR(z), and hence both strategies are equilibrium strategies. In the case of when the GSH is a pure GSS, we have that y ∈ BR(z) and z ∈ BR(y), and hence both strategies are simply rationalizable,8 but need not be equilibrium strategies. In a general GSH, we see that both y and z are rationalizable, but need not be either equilibrium or simply rationalizable strategies. However, Lemmas 4 and 5 show that y and z are the smallest and largest serially undominated (and hence rationalizable) strategies, respectively, which, by Lemma 3 below, provide a bound on the limit of any
adaptive dynamic k ∞ . Notice that together, these results draws a tight connection (a )k=0 between two seemingly distinct solution concepts in a GSH. That is, the limit of any adaptive dynamic, which can be seen as the result of the "interactive" process of learning in a repeated game, must lie within the interval determined by the largest and smallest rationalizable strategies, which are the result of the "introspective" process of deleting strategies that are never best responses.
Lemma 3. Suppose that is a GSH, and let k ∞ and k ∞ be the upper and lower Γ (z )k=0 (y )k=0 dynamics, respectively. Let k ∞ be any adaptive dynamic. Then (a )k=0
k N N 1. ∀N ≥ 0, ∃KN ≥ 0, ∀k ≥ KN , a ∈ [y , z ].
2. y lim inf(ak) lim sup(ak) z.
Proof. To prove Claim 1, notice that the statement holds trivially for N = 0. Suppose
that this statement holds for N − 1, so that there is a KN−1 such that for all k ≥ KN−1, ak ∈ [yN−1, zN−1]. To see that this holds for N, rst note that the previous fact implies that
8We will say that a strategy a ∈ A is simply rationalizable if there exists a0 ∈ A such that a ∈ BR(a0) and a0 ∈ BR(a). That is, a can be rationalized by a short cycle of conjectures.
12 N−1 N−1 for all k ≥ KN−1, [∧P (KN−1, k), ∨P (KN−1, k)] ⊆ [y , z ]. Then, by the de nition of an
k adaptive dynamic, let KN be such that ∀k ≥ KN , a ∈ URd([∧P (KN−1, k), ∨P (KN−1, k)]).
Then, for alll k ≥ KN ,
k N−1 N−1 a ∈ URd([∧P (KN−1, k), ∨P (KN−1, k)] ⊆ URd([y , z ]) =
= [(∧BRS(zN−1), ∧BRC (yN−1)), (∨BRS(yN−1), ∨BRC (zN−1))] = [yN , zN ],
where the inclusion follows from the monotonicity of URd, and the rst equality follows from Lemma 1, proving Claim 1. The Claim 2 follows immediately from this observation.
We now show that y and z are themselves serially undominated, and that they provide a bound for the set of serially undominated strategies.
Lemma 4. Let Γ be a GSH. Then
1. SU ⊆ [y, z].
2. y, z ∈ SU.
Proof. To prove Claim 1, note that S0 ⊆ [∧A, ∨A] = [y0, z0]. Suppose by way of induction that for k ≥ 0, Sk ⊆ [yk, zk]. Then
Sk+1 = UR(Sk) ⊆ UR([yk, zk]) ⊆ [∧UR([yk, zk]), ∨UR([yk, zk])] ≡ URd([yk, zk]).
By Lemma 1, URd([yk, zk]) = [(∧BRS(zk), ∧BRC (yk)), (∨BRS(yk), ∨BRC (zk))] = [yk+1, zk+1], completing the induction step. Hence, SU = ∩ Sk ⊆ ∩ [yk, zk] = [y, z], proving Claim 1. k≥0 k≥0
Claim 2 follows immediately from Claim 3 of Lemma 2. That is, because each yi and zi
are best responses to either y−i or z−i, then y and z survive the process of iteratively deleting strictly dominated strategies, and are hence serially undominated.
