Corrigendum to “Rationality and coherent theories of strategic behavior” [J. Econ. Theory 70 (1) (1996) 1–31]

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Corrigendum to “Rationality and coherent theories of strategic behavior” [J. Econ. Theory 70 (1) (1996) 1–31] ✩ Xiao Luo ∗ , Yongchuan Qiao Department of Economics, National University of Singapore, Singapore 117570, Singapore Received 15 December 2015; final version received 2 November 2016; accepted 11 November 2016 Available online 30 November 2016

Abstract We show, by means of a counterexample, that Propositions 3, 4, and 7 in Gul (1996) [G96] are incorrect. Our example is also a counterexample to Lemma 1 in G96, which is the basis for the proofs of these propositions. We provide insight for the source of the problem in the definition of a perfect τ -theory in G96, and then offer a modification that restores these results in G96. © 2016 Elsevier Inc. All rights reserved. JEL classification: C72 Keywords: Game theory; Perfect τ -theories

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We are grateful to the editor, an associate editor, and two anonymous referees for very useful comments and constructive suggestions that led to a significant improvement of this note. We thank Yi-Chun Chen, Takashi Kunimoto, Shravan Luckraz, Andrés Perea, Xuewen Qian, Chen Qu, Yang Sun, and Satoru Takahashi for comments and discussions. We are especially indebted to Yossi Greenberg for thoughtful advice and suggestions. The usual disclaimer applies. Financial support from the National University of Singapore (grant No: R122000246115) is gratefully acknowledged. * Corresponding author. E-mail addresses: [email protected] (X. Luo), [email protected] (Y. Qiao). http://dx.doi.org/10.1016/j.jet.2016.11.002 0022-0531/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction Gul (1996), henceforth G96, introduces the solution concept of τ -theory. G96 establishes the relationship between this solution concept and several others, including rationalizability, trembling-hand perfection, iterated weak dominance, and backward induction. Propositions 3–4 in G96 claim the weakest perfect τ -theory yields the outcome set of the Dekel–Fudenberg (DF) procedure.1 Proposition 7 in G96 claims an extensive-form perfect τ -theory satisfies the invariance requirement; that is, a perfect τ -theory in the extensive-form is equivalent to a perfect τ -theory in its normal-form representation. In this note, we show by means of a counterexample that Propositions 3, 4, and 7 in G96 are incorrect. Our example is also a counterexample to Lemma 1 in G96, which is the basis for the proofs of these propositions. We provide insight for the source of the problem in the definition of a perfect τ -theory in G96, and then offer a minor modification that restores the aforementioned results. 2. Gul’s perfect τ -theory We adopt the notation in G96. Consider a finite n-person game G = (Ai , ui )ni=1 , where Ai is the finite set of player i’s actions and ui : ni=1 Ai → R is player i’s payoff function. Let Si and S−i denote the sets of all probability distributions on Ai and A−i = j =i Aj , respectively. For any set Zj ⊆ Sj , let Zj denote the convex hull of Zj . Denote τ = ((Ri )ni=1 , (i )ni=1 ), where Ri ⊆ Si is a set of player i’s “rational” (mixed) strategies and i ⊆ Si is a set of player i’s “irrational” (mixed) strategies for i = 1, 2, . . . , n. (The notions of “rational” and “irrational” will be defined implicitly in the definitions that come.) For ε any ε ∈ (0, 1), G96 defines the set C−i (τ ) of ε-allowable conjectures for player i as follows:  ε C−i (τ ) = s−i ∈ S−i | πj (s−i ) = αj sj1 + (1 − αj )sj2 for all j = i,  where αj ∈ [1 − ε, 1), sj1 ∈ R j and sj2 ∈  j , where πj (s−i ) is the marginal distribution of s−i on Aj . That is, an ε-allowable conjecture for player i is a probability distribution in S−i , which has a marginal probability of at least 1 − ε on playing some strategy in the set Rj and a marginal probability of at most ε on playing some strategy in the set j for each player j = i. We follow G96 to define the notions of a τ -theory and a perfect τ -theory as follows. Definition 1. τ = ((Ri )ni=1 , (i )ni=1 ) is a τ -theory if, for i = 1, 2, · · · , n, Ri = ∩ε>0 Bεi (τ ), where ε Bεi (τ ) denotes the set of i’s best responses to some conjecture in C−i (τ ). Definition 2. A τ -theory τ = ((Ri )ni=1 , (i )ni=1 ) is a perfect τ -theory if i ⊆ Int Si for i = 1, 2, · · · , n, where Int Si denotes the interior of Si (i.e., the set of full-support probability distributions on Ai ). Definition 2 requires that every player must assign a positive marginal probability to every action of opponents. Having introduced the notion of a perfect τ -theory, G96 then relates it to 1 Dekel and Fudenberg (1990) proposed an iterative procedure: removing all weakly dominated strategies in the first round and then removing only strictly dominated strategies in the subsequent rounds.

