On non-monotonic strategic reasoning

Games and Economic Behavior 120 (2020) 209–224

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

On non-monotonic strategic reasoning ✩ Emiliano Catonini National Research University Higher School of Economics, Russian Federation - International College of Economics and Finance, Russian Federation

a r t i c l e

i n f o

Article history: Received 15 January 2019 Available online 27 January 2020 Keywords: Strong Rationalizability Strong--Rationalizability Path restrictions Epistemic priority Order independence Backward induction

a b s t r a c t Strong--Rationalizability introduces first-order belief restrictions in the analysis of forward induction reasoning. Without actual restrictions, it coincides with Strong Rationalizability (Battigalli, 2003; Battigalli and Siniscalchi, 2003). These solution concepts are based on the notion of strong belief (Battigalli and Siniscalchi, 2002). The non-monotonicity of strong belief implies that the predictions of Strong--Rationalizability can be inconsistent with Strong Rationalizability. I show that Strong--Rationalizability refines Strong Rationalizability in terms of outcomes when the restrictions correspond to belief in a distribution over terminal nodes. Moreover, under such restrictions, the epistemic priority between rationality and belief restrictions is irrelevant for the predicted outcomes. © 2020 Elsevier Inc. All rights reserved.

1. Introduction Strong Rationalizability (Battigalli, 2003; Battigalli and Siniscalchi, 2003) is a form of rationalizability for dynamic games originally called Extensive-Form Rationalizability (Pearce, 1984; Battigalli, 1997).1 Specifically, it is the iterated elimination of “never sequential best replies” under strong belief (i.e., belief as long as not contradicted by observation) in the opponents’ strategies that survive the previous step of elimination. Strong--Rationalizability (Battigalli, 2003; Battigalli and Siniscalchi, 2003) introduces first-order belief restrictions in the same reasoning scheme: only belief systems in an exogenously given set are allowed at all steps. It is well-known that the introduction of belief restrictions can let the elimination procedure depart from Strong Rationalizability in unintuitive ways: some outcomes that are eliminated without belief restrictions may instead survive the elimination procedure under some belief restrictions. This is due to the non-monotonicity of strong belief, that is, strong belief in an event does not imply strong belief in a larger event. Even in a perfect information game without relevant ties, the introduction of first-order belief restrictions can induce different outcomes with respect to the unique strongly rationalizable/backward induction outcome (see, e.g., the introductory example in Catonini, 2019). Are there interesting belief restrictions under which Strong--Rationalizability refines the set of strongly rationalizable outcomes? Under such restric-

✩ I want to thank Pierpaolo Battigalli, Enzo di Pasquale, Carlo Cusumano, Andres Perea, Giulio Principi, two anonymous referees, three anonymous referees of LOFT 2018, and all the attendants of my presentation at the conference. The study has been funded within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE) and by the Russian Academic Excellence Project ’5-100. E-mail address: [email protected]. 1 Strong Rationalizability is equivalent to Extensive-Form Rationalizability of Pearce (1984) and Battigalli (1997) in games with two players and no chance move. Battigalli and Siniscalchi (2002) provide a definition of extensive-form rationalizability which is equivalent to Strong Rationalizability in all games. The Strong Rationalizability terminology has been adopted by Battigalli (2003) because there are different legitimate extensions of the rationalizability solution concept from simultaneous-move games to dynamic games represented in their extensive form, and this one — based on the strong belief concept — yields the strongest predictions (cf. Chen and Micali, 2013, and Perea, 2018a).

https://doi.org/10.1016/j.geb.2020.01.004 0899-8256/© 2020 Elsevier Inc. All rights reserved.

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E. Catonini / Games and Economic Behavior 120 (2020) 209–224

tions, the predictions of Strong--Rationalizability would be reassuringly compatible with common strong belief in rationality (Battigalli and Siniscalchi, 2002), whose behavioral implications are captured by Strong Rationalizability. It turns out that, in all games with observable actions,2 the outcomes predicted by Strong--Rationalizability are always strongly rationalizable when the restrictions correspond to initial belief in an outcome (distribution), or in a set of outcomes that all receive positive probability.3 I will refer to both as “path restrictions”. Path restrictions are important both for theory and applications. Agreements among real players often specify only one or more outcomes to achieve and fall through if a player deviates, i.e., they do not specify off-path behavior — see Catonini (2020). Or, if the source of the belief restrictions is learning, players are likely to have a rich record of observations of the on-path play, but limited or no experience of what would happen off-path — a motivation used by Sobel et al. (1990). Theoretically, path restrictions can be used to test the compatibility of an outcome distribution with a kind of forward induction reasoning, whereby a deviation from the path(s) signals optimistic beliefs about the opponents’ reaction, rather being interpreted as a mistake or as disbelief that the opponents will stay on path themselves. In Catonini (2019) I elaborate on this use of path restrictions and on the connection with strategic stability (Kohlberg and Mertens, 1986). Equilibrium refinements related to strategic stability have been motivated informally with the idea that a deviator will be expected to aim for a higher payoff than the equilibrium payoff. Path restrictions have been used to make this idea precise: see Battigalli and Siniscalchi (2003) for the Iterated Intuitive Criterion (Cho and Kreps, 1987), Catonini (2020) for the elimination of “equilibrium paths that can be upset by a convincing deviation” (Osborne, 1990), and (with different but related techniques) Sobel et al. (1990) for Divine Equilibrium (Banks and Sobel, 1987). Why do path restrictions refine the set of strongly rationalizable outcomes? Note that, also under path restrictions, Strong--Rationalizability forces players to give up orders of belief in rationality that are per se compatible with the observed behavior, in order to keep orders of belief in the path. A more general result will highlight that the outcome inclusion between Strong--Rationalizability and Strong Rationalizability is preserved as long as the belief restrictions “never bite off-path”. With this, I refer to restrictions that exclude belief systems only according to the probabilities they assign to the opponents’ moves along the paths that survive all steps of Strong--Rationalizability itself. This self-referential condition is of little practical use but sheds light on the root of the problem. Under such restrictions, when a deviation from the surviving paths is eliminated, players have no restriction to beliefs about the continuation play after the deviation. Hence, the deviation is unprofitable also for beliefs that are compatible with all orders of belief in rationality. On the contrary, in presence of off-path restrictions, the deviation may be eliminated only because the beliefs are constrained on a particular continuation play, which may not be compatible with some order of belief in rationality. In Catonini (2019) I define an elimination procedure with first-order belief restrictions, Selective Rationalizability, that refines Strong Rationalizability and thus guarantees common strong belief in rationality. Selective Rationalizability is based on the idea that all orders of belief in rationality receive epistemic priority over all orders of belief in the restrictions. For instance, when a player displays behavior that cannot be rational under her belief restrictions, Selective Rationalizability requires to keep the belief that this player is rational (if per se compatible with the observed behavior) and drop the belief that her restrictions hold, while Strong--Rationalizability captures the opposite epistemic priority choice.4 Selective Rationalizability and Strong--Rationalizability can even predict non-empty, disjoint outcome sets for the same belief restrictions (see Catonini, 2019). But in light of the main monotonicity result, one expects the two solution concepts to give the same predictions under path restrictions. I prove that this is the case. Hence, path restrictions have the further advantage of giving predictions that are robust to this epistemic priority choice. The workhorse lemma of the paper also yields as a corollary the following result, originally proven by Perea (2018a): in games with observable actions, the iterated deletion of never sequential best replies under the strong belief operator (of which Strong Rationalizability is the maximal elimination order) is order independent in terms of predicted outcomes. Chen and Micali (2013) characterize Strong Rationalizability with the maximal iterated elimination of distinguishably dominated strategies,5 which they prove to be order independent in terms of outcomes in all games with perfect recall. Here, like in the recent work of Perea (2018a, 2018b), I do not use dominance characterizations. Finally, I use the order independence result to show that Strong Rationalizability refines (in terms of outcomes) a generalization of backward induction to games without perfect information, Backwards Extensive-Form Rationalizability of Penta (2011, 2015). Perea (2018a) provides a similar result for the Backwards Dominance Procedure (Perea, 2014). Perea (2018b) had already shown that Strong Rationalizability refines backward induction in games with perfect information, to shed new light on their outcome equivalence in absence of relevant ties — a result originally proven by Battigalli (1997), and then by Heifetz and Perea (2015) in a more direct way. I obtain this classical result as a corollary as well. The rest of the paper is structured as follows. Section 2 introduces the formal framework for the analysis. Section 3 defines elimination procedures and introduces the workhorse lemma. Section 4 illustrates the results on outcome mono-