We now come to the rst main result, which summarizes the above observations by showing that dominance solvability is equivalent to the convergence of all adaptive dynamics to a unique Nash equilibrium. Note that this automatically implies that a dominance solvable Nash equilibrium will be stable under all adaptive dynamics.
13 Theorem 1. Let Γ be a GSH. Then the following are equivalent:
1. Γ is dominance solvable, or y = z.
2. Every adaptive dynamic converges.
3. Every adaptive dynamic converges to the same Nash equilibrium.
In Roy and Sabarwal [9] and Milgrom and Roberts [4], the upper and lower dynamics
were derived from best responses sequences starting from ∨A and ∧A, which are themselves adaptive dynamics, in which case the implication (2) ⇒ (1) in Theorem 1 above follows immediately. However, our upper and lower dynamics are not constructed from such best response sequences, and hence we make use of the following Lemma:
Lemma 5. De ne the sequence k ∞ by the following: (a )k=0
k y 2 , k even ak = . k−1 z 2 , k odd
Then k ∞ is an adaptive dynamic. Furthermore, if this sequence converges, we have that (a )k=0 y = z.
Proof. See Appendix.
We now prove Theorem 1.
Proof. (of Theorem 1) It is immediate that (3) implies (2), and from Lemma 3, that (1) implies (2) and (3). It only remains to verify that (2) implies (1). Suppose that (2) holds,
and consider the adaptive dynamic k ∞ de ned in Lemma . Then, since all adaptive (a )k=0 5 dynamics converge, by Lemma 5 we have that y = z, establishing (1).
Following Roy and Sabarwal [9], we say that a pure strategy Nash equilibrium a∗ ∈ A is globally stable if every adaptive dynamic converges to a∗. It then follows from Theorem 1 that every dominance solvable GSH has a unique Nash equilibrium which is also globally stable. In fact, the existence of a globally stable equilibrium is equivalent to dominance solvability. We now present an example:
14 Example 2. (Cournot-Bertrand Duopoly, continued) Consider a more general version of the Cournot-Bertrand Duopoly from Example 1, where the demand system is given by 1 p = a − q − q 1 1 2 2 1 q = a − p − q . 2 2 2 1
for some constant a > 0, and suppose that marginal costs are given by c > 0. Best responses can then be written as a − 2c 1 q∗ = + p 1 3 3 2 a + c 1 p∗ = − q . 2 2 4 1
Suppose also that A1 = [0, a¯1] and A2 = [0, a¯2]. We will use Theorem 1 to show the existence of a globally stable dominance solvable Nash equilibrium by showing that the upper and lower dynamics converge to the same strategy pro le. To that end, let a¯ = max{a¯1, a¯2}, and note that for k = 0, it is trivially true that
0 0 zi − yi ≤ a.¯ for each player i = 1, 2. Suppose by way of induction that for each k ≥ 0 we have that
a¯ zk − yk ≤ . i i 2k
Then, since player 1 is a complements player, we have by the de nition of upper and lower dynamics that
a − 2c 1 a − 2c 1 1 1 a¯ a¯ zk+1 − yk+1 = ( + zk) − ( + yk) = (zk − yk) ≤ ≤ . 1 1 3 3 2 3 3 2 3 2 2 3 2k 2k+1
Also, since player 2 is a substitutes player, we have by the de nition of upper and lower
15 dynamics that
a + c 1 a + c 1 1 1 a¯ a¯ zk+1 − yk+1 = ( − yk) − ( − zk) = (zk − yk) ≤ ≤ . 2 2 2 4 1 2 4 2 4 1 1 4 2k 2k+1
Therefore, it is clear that as k → ∞, both zk and yk converge to the same strategy pro le a∗. By Theorem 1, a∗ is a dominance solvable Nash equilibrium which is also globally stable.
