Rationalizable strategies in random games

Games and Economic Behavior 118 (2019) 110–125

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Games and Economic Behavior www.elsevier.com/locate/geb

Rationalizable strategies in random games Ting Pei ∗ , Satoru Takahashi Department of Economics, National University of Singapore, Singapore

a r t i c l e

i n f o

Article history: Received 8 September 2018 Available online 2 September 2019 Keywords: Random games Rationalizability Point rationalizability Pure dominance Random mappings

a b s t r a c t We study point-rationalizable and rationalizable strategies in random games. In a random n × n symmetric game, an explicit formula is derived for √ the distribution of the number of point-rationalizable strategies, which is of the order n in probability as n → ∞. The number of rationalizable strategies depends on the payoff distribution, and is bounded by the number of point-rationalizable strategies (lower bound), and the number of strategies that are not strictly dominated by a pure strategy (upper bound). Both bounds are tight in the sense that there exists a payoff distribution such that the number of rationalizable strategies reaches the bound with a probability close to one. We also show that given a payoff distribution with a finite third moment, as n → ∞, all strategies are rationalizable with probability one. Our results qualitatively extend to two-player asymmetric games, but not to games with more than two players. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Point-rationalizable strategies are strategies surviving iterated elimination of never best responses against pure strategies. By this definition, strategies that form pure-strategy Nash equilibria must be point rationalizable, but not every pointrationalizable strategy appears in a pure-strategy Nash equilibrium. Hence in any two-player game with no tie, the number of point-rationalizable strategies for each player is weakly larger than the number of pure-strategy Nash equilibria.1 The difference in the two numbers can vary from zero to the size of the strategy space depending on the setting of the game. One extreme case is a game like matching pennies that has no pure-strategy Nash equilibrium, and all strategies are point rationalizable. The other extreme case is where all point-rationalizable strategies can form pure-strategy Nash equilibria; examples include common interest games and two-player games with weakly increasing best response functions. A natural question arises: how large is the difference between the number of point-rationalizable strategies and the number of purestrategy Nash equilibria in “typical” games? Similarly, since the number of rationalizable strategies is weakly larger than the support size of mixed-strategy Nash equilibria, we ask: how large is the difference between the number of rationalizable strategies and the support size of mixed-strategy Nash equilibria? To formalize and answer these questions, we study (point) rationalizability in random games. A random game is a normal-form game where all payoffs are drawn independently from some identical continuous distribution. For any realized game, we can compute the number of strategies that satisfy a given solution concept (such as point rationalizability and pure-strategy Nash equilibrium). Hence with the random game framework, we can compute the distribution of the number of point-rationalizable strategies, and compare that with the distribution of the number of

*

Corresponding author. E-mail addresses: [email protected] (T. Pei), [email protected] (S. Takahashi). 1 In a two-player game with no tie, both players have the same number of point-rationalizable strategies. Throughout this paper except Section 4.5, we focus on two-player games with no tie. https://doi.org/10.1016/j.geb.2019.08.011 0899-8256/© 2019 Elsevier Inc. All rights reserved.

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pure-strategy Nash equilibria. Similarly, we can compute the distribution of the number of rationalizable strategies, and compare that with the distribution of the support size of mixed-strategy Nash equilibria. We can quantify differences in solution concepts according to differences in the distributions of the number of strategies satisfying these solutions. First, we apply the random game framework to point rationalizability. Our results provide both the exact and the asymptotic distributions of the number of point-rationalizable strategies as well as the correlation between the number of point-rationalizable strategies and the number of pure-strategy Nash equilibria. √  In a random n × n symmetric game, asymptotically, the number of point-rationalizable strategies follows Rayleigh n (Theorem 3).2 The result is qualitatively the same for asymmetric games, where each player’s payoffs are drawn independently across players (Theorem 10). These results allow us to compare the number of point-rationalizable strategies and the number of pure-strategy Nash equilibria. Indeed, it is known that the asymptotic distribution of the number of pure-strategy Nash equilibria follows a Poisson distribution (Stanford, 1995, 1996). Hence a typical game has many more point-rationalizable √ strategies than pure-strategy Nash equilibria. Specifically, the number of point-rationalizable strategies is of the order n in probability, and the number of pure-strategy Nash equilibria is of the order 1 in probability.3 In addition, we show that the number of point-rationalizable strategies and the number of pure-strategy Nash equilibria are asymptotically independent (Propositions 4 and 11); however, they are slightly correlated when n is finite. Second, we study strategies that survive iterated elimination of strategies strictly dominated by pure strategies.4 We show that with a probability close to one as the size of the strategy space becomes large, no strategy is strictly dominated by pure strategies. Hence with the same probability, all strategies survive iterated elimination of strategies that are strictly dominated by pure strategies (Proposition 5). Third, we study rationalizable strategies. Note that the payoffs of mixed strategies depend on the cardinal payoffs of pure strategy profiles. Hence the distribution of the number of rationalizable strategies and the distribution of the support size of mixed-strategy Nash equilibria are sensitive to the payoff distribution. Our study on rationalizability is asymptotic and based on different orders of limits on the distribution of payoffs, F , and the size of the strategy space, n. One result on rationalizability is derived under a fixed n and a varying F . By definition, the number of rationalizable strategies is bounded below by the number of point-rationalizable strategies (lower bound), and bounded above by the number of strategies that survive iterated elimination of strategies strictly dominated by pure strategies (upper bound). Note that both the lower and upper bounds are determined by the ordinal rankings on the player’s pure strategies against the opponent’s pure strategies. We show that both bounds are tight, i .e., for a given n, there exists a payoff distribution such that the number of rationalizable strategies equals the lower bound with probability close to one, and there exists a payoff distribution such that the number of rationalizable strategies equals the upper bound with probability close to one (Theorem 6). This result is reminiscent of Börgers (1993) and Weinstein (2016). Börgers (1993) shows that a strategy is a best response against some conjecture about the opponents’ pure strategy profiles under some cardinal utility function if and only if it is not dominated in his sense. Weinstein (2016) extends Börgers (1993) to show that the set of rationalizable strategies under a sufficiently risk-averse utility function equals the set of strategies that survive iterated elimination of strategies dominated in Börgers’ sense, and that the set of rationalizable strategies under a sufficiently risk-loving utility function equals the set of point-rationalizable strategies. In order to apply Weinstein’s results to the random game framework, we use Lemma 7 to show the existence of payoff distribution with which his risk-loving (risk-averse) condition is satisfied with a large probability. And then our results and proofs of Theorem 6 for the lower and upper bounds correspond to Weinstein’s for sufficiently risk-loving and risk-averse utility functions. The other result on rationalizability is derived under a fixed F and a varying n. We show that for a given F with a finite third moment, when n goes to infinity, all strategies are rationalizable with probability one (Theorem 8). This result allows us to compare rationalizable strategies and the support of mixed-strategy Nash equilibria. In a random n × n asymmetric game where the payoffs are normally distributed, if n is large, then the expected number of mixed-strategy Nash equilibria is ≈ 1.33n , and most mixed-strategy Nash equilibria assign positive probabilities to ≈ 0.316n pure strategies for each player (McLennan and Berg, 2005). Theorem 8 applies to this case, implying that all strategies are rationalizable with probability close to one. Therefore, with normally distributed payoffs, the number of rationalizable strategies and the support size of all mixed-strategy Nash equilibria are of the same order, n, in probability. (See Table 1 for a summary of our results.) We discuss point rationalizability in subclasses of games: zero-sum games, common interest games, games with arbitrary correlation between players’ payoffs, and games with weakly increasing best responses, where there are additional restrictions on the payoffs. We conjecture that the asymptotic distributions of the number of point-rationalizable strategies in random two-player games are the same in zero-sum games and in games where the two players’ payoffs are independent. For both two-player common interest games and two-player games with weakly increasing best responses, only strategies forming pure-strategy Nash equilibria are point rationalizable. Hence the number of point-rationalizable strategies is the same as the number of pure-strategy Nash equilibria, which is of the order min {m, n} in probability in a random m × n  2  2  with mean √ The Rayleigh distribution with scale parameter σ , denoted by Rayleigh(σ ), has cumulative distribution F (x; σ ) = 1 − exp −x / 2σ σ π /2. 3 For a sequence of random variables X n and a corresponding sequence of constants an , we say that X n is of the order an in probability if for any ε > 0, there exist a finite M > 0 and a finite N > 0 such that P ( M > | X n /an | > 1/ M ) > 1 − ε for all n > N. 2