2

That is, games where, allowing for simultaneous moves, every player knows the current history of the game. These are two extremes of the spectrum of sets of outcome distributions with the same support. The whole spectrum will be considered by the formal analysis. 4 This epistemic justification of Strong--Rationalizability is provided by Battigalli and Prestipino (2013). See Battigalli and Siniscalchi (2007) for an alternative justification where rationality and the restrictions are at the same epistemic priority level. 5 They do so by proving the equivalence between distinguishable and conditional dominance, whereby the maximal iterated elimination of conditionally eliminated strategies was proven to be equivalent to extensive-form rationalizability by Shimoji and Watson (1998). 3

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tonicity with respect to first-order belief restrictions and outcome equivalence with respect to the epistemic priority choice. Section 5 presents the results on order independence and backward induction. Section 6 provides an example of outcome inclusion and the sketch of the proof of the workhorse lemma. In Section 7 I elaborate on similarities and differences between my approach and Perea’s. The Appendix contains the proof of the workhorse lemma. 2. Preliminaries Primitives of the game.6 Let I be the finite set of players. For any profile of sets ( X i )i ∈ I and any non-empty subset of players J ⊆ I , I write X J := × j ∈ J X j , X := X I , X −i := X I \{i } . Let ( A i )i ∈ I be the finite sets of actions potentially available

 

t

to each player. Let H ⊆ ∪t =1,..., T A ∪ h0

be the set of histories, where h0 ∈ H is the empty, initial history and T is the

finite horizon. The set H must have the following properties. First property: For any h = (a1 , ..., at ) ∈ H and l < t, it holds h = (a1 , ..., al ) ∈ H , and I write h ≺ h.7 Let Z := { z ∈ H : h ∈ H , z ≺ h} be the set of terminal histories (henceforth, outcomes or paths),8 and H := H \ Z the set of non-terminal histories (henceforth, just histories). Second property: For every h ∈ H , there is a Cartesian set of action profiles A (h) such that (h, a) ∈ H if and only if a ∈ A (h). For each i ∈ I , let A i (h) denote 9 the set  of actions  available at h, so that ×i ∈ I A i (h) = A (h). For each i ∈ I , let u i : Z → R be the payoff function. The list  = I , H , (u i )i ∈ I is a finite game with complete information and observable actions. Derived objects. A strategy of player i is an element of ×h∈ H A i (h). Let S i denote the set of all strategies of i. A strategy profile s ∈ S naturally induces a unique outcome z ∈ Z . Let ζ : S → Z be the function that associates each strategy profile with the induced outcome. For any history or path h ∈ H , and for every subset of players J ⊆ I , the set of strategy sub-profiles consistent with h is10 :





S J (h) := s J ∈ S J : ∃ z h, ∃s I \ J ∈ S I \ J , ζ (s J , s I \ J ) = z . By observable actions, we have S J (h) = × j ∈ J S j (h). For any S J ⊂ S J , let S J (h) := S J (h) ∩ S J . Let





H ( S J ) := h ∈ H : S J (h) = ∅

 

denote the set of histories consistent with S J . For any h = (h , a) ∈ H \ h0 , let p (h) denote the immediate predecessor h of h. Since the game has observable actions, each history h ∈ H is the root of a subgame (h). If h = h0 , all the objects in (h) will be denoted with h as superscript, except for histories and outcomes, which will be identified with histories and outcomes of the whole game, and not redefined as shorter sequences of action profiles. For any h ∈ H , shi ∈ S hi = ×h h A i (h ),





and  h ∈ H h = h ∈ H : h  h , shi | h will denote the projection of strategy shi in the subgame with root  h; that is, the strategy 





shi ∈ S hi such that shi ( h) = shi ( h) for all  h  h. For any S i ⊆ S hi , S i | h will denote the set of projections of the strategies of S i h



h



h



in the subgame with root  h; that is, all the strategies shi ∈ S hi such that shi = shi | h for some shi ∈ S i . h

Beliefs. In this dynamic framework, beliefs are modeled as Conditional Probability Systems (Renyi, 1955; henceforth, CPS). Definition 1. Fix i ∈ I . An array of probability measures (μi (·|h))h∈ H over co-players’ strategies S −i is a Conditional Probability System if for each h ∈ H , μi ( S −i (h)|h) = 1, and for every h  h and S −i ⊆ S −i (h ),

μi ( S −i |h) = μi ( S −i (h )|h) · μi ( S −i |h ). The set of all CPS’s on S −i is denoted by  H ( S −i ). In words, a player endowed with a CPS about the opponents’ strategies updates her beliefs according to the rules of conditional probabilities whenever possible, even after she has already been “surprised”, in which case she is updating a revised belief. For any subset of opponents’ strategies S −i ⊆ S −i , I say that a CPS μi ∈  H ( S −i ) strongly believes S −i if, for all h ∈ H ( S −i ), q μi ( S −i |h) = 1. I say that a CPS strongly believes a sequence of events ( S −i )q∞=0 when it strongly believes each event in the sequence. I fix the following convention: H (∅) = ∅. With this, the empty set is always strongly believed, because the condition is vacuously satisfied.

6 7 8 9 10

The main notation is almost entirely borrowed from Osborne and Rubinstein (1994). Then, H endowed with the precedence relation ≺ is a tree with root h0 . “Path” will be used with emphasis on the moves, and “outcome” with emphasis on the end-point of the game. When player i is not truly active at history h, A i (h) consists of just one “wait” action. For J = I , s I \ J must be omitted from the definition of S J (h).

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Rationality. I consider players who reply rationally to their conjectures. Rationality here means that players, at every history, choose an action that maximizes expected payoff given the belief about how the opponents will play and the expectation to play rationally again in the continuation of the game. By standard arguments, this is equivalent to playing a sequential best reply to the CPS. Definition 2. Fix μi ∈  H ( S −i ). A strategy si ∈ S i is a sequential best reply to best reply to μi (·|h), i.e., for every  si ∈ S i (h),



u i (ζ (si , s−i ))μi (s−i |h) ≥

s−i ∈ S −i (h)



μi if for every h ∈ H (si ),11 si is a continuation

u i (ζ ( si , s−i ))μi (s−i |h).

s−i ∈ S −i (h)

μi is denoted by ri (μi ). For each h ∈ H , the set of continuation best replies to μi (·|h)      μhi , rih (μhi ), and  r ih (μhi , h) ( h  h) will denote, respectively, a CPS for the subgame with   root h, its sequential best replies, and the continuation best replies to μhi (·|h). The set of sequential best replies to

is denoted by  r i (μi , h). Analogously,

3. Elimination procedures and the workhorse lemma I provide a very general notion of elimination procedure, which encompasses all the procedures I am ultimately interested in, or that will be used in the proofs. Definition 3. Fix h ∈ H . An elimination procedure in (h) is a sequence (( S hi,q )i ∈ I )q∞=0 where, for every i ∈ I , EP1 S hi,0 = S hi ; EP2 S hi,n−1 ⊇ S hi,n for all n ∈ N ;

EP3 for every shi ∈ S hi,∞ := ∩n∈N S hi,n , there exists

μhi that strongly believes ( S h−i,q )q∞=0 such that shi ∈ rih (μhi ) ⊆ S hi,∞ .

Note three things. First, the fact that shi is a sequential best reply to some

h that strongly believes ( S h−i ,q )nq=0 does not i h imply that ∈ S i ,n+1 , and vice versa; this allows to encompass first-order belief restrictions and “slow” elimination orders. Second, EP2 allows S nh = S nh+1 , and yet S h∞  S nh+1 ; that is, the eliminations can stop for all players and then resume. Third, S hi,n = ∅ implies S hi,m = ∅ for all m > n, but does not imply S hj,∞ = ∅ for j = i: as already established, the empty set is always

μ

shi

strongly believed, hence EP3 can be satisfied for j. These three facts allow Definition 3 to encompass what I will call the “projection” of an elimination procedure in a subgame. Remark 1. For every elimination procedure (( S hi,q )i ∈ I )q∞=0 and every  h  h, (( S hi,q ( h)| h)i ∈ I )q∞=0 is an elimination procedure. Proof. See the Appendix.