3.2. Monotonic Embeddings
One reasonable way to de ne a GSH would have been to have pairwise complements or substitutes. Theorem 2 shows that in many cases, such games can be analyzed as a GSH as de ned in De nition 1, which greatly broadens the applicability of the results in Section 3.1. In order to do so, we introduce the following notion:
De nition 3. Let Γ = {I, (Ai, πi)i∈I } be a game (not necessarily a GSH), where each is a complete lattice and linearly ordered by . Then is a Ai i Γe = {I, (Aei, πei, fi)i∈I } monotonic embedding of Γ if, for each i ∈ I, the following hold:
1. Aei is a set of actions for player i, which is a complete lattice and linearly ordered by . e i 2. is continuous in , satis es quasisupermodularity in , and satis es either uei : Ae → R ea eai the single crossing or decreasing single crossing property in . (eai; ea−i) 9 3. fi : Ai → Aei is bijective, and either strictly increasing or strictly decreasing. 4. For each , we have that .10 a ∈ A uei(f(a)) = ui(a)
Note that by the bijectiveness of each , we can describe an element as fi ea ∈ Ae ea = f(a) for the appropriate a = (ai)i∈I ∈ A without loss of generality. Conditions 1 and 2 in the above de nition imply that Γe is itself a GSH. Conditions 3 and 4 ensure that Γ can be embedded into Γe in a monotonic way through the fi. Notice that Condition 4 implies the
9 is strictly increasing if 0 0 and 0 0 , and strictly decreasing fi ai i ai ⇒ fi(ai)e ifi(ai) ai i ai ⇒ fi(ai) e ifi(ai) if 0 0 and 0 0 . ai i ai ⇒ fi(ai)e ifi(ai) ai i ai ⇒ fi(ai) e ifi(ai) 10 For each a ∈ A, f(a) is de ned as (fi(ai))i∈I ∈ Ae.
16 following: For each , each 0 , and each , i ∈ I ai, ai ∈ Ai a−i ∈ A−i
0 if and only if 0 πi(ai, a−i) ≥ πi(ai, a−i) πei(fi(ai), f−i(a−i)) ≥ πei(fi(ai), f−i(a−i)).
Therefore, a ∈ A is a Nash equilibrium in Γ if and only if f(a) ∈ Ae is a Nash equilibrium in Γe, and a ∈ A is serially undominated in Γ if and only if f(a) ∈ Ae is serially undominated in Γe. Because upper and lower serially undominated strategies are guaranteed to exist in a GSH, it follows that the set of serially undominated strategies in Γ is non-empty as long as there exists a monotonic embedding Γe of Γ. Proposition 1 below allows us to say even more concerning structure of the set of serially undominated strategies as well as adaptive dynamics when an embedding exists. Speci cally, if a monotonic embedding Γe of Γ exists, then we can draw conclusions about the set of serially undominated strategies in Γ by studying the corresponding GSH Γe. To that end, we will often describe a strategy as , where (resp. ) are those a = ((ai)i∈ID ), (ai)i∈II )) i ∈ ID i ∈ II players whose fi is strictly the decreasing (resp. increasing) according to De ntion 3.
Proposition 1. Let Γ be a game, and Γe a monotonic embedding of Γ. Then we have the following:
1. There exist smallest and largest serially undominated strategies a = ((ai)i∈ID , (ai)i∈II ) and in , where and a¯ = ((¯ai)i∈ID , (¯ai)i∈II ) Γ ((fi(¯ai)i∈ID ), (fi(ai)i∈II )) ((fi(ai)i∈ID ), (fi(¯ai)i∈II )) correspond to the smallest and largest serially undominated strategies in Γe, respectively. In speci c, Γ is dominance solvable if and only if Γe is dominance solvable.
2. k ∞ is an adaptive dynamic in if and only if k ∞ is an adaptive dynamic (a )k=0 Γ (f(a ))k=0 in Γe.
Proof. See Appendix.