4 If all payoffs are distinct, then the notion of strict dominance by pure strategies is equivalent to the notion of dominance introduced by Börgers (1993). Hence the two notions are almost surely equivalent in the random game setting with a continuous payoff distribution.

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Table 1 Asymptotic results in random n × n symmetric games.

Pure strategy Nash equilibria Point rationalizable Rationalizable∗∗ Undominated by pure strategies

Asymptotic distribution

Mean

X + 2Y ∗ √  Rayleigh n Depends on the distribution of payoffs n with probability one

2

√ √π n/2  π n/2, n n

∗ X ⊥ Y , X ∼ Pois (1), and Y ∼ Pois (1/2). ∗∗ Details in Theorems 6 and 8.

√

asymmetric common interest game (Roberts, 2005), and of the order n in probability in a random two-player game with weakly increasing best responses (Takahashi, 2008). In games with arbitrary correlation between players’ payoffs, we show by simulation the relation between the expected number of point-rationalizable strategies and the correlation coefficient. Finally, we show that in random games with three or more players, all strategies are point rationalizable with probabilities close to one. To be clear, we do not claim mathematical novelty in computing the number of point-rationalizable strategies in random games. In fact, we are aware of the vast literature on random mappings (Flajolet and Odlyzko, 1990). Our results on point rationalizability, in particular Theorems 3 and 10, are restatements of the distribution of cyclic points of a random mapping (see Rubin and Sitgreaves (1954, Equation 3.8), Harris (1960, Equation 3.12), and Jaworski (1985, Corollary 3)). Our contribution is to extend these results to the context of random games, and to compare the distribution of the number of point-rationalizable strategies to the number of pure-strategy Nash equilibria, rationalizable strategies, and strategies surviving iterated elimination of strategies strictly dominated by a pure strategy. We believe that such comparison improves our understanding of the relationship among various solution concepts in game theory. 2. Random two-player symmetric games 2.1. Notation and formulation A random n × n symmetric game can be represented by U with the following characteristics: 1. {1, 2, . . . , n} is the strategy space for both players. 2. U is an n × n matrix representing the payoffs of both players where u i j is the payoff for a player when she plays strategy i and her opponent plays strategy j. 3. The n2 payoffs u i j are real-valued independent random variables with an identical continuous distribution. Due to the continuity of the payoff distribution, the n2 payoffs are distinct with probability one. Hence, each player has a unique best response against each strategy of her opponent, i .e., the best response function is well defined. Let b : {1, 2, . . . , n} → {1, 2, . . . , n} be the best response function, where b ( j ) denotes the best response to the opponent’s strategy j. The payoffs are i.i.d. continuous random variables, b (1) , b (2) , . . . , b (n) are independent, and each b ( j ) is uniformly distributed on {1, . . . n}. Therefore, b is uniformly distributed on the set of all functions from {1, 2, . . . , n} to itself.5 2.2. Point rationalizability We say a strategy is point rationalizable if it survives iterated elimination of never  best responses against pure strategies ∞ (Bernheim, 1984). In our notation, the set of point-rationalizable strategies is given by t =1 bt ({1, 2, . . . , n}).6 Let PRn denote the random variable that  represents the number of point-rationalizable strategies in a game. We say a sequence i , b (i ) , b2 (i ) , . . . is a best response path. Each best response path will enter into a cycle since the strategy space is finite. Let λ (i ) denote the first strategy that appears twice in the best response path from i. We call λ (i ) the terminal strategy of i. Let  be the set of all λ (i ) , i = 1, . . . , n. Lemma 1.  is the set of all point-rationalizable strategies. Proof. Note that  is the collection of all strategies in best response cycles. The strategies in best response cycles will not be eliminated at any step of the elimination process, i .e., all strategies in  are point rationalizable. Also, in the set of point-rationalizable strategies, each strategy is the best response to another strategy in the set. The restriction of the best response function to the set of point-rationalizable strategies is a bijection to itself. Hence, all point-rationalizable strategies form disjoint best response cycles, i .e., they are all in . 2 5 Note that b (1) , . . . , b (n) would be independently and uniformly distributed even if payoffs u i j were drawn from different distributions across different columns. 6 The function bt means applying the best response function t times to the strategies.

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Table 2 Strategies and their best responses and terminal strategies. i b (i ) λ (i )

1 7 10

2 6 15

3 8 8

4 12 4

5 10 10

6 15 15

7 10 10

8 4 8

9 6 15

10 10 10

11 15 15

12 8 12

13 8 8

14 15 14

15 14 15

16 12 12

Fig. 1. Graph representation of strategies and the best response function.