2

At each step of reasoning q and for each player i, S hi,q ( h)| h is constructed by taking the strategies in S hi,q that allow  h (that is, S hi,q ( h)) and restricting their domain to the histories that weakly follow  h; that is, the histories of the subgame

 ( h). Strictly speaking, each S hi,q ( h)| h is the projection of S hi,q ( h), and not of S hi,q , over the subspace × h  h A i (h ) (the space h ∞ of strategies of the subgame ( h )); however, with an abuse of terminology, I will call (( S ( h)| h )i ∈ I ) “projection of i ,q

(( S hi,q )i ∈ I )q∞=0 in ( h).”

q=0



It will be important to keep in mind that a strategy shi can be eliminated from S hi,n ( h)| h “exogenously”: it may be a  h h n   sequential best reply to some μi that strongly believes ( S −i ,q (h)|h)q=0 , and yet not belong to S hi,n+1 ( h)| h, because no μhi that strongly believes ( S h−i ,q )nq=0 and induces



μhi in ( h) has sequential best replies that allow  h.12

h

The workhorse lemma of the paper claims the outcome inclusion between two elimination procedures, (( S i ,q )i ∈ I )q∞=0 h

(Procedure 1) and (( S hi,q )i ∈ I )q∞=0 (Procedure 2), with the following features. For each player i and each strategy shi ∈ S i ,∞ , fix h

h

a CPS λhi (shi ) that satisfies EP3, i.e., it strongly believes ( S −i ,q )q∞=0 , and shi ∈ r ih (λhi (shi )) ⊆ S i ,∞ . Say that a CPS

11

It would be immaterial for the analysis to require si to be optimal also at the histories precluded by si itself.

12

I say that





μhi “mimics”

μhi induces μhi in ( h) when, for each  h  h, μhi (·| h) is the pushforward of μhi (·| h) through the map that associates each shi ∈ S hi ( h) with shi | h.

E. Catonini / Games and Economic Behavior 120 (2020) 209–224

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h

λhi (shi ) along the paths predicted by S ∞ when, at every history along these paths, μhi and λhi (shi ) assign the same probabilih h) = λh (sh )( S −i ( z)| h) for all  h ∈ H ( S ) and ties to (the opponents playing compatibly with) each of these paths: μh ( S −i ( z)| i

h

i

∞

i

h

μhi mimics some λhi (shi ) along ζ ( S ∞ ) and h h −1 h h strongly believes ( S −i ,q )m q=0 , then its sequential best replies survive step m of elimination of Procedure 1: r i (μi ) ⊆ S i ,m ; if all z ∈ ζ ( S ∞ ). Suppose that, at every step m ∈ N , the following is true: if a CPS

−1 μhi mimics some λhi (shi ) along ζ ( S ∞ ) and strongly believes ( S h−i,q )m q=0 , then its sequential best replies survive the step m of elimination of Procedure 2: r ih (μhi ) ⊆ S hi,m . Under these two conditions, the lemma claims that Procedure 2 predicts a h

a CPS

h

superset ζ ( S h∞ ) ⊇ ζ ( S ∞ ) of the outcomes predicted by Procedure 1. h

Lemma 1. Fix h ∈ H and two elimination procedures (( S i ,q )i ∈ I )q∞=0 , (( S hi,q )i ∈ I )q∞=0 . h

h

h

For every i ∈ I , fix a map λhi : S i ,∞ → iH ( S h−i ) such that, for each shi ∈ S i ,∞ , λhi (shi ) strongly believes ( S −i ,q )q∞=0 , and shi ∈ h

h

r ih (λhi (shi )) ⊆ S i ,∞ . Suppose that the two procedures satisfy the following property: h

h

−1 m−1 A0 for each i ∈ I , shi ∈ S i ,∞ , m ∈ N , and for every μhi that strongly believes ( S h−i ,q )m q=0 (resp., ( S −i ,q )q=0 ) and satisfies h h ∀ h ∈ H ( S ∞ ), ∀ z ∈ ζ ( S ∞ ), μhi ( S −i ( z)| h) = λhi (shi )( S −i ( z)| h),

(1)

h

we have r ih (μhi ) ⊆ S hi,m (resp., r ih (μhi ) ⊆ S i ,m ). h

Then, ζ ( S ∞ ) ⊆ ζ ( S h∞ ). The proof of Lemma 1 is in the Appendix. In Section 6, I provide an example of outcome inclusion between two relevant elimination procedures, Strong--Rationalizability and Strong Rationalizability, and I illustrate the main intuition for it. Then, I sketch the proof of Lemma 1 in its generality. 4. Belief-restrictions and monotonicity In this section, I am going to focus on the following elimination procedures (for the whole game).13 Definition 4. An elimination procedure (( S i ,q )i ∈ I )q∞=0 is “unconstrained” when for every n ∈ N , i ∈ I , and

μi that strongly

−1 believes ( S −i ,q )nq= 0 , r i (μi ) ⊆ S i ,n .

Definition 5. Strong Rationalizability is the unconstrained elimination procedure (( R i ,q )i ∈ I )q∞=0 such that for every n ∈ N , i ∈ I , and si ∈ R i ,n , there is

−1 μi that strongly believes ( R −i,q )nq= 0 with s i ∈ r i (μi ).

∞ Definition 6. For each i ∈ I , fix i ⊆  H ( S −i ). Strong--Rationalizability is the elimination procedure (( R  i ,q )i ∈ I )q=0 such

that, for every n ∈ N , i ∈ I , and si ∈ S i , si ∈ R  if and only if si ∈ r i (μi ) for some i ,n

n−1 μi ∈ i that strongly believes ( R  −i ,q )q=0 .

Definition 7. For each i ∈ I , fix i ⊆  H ( S −i ). Selective Rationalizability is the elimination procedure (( R iS,q )i ∈ I )q∞=0 such that: 1. ( R qS )qM=0 = ( R q )qM=0 , where M is the smallest n ≥ 0 such that R n+1 = R n ;

2. for every n > M, i ∈ I , and si ∈ S i , si ∈ R iS,n if and only if si ∈ r i (μi ) for some

13

−1 14 μi ∈ i that strongly believes ( R −S i,q )nq= 0.

The definitions of Strong Rationalizability and Strong--Rationalizability are the ones of Battigalli (2003). Selective Rationalizability in Catonini (2019) is initialized with R ∞ and strong belief in ( R −i ,q )q∞=0 . It is easy to see that Definition 7 is equivalent in finite games. Moreover, in Catonini (2019) Selective Rationalizability is defined under the hypothesis of independent rationalization. That is, a valid μi is required to −1 n−1 S strongly believe ( R Sj,q )nq= 0 for all j = i, in place of just ( R −i ,q )q=0 . Although it is natural to assume independent rationalization together with stochastic independence of beliefs across opponents (as in Battigalli, 1996, and Battigalli and Siniscalchi, 1999) one hypothesis does not entail the other, and Selective Rationalizability does not adopt stochastic independence for simplicity. The result on Selective Rationalizability of this paper (Theorem 2) is robust to any combination of these hypotheses — a proof is available upon request. 14

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Consider first-order belief restrictions (i )i ∈ I with the following property: for each player i and CPS μi , all that matters to determine whether μi belongs to i are the probabilities μi ( S −i ( z)|h) assigned at each strongly--rationalizable history  h ∈ H (R ∞ ) to (the opponents playing compatibly with) each strongly--rationalizable path z ∈ ζ ( R ∞ ). This property is self-referential, in that it requires to scrutinize the final output of Strong--Rationalizability itself to check whether the given belief restrictions satisfy it. Therefore, it cannot be used to assess at the outset whether the restrictions guarantee outcome monotonicity. Nonetheless, this property identifies a class of belief restrictions for which Strong--Rationalizability satisfies the hypotheses of Lemma 1 in the role of first elimination procedure. Strong Rationalizability can always be assigned the role of second elimination procedure. This is because it is an unconstrained procedure, and thus it saves at each step all the sequential best replies to every CPS that strongly believes in the previous steps, regardless of whether the CPS satisfies (1) or not. The desired outcome inclusion with respect to belief restrictions that “do not end up off-path” obtains. Lemma 2. For each i ∈ I , fix i ⊆  H ( S −i ). Suppose that for each μi ∈ i and μi ∈  H ( S −i ),



   ∀ h ∈ H (R ∞ ), ∀ z ∈ ζ ( R ∞ ), μi ( S −i ( z)|h ) = μi ( S −i ( z)|h ) ⇒ (μi ∈ i ) .