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the DF procedure. In particular, Proposition 3 in G96 claims the outcome set of the DF procedure can be generated by a perfect τ -theory; Proposition 4 in G96 claims that the outcome set of the DF procedure includes all the strategies that can be generated by a perfect τ -theory. That is, the outcome set of the DF procedure can be generated by the weakest perfect τ -theory. (A perfect τ -theory τ ∗ = ((Ri∗ )ni=1 , (Int Si )ni=1 ) is the weakest perfect τ -theory if τ = ((Ri )ni=1 , (i )ni=1 ) is a perfect τ -theory implies Ri ⊆ Ri∗ for i = 1, 2, · · · , n.) The example in the following section shows these two propositions are wrong. We postpone our discussion of G96’s extensive-form perfect τ -theory to the conclusion. 3. A counterexample Example 1. Consider the following three-person game where player 1 chooses a row, player 2 chooses a column, and player 3 chooses a box. The players’ payoffs are shown in the tables below (in which the payoffs for players 1 and 2 are constantly zero). y1 y2 x1 0, 0, 1 0, 0, 1 x2 0, 0, 1 0, 0, 1 z1

y1 y2 x1 0, 0, 1 0, 0, 1 x2 0, 0, 1 0, 0, 0 z2

The set of outcomes (for player 3) resulting from a perfect τ -theory is {z1 , z2 }. In particular, because player 3 can hold the allowable action z2 cannot be excluded  by  any perfect   τ -theory,  ∗ ≡ 1 x , y + 1 x , y + 1 x , y that supports playing z .2 But, z is excluded conjecture s−3 1 1 1 2 2 1 2 2 3 3 3 by the DF procedure, because z2 is weakly dominated by z1 . Thus, contrary to the assertion of Proposition 3 in G96, no perfect τ -theory yields the outcome set of the DF procedure. For the same reason, namely, that z2 belongs to the perfect τ -theory but is ruled out by the DF procedure, our example shows Proposition 4 in G96 is wrong too; a perfect τ -theory can include an outcome, which is outside the outcome set of the DF procedure. Our example is also a counterexample to Lemma 1 in G96, which is the basis for the proofs of Propositions 3 and 4. The second part of Lemma 1 states that “si is weakly dominated if and only if it is not a best response to some conjecture s−i such that πj (s−i ) ∈ Int Sj ∀j = i.” However, ∗ ) on A is 2 x + 1 x this claim is not correct. In our example, the marginal distribution π1 (s−3 1 3 1 3 2 2 1 ∗ that lies in the interior of S1 , and the marginal distribution π2 (s−3 ) on A2 is 3 y1 + 3 y2 that lies in the interior of S2 . But, the weakly dominated strategy z2 is a best response to the allowable ∗ . It is Lemma 1 that is the culprit for the mistakes in G96. conjecture s−3 2 Because the payoffs for players 1 and 2 are constantly zero, any perfect τ -theory τ = ((R )3 , ( )3 ) satisfies i i=1 i i=1

Ri = Si for i = 1, 2. Consider ax1 + (1 − a) x2 in  1 ⊆ Int S1 and by1 + (1 − b) y2 in  2 ⊆ Int S2 . For any ε ∈ (0, 1), we can find α1 and α2 in [1 − ε, 1) such that ⎧    

 2−3a 1−α1 1−3(1−a) 1−α1 ⎪ ∗ ⎪ = 23 x1 + 13 x2 = α1 x + x2 + (1 − α1 ) (ax1 + (1 − a) x2 ) ⎨ π1 s−3 1 3α1 3α1    

   2−3b 1−α 1−3(1−b) 1−α ⎪ 2 2 ∗ ⎪ = 23 y1 + 13 y2 = α2 y1 + y2 + (1 − α2 ) by1 + (1 − b) y2 , ⎩ π2 s−3 3α 3α 2

2

        2−3a 1−α1 1−3(1−a) 1−α1 2−3b 1−α2 1−3(1−b) 1−α2 ∗ is an where x1 + x2 ∈ R 1 and y1 + y2 ∈ R 2 . Therefore, s−3 3α1 3α1 3α2 3α2

ε-allowable conjecture for all ε ∈ (0, 1).