To elaborate, we introduce some concepts in graph theory and show the relation between a best response function and a graph.7 An undirected graph consists of a finite set of nodes, of which some pairs are connected by a single edge. A path is a finite sequence of nodes and distinct edges that are linked. A cycle is a path of edges and nodes where every node is reachable from itself. A graph is connected if every pair of nodes is linked by a path. A tree is a connected graph that has no cycles. Given a tree, if we choose some specific node of a tree, and call it root, then there exists a unique path from any other node to the root. A forest is a union of disjoint trees. A directed graph consists of a set of nodes and a set of directed edges. A directed edge pointing from node p to node q is denoted as the ordered pair ( p , q). A rooted tree is considered directed if all its edges point towards the root. Given the best response function b, we can draw the corresponding directed graph with n nodes representing the strategies and edges (i , b (i )). For any λ ∈ , the collection of all nodes i with λ (i ) = λ and edges (i , b (i )) except (λ, b (λ)) is a (directed) tree rooted at λ. The whole directed graph can be decomposed into two parts: (i) a forest with roots being strategies in  and (ii) disjoint cycles in . The following example illustrates the relation between the best response function and the rooted forest: Example 2. Consider a set of strategies {1, . . . , 16} and the corresponding best response function in Table 2. From the best response function, we know that the set of point-rationalizable strategies is {4, 8, 10, 12, 14, 15}. We now translate the best response function to a directed graph. In Fig. 1, each node is either in a cycle or connected to a cycle with some path. The set of terminal strategies is  = {4, 8, 10, 12, 14, 15}. Fig. 2 shows the edges outside . The graph is a directed forest with 6 rooted trees: the tree rooted at 10 consisting of nodes {1, 5, 7, 10} and edges {(1, 7) , (5, 10) , (7, 10)}; the tree rooted at 15 consisting of nodes {2, 6, 9, 11, 15} and edges {(2, 6) , (6, 15) , (9, 6) , (11, 15)}; the tree rooted at 8 consisting of nodes {3, 8, 13} and edges {(3, 8) , (13, 8)}; the tree rooted at 12 consisting of nodes {12, 16} and edge {(16, 12)}; and two single node trees consisting of nodes {4} and {14}. Fig. 3 displays the edges within . Since the restriction of the best response function to the set of point-rationalizable strategies is a bijective mapping from the set to itself, it forms a permutation on . In the graph, they form disjoint cycles. Note that only strategies in cycles of length one and two determine pure-strategy Nash equilibria. In the current example, (10, 10), (14, 15), and (15, 14) are three pure-strategy Nash equilibria. As in Example 2, a best response function determines three objects: (i) a set of point-rationalizable strategies ; (ii) a forest rooted at ; (iii) a permutation on . Conversely, given , a forest with roots in  determines the restriction

7

A detailed discussion of the concepts in graph theory can be found in Bondy and Murty (1976).

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Fig. 2. The edges outside .

Fig. 3. The edges within .

of player’s best response function to {1, 2, . . . , n} − , and a permutation on  determines the restriction of player’s best response function to strategies in . Using the bijection between a best response function and the three objects, Theorem 3 expresses an explicit formula and its asymptotic analysis for the distribution of the number of point-rationalizable strategies. Theorem 3. In a random n × n symmetric game, for any 1 ≤ k ≤ n, we have

  n P (PRn = k) = k! kn−k−1 , k

P (PRn ≥ k) =

n! . (n − k)!nk

And for every real number x > 0, we have



√ 



lim P PRn ≥ x n = exp −

n→∞

√

x2 2

 ,

i.e., PRn / n converges in distribution to Rayleigh(1) with mean

√

π /2.

Proof. Consider the product set Tn,k × Pk , where Tn,k is the set of all n-node forests with k rooted trees, and Pk is the set of permutations of k elements. There is a bijection between the set of best response functions with k point-rationalizable n strategies and Tn,k × Pk . Hence we only need to compute the number of forests with k rooted Trees, which is k knn−1−k

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(Moon, 1970, Section 4.2), and the number of permutations with k elements, which is k!. These numbers, combined with the n fact that each best response function appears with probability 1/n probability that a random n × n symmetric  the ,n givenus n−1−t game has k point-rationalizable strategies. By induction, we have t ! tn = n!nn−k / (n − k)!. Thus we can compute t =k t the cumulative distribution. The asymptotic result follows from Stirling’s formula, for example, Sedgewick and Flajolet (1996, Section 4.6, Theorem 4.4). 2 When we translate the best response function into a graph, the study of point-rationalizable strategies in random games is equivalent to the study of cyclic points in random mappings. Hence, Theorem 3 can be viewed as a restatement of Equation 3.8 in Rubin and Sitgreaves (1954) (Equation 3.12 in Harris (1960)), which computes the distribution of cyclic points in a random mapping. Note that PRn = 1 means that the realized game is solvable under the elimination of never best response against pure strategies. By the celebrated Cayley formula, the number of trees on n nodes is nn−2 , and hence the number of rooted trees is nn−1 . Thus PRn = 1 with probability 1/n. In other words, in a random two-player symmetric game with strategy size n, there is 1/n chance that the game is solvable. On the other hand, if PRn = n, then all strategies are point rationalizable. This case occurs when the best response function forms a permutation on all n strategies. Here the probability is n!/nn . Theorem 3 generalizes these observations. PRn takes n anyn!integer value between 1 and n, its distribution is known as the Ramanujan Q -distribution, and the expectation k is the so-called Ramanujan Q -function. k=1 (n−k)!n

√

Theorem 3 implies that PRn is of the order n in probability. Let En be the random variable that represents the number of pure-strategy Nash equilibria in a random n × n symmetric game. By Stanford (1996), we know that En is asymptotically X + 2Y in distribution, where X , Y are two independent random variables with Poisson distributions of rates 1 and 1/2, respectively. Hence En is of the order 1 in probability. Therefore, the number of point-rationalizable strategies is much larger than the number of pure-strategy Nash equilibria with a probability close to one. 2.3. Asymptotic independence between point-rationalizable strategies and pure-strategy Nash equilibria In this section, we study the relation between the number of point-rationalizable strategies, PRn , and the number of pure-strategy Nash equilibria, En , in random games. The fact that En ≤ PRn with probability one suggests that PRn and En are in general correlated. This is true for any finite n. Are PRn and En correlated when n is large? The answer is no. A more precise statement is given below. Proposition 4. The asymptotic distribution of En conditioning on PRn = k is given by8

lim P (En = l| PRn = k) = exp (−1.5)

k,n→∞



l/2 t =0

1

(l − 2t )!2t t !

,

i.e., as k → ∞, the conditional distribution of En given PRn = k converges to the distribution of X + 2Y where X ⊥ Y , X ∼ Pois (1), and Y ∼ Pois (1/2). Stanford (1996) shows that the unconditional distribution of En is asymptotically equal to the independent sum, X + 2Y , in distribution. Proposition 4 shows that the conditional distribution of En given PRn is also asymptotically equal to the independent sum, X + 2Y , in distribution. As long as PRn is large, the precise value of PRn has little effect on the distribution of En . Therefore, knowing the value of PRn does not improve our prediction of En , except if PRn is small, which is unlikely. 2.4. Strict dominance by pure strategies We study the set of strategies that survive iterated elimination of strategies strictly dominated by pure strategies. In our setting, since all the payoffs are distinct with probability one, the notion of strict dominance by pure strategies is equivalent to the notion of dominance in the sense of Börgers (1993). Let Dn denote the number of strategies that survive iterated elimination of strategies strictly dominated by pure strategies. Note that Dn depends on the entire ordinal rankings of the player’s pure strategies against the opponent’s pure strategies. Nonetheless, the distribution of Dn is independent of the payoff distribution as long as the payoff distribution is continuous. Proposition 5 shows that all strategies survive iterated elimination of strategies strictly dominated by pure strategies with probability close to one when the strategy space is large. Proposition 5. In a random n × n symmetric game, we have

lim P (Dn = n) = 1.

n→∞

8

x denotes the largest integer that is less than or equal to x.