(2)

Then, ζ ( R  ∞ ) ⊆ ζ ( R ∞ ). ∞ μi ∈ i that strongly believes ( R  −i ,q )q=0 such that s i ∈ r i (μi ) ⊆ n R . For each μi that satisfies (1) with μi , by (2) we have μi ∈ i . So, for each n ∈ N , if μi strongly believes ( R  i ,∞ −i ,q )q=0 ,  then r i (μi ) ⊆ R i ,n+1 . Thus, A0 is satisfied for Strong--Rationalizability, while, as observed, it is always satisfied for Strong

Proof. Fix i ∈ I . For each si ∈ R  , by Definition 6 there is i ,∞

Rationalizability. Hence, by Lemma 1, ζ ( R  ∞ ) ⊆ ζ ( R ∞ ).

2

Lemma 2 provides theoretical insight on what determines the non-monotonicity of the predicted outcome set with respect to the belief restrictions: the presence of (what end up being) off-path restrictions. However, as already observed, whether the restrictions are off-path or not refers to the final output of Strong--Rationalizability itself. Therefore, I am going to identify a class of belief restrictions that, whenever Strong--Rationalizability yields a non-empty output, cannot end up off-path. Consider first-order belief restrictions that correspond to the belief in a set of outcome distributions with the same support. To formalize, fix a set Z ⊂ ( Z ) of probability measures with the same support Z that can be induced by a product measure ×i ∈ I νi ∈ ( S ) (νi ∈ ( S i )) over strategy profiles. The focus on product measures is of course restrictive (also for the possible supports Z ), but it greatly simplifies exposition. Every player i initially believes that the opponents will play compatibly with an outcome distribution in Z . With this, I define the notion of “path restrictions”. Definition 8. Fix a set Z ⊂ ( Z ) of outcome distributions with support Z ⊂ Z that can be induced by a product measure over strategy profiles. The corresponding path restrictions of player i are given by the set of CPS’s

 i := μi ∈  H ( S −i ) : ∃η ∈ Z , ∃νi ∈ ( S i ), ∀ z ∈ Z , νi ( S i ( z))μi ( S −i ( z)|h0 ) = η( z) . If Z is a singleton, the corresponding path restrictions represent the belief in an outcome distribution; if Z is a singleton, they represent the belief in a specific path z: μi ( S −i ( z)|h0 ) = 1.15 When Z is not a singleton and Z is the set of all probability measures with support Z , the corresponding path restrictions represent a “restricted full support” condition with respect to a subset of outcomes. Under path restrictions, when Strong--Rationalizability yields a non-empty set, all the paths in Z must be strongly--rationalizable. To see this, suppose by contraposition that, at some step n, player i eliminates all the strate∩ S i (z) = ∅. Then, it is easy to check that R j ,n+1 = ∅ for each gies that are consistent with some path z ∈ Z , that is, R  i ,n j = i.  Remark 2. Under path restrictions, if R  ∞ = ∅, then Z ⊆ ζ ( R ∞ ).

Given this, the belief restrictions never “end up off-path”, and Lemma 2 can be applied. Theorem 1. Fix path restrictions (i )i ∈ I . We have ζ ( R  ∞ ) ⊆ ζ ( R ∞ ). 15 For Strong--Rationalizability and Selective Rationalizability, initial belief in S −i ( z) is equivalent to strong belief in S j ( z) for all j = i — the belief in the (path) agreement as modeled in Catonini (2020). The reason is that after a deviation from the path by a player different than j, believing that j would have stayed on path is not restrictive for the expected behavior of j after the deviation. A formal proof is available upon request.

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 H Proof. If R  ∞ = ∅, the statement is trivially true, so suppose R ∞ = ∅. Fix i ∈ I , μi ∈ i , and μi ∈  ( S −i ). By Definition 8, 0 there are η ∈ Z and νi ∈ ( S i ) such that νi ( S i ( z))μi ( S −i ( z)|h ) = η( z) for all z ∈ Z . By Remark 2, Z ⊆ ζ ( R  ∞ ). Hence, we have



   ∀ h ∈ H (R ∞ ), ∀ z ∈ ζ ( R ∞ ), μi ( S −i ( z)|h ) = μi ( S −i ( z)|h) ⇒ 

∀ z ∈ Z , μi ( S −i ( z)|h0 ) = μi ( S −i ( z)|h0 ) ⇒

 ∀ z ∈ Z , νi ( S i ( z))μi ( S −i ( z)|h0 )) = η( z) ⇒ (μi ∈ i ) .

Thus, (2) holds, and Lemma 2 yields ζ ( R  ∞ ) ⊆ ζ ( R ∞ ).

2

Corollary 1. Fix z ∈ Z . Let i be the set of all μi ∈  H ( S −i ) such that μi ( S −i ( z)|h0 ) = 1. Then, ζ ( R  ∞ ) ⊆ ζ ( R ∞ ). Also Selective Rationalizability eventually saves only strategies that are sequential best replies to CPS’s in the restricted set. Therefore, for path restrictions, Selective Rationalizability and Strong--Rationalizability satisfy the hypotheses of Lemma 1 regardless of the roles assigned to the two procedures. Then, the outcome equivalence of the two procedures under path restrictions obtains. S Theorem 2. Fix path restrictions (i )i ∈ I . We have ζ ( R  ∞ ) = ζ ( R ∞ ). S  Proof. I show that ζ ( R  ∞ ) ⊆ ζ ( R ∞ ) — the proof of the opposite inclusion is identical. If R ∞ = ∅, the inclusion holds trivially,  ∞  so suppose R ∞ = ∅. Fix i ∈ I . For each si ∈ R i ,∞ , by Definition 6 there is μi ∈ i that strongly believes ( R  −i ,q )q=0 such that

si ∈ r i (μi ) ⊆ R  . For each i ,∞

So, for each n ∈ N , if

μi that satisfies (1) with μi , proceeding like in the proof of Theorem 1, we find that μi ∈ i . n S n  μi strongly believes ( R  −i ,q )q=0 , then r i (μi ) ⊆ R i ,n+1 , and if μi strongly believes ( R −i ,q )q=0 , then

r i (μi ) ⊆ R iS,n+1 . Thus, A0 is satisfied for both Strong--Rationalizability and Selective Rationalizability. Hence, Lemma 1 S yields ζ ( R  ∞ ) ⊆ ζ ( R ∞ ). 2 S Corollary 2. Fix z ∈ Z . Let i be the set of all μi ∈  H ( S −i ) such that μi ( S −i ( z)|h0 ) = 1. Then ζ ( R  ∞ ) = ζ ( R ∞ ).

An example of outcome inclusion between Strong--Rationalizability with path restrictions and Strong Rationalizability is provided in Section 6. 5. Order independence and backward induction For any unconstrained elimination procedure, exactly as for Strong Rationalizability, A0 holds. An unconstrained elimination procedure is what I referred to in the Introduction as an order of elimination of never sequential best replies. Thus, using Lemma 1 in both directions with Strong Rationalizability and any other unconstrained elimination procedure, the order independence (in terms of outcomes) of the iterated elimination of never sequential best replies obtains. Theorem 3. For every unconstrained elimination procedure (( S i ,q )i ∈ I )q∞=0 , ζ ( S ∞ ) = ζ ( R ∞ ). Proof. Any two unconstrained elimination procedures, taken in both orders, satisfy A0. The results follows then from Lemma 1. 2 In games with observable actions, the well-known backward induction procedure for games with perfect information has been generalized by Penta (2011, 2015) and Perea (2014) in the following way. Starting from the bottom of game, an action of a player at a history is eliminated when it is not “folding-back optimal” under any conjecture over the surviving actions of the opponents at the same history and at the future histories. Here I adopt Backwards Extensive-Form Rationalizability of Penta (2011), because it is formulated in the language of extensive-form rationalizability, i.e., as a procedure of elimination of strategies that are not sequentially optimal for any viable conditional probability system. The following is a simplification of Penta’s definition for games with complete information. Definition 9. Backwards Extensive-Form Rationalizability is a sequence (( R iB,q )i ∈ I )q∞=0 where, for every i ∈ I , BR1 R iB,0 = S i ; BR2 for each n ∈ N and si ∈ S i , si ∈ R iB,n if and only if there exists (i) there is  si ∈ r i (μi , h) such that  si |h = si |h;

μi ∈  H ( S −i ) such that, for each h ∈ H ,

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(ii) for each  s−i with

B μi ( s−i |h) > 0, there is s−i ∈ R − such that  s−i |h = s−i |h. i ,n−1

B Condition BR2.(ii) does not discriminate between strategies in R − that allow h or not; for this reason, it does not i ,n−1 capture forward induction reasoning. Condition BR2.(i) requires si to be optimal given μi (·|h) from any history h onwards, and not only when h ∈ H (si ) (cf. Definition 2 of sequential best reply). This entails that players’ strategies are further refined also at histories that are not allowed anymore by some player. However, being such histories off-path, each step of Backwards Extensive-Form Rationalizability is outcome-equivalent to a step of an unconstrained elimination procedure. Moreover, Backwards Extensive-Form Rationalizability stops when the unconstrained procedure may not yet be allowed to, because EP3 is not satisfied.