In the case when each fi is continuous and has a continuous inverse, Proposition 1 allows
us to fully extend Theorem 1 to games Γ which have a monotonic embedding Γe, which is summarized in the following Theorem:
17 Theorem 2. Let Γ be a game, and Γe a monotonic embedding of Γ according to De nition
3. Suppose that each fi : Ai → Aei is a homeomorphism. Then the following are equivalent statements about the game Γ:
1. Γ is dominance solvable.
2. Every adaptive dynamic converges.
3. Every adaptive dynamic converges to the same Nash equilibrium.
Proof. Suppose property (1) holds, suppose k ∞ is an adaptive dynamic in . By Propo- (a )k=0 Γ sition 1, k ∞ is an adaptive dynamic in , which is dominance solvable. Since is (f(a ))k=0 Γe Γe a GSH, it follows from Theorem 1 that k ∞ converges to a Nash equilibrium. Since (f(a ))k=0 each has a continuous inverse, this implies that k ∞ converges to a Nash equilibrium. fi (a )k=0 Thus (1) implies (2) and (3). Now suppose that property (2) holds. Notice that if k ∞ (f(a ))k=0 is an adaptive dynamic in , by Proposition 1, k ∞ is a convergent adaptive dynamic. Γe (a )k=0 By the continuity of each , k ∞ converges as well. Thus, every adaptive dynamic in fi (f(a ))k=0 Γe is convergent, and hence by Theorem 1, Γe is dominance solvable. By Proposition 1, Γ is dominance solvable as well. Hence, (2) implies (1) and (3). The fact that (3) implies (1) and (2) follows by the same arguments above.
We now present an example:
Example 3. (3 Player Counot-Bertrand) Consider a 3 player Bertrand-Cournot model, where each rm's pro ts can be described as πi = (pi − c)qi, and each rm faces a demand for its product given by 11 1 2 1 1 p = a − q + p + p 1 3 3 1 3 2 3 3 2 1 4 2 q = a − q − p + p 2 3 3 1 3 2 3 3 2 1 2 4 q = a − q + p − p , 3 3 3 1 3 2 3 3
11Note that this demand system is derived by rewriting the demand system given by 1 P pi = a − qi − 2 qj j6=i for each i = 1, 2, 3 in terms of each opponent's strategic variable.
18 where rm 1 competes in quantity, and rms 2 and 3 compete in price. Also assume that
each Ai = [0, a¯i] for some a¯i > 0. Then each rm's best response can be expressed as
a − 3c 1 1 q∗ = + p + p 1 4 4 2 4 3
a + 2c 1 1 p∗ = − q + p 2 4 8 1 4 3 a + 2c 1 1 p∗ = − q + p , 3 4 8 1 4 2
Notice that both Bertrand rms have best response functions which are increasing in one opponent's strategy and decreasing in the other opponent's strategy, and hence this game is not a GSH. To see how this game can be embedded into a GSH according to De nition
3, for player 1 de ne Ae1 = −A1 and f1(q1) = −q1, and for i = 2, 3, de ne Aei = Ai and
fi(pi) = pi. Lastly, for each player i = 1, 2, 3, de ne
πei(q1, p2, p3) = πi(−q1, p2, p3).
Then
πei(f(q1), f2(p2), f3(p3)) = πi(q1, p2, p3),
and it is readily checked that ∂πe1 = − 1 for j = 2, 3, and ∂πei = 1 , ∂πei = 2 for i, j = 2, 3, ∂pej ∂qe1 3 ∂qe1∂pei 3 ∂pej ∂pei 3 . Hence each satis es the single-crossing property between own action and opponents' i 6= j πe1 action, while and satisfy the decreasing single-crossing property between own action πe2 πe3 and opponents' action. Thus the original game can be embedded into a GSH accroding to De nition 3. Notice that each player's best response function in the transformed game can be written as 3c − a 1 1 q˜∗ = − p˜ − p˜ 1 4 4 2 4 3 a + 2c 1 1 p˜∗ = + q˜ + p˜ 2 4 8 1 4 3 a + 2c 1 1 p˜∗ = + q˜ + p˜ , 3 4 8 1 4 2
19 where each player now has a monotone best response function. Proceeding as in Example 2,
by de ning a¯ = max{a¯1, a¯2, a¯3}, it is straightforward to show that for each i = 1, 2, 3, and each k ≥ 0, 1k zk − yk ≤ a,¯ i i 2
so that zk − yk → 0 as n → ∞. By Proposition 1, the original game has a unique dominance
solvable Nash equilibrium. Furthermore, since each each fi is a homeomorphism, by Theorem 2 we have that the equilibrium is also globally stable.