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Proof. Note that the probability that strategy i is strictly dominated by strategy j is 1/2n . There are total n (n − 1) ordered pairs. Thus, the probability that some strategy is strictly dominated by another strategy is less than n (n − 1) /2n . Hence we have

P (Dn = n) ≥ 1 − which converges to one.

n (n − 1) 2n

,

2

2.5. Rationalizability Rationalizable strategies are strategies that survive iterated elimination of never best responses (Bernheim, 1984). In two-player finite games, by Pearce’s Lemma (Pearce, 1984, Lemma 3), rationalizable strategies can be equivalently viewed as strategies that survive iterated elimination of strategies strictly dominated by mixed strategies. Let Rn denote the number of rationalizable strategies in a random n × n symmetric game. The elimination process is slower than eliminating the never best responses against pure strategies, and faster than eliminating the strategies strictly dominated by pure strategies. Therefore, we have PRn ≤ Rn ≤ Dn with probability one. Thus far, we have analyzed En , PRn , and Dn , all of which depend only on the ordinal rankings of pure strategies. In particular, En and PRn are determined by the best responses, i .e., the strategies with the highest payoffs against the opponent’s pure strategies. But for rationalizable strategies, since the elimination process involves mixed strategies, Rn depends on the cardinal payoffs of pure strategy profiles. Therefore, the distribution of Rn depends on the payoff distribution. Rn turns out to have two different asymptotic analyses, depending on the order of limits. Theorem 6 deals with one such order: we fix n (and ε ) first, and then choose the distribution of payoffs. Theorem 8 deals with another order: we fix the payoff distribution first, and then take n to infinity. Let us fix n first. Note that varying the distribution of payoffs affects the distribution of Rn . Recall that the distributions of PRn and Dn are independent of the payoff distribution as long as it is continuous. By definition, we know that PRn ≤ Rn ≤ Dn for any realized game. Are PRn and Dn tight bounds for Rn ? More precisely, for a fixed ε > 0, is there some payoff distribution such that the probability that Rn equals its lower bound PRn (upper bound Dn ) is larger than 1 − ε in a realized game that is generated by this distribution? Theorem 6 confirms that PRn and Dn are indeed tight bounds for Rn . Theorem 6. Fix n and ε > 0. 1. There exists a continuous payoff distribution with bounded support such that P (Rn = PRn ) > 1 − ε . 2. There exists a continuous payoff distribution with bounded support such that P (Rn = Dn ) > 1 − ε . Before proceeding to the proof of Theorem 6, we introduce a lemma. Lemma 7. For any n, M > 0, and ε > 0, there exists a positive-valued continuous distribution F n, M ,ε with bounded support such that

 P

x(k) x(k+1)

 > M for all k = 1, . . . , n − 1 > 1 − ε ,

( )

where xi are independently generated by F n, M ,ε for all i = 1, . . . , n, and the order statistic x(k) is the k-th largest value among x1 , . . . , xn . In the proof of Lemma 7, which is relegated to the Appendix, we show that if x1 , x2 , . . . , xn are independently drawn from the Pareto distribution, Pareto(1, α ), with sufficiently small α , () is satisfied.9 Note that Pareto(1, α ) is positive-valued and continuous, but it has unbounded support. Nonetheless, we can truncate the distribution at a sufficiently high value such that () still holds with the truncated distribution. Proof of Theorem 6. For each j, define the order statistics u (k) j as the k-th largest value among u 1 j , . . . , unj . First, we prove part 1. Suppose all u i j are generated by F n,n−1,ε/n in Lemma 7. Then u i j > 0 and P ( E ) > 1 − ε , where E is the event that u (k) j /u (k+1) j > n − 1 for all k = 1, . . . , n − 1 and j = 1, . . . , n. Since P (PRn ≤ Rn ) = 1, we only need to show P (PRn ≥ Rn ) > 1 − . Hence, it is sufficient to show that when event E occurs in a realized game, PRn ≥ Rn . Suppose in a realized game where event E occurs, we have PRn < Rn . Then there exists a rationalizable strategy i that is a never best response against any rationalizable strategy of the opponent. Consider the mixed strategy σ that takes the

9 The Pareto distribution with scale parameter xm and shape parameter which has support [xm , ∞).

α , denoted as Pareto(xm , α ) has cumulative distribution F (x; xm , α ) = 1 −(xm /x)α ,

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uniform mixture of all strategies except i. For any rationalizable strategy j of the opponent, there exists a rationalizable strategy i  such that u i  j > u i j . Then we have u i  j /u i j > n − 1. Hence, the payoff of σ is strictly higher than i against j since

i  =i

1 n−1

u i  j >

1 n−1

u i j > u i j .

Therefore, σ strictly dominates i against all rationalizable strategies of the opponent, which is a contradiction since i is rationalizable. Hence, we have PRn ≥ Rn .   Now, we prove part 2. Suppose all −u i j are generated by F n,n−1,ε/n in Lemma 7. Then u i j < 0 and P E  > 1 − ε , where E  is the event that u (k+1) j /u (k) j > n − 1 for all k = 1, . . . , n − 1 and j = 1, . . . , n. Since P (Rn ≤ Dn ) = 1, we only need to show P (Rn ≥ Dn ) > 1 − ε . Hence, it is sufficient to show that when event E  occurs in a realized game, Rn ≥ Dn . Suppose in a realized game where event E  occurs, we have Rn < Dn . By definition, in the set of strategies surviving iterated elimination of strategies strictly dominated by pure strategies, there exists a strategy i that is not rationalizable, i .e., i is strictly dominated by some mixed strategy σ  whose support is a subset of all surviving strategies except i.10 Note that the number of surviving strategies is no larger than n. Hence, there exists i  in the support of σ  receiving a probability   σ  i  ≥ 1/ (n − 1). Suppose i  does not strictly dominate i. Then u i j > u i j for some surviving strategy j. Then we have u i  j /u i j > n − 1. Hence, the payoff of σ  is strictly less than i against j, since all payoffs are negative and

 

σ  i u i j ≤ Therefore,

1 n−1

u i j < u i j .