Lemma 3. Let N be the smallest n such that R nB = R nB+1 . There exists an unconstrained elimination procedure (( S i ,q )i ∈ I )q∞=0 such that for each n = 1, ..., N,





S i ,n = S i ,n−1 ∩ si ∈ S i : ∃si ∈ R iB,n , ∀h ∈ H ( R nB ) ∩ H (si ), si (h) = si (h) . Proof. See the Appendix.

2

Hence, Backwards Extensive Rationalizability predicts a superset of the outcomes predicted by Strong Rationalizability. B Theorem 4. Every strongly rationalizable outcome is a backwards extensive-form rationalizable outcome: ζ ( R ∞ ) ⊆ ζ ( R ∞ ).

Proof. Immediate from Lemma 3 and Theorem 3.

2

In perfect information games without relevant ties, the backward induction outcome is unique. Thus, the following obtains. Corollary 3 (Battigalli, 1997). In every perfect information game without relevant ties, Strong Rationalizability and backward induction yield the same unique outcome. 6. Example and sketch of the proof of the workhorse lemma In this section, I provide an example of outcome inclusion between Strong--Rationalizability with path restrictions and Strong Rationalizability, and I illustrate the rough intuition behind it. Then, I generalize these ideas to sketch the proof of the workhorse lemma. Consider the following game.

A\B N S

W 2, 2 0, 0

E

·− 2, 2

→

A\B U M D

L 1, 1 0, 0 0, 0

C 1, 0 0, 1 0, 0

R 0, 0 1, 0 0, 3

For brevity, I am going to use plans of actions (also known as reduced strategies) in place of strategies: given Definition 2, if a strategy is a sequential best reply to a CPS, all other strategies that correspond to the same plan of actions are as well. The plans of actions of Ann will be written as S, N .U , N . M, N . D; likewise for Bob. Strong Rationalizability goes as follows. At the first step, Ann eliminates N . D, which is dominated by 1/2( N .U ) + 1/2( N . M ) in the subgame. At the second step, Bob eliminates E . R, now dominated by 1/2( E . L ) + 1/2( E .C ) in the subgame. At the third step, Ann eliminates N . M, dominated by N .U in the subgame. At the fourth step, Bob eliminates E .C , dominated by E . L in the subgame. The final output is R ∞ = { S , N .U } × { W , E . L }. Now, fix path z := ( N , W ); thus, S A ( z) = { N .U , N . M , N . D } and S B ( z) = { W }. The corresponding path restrictions are defined as

 i := μi ∈ iH ( S −i ) : μi ( S −i ( z)|h0 ) = 1 , i = A , B.

Strong--Rationalizability goes as follows. At the first step, Ann eliminates S, which is dominated by N under belief in W , and N . D, which is dominated by 1/2( N .U ) + 1/2( N . M ) in the subgame; Bob eliminates E . L and E .C , because they are both dominated by W under belief in N. At the second step, Ann eliminates N .U , now dominated by N . M in the subgame, and Bob eliminates E . R, now dominated by W . The final output is: R  ∞ = {( N . M , W )}.  Note that ζ ( R  ∞ ) = { z} ⊆ ζ ( R ∞ ), although R ∞ ∩ R ∞ = ∅. Why does the outcome inclusion hold, despite of the disjoint strategy sets? At every step of Strong Rationalizability, Ann can play N as long as Bob may play W . For Bob, it is a bit more

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217

complicated. To play W , he needs at the same time to believe in N and have a belief for the subgame that deters a deviation to E. This is far from guaranteed, precisely because Strong Rationalizability and Strong--Rationalizability depart off the strongly--rationalizable path: the former ends up predicting (U , L ), the latter M for Ann and R as the best rationalizable action for Bob. But note two things. First: Ann can play N while believing in W and thus being surprised by E. So, after ( N , E ), she must come up with a new belief, and then, at each step of reasoning, she can combine N with any action that can be justified after E. Second: for Bob to abandon W at a step of reasoning, he must be willing to deviate to E for every belief after E he can associate with the belief in N. By the argument about Ann, he can associate with the belief in N any belief after E that is compatible with the step of reasoning. Thus, if he abandons W , he can combine E with any action that can be justified after E as well. Therefore, if Bob abandons W at step n, the set of action profiles R n−1 (( N , E ))|( N , E ) must feature all the best replies of both players to beliefs over the set. This allows to refine the set by iteratively eliminating dominated actions and find a non-empty best response set. Now, by the absence of off-path restrictions and by the same combination arguments, this best response set would survive and keep supporting E throughout Strong--Rationalizability. But E does not survive Strong--Rationalizability. This tells us that Bob cannot abandon W in Strong Rationalizability. The workhorse lemma generalizes these intuitions to all elimination procedures, predicting a set of paths, with many possible deviations, followed by a non-static subgame. That deviations can be followed by a dynamic game is the real challenge that the workhorse lemma has to face; generalizing in all other dimensions complicates the exposition but does not present interesting challenges. Therefore, I sketch here the proof of the workhorse lemma in its generality, but with particular focus on how the issue of dynamic subgames is tackled. Essentially, for a subgame that follows a hypothetical deviation at some step of Procedure 2 from the paths predicted by Procedure 1, the task is to generate an Extensive-Form Best Response Set (Battigalli and Friedenberg, 2012) of substrategies that make the deviation profitable, and prove that its sub-paths should have survived the projection of the Procedure 1 in the subgame, a contradiction to the fact that the subgame follows a deviation from the paths predicted by Procedure 1 (henceforth, just “the paths”). As long as the paths survive also Procedure 2, players can form beliefs that, along the paths, mimic the beliefs that justify the output of Procedure 1. The sequential best replies to these beliefs survive Procedure 2 by A0. Then, the only way one of the paths can be abandoned is that, at some step n + 1, for all these beliefs, a player finds a particular deviation outside of the paths more profitable, no matter what she believes the reactions of the opponents to the deviation will be. The opponents may be surprised by the deviation, hence they may react with any continuation plan that survives until step n. This is because until step n they followed the paths while believing that the others would follow them as well. Suppose for simplicity that the deviator has a deterministic belief as to which subgame the deviation will lead to. So, the projection of Procedure 2 in the subgame at step n features, both for the deviator and the opponents, all the sequential best replies to all the beliefs over step n itself. Then, the output of the following, auxiliary elimination procedure in the subgame is non-empty: it coincides with the projection of Procedure 2 until step n, and iteratively eliminates the substrategies that are never sequential best replies afterwards. Take the subpaths induced by this auxiliary procedure. I want to show that they survive the projection of Procedure 1 in the subgame, the desired contradiction. Note first that believing in these subpaths incentivizes our player to deviate from the paths also along Procedure 1. Given this, if the subgame is a static game, it is easy to observe that they do survive — see the example above. If the subgame has length higher than 1, then suppose by induction that the lemma is true in games of smaller length than the one under analysis — as a basis step, it is easy to see that the lemma holds in static games. By the absence of off-path restrictions (i.e., by A0 for Procedures 1 and 2) and by the deviation incentives, the projection of Procedure 1 in the subgame and the auxiliary procedure both satisfy A0 with respect to the subpaths induced by the auxiliary procedure, thus with inverted roles with respect to Procedures 1 and 2 that generated them. Then, by the induction hypothesis, the projection of Procedure 1 induces a superset of those subpaths. 7. Discussion – comparison with Perea (2018a, 2018b) To facilitate the comparison between my methodology and Perea’s, I refer to the order independence problem tackled by Perea (2018a).16 The fundamental problem is the need to overcome the non-monotonicity of the strong belief, which implies that delaying the elimination of some strategy can provoke the elimination of another strategy that would have not been eliminated otherwise. Consider two nested, Cartesian sets of strategy profiles, S = ×i ∈ I S i ⊂  S = ×i ∈ I  S i . Fix a player i ∈ I and consider the sets ∗ ∗ Si , S i of sequential best replies to CPS’s that strongly believe, respectively,  S −i and S −i . By non-monotonicity of strong ∗

belief, it needs not be the case that S i ⊂  S ∗i . However, for every CPS that strongly believes S −i , there is one that strongly