4. Appendix
We prove Lemma 5 below:
Proof. (Lemma ) First, because it is clear that both k ∞ and k ∞ are subsequences 5 (y )k=0 (z )k=0 of k ∞ , then if k ∞ converges, we have that . (a )k=0 (a )k=0 y = z By Lemma , in order to show that k ∞ is an adaptive dynamic, we must show that 1 (a )k=0 ∀K ≥ 0, ∃K0 ≥ 0, ∀k ≥ K0,
ak ∈ [(∧BRS(∨P (K, k)), ∧BRC (∧P (K, k))), (∨BRS(∧P (K, k)), ∨BRC (∨P (K, k)))] (1)
Let K ≥ 0 be given. We will show that by de ning K0 = K + 2, the above inclusion holds by considering two cases.
t K K 1. Case 1: K is even. Recall that P (K, k) = {a | K ≤ t < k}, and note that y 2 = a ∈
. Suppose that . By the increasingness of k ∞ , we have that for P (K, k) k ≥ K + 2 (y )k=0 all k odd,
k k−1 k−1 K a = z 2 y 2 y 2 .
Similarly, for all k even, we have that
k k K a = y 2 y 2 .
K Hence ∧P (K, k) = y 2 .
20 K K+1 Likewise, for all k ≥ K + 2, z 2 = a ∈ P (K, k). Then, for all k odd, by the
decreasingness of k ∞ , we have that (z )k=0
k k−1 K a = z 2 z 2 ,
and for all k even, we have that
k k k K a = y 2 z 2 z 2 .
K Hence ∨P (K, k) = z 2 . Therefore, the inequality in (1) holds as long as
k S K C K S K C K K+2 K+2 a ∈ [(∧BR (z 2 ), ∧BR (y 2 )), (∨BR (y 2 ), ∨BR (z 2 ))] = [y 2 , z 2 ].
K+2 k k K+2 Because for any k ≥ K+2, we have that z 2 ≥ z y y 2 , then by the de nition
K+2 K+2 of k ∞ , we will have that 2 k 2 , giving the result. (a )k=0 z a y
2. Case 2: K is odd. Using similar arguments as above, we can show that for k ≥ K + 2,
K+1 K−1 ∧P (K, k) = y 2 and ∨P (K, k) = z 2 . Therefore, the inclusion in (1) will hold as long as
k S K−1 C K+1 S K+1 C K−1 a ∈ [(∧BR (z 2 ), ∧BR (y 2 )), (∨BR (y 2 ), ∨BR (z 2 ))]. (2)
By the increasingness and decreasingness of k ∞ and k ∞ , respectively, and by (y )k=0 (z )k=0
S K−1 S K+1 strategic complements and substitutes, we have that ∧BR (z 2 ) ∧BR (z 2 ) and
C K+1 C K−1 ∨BR (z 2 ) ∨BR (z 2 ). Therefore, the inclusion in (2) will hold as long as
k S K+1 C K+1 S K+1 C K+1 K+3 K+3 a ∈ [(∧BR (z 2 ), ∧BR (y 2 )), (∨BR (y 2 ), ∨BR (z 2 ))] = [y 2 , z 2 ].