σ  does not strictly dominate i, which is a contradiction. Hence, we have Rn ≥ Dn . 2

In the context of deterministic games, Weinstein (2016, Proposition 2) shows that the set of rationalizable strategies equals the set of point-rationalizable strategies under a risk-averse condition   on payoffs. If we  state his result for two-player games with no tie, the condition is equivalent to that u (k+1) j − u (k+2) j / u (k) j − u (k+1) j is arbitrarily close to zero for any k and j. In the context of random games, Lemma 7 implies the existence of a payoff distribution such that if payoffs are independently drawn from this distribution, the realized game satisfies Weinstein’s risk-averse condition with a high probability. Hence Theorem 6 part 1 holds. On the other hand, Weinstein (2016, Proposition 3) shows that the set of rationalizable strategies equals the set of strategies surviving iterated elimination of strategies dominated in Börgers’ sense under a risk-loving condition on payoffs. Hence, Theorem 6 part 2 follows from the existence of a payoff distribution such that the realized game satisfies Weinstein’s risk-loving condition with a high probability.11 We now consider the other order of limits, i .e., fix the distribution of payoffs and take n to infinity. Theorem 8 shows that for a fixed distribution of payoffs with a finite third moment, when n approaches infinity, it is almost impossible for a strategy to be strictly dominated by mixed strategies. Hence all strategies are rationalizable with probability converging to one. Theorem 8. Fix a payoff distribution that has a finite third moment. Then

lim P (Rn = n) = 1.

n→∞

Proof. Let P n be the probability that no pure strategy is strictly dominated by mixed strategies. It is sufficient to show that P n → 1 when n → ∞. Let h (x) be a positive bounded strictly increasing function (e.g., h (x) = arctan (x) + π ). Fix strategy i and define

1   h u i j (u i  j − u i j ) n n

dii  =

for each i  = i .

j =1

Since h (x) is positive, we can construct a mixed strategy

σ i ∈ ({1, . . . , n}), given by σ ji =





h ui j





j

h u i j

. Then dii 

≤ 0 is

equivalent to strategy i having a weakly better payoff than strategy i  against the mixed strategy σ i . Therefore, if dii  ≤ 0 for all i  = i, strategy i is a best response to σ i . Hence strategy i is not strictly dominated by mixed strategies. Fix i and i  with i = i  . We now estimate P (dii  ≤ 0). The term dii  is the empirical mean of i.i.d. random variables    {Y j : Y j = h u i j u i  j − u i j , j = 1, . . . , n}. Since h is strictly increasing, we have

10 If a mixed strategy that strictly dominates i assigns positive probability to i, we can always reassign that probability from i to all other supporting strategies proportionally. 11 Recall that if there is no tie, then dominance by pure strategies coincides with dominance in Börgers’ sense.

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Fig. 4. P (Rn = n) (the dashed line), P (PRn = n) (the solid line), and P (Dn = n) (the dash-dot line) in random symmetric games with normally distributed payoffs.

      1    E Y j = E h u i j − h u i  j u i  j − u i j < 0. 2







Consider random variables X j = Y j − E Y j . Note that X j are i.i.d. random variables as well. Since h is bounded and all u i j have a finite third moment, all X j have a finite third moment. Therefore, the distribution of X j satisfies F (x) =       n 2 1 − o x−3 .12 Note that E X j = 0 and j =1 E X j = nVar Y j . By Hoeffding’s inequality for unbounded random variables (e.g., Theorem 2.18 in Peña et al. (2009)), for any c > 0 and δ > 0,





P dii  − E Y j



⎛ ⎞    n

⎝ ⎠ ≥c =P X j ≥ cn ≤ P max X j > δ cn +





1≤ j ≤n

j =1

3nVar Y j







nVar Y j + δ c 2 n2

1 δ

.

Note that



 P





max X j > δ cn = 1 − F (δ cn) = 1 − 1 − o

1≤ j ≤n

n

1

n

(δ cn)

3

 = o n −2 .

Taking δ = 13 ,





3nVar Y j







1 δ



=

nVar Y j + δ c 2 n2



3Var Y j





Var Y j +



1 2 c n 3

3

 = o n −2 .

            P dii  − E Y j ≥ c = o n−2 . Since E Y j < 0, taking c = − 12 E Y j , we have P (dii  > 0) = o n−2 . Thus we show that

Pn ≥ 1 −



 P (dii  > 0) = 1 − n(n − 1)o n−2 → 1

as n → ∞.

2

i =i 

We see a finite third moment as a weak assumption, which is satisfied by all distributions with bounded supports and many other distributions such as normal distributions. Nonetheless, we do not know if Theorem 8 holds if this assumption is relaxed. As long as we accept the finite third moment condition, the only difference between Theorems 6 and 8 is the order of limits. For Theorem 6, we first fix n, and then choose a truncated Pareto distribution with a bounded support, and hence all moments are finite. In contrast, for Theorem 8, we first pin down a payoff distribution, and then take n to infinity. We show that the finite third moment condition for the payoff distribution is sufficient for Theorem 8 to hold. To illustrate Theorem 8, we run a simulation using the standard normal distribution. In Fig. 4, we show the simulation results on the probabilities that all strategies are rationalizable (P (Rn = n)) in random symmetric games where the payoffs are independently drawn from the standard normal distribution. We also plot the probabilities that all strategies are point rationalizable (P (PRn = n)), and the probabilities that all strategies are not strictly dominated (P (Dn = n)).13

f ( x)

We write f = o ( g ) if limx→∞ g (x) = 0. For Rn and Dn , we generate 10,000 payoff matrices by the standard normal distribution, and simulate the elimination process. For PRn , we use the result in Theorem 3, i .e., P (PRn = n) = n!/nn . 12 13

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119

Fig. 5. Bipartite graph and best response functions.

3. Random two-player asymmetric games In this section, we focus on point rationalizability in random two-player asymmetric games. The results on strict dominance by pure strategies and rationalizability can be easily replicated in random two-player asymmetric games. To see this, note that in the proofs of Proposition 5 and Theorem 8, with probability close to one, the elimination never happens. In Theorem 6, the same arguments apply to the asymmetric case. Hence, when the sizes of the two players’ strategy spaces are of the same order, there is no difference between the symmetric and the asymmetric cases in these analyses. 3.1. Notation and formulation A random m × n asymmetric game can be represented by U 1 , U 2 with the following characteristics: 1. Player 1 has strategy space {1, 2, . . . , m} and player 2 has strategy space {m + 1, m + 2, . . . , m + n}. 2. U 1 is an m × n matrix representing the payoffs of player 1 where u 1i j is the payoff for player 1 when she plays strategy i and player 2 plays strategy m + j. U 2 is an m × n matrix representing the payoffs of player 2 where u 2i j is the payoff for player 2 when she plays strategy m + j and player 1 plays strategy i. 3. The mn payoffs u 1i j (u 2i j ) are real-valued independent random variables with an identical continuous distribution. By the same argument as in Section 2.1, we know that the best response functions of two players, b1 and b2 , are well defined and independent. Also, b1 is uniformly distributed on the set of functions from {m + 1, m + 2, . . . m + n} to {1, 2, . . . , m}, and b2 is uniformly distributed on the set of functions from {1, 2, . . . , m} to {m + 1, m + 2, . . . , m + n}. Note that both players have the same number of point-rationalizable strategies. Let PRm,n denote the random variable that represents the number of point-rationalizable strategies for a player. Without loss of generality, we assume n ≤ m. 3.2. Point rationalizability Note that in random m × n asymmetric games, the payoff matrices U 1 , U 2 can be generated by different distributions. The best response functions b1 and b2 are independent in the asymmetric case, and perfectly positively correlated in the symmetric case. Hence in the asymmetric case, even if m = n, the independence between b1 and b2 makes the distribution of PRn,n differ from the distribution of PRn . Nonetheless, similar techniques apply, and the results hold. For a player’s strategy i ∈ {1, . . . , m + n}, we alternately apply her opponent’s best response function and her own best response function to i to get the best response path of i. Let λ (i ), the terminal strategy of strategy i, denote the first strategy that appears twice in the best response path from i. Let  be the set of all λ (i ), i being the strategy of player 1 or player 2. Lemma 9.  is the set of all point-rationalizable strategies of both players. Proof. The proof is similar to the proof of Lemma 1.