∗ believes  S −i and is identical to the first at all histories consistent with S . Then, for every si ∈ S i , there is si ∈  S ∗i that is

16 Perea formulates the problem for iterated eliminations under the strong belief reduction operator, i.e., the elimination at each step of some strategies that are not sequential best replies to CPS’s that strongly believe in the opponents’ strategies that survived the last step, without memory of the previous steps. (This is probably the most interesting formulation of the problem, because slow reduction orders can be seen as heuristic procedures.) This makes an iterated deletion of never sequential best replies as defined in this paper not necessarily an order of elimination under the strong belief reduction operator, and vice versa, although both maximal elimination orders coincide with Strong Rationalizability. However, this subtle difference is immaterial for this discussion.

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identical to si at each h ∈ H ( S ). So, if we focus on the paths induced by S,  S ∗ must induce a (weakly) larger subset of them ∗ ∗ with respect to S . If S has been obtained by iterated elimination of never sequential best replies, we have ζ ( S ) ⊆ ζ ( S ), ∗ ∗ and then ζ ( S ) ⊆ ζ ( S ) as well. ∗ and S ∗ , However, if one wants to iterate further and find the sequential best replies to CPS’s that strongly believe  S− −i i

there is the problem that these two sets are no more nested. So, let us restart with two sets S ,  S where the projection of S on H ( S ) is smaller than that of  S (on H ( S ) as well). Now,  S −i may feature fewer reactions than S −i to a deviation by player i from the paths induced by S, and this may induce i to leave one of these paths under strong belief in  S −i , but not under strong belief in S −i . The challenge is proving that this possibility does not arise when S ,  S have been derived from the iterated elimination of never sequential best replies. Perea does it by restricting the definition of “monotonicity on reachable histories” to sets S ,  S where the projection of  S ∗i on H ( S ) is smaller than that of S i , which excludes the presence of such deviations. Then, he divides the order independence problem into a chain of pairwise comparisons between iterated eliminations ( S n )n≥0 , ( S n )n≥0 that are identical up to the step m, after which the first becomes maximal, and so does the second one step later. In this way, for each n > m, S n and  S n , and with inverted roles S n and  S n+1 are shown to satisfy the requirements. In this paper, I observe that if the deviation by player i depicted above was to arise, there would be an Extensive-Form Best Response Set of the subgame that follows the deviation which justifies it; but then, the sub-paths induced by this set and the deviation should have survived also the procedure that generated S , a contradiction. In my view, the approach of Perea is better suited than the approach of this paper for order independence problems, because it teaches why delaying some eliminations does not matter. The approach of this paper is inspired by, and designed for, outcome monotonicity problems, and it aims to shed light on their roots by comparing directly maximal elimination procedures under belief restrictions, showing when and why deviations from the paths induced by the more restrictive procedure cannot occur in the less restrictive one. The identification of path restrictions as class of belief restrictions that preserve outcome monotonicity was indeed triggered by this view and by the workhorse lemma that captures the main conceptual insight. However, the outcome equivalence of Strong--Rationalizability and Selective Rationalizability under such restrictions, which implies the outcome inclusion with Strong Rationalizability, can also be seen as an order independence problem (while the workhorse lemma in its generality cannot). Indeed, in finite games, Selective Rationalizability can be seen as a slow elimination order of strategies that are not sequential best replies under the belief restrictions, where the first steps coincide with Strong Rationalizability. So, focusing for simplicity on a single path z, one could: – define a reduction operator r z that takes a Cartesian set of strategy profiles S = ×i ∈ I S i and, for each player i, returns the strategies in S i that are sequential best replies to a CPS that strongly believes S −i and initially believes in S −i ( z); – invoke Proposition 1 in Battigalli and Prestipino (2013) to claim that Strong--Rationalizability coincides with the maximal elimination order under the r z operator17 ; – show that Selective Rationalizability can be written as an elimination order under the r z operator that coincides with Strong Rationalizability until convergence, and then proceeds at “full speed”18 ; – show that r z is monotone on reachable histories (Perea, 2018b); – invoke Theorem 3.2 in Perea (2018b) to claim the outcome equivalence of Selective Rationalizability and Strong-Rationalizability. That r z is monotone on reachable histories is so far only a conjecture, but I expect the proof to follow the same lines of the proof of Perea that the strong belief operator is monotone on reachable histories (Theorem 3.1 in Perea, 2018b). Exploiting the existing results and proofs, the roadmap above would probably result into a shorter proof of the outcome equivalence and outcome monotonicity results under path restrictions. 8. Appendix 8.1. Proofs omitted from Sections 4 and 5 Proof of Remark 1. That the projection of an elimination procedure in a subgame satisfies EP1 and EP2 is obvious.

     To show EP3, fix i ∈ I and shi ∈ S hi,∞ ( h)| h. I want to find hi that strongly believes ( S h−i ,q ( h)| h))q∞=0 such that shi ∈ r ih ( hi ) ⊆ h S i ,∞ ( h)| h.  h h) such that shi | h = shi . Since (( S hi,q )i ∈ I )q∞=0 satisfies EP3, there is hi that strongly believes ( S h−i ,q )q∞=0 such Fix si ∈ S hi,∞ (   that shi ∈ r ih ( hi ) ⊆ S hi,∞ . Construct hi as follows: for each  h  h, define hi (·| h) as the pushforward of hi (·| h) through the  h h  h  h map that associates each s−i ∈ S −i (h) with s−i |h. It is easy to see that i is a CPS that strongly believes ( S h−i ,q ( h)| h)q∞=0 .

μ

μ

μ

μ

17

μ

μ μ

μ

It is easy to see that path restrictions are closed under composition (Battigalli and Prestipino, 2013). Under the hypotheses of strategic independence (Battigalli, 1996), or just independent rationalization (Catonini, 2019), Strong--Rationalizability and Selective Rationalizability cannot be written as reduction procedures, not even under path restrictions. However, for path restrictions, outcome equivalences hold. A proof is available upon request. 18

E. Catonini / Games and Economic Behavior 120 (2020) 209–224

219

  h  (·|h) and hi (·| h) induce the same distribution over terminal histories when coupled with any  shi and  shi i    h h h h h such that  si | h = si . Therefore, that si is a continuation best reply to i (·| h) implies that si is a continuation best reply to     h h h h . Thus, si ∈ r i ( i ). i    shi to hi . It is easy to see that the strategy  shi with  shi | h = shi and  shi ( h) = shi ( h) for each Finally, fix a sequential best reply     h ∈ H \ H h is a sequential best reply to hi . Thus, by r ih ( hi ) ⊆ S hi,∞ ,  shi ∈ S hi,∞ ( h), and then  shi ∈ S hi,∞ ( h)| h. 2

For each  h  h,

μ

μ

μ

μ

μ

μ

μ





μ

N

S i ,n i ∈ I n=0 as in the statement of the lemma, and for each n > N and i ∈ I , let si ∈ S i ,n if

n−1 and only if there exists μi that strongly believes S −i ,q q=0 such that si ∈ r i (μi ). It is immediate to see that (( S i ,q )i ∈ I )q∞=0 is an elimination procedure. Now I show it is unconstrained. Fix n = 1, ..., N, and if n > 1, suppose by way of inducm−1 tion that for each m = 1, ..., n − 1, i ∈ I , and μi that strongly believes S −i ,q q=0 , we have r i (μi ) ⊆ S i ,m . Fix μi that Proof of Lemma 3. Define



n−1

strongly believes S −i ,q q=0 . I show that r i (μi ) ⊆ S i ,n . By definition of S −i ,n−1 , I can construct that

μi that satisfies BR2.(ii) such

∀h ∈ H ( S n−1 ) , ∀ z ∈ ζ ( S n−1 ) , μi ( S −i ( z)|h) = μi ( S −i ( z)|h).