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4. A modification that rescues the results in G96 To gain deeper insight as to why a perfect τ -theory may contain weakly dominated strategies, we need to realize that for games with more than two players, the requirement that the marginal distribution πj (s−i ) on action space Aj lies in the interior of Sj for all j = i does not imply ∗ has the marginal s−i lies in Int S−i . As Example 1 demonstrates, although the conjecture s−3 ∗ ) in the interior of S for j = 1, 2, this conjecture does not lie in the interior distribution πj (s−3 j   of the joint probability space S−3 because it puts zero probability on x2 , y2 . As a consequence, G96’s notion of a perfect τ -theory fails to rule out all the weakly dominated strategies. To rectify these propositions in G96, we offer a modified notion of admissible conjectures so that every conjecture permitted to a perfect τ -theory must lie in the interior of the joint probability ε (τ ) space. Specifically, within the analytical framework proposed by G96, we define the set C −i of ε-admissible conjectures for player i as follows:  ε 1 2 −i + s−i C (τ ) = s−i ∈ S−i | s−i = (1 − )s−i  1 2 for some  ∈ (0, ε], s−i ∈ j =i Rj and s−i ∈ j =i j . That is, an ε-admissible conjecture for player i assigns a probability of at least 1 − ε on playing some strategy profile in the set j =i Rj and a probability of at most ε on playing some strategy profile in the set j =i j . Under this alternative notion of admissible conjectures, we obtain a modified notion of a perfect τ -theory that possesses all desirable properties in G96, because Definition 2 implies ε (τ ) ⊆ Int S−i . (In our example, z2 is excluded by the modified perfect τ -theory, because C −i ∗ is no longer an admissible conjecture: s ∗ ∈ C ε (τ ) \C ε (τ ) for the allowable conjecture s−3 −3 −3 −3 all ε ∈ (0, 1).) Subsequently, we can use Pearce’s (1984) Lemma 4 instead of the second part of G96’s Lemma 1 to prove Propositions 3 and 4; hence, the modified perfect τ -theory yields the outcome set of the DF procedure. Note that in the case of two-person games, G96’s results on the perfect τ -theory hold true because the modified notion of a perfect τ -theory coincides with the original definition.3 5. Conclusion Definition 10 in G96 offers an extensive-form version of a (perfect) τ -theory by imposing an additional restriction Rix (τ ), which requires that each rational player should choose an optimal strategy at every information set that is not precluded by the strategy. Proposition 7 in G96 states that a perfect τ -theory in an extensive game is equivalent to a perfect τ -theory in its normal-form representation. However, the following example shows that an extensive-form perfect τ -theory is more restrictive than the corresponding normal-form perfect τ -theory. Consider the following extensive game .

3 We thank the associate editor for pointing this out to us.

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The following is the normal form representation, G( ), of . y1 y2 x1 0, 0, 1 0, 0, 1 x2 0, 0, 1 0, 0, 1 z1

y1 y2 x1 0, 0, 1 0, 0, 1 x2 0, 0, 1 0, 0, 0 z2

In the extensive game , player 3’s action z2 is strictly dominated at his decision node, and thus z2 is excluded by the additional requirement R3x (τ ). That is, the extensive-form perfect τ -theory excludes z2 . But, as argued in Example 1, no perfect τ -theory for the normal-form game G( ) excludes z2 . Hence, Proposition 7 in G96 is incorrect. Similarly, G96’s extensive-form notion of a perfect τ -theory cannot yield an extensive-form analog of the DF procedure in Ben-Porath (1997) (although in “generic” extensive games, the extensive-form notion of a τ -theory yields the outcome set of the DF procedure; cf. Dekel and Gul (1997, Sec. 5.4) for related discussions). ε (τ ) also rescues the result in G96 The modified definition of a perfect τ -theory by using C −i about extensive games. References Ben-Porath, E., 1997. Rationality, Nash equilibrium, and backward induction in perfect information game. Rev. Econ. Stud. 64 (1), 23–46. Dekel, E., Fudenberg, D., 1990. Rational behavior with payoff uncertainty. J. Econ. Theory 52 (2), 243–267. Dekel, E., Gul, F., 1997. Rationality and knowledge in game theory. In: Kreps, D.M., Wallis, K.F. (Eds.), Advances in Economics and Econometrics: Theory and Applications, vol. 1. Cambridge University Press, pp. 87–172. Gul, F., 1996. Rationality and coherent theories of strategic behavior. J. Econ. Theory 70 (1), 1–31. Pearce, D.G., 1984. Rationalizable strategic behavior and the problem of perfection. Econometrica 52 (4), 1029–1051.