2

 can be divided into 1 and 2 — sets of point-rationalizable strategies of players 1 and 2, respectively. We know that PRm,n = #1 = #2 . We apply the same method in Section 2.2 to translate the best response functions to a directed graph. In order to represent the two functions, we introduce the notion of bipartite graphs. An m × n bipartite graph is a graph with two node sets, m “dark” nodes and n “light” nodes, such that every edge of the graph connects a dark node with a light node. The best response function b1 determines the directed edges pointing from light nodes to dark nodes, and the best response function b2 determines the directed edges pointing from dark nodes to light notes (see Fig. 5).

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The best response functions b1 , b2 determine five objects: (i-1) player 1’s set of point-rationalizable strategies 1 ; (i-2) player 2’s set of point-rationalizable strategies 2 ; (ii) a bipartite forest rooted at ; (iii-1) a one-to-one function from 2 to 1 ; (iii-2) a one-to-one function from 1 to 2 . Conversely, given 1 and 2 , a bipartite forest rooted at  determines the restrictions of best response functions for players 1 and 2 to {m + 1, m + 2, . . . , m + n} − 2 and {1, 2, . . . , m} − 1 , respectively. A one-to-one function from 2 to 1 determines the restriction of player 1’s best response function to 2 , and a one-to-one function from 1 to 2 determines the restriction of player 2’s best response function to 1 . Theorem 10 gives an explicit formula and asymptotic analyses for the distribution of the number of point-rationalizable strategies in a random m × n asymmetric game. Theorem 10. In a random m × n two-player asymmetric game, for any 1 ≤ k ≤ n

  





2

P PRm,n = k = (k!)   P PRm,n ≥ k =

m

n

k

k

(m + n − k) km−k−1n−k−1 ,

m! n! . (m − k)!mk (n − k)!nk

And for every real number x > 0, if m/n → ∞, then

√ 





lim P PRm,n ≥ x n = exp −

m,n→∞

x2 2

 ,

√

i.e., PRm,n / n converges in distribution to Rayleigh(1) with mean

√ 





lim P PRm,n ≥ x n = exp −

m,n→∞

(c + 1) x2

,

√

√

π /2; if m/n → c where c ≥ 1 is a real number, then



2c

i.e., PRm,n / n converges in distribution to Rayleigh

√



c / (c + 1) with mean

√

c π / (2c + 2).

Proof. Consider set Tm,n,k × Pk × Pk , where Tm,n,k is the set of all m × n bipartite forests with k trees rooted at dark nodes and k trees rooted at light nodes, and Pk is the set of permutations of k elements. There is a bijection between the set of best response function pairs with k point-rationalizable strategies for each player to Tn,k × Pk × Pk . (Observe that the number of one-to-one functions from a k-element set to a k-element set is the same as Pk .) Note that #Pk = k!, mn and by Moon (1970, Section 4.4), #Tm,n,k = k k (m + n − k) kmn−k−1 nm−k−1 . These numbers, combined with the fact that  each best response function appears with probability that each best response function pair appears with probability 1/ mn nm , give us the probability that a random m × n asymmetric game has k point-rationalizable strategies. By induction,

n

t =k

(k!)2

mn k

k

(m + n − k) km−k−1 n−k−1 =

m! n! . Thus we can compute the cumulative distribution. (m−k)!mk−n (n−k)!nk−m

The asymptotic results are directly from Stirling’s formula.

2

Theorem 10 is a restatement of Corollary 3 in Jaworski (1985), which computes the distribution of cyclic points in random bipartite mappings. Now we evaluate the conditional distribution of the number of pure-strategy Nash equilibria on PRm,n . Let Em,n denote the random variable that represents the number of pure-strategy Nash equilibria in a random m × n asymmetric game. Note that the asymptotic distribution of PRm,n depends on the relative speed at which m and n approach infinity. But for Em,n , the asymptotic distribution follows the Poisson distribution with mean 1 as long as min {m, n} approaches infinity (Stanford, 1995). Proposition 11 computes the conditional distribution of the number of pure-strategy Nash equilibria given the number of point-rationalizable strategies: Proposition 11. In a random m × n asymmetric game, the asymptotic distribution of Em,n conditioning on PRm,n = k is given by

lim

k,m,n→∞

 exp (−1)  P Em,n = l| PRm,n = k = , l!

i.e., as k → ∞, the conditional distribution of Em,n given PRm,n = k converges to Pois (1). Note that the asymptotic conditional distribution is the same as the asymptotic unconditional distribution of Nash equilibria shown in Stanford (1995). As in the symmetric case, Em,n and PRm,n are asymptotically independent.

T. Pei, S. Takahashi / Games and Economic Behavior 118 (2019) 110–125



121



Fig. 6. P PRnz = k (the dashed line) and P (PRn = k) (the solid line) for n = 100.

Fig. 7. A simulation mean of PRnz (the fluctuated curve) and

√

π n/2 (the smooth curve).

4. Discussions We first discuss subclasses of random games with restrictions on payoffs or best response functions. Two interesting classes of games are zero-sum games and common interest games. The two players’ payoff profiles are perfectly negatively correlated in zero-sum games, and perfectly positively correlated in common interest games. We also discuss games where the correlation between the two players’ payoffs is arbitrary. Another interesting class of games is games with weakly increasing best responses where the best response is no longer independently drawn from the strategy space. We conclude by discussing random games with more than two players. 4.1. Zero-sum games We first consider random zero-sum games. We focus on a random n × n symmetric zero-sum game, generated by a skew-symmetric payoff matrix U = M − M T , where all mi j are real-valued independent random variables with an identical continuous distribution. Hence all diagonal payoffs u ii are zero, all payoffs above the diagonal u i j with i < j are real-valued independent random variables with an identical, symmetric, and continuous distribution, and u ji = −u i j with i < j. Let PRnz denote the number of point-rationalizable strategies in a random n × n symmetric zero-sum game. Unfortunately we know neither the exact nor the asymptotic distribution of PRnz due to complications arising from the correlation between payoffs. We propose a conjecture based on simulations. √  Our simulation results suggest that the distribution of PRnz asymptotically coincides with Rayleigh n . Recall that √  Rayleigh n is also the asymptotic distribution of PRn (Theorem 3). We present the results of several simulations. Fig. 6 shows the probability distribution of PRnz in simulation and the exact probability distribution of PRn when n = 100. Fig. 7 shows a simulation mean of PRnz and the exact mean of PRn for n = 1, . . . , 400.14 Note that PRn and PRnz have similar distributions, especially when the number of point-rationalizable strategies is greater than or equal to 2. In a random symmetric zero-sum game, the probability of having only one point-rationalizable strategy is lower than the probability of having a pure-strategy Nash equilibrium. Hence P PRnz = 1 is close to zero while P (PRn = 1) = 1/n. But these differences seem negligible when n becomes large. These observations lead to Conjecture 12.