(3)

For each si ∈ r i (μi ), there is a realization-equivalent si that satisfies BR2.(i), so that si ∈ R iB,n ⊆ R iB,n−1 . Then,







B ζ r i (μi ) × R − ⊆ ζ R nB−1 . By definition of S n−1 , ζ ( S n−1 ) = ζ R nB−1 . Thus, (i) ζ r i (μi ) × S −i ,n−1 ⊆ ζ ( S n−1 ). For i ,n−1

each si ∈ r i (μi ), by the induction hypothesis we have si ∈ S i ,n−1 . Thus, (ii) ζ r i (μi ) × S −i ,n−1 ⊆ ζ ( S n−1 ). Together with (3), (i) and (ii) imply that for each si ∈ r i (μi ), there is si ∈ r i (μi ) such that si (h) = si (h) for all h ∈ H ( S n−1 ). As observed, there is si ∈ R iB,n realization-equivalent to si , thus si (h) = si (h) = si (h) for all h ∈ H ( S n−1 ) ∩ H (si ). Hence, by definition of S i ,n , we have si ∈ S i ,n , as desired. 2 8.2. Proof of the workhorse lemma h

The proof of Lemma 1 is by induction. Let Z := ζ ( S ∞ ). The induction hypothesis claims that S hi,n contains strategies h

that imitate those in S i ,∞ along the Z paths. This implies the desired outcome inclusion, and it also allows to construct beliefs for step n + 1 that satisfy (1): for each shi ∈ S i ,∞ , for each  h ∈ H ( S ∞ ), one can substitute in the support of λhi (shi )(·| h) h

h

h

the strategies in S −i ,∞ with their imitations in S h−i ,n , and complete the new μhi as to strongly believe ( S h−i ,q )nq=0 . By A0, r ih (μhi ) ⊆ S hi,n+1 . If player i has no incentive to move out of Z under μhi , there is a sequential best reply to it that imitates shi along Z : at any  h ∈ H ( S ∞ ), with all strategies si ∈ S i ( h) such that ζ ({si } × S −i ,∞ ( h)) ⊆ Z , μhi (·| h) and λhi (shi )(·| h) induce the same outcome distribution, hence justify the same moves if the other strategies are suboptimal. But there is no guarantee of this. Before tackling this fundamental issue, I formalize the induction hypothesis. It will come in handy to formalize it in terms also of the beliefs that mimic the λhi (shi )’s along Z , beside the strategies that imitate the shi ’s. To shorten the formulation of these concepts, I introduce some additional notation. h

h





For any  h h, (shj ) j ∈ I ∈ S h , (shj ) j ∈ I ∈ S h ,

 h







μhi ∈  H ( S h−i ), μhi ∈  H ( S h−i ),  Z ⊆ Z h , and J ⊆ I , let: h

 



• shJ = Z shJ if for each z ∈  Z and  h with  h  h ≺ z, shJ ( h) = shJ ( h );    h h h h Z  • μi = μi if for each z ∈  Z and  h with  h  h ≺ z, μi ( S −i ( z)| h) = μhi ( S h−i ( z)| h );  



 h





 h



• shJ =h shJ and μhi =h μhi if, respectively, shJ = Z shJ and μhi = Z μhi . h

h

ih : S i ,∞ → S hi such that for each λhi : S i ,∞ →  H ( S h−i ) and ς Induction Hypothesis (n): For each i ∈ I , there exist maps  h shi ∈ S i ,∞ : h

−1 Z h h h h h h λhi (shi ) strongly believes ( S h−i ,q )nq= IH1.  0 , and λi (s i ) = λi (s i ) (i.e., λi (s i ) satisfies (1)); ih (shi ) = Z shi and ς ih (shi ) ∈ r ih ( ih (shi ) ∈ S hi,n )). IH2. ς λhi (shi )) (so, by A0, ς

ih (·) the identity map. λhi (·) = λhi (·) and ς Basis step (1): For all i ∈ I , the Induction Hypothesis holds with 

h λhi that satisfies IH1 at step n + 1 by doing, for each shi ∈ S i ,∞ As anticipated, it is always possible to construct a map 

hj constructed at step n. The problem could and at each  h ∈ H ( S ∞ ), the pushforward of λhi (shi )(·| h) through the map × j =i ς h

h

be that, for some l ∈ I and some slh ∈ S l,∞ , every slh

along Z .

μlh that satisfies IH1 at step n + 1 does not justify a strategy that imitates

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Negation of the induction hypothesis at step n + 1: h

NIH. there exist l ∈ I and slh ∈ S l,∞ such that for every with

μlh = Z λlh (slh ) that strongly believes ( S h−l,q )nq=0 , there is no slh ∈ rlh (μlh )

slh = Z slh .19

I am going to claim that, if this was the case, there would be a unilateral deviation by player l out of Z with the following property: every belief over the reactions of the opponents consistent with step n is induced by a CPS that mimics λlh (slh ) along Z and incentivizes player l to do that particular deviation. From here, I will eventually arrive to the conclusion h

that some reactions that justify the deviation should have survived throughout (( S i ,q )i ∈ I )q∞=0 as well, a contradiction. Additional notation is needed. Let

D l := { h ∈ H ( S −l,∞ )\ H ( S ∞ ) : p ( h) ∈ H ( S ∞ )} h

h

h

be the set of histories that immediately follow a unilateral deviation by player l from the paths. For every  h ∈ D l and  h



h m ∈ N , call M m (resp., M m ) the set of all





 m μlh that strongly believe ( S h−l,q ( h)| h)m q=0 (resp., ( S −l,q (h )|h )q=0 ) with the following h



lh that strongly believes ( S h−l,q ( lh property: There exists μ h)| h)nq=0 such that the initial expected payoff of player l under μ   lh (·| (i.e., the expected payoff under μ h)) is not higher than under μlh .20 Claim 1. There exists  h ∈ D l such that: 





 h









C1. for every m ≤ n and

h h h h h h μlh ∈ Mm , there exists μlh = Z λlh (slh ) that strongly believes ( S h−l,q )m q=0 such that μl = μl and rl (μl ) ∩

C2. for every p ∈ N and

μlh ∈ M p , there exists μlh = Z λlh (slh ) that strongly believes ( S −l,q )qp=0 such that μlh =h μlh and rlh (μlh ) ∩

S lh ( h) = ∅;

S lh ( h) = ∅.21

h

C1 with m = n is the result described in words after NIH. C1 also extends the claim to the previous steps of reasoning, focusing on the beliefs about the reactions that are at least as optimistic as those of step n + 1. C2 extends the claim to the other procedure for all steps of reasoning. It will become clear later why these additional claims are needed. The proof of Claim 1, deferred to the end of this appendix, is by contraposition. I illustrate it for C1 with m = n; it can be easily extended to all other cases. Suppose that for every  h ∈ D l , there is a belief over S h−l,n ( h)| h for which there is no CPS in (h) that (i) induces such belief after  h, (ii) strongly believes ( S h−l,q )nq=0 , (iii) mimics λlh (slh ) along Z , and (iv) incentivizes player l to deviate towards  h.22 But all such beliefs (one for each  h ∈ D l ) can be induced by the same CPS μlh that strongly

believes ( S h−l,q )nq=0 and mimics λlh (slh ) along Z , thus satisfying (i)-(ii)-(iii). So, μlh violates (iv) for every  h ∈ D l . But then, under μlh player l has no incentive to do any deviation from Z , and so there is a sequential best reply to μlh that imitates slh along Z , contradicting NIH. The proof that such

hypothesis, for any player i = l, the strategies in

μlh can be constructed is based on the following idea. By the induction

S hi,n

h

that imitate those in S i ,∞ along Z are sequential best replies to CPS’s that assign probability 0 to deviations by the opponents from Z until they happen. Being surprised by each deviation, player i must come up with a new belief afterwards. For each  h ∈ D l , these new beliefs can justify the strategies in S h−l,n ( h)| h that h

player l has to believe in. Thus, there are strategies in S h−l,n that imitate those in S −l,∞ along Z and react to player l’s deviation in any way that is consistent with step n. These strategies support the required combination of beliefs. Now it is time to use the negation of the induction hypothesis and Claim 1 to arrive to a contradiction and thus prove the lemma. C1 with m = n and A0 imply that S lh,n+1 ( h)| h (and thus S lh,n ( h)| h) contains all the sequential best replies to all 

the beliefs in M nh , i.e., to all



μlh that strongly believe ( S h−l,q ( h)| h)nq=0 . The same holds for S hi,n ( h)| h with i = l, because by

the induction hypothesis player i can allow  h while assigning probability 0 to reaching it until it is actually reached, and then she can come up with any new belief and best reply to it. So, for each player j, S hj,n ( h)| h contains all the sequential