K+3 k k K+3 Once again, since for any k ≥ K + 2, we have that z 2 z y y 2 , then by
K+3 K+3 the de nition of k ∞ , we will have that 2 k 2 , giving the result. (a )k=0 z a y
This establishes the result.
21 In order to prove Proposition 1, we will make use of the following Lemma. To establish
notation, suppose that Γ is a game and Γe is a corresponding monotonic embedding according to De nition 3. Given a sequence of play
P (K, k) = {at | K ≤ t < k} in Γ, the corresponding sequence of play in Γe is given by
t t Pf (K, k) = f({a | K ≤ t < k}) = {f(a ) | K ≤ t < k}.
Lastly, given a set of actions S ⊂ Ae−i, we will denote the set of player i's undominated responses in Γe as URgi(S).
Lemma 6. Let be a game, a corresponding monotonic embedding, and let k ∞ be a Γ Γe (a )k=0 sequence of actions in Γ. Then we have the following:
1. For each a ∈ A, a ∈ [∧P (K, k), ∨P (K, k)] if and only if f(a) ∈ [∧Pf (K, k), ∨Pf (K, k)].
2. For each i ∈ I and ai ∈ Ai,
ai ∈ URi([∧P (K, k), ∨P (K, k)]) if and only if fi(ai) ∈ URgi([∧Pf (K, k), ∨Pf (K, k)]).
Proof. It is straightforward to check that if fi is strictly increasing and injective, then for each S ⊂ Ai, we have that fi(ai) = ∨fi(S) (resp. ∧fi(S)) if and only if ai = ∨S (resp. ai = ∧S). Alternatively, if fi is strictly decreasing and injective, then fi(ai) = ∨fi(S) (resp.
∧fi(S)) if and only if ai = ∧S (resp. ai = ∨S). To prove Claim 1, suppose that a ∈ [∧P (K, k), ∨P (K, k)], or, equivalently, for each i ∈ I,
j j ∧{ai }K≤j If i ∈ ID, this implies j j fi(∨{ai }K≤j 22 By the above observation, this implies that j j ∧{fi(ai )}K≤j and likewise for i ∈ II . Therefore, f(a) ∈ [∧Pf (K, k), ∨Pf (K, k)]. Because the converse can be proven similiary, this establishes Claim 1. To establish Claim 2, suppose that ai ∈ URi([∧P (K, k), ∨P (K, k)]). (3) Consider , and let 0 be arbitrary. By equation 3 above, there exists some fi(ai) fi(ai) ∈ Aei a−i ∈ A−i within the interval de ned by the highest and lowest strategies played between periods K and k such that 0 πi(ai, a−i) ≥ πi(ai, a−i). By Condition 4 in De nition 3, this implies that 0 πei(fi(ai), f−i(a−i)) ≥ πei(fi(ai), f−i(a−i)). By Claim 1, since a−i falls between the highest and lowest strategies played in Γ between periods K and k, then f−i(a−i) does so as well in Γe, and hence by de nition, we have that fi(ai) ∈ URgi([∧Pf (K, k), ∨Pf (K, k)]). Because the converse can be proven similarly, Claim 2 is established, completing the proof. We now prove Proposition 1: Proof. (Proposition 1) To prove Claim 1, note that because Γe is a GSH, we have by Lemma 4 that there exist lowest and highest serially undominated strategies, which we write respec- tively as and . Then ((fi(¯ai)i∈ID ), (fi(ai)i∈II )) ((fi(ai)i∈ID ), (fi(¯ai)i∈II )) a = ((ai)i∈ID , (ai)i∈II ) 23 and are serially undominated strategies in , which we now show de- a¯ = ((¯ai)i∈ID , (¯ai)i∈II ) Γ ne lower and upper serially undominated strategies. To that end, suppose that a ∈ A is serially undominated in Γ, and suppose that a 6 a. Since each i is a linear order, this implies that for some , we have that . If , then by the strict decreasingness