14

For each n = 1, 2, . . . , 400, we generate 10,000 random matrices, and compute the number of point-rationalizable strategies using MATLAB.

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Fig. 8. Simulation means of the number of point-rationalizable strategies (the dashed line) and that of pure-strategy Nash equilibria (the solid line) for a random 100 × 100 asymmetric game with payoff correlation ρ from −1 to 1 with grid 0.05.

√

Conjecture 12. PRnz / n converges in distribution to Rayleigh(1). We also have a similar conjecture for the number of point-rationalizable strategies in random asymmetric zero-sum games, i .e., its asymptotic distribution is the same as the asymptotic distribution of PRm,n . 4.2. Common interest games Here, we consider random common interest games. Note that in a common interest game with no tie, there does not exist any best response cycle except pure-strategy Nash equilibria because players’ payoffs increase strictly along any best response path. Therefore, the number of point-rationalizable strategies is equal to the number of pure-strategy Nash equilibria in a two-player common interest game with no tie. A random m × n asymmetric common interest game has payoff matrix U , where u i j represents the payoffs for both players when the strategy profile is (i , j ). Again, all u i j are real-valued independent random variables with an identical continuous distribution. Stanford (1999) gives the expected number of pure-strategy Nash equilibria, and Roberts (2005) extends the result and gives an explicit formula for the probability of a random m × n asymmetric common interest game having exactly k pure-strategy Nash equilibria. Hence the probability of mn k(m−1)!(n−1)! a random m × n asymmetric common interest game having exact k point-rationalizable strategies is k k (m+n−1)! , and the expectation is mn/ (m + n − 1). Note that the expectation is of the order min {m, n}. Consequently, a common interest game has many more point-rationalizable strategies than a non-common interest game. 4.3. Games with arbitrary correlations between players’ payoffs We briefly summarize the results of simulations on random games with arbitrary correlations between player’s payoffs. Following Rinott and Scarsini (2000), we consider a random m × n asymmetric game, where the pair of payoffsfor each    0 1 ρ strategy profile is drawn from the bivariate normal distribution with mean vector and covariance matrix . 0 ρ 1 We focus on the case of m = n = 100. As shown in Fig. 8, the simulation mean of the number of point-rationalizable strategies is almost constant for −1 ≤ ρ ≤ 0, increases with ρ for 0 ≤ ρ ≤ 1, and converges to a value around 50 as ρ → 1. These observations in simulations are consistent with √ our analytical results in Theorem 10 for random games with independent payoffs between the players (asymptotically 25π ≈ 8.86 at ρ = 0). These observations are also consistent with our results in Section 4.2 for random common interest games (1002 /199 ≈ 50.25 at ρ = 1). For comparison, we also report simulation results on the expected number of pure-strategy Nash equilibria. It is easy to show that the exact mean is (100!)2 /199! ≈ 2.2 × 10−57 at ρ = −1, 1 at ρ = 0, and ≈ 50.25 at ρ = 1. Rinott and Scarsini (2000) also show that the mean converges to zero in probability as n → ∞ when −1 ≤ ρ < 0, and increases with ρ when 0 ≤ ρ ≤ 1. As in the figure, the expected number of pure-strategy Nash equilibria is smaller than the number of point-rationalizable strategies for ρ < 1, and reaches equality at ρ = 1. 4.4. Two-player games with weakly increasing best responses We consider a random two-player game with weakly increasing best responses. We follow the setting in Takahashi (2008) and restrict the best response function b to be uniformly chosen from the set of weakly increasing functions from {1, 2, . . . , n} to itself. In the symmetric game setting, we draw such a best response function only once, and apply the function to both players. In the asymmetric game setting, we draw two such best response functions independently. Note that when both players have weakly increasing best response functions, each best response path ends with pure-strategy Nash equilibria (Kukushkin et al., 2005). Therefore, in a game with weakly increasing best responses, each point-rationalizable strategy is in a pure-strategy Nash equilibrium profile, i .e., the number of point-rationalizable strategies is the same as the

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123

number of pure-strategy Nash equilibria. Hence, by Takahashi (2008), in the n × n symmetric case, the probability that √ the  (n−1)!n! number of point-rationalizable strategies is no less than k is (n−k)!(n+k−1)! , whose asymptotic distribution is Rayleigh n/2 . In the m × n asymmetric case, the expected number of point-rationalizable strategies is

√

2m+2n−2  m+n−1m+n−1 , / 2 m−1 n−1 2m−1

which converges to π mn/ (m + n)/2. Another interesting “solution concept” for games with weakly increasing best responses is the set of strategies bounded by the smallest and the largest pure-strategy Nash equilibria. This set (or interval) is useful especially when the strategy space is large since the smallest and the largest Nash equilibria are easy to compute by iteration of best responses from the bottom and the top, respectively. Note that in supermodular games, this interval contains all rationalizable strategies and all limit strategies of adaptive learning (Milgrom and Roberts, 1990). However, Milgrom and Roberts (1990, p. 1258) point out that “for some games, these bounds are so wide that our result is of little help” and “the bounds are quite narrow” for some other cases. We are interested in the bounds of games with weakly increasing best responses. With the random game framework, we can evaluate the width of these bounds by computing the distribution of the difference between the smallest and the largest pure-strategy Nash equilibria. We denote the difference by Diffn in a random n × n symmetric games with weakly increasing best responses. Hence, the largest possible Diffn is n − 1 when both (1, 1) and (n, n) are pure-strategy Nash equilibria, and the smallest possible Diffn is 0 when the game has a unique pure-strategy Nash equilibrium. Proposition 13 provides an explicit formula for the distribution of the difference between the smallest and largest purestrategy Nash equilibria in a random n × n symmetric game with weakly increasing best responses: Proposition 13. For 1 ≤ t ≤ n, the distribution of difference between the largest and smallest pure-strategy Nash equilibria in a random n × n symmetric two-player game with weakly increasing best responses follows15

P (Diffn = n − t ) =

1 t+1

  2n−2t −1 2t n−t 2n−1 , t

n −1

and

lim P (Diffn = n − t ) =

n→∞

1 t +1

  2t

1

t

4t

.

Proposition 13 implies that when n is large,√n − Diffn is of order 1 in probability. Compared to the number of purestrategy Nash equilibria, which is of the order n in probability, Diffn reflects that the bounds are indeed wide for most realizations of a random game with weakly increasing best responses. 4.5. Games with more than two players We now extend our analysis to games with three or more players. For simplicity, we consider random three-player symmetric games where each player has n strategies. Each player’s payoffs are drawn independently across players. Then, the best response function b is uniformly distributed on the set of functions from a set of n2 strategy profiles to the 2

strategy space. The probability that a strategy is not a best response against any of n2 strategy profiles is (1 − 1/n)n . Then n2

the probability that all strategies are point rationalizable is at least 1 − n (1 − 1/n) , which converges to 1 as n → ∞.16 This idea can be easily extended to the general case as long as each player’s strategy space is sufficiently small compared to the space of her opponents’ strategy profiles. Therefore, in games with more than two players, all strategies are point rationalizable with a probability close to one. 5. Conclusion

√ We show that in random two-player games, the distribution of the number of pint rationalizable strategies is of the order n in probability, and the distribution of the number of rationalizable strategies under the finite third moment condition is of the order n. We also show the dependence of rationalizability on the payoff distribution in random games. We leave as open questions the analyses of the distribution of the numbers of point-rationalizable strategies in random zero-sum games and in games with payoffs that are arbitrarily correlated.

15 16

−1

When t = n, let 0 = 1. We can compute the exact probability that all strategies are point rationalizable, which is equal to the probability that the best response function

is a surjective mapping. Note that the number of surjective mapping is

   k n2 n−k n . k=0 (−1) k n

n

n

k=0

(−1)n−k

n k

2

kn . Therefore, the probability that b is a surjective function is

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Acknowledgment We are grateful to Yeneng Sun for suggesting the theorem (Peña et al., 2009, Theorem 2.18) used in the proof of Theorem 8. We are grateful to Andy McLennan and Junjie Zhou for their comments and suggestions. Appendix A Proof of Proposition 4. Note that with k point-rationalizable strategies, each permutation appears with probability 1/k!. In order to have l pure-strategy Nash equilibria, we must have l strategies in cycles of length one or two, and the other k − l strategies in cycles of length greater than two. The number of permutations with no cycles of length greater than two is   determined by the exponential generating function f ( z) = exp z + z2 /2 , and the number of permutations with all cycles’   of length greater than two is determined by the exponential generating function g ( z) = exp − z − z2 /2 / (1 − z). Applying Taylor’s theorem and combining these numbers with

 

P (En = l| PRn = k) = Taking k → ∞,

g (k−l) (0) (k−l)!

1 k k! l

k

f (l) (0) g (k−l) (0) =

l

possible choices of pure-strategy Nash equilibria gives us17

f (l) (0) g (k−l) (0) l! (k − l)!

approximates to e −1.5 . Also note that

f (l) (0) l!

=

.

l/2

1 t =0 (l−2t )!2t t ! .

For all the results for permutations, see Sedgewick and Flajolet (1996, Section 7.4).

Thus we obtain the asymptotic result.

2

− log(1−ε )

Proof of Lemma 7. Fix n, M > 0, and ε > 0 throughout this proof. Let α = n(n−1) log M . Consider F n, M ,ε (x) = 1 − x−α , i .e., F n, M ,ε is Pareto(1, α ). Suppose x1 , . . . , xn are independently generated by F n, M ,ε . For any k = 1, . . . , n − 1, let A k be the event that x(k) /x(k+1) > M and A = k A k . Let P be the probability measure generated by F n, M ,ε . We will show that () holds with F n, M ,ε , i .e ., P ( A ) > 1 − ε . Let y i be random variables with y i = log xi and let the corresponding y (k) be the k-th largest value among y 1 , . . . , yn . Note that y i are i.i.d. random variables following the exponential distribution with parameter α , where P ( y i > y ) = exp (−α y ).18 Therefore y (1) − y (2) , . . . , y (n−1) − y (n) are independent with y (k) − y (k+1) following the exponential distribution with parameter kα . Then event A k occurs with probability

  P ( A k ) = P y (k) − y (k+1) > log M = exp (−kα log M ) = M −kα . Since all the events A 1 , . . . , A n−1 are independent, the probability that they occur simultaneously is

P ( A) =

n −1

P ( Ak ) = M −

n(n−1) α 2

=

√

1−ε>1−ε

k =1

− log(1−ε )

given α = n(n−1) log M . Note that F n, M ,ε is positive-valued and continuous with support [1, ∞). We now truncate F n, M ,ε at x, where x satisfies 1/n    F n, M ,ε x ≤ x ≤ 1 − P ( A ) + (1 − ε ) . Denote the truncated distribution by F˜ n, M ,ε , i .e.,

F˜ n, M ,ε (x) =

 F n,M ,ε (x)

 

F n, M ,ε x

1

if x ≤ x, if x > x.

˜ be the probability measure generated by F˜ n, M ,ε . It is clear that F˜ n, M ,ε is positive-valued and continuous with support Let P [1, x]. We show below that with F˜ n, M ,ε , () holds, i .e .,

P˜ ( A ) =

P

>P



P 



xi ≤ x, i = 1, . . . , n ∩ A



xi ≤ x, i = 1, . . . , n







xi ≤ x, i = 1, . . . , n ∩ A



 n  ≥ P ( A ) − 1 − P xi ≤ x

≥ P ( A ) − (1 − (1 − P ( A ) + (1 − ε ))) = 1 − ε . 2 17 Recall that pure-strategy Nash equilibria are determined by the restriction of best response function to , and all strategies forming pure-strategy Nash equilibria are in cycles of length one or two within the permutation of all point-rationalizable strategies. This implies that given PRn , the conditional distribution of En is independent of n. 18 The exponential distribution with rate parameter λ has cumulative distribution F (x; λ) = 1 − exp (−λx), which has support [0, ∞).

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125

Proof of Proposition 11. Note that for any best response function of player 1 on her k rationalizable strategies, the best response function of player 2 on his k rationalizable strategies determines a bijective mapping to a permutation on player 1’s k rationalizable strategies. The number of pure-strategy Nash equilibria is the number of fixed points in the permutation.  Therefore, the probability, P Em,n = l| PRm,n = k , is the product of two probabilities: (i) the probability that there are l fixed points and (ii) the probability that the other k − l strategies are derangement. The first probability is 1/l!, and the second t probability is 0≤t ≤k−l (−1) /t ! by Sedgewick and Flajolet (1996, Section 7.4). Therefore,

 1  P Em,n = l| PRm,n = k = l!

0≤t ≤k−l

(−1)t , t!

and the asymptotic result is directly from the exact distribution.

2

Proof of Proposition 13. The best response function of a game with Diffn = k can be divided into two parts. First, each best response function contains a weakly increasing mapping from the set of strategies between the smallest and largest pure-strategy Nash equilibria to itself, where the starting and ending points are fixed points. The total number of such 2k−1 mappings is . Second, if we compress the whole part starting from the smallest pure-strategy Nash equilibrium and k ending with the largest pure-strategy Nash equilibrium to one point, then we obtain a weakly increasing mapping with 2n−2k−1 2n−2k−1 a unique fixed point with the remaining n − k points. The total number of such mappings is n−k−1 − n−k−2 . Then

2n−1

among n−1 weakly increasing best response functions, the number of best response functions where Diffn = k is equal to the product of the numbers of two such mappings. Replacing k with n − t, we have the exact distribution. The asymptotic result is obvious from the exact distribution. 2 Appendix B. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.geb.2019.08.011. References

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