i ∈ I ai i ai i ∈ ID of , we have . However, because is serially undominated in , this con- fi fi(ai) e ifi(ai) f(a) Γe tradicts the fact that is the highest serially undominated strategy ((fi(ai)i∈ID ), (fi(¯ai)i∈II )) in Γe. The case when i ∈ II follows similarly. Hence, a is the lowest serially undominated strategy in Γ. Because the proof showing that a¯ is the highest serially undominated strategy in Γ follows along the same lines, Claim 1 is established. For Claim 2, suppose that k ∞ is an adaptive dynamic. To see that k ∞ is an (a )k=0 (f(a ))k=0 adaptive dynamic, let K ≥ 0 be given. Then there exists K0 ≥ 0 such that for all k ≥ K0, k ∧URi([∧P (K, k), ∨P (K, k)]) ai ∨URi([∧P (K, k), ∨P (K, k)]). If i ∈ ID, this implies that k fi(∨URi([∧P (K, k), ∨P (K, k)])) e fi(ai ) e fi(∧URi([∧P (K, k), ∨P (K, k)])), which, by the observation in the beginning of the proof of Lemma 6 implies that k (4) ∧fi(URi([∧P (K, k), ∨P (K, k)])) e fi(ai ) e ∨ fi(URi([∧P (K, k), ∨P (K, k)])). A similar argument shows that equation 4 holds if as well. Notice that i ∈ II eai ∈ if and only if for some , fi(URi([∧P (K, k), ∨P (K, k)])) eai = fi(ai) ai ∈ URi([∧P (K, k), ∨P (K, k)]) which, by Claim 2 in Lemma 6, is equivalent to . eai = fi(ai) ∈ URgi([∧Pf (K, k), ∨Pf (K, k)]) Hence, fi(URi([∧P (K, k), ∨P (K, k)])) = URgi([∧Pf (K, k), ∨Pf (K, k)]), 24 so that equation 4 becomes k ∧URgi([∧Pf (K, k), ∨Pf (K, k)]) e fi(ai ) e ∨ URgi([∧Pf (K, k), ∨Pf (K, k))), establishing that k ∞ is an adaptive dynamic. The converse can be proven similarly, (f(a ))k=0 establishing Proposition 1. References [1] Amir, R. (1996). Cournot oligopoly and the theory of supermodular games. Games and Economic Behavior, 15(2), 132148. [2] Frankel, D. M., Morris, S., & Pauzner, A. (2003). Equilibrium selection in global games with strategic complementarities. Journal of Economic Theory, 108(1), 144. [3] Mathevet, L. (2010). Supermodular mechanism design. Theoretical Economics, 5(3), 403443. [4] Milgrom, P. & Roberts, J. (1990). Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, 58, 12551277. [5] Milgrom, P. & Shannon, C. (1994). Monotone comparative statics. Econometrica, 62, 157180. [6] Monaco, A. J. & Sabarwal, T. (2015). Games with strategic complements and substitutes. Economic Theory, (pp. 127). [7] Naimzada, A. & Tramontana, F. (2012). Dynamic properties of a cournotbertrand duopoly game with dierentiated products. Economic Modelling, 29(4), 14361439. [8] Roy, S. & Sabarwal, T. (2010). Monotone comparative statics for games with strategic substitutes. Journal of Mathematical Economics, 46(5), 793806. [9] Roy, S. & Sabarwal, T. (2012). Characterizing stability properties in games with strategic substitutes. Games and Economic Behavior, 75(1), 337353. 25 [10] Topkis, D. M. (1979). Equilibrium points in nonzero-sum n-person submodular games. Siam Journal on control and optimization, 17(6), 773787. [11] Topkis, D. M. (1998). Supermodularity and complementarity. Princeton University Press. [12] Vives, X. (1990). Nash equilibrium with strategic complementarities. Journal of Math- ematical Economics, 19(3), 305321. 26