19 20 21

μlh = Z λlh (slh ) that strongly believes ( S h−l,q )nq=0 is assumed to exist yet. h  l strongly believes ( S h−l,q ( Note: μ h)| h)nq=0 also when μlh strongly believes ( S −l,q ( h)| h)m q=0 . Note that, to be rigorous, no  h



h h The statement must hold vacuously for some p ∈ N (i.e., M p = ∅): since  h∈ / H ( S l,∞ ), there cannot be

h

μlh = Z λlh (slh ) that strongly believes ( S −l,q )q∞=0

such that ρl (μlh ) ∩ S lh ( h) = ∅ (recall that ρl (μlh ) ⊂ by A0). 22 In presence of probabilistic beliefs along the paths, player l can be unsure as to which h ∈ D l will realize after the deviation, but this is immaterial for the argument. h S l,∞

E. Catonini / Games and Economic Behavior 120 (2020) 209–224

221

best replies to CPS’s that strongly believe ( S h− j ,q ( h)| h)nq=0 . Then, we can refine S nh ( h)| h with an iterated deletion of never sequential best replies (from n onwards) and obtain an elimination procedure with non-empty output. Consider now the projection of (( S i ,q )i ∈ I )q∞=0 in ( h). By Remark 1 it is an elimination procedure as well. If we conclude that it yields a non-empty output, we contradict that  h is a deviation from Z . How to do that? By showing two things. First, that the lemma holds in games of smaller length. This can be assumed by induction, because the lemma obviously holds in static games. (The formal argument will be that  h cannot exist in static games.) Second, the two elimination procedures in ( h ), with inverted roles with respect to those in (h) that originated them, satisfy A0. Here is where the rest of Claim 1 kicks in. The whole argument is now formalized. h

h

Proof that the negation of the induction hypothesis leads to contradiction. If (h) is a static game, D l ( S ∞ ) = ∅, so the existence of  h by Claim 1 is already a contradiction. This allows to assume by way of induction that Lemma 1 holds in games of smaller length than (h), thus in ( h ).  h

 h

 h



Define (( S i ,q )i ∈ I )q≥0 as follows: for every i ∈ I and m ≤ n, S i ,m = S hi,m ( h)| h; for every m > n, shi ∈ S i ,m if and only if  h







 h



−1 h h h there exists μhi that strongly believes ( S −i ,q )m q=0 such that s i ∈ r i (μi ). I want to show that (( S i ,q )i ∈ I )q≥0 is an elimination procedure with non-empty output, and that it satisfies A0. h h h For every i = l, since  h ∈ D l ( S ∞ ), S i ,∞ ( h) = ∅. So, fix shi ∈ S i ,∞ ( h). For every m ≤ n, the Induction Hypothesis provides h h m − 1 λhi (si ) that strongly believes ( S h−i ,q )q=0 . Note that  λhi (shi ) = Z λhi (shi ) implies  λhi (shi )( S h−i ( ςih (si ) ∈ S hi,m ( h) and  h)| p ( h)) = 0.

 h







−1 m−1 h h h h μhi that strongly believes ( S −i,q )m q=0 , I can construct μi = μi that strongly believes ( S −i ,q )q=0 such that h h h  h h h h h h 23 Z μi (·|h) =  λi (si )(·| h) for all  h  h. Thus, r i (μi ) ∩ S i ( h) = ∅, and by μi = λi (si ) and A0 (referred to (( S hj,q ) j ∈ I )q≥0 ),

Hence, for every



 h



r ih (μhi ) ⊆ S hi,m . So, r ih (μhi ) ⊆ S i ,m . Fix

 h







μlh that strongly believes ( S −l,q )nq=0 ; trivially, μlh ∈ Mnh . Hence, by C1, there exists μlh = Z λlh (slh ) that strongly believes 



h h     ( S h−l,n )nq=0 such that μlh =h μlh and rlh (μlh ) ∩ S lh ( h) = ∅. By A0, rlh (μlh ) ⊆ S lh,n . So, rlh (μlh ) ⊆ S l,n+1 ⊆ S l,n .

Then, for every i ∈ I and  h

 h







 h

 h

 h

μhi that strongly believes ( S −i,q )nq=0 ,24 rih (μhi ) ⊆ S i,n = ∅. So, S i,n ⊇ S i,n+1 = ∅. Therefore,  h

(( S i ,q )i ∈ I )q≥0 is an elimination procedure with S ∞ = ∅. Fix m ≤ n, 

 h



 h



 h



 h



 h



−1 h ζ (S∞) μlh that strongly believes ( S −l,q )q∞=0 , and μlh =ζ ( S ∞ ) μlh that strongly believes ( S −l,q )m q=0 . Since (i) μl =  h





 h



μlh , (ii) μlh strongly believes S −l,∞ , and (iii) rlh (μlh ) × S −l,∞ ⊆ S ∞ , player l initially expects a non lower payoff under μlh than under



 h



−1 that strongly believes ( S h−l,q )m q=0 such that

Then, for every m ∈ N , i ∈ I , 







h μlh . So, since μlh strongly believes ( S −l,q )nq=0 = ( S h−l,q ( h)| h)nq=0 , μlh ∈ M m . Thus, by C1 there exists μlh = Z λlh (slh )

 h

 h







 h



μlh =h μlh and rlh (μlh ) ∩ S lh ( h) = ∅. By A0, rlh (μlh ) ⊆ S lh,m . So rlh (μlh ) ⊆ S l,m .  h





 h

 h



−1 μhi that strongly believes ( S −i,q )q∞=0 and μhi =ζ ( S ∞ ) μhi that strongly believes ( S −i,q )m q=0 ,

r ih (μhi ) ⊆ S i ,m . Thus, (( S i ,q )i ∈ I )q≥0 satisfies A0. 

Define now (( S hi,q )i ∈ I )q≥0 as (( S i ,q ( h)| h)i ∈ I )q≥0 . By Remark 1 it is an elimination procedure. I want to show that it satisfies A0. h  Fix i = l and m ∈ N . Since λhi (shi ) strongly believes S −i ,∞ , λhi (shi )( S h−i ( h)| p ( h)) = 0. Hence, for every μhi that strongly h



−1 believes ( S h−i ,q )m q=0 , I can construct

Thus, r ih (μhi ) ∩ S hi ( h) = ∅, and by





−1 h  h h    μhi =h μhi that strongly believes ( S −i,q )m q=0 such that μi (·|h ) = λi (s i )(·|h ) for all h  h. h

μhi = Z λhi (shi ) and A0 (referred to

h (( S j ,q ) j ∈ I )q≥0 ),

 h   h  For every m ∈ N , μlh that strongly believes ( S −l,q )q∞=0 , and μlh =ζ ( S ∞ )  h h Z h h

the same argument used for M m . Thus, by C2 there exists and rlh (μlh ) ∩ S lh ( h) = ∅. By A0, rlh (μlh ) ⊆ Then, for every m ∈ N , i ∈ I ,  h

 h

r i (μi ) ⊆

 S hi,m . Thus,

h S l,m .















 h

−1 h μlh that strongly believes ( S h−l,q )m q=0 ,μl ∈ M m by 

h



−1 h h h μl = λl (sl ) that strongly believes ( S −l,q )m q=0 such that μl = μl 

So, rlh (μlh ) ⊆ S lh,m .





h

r ih (μhi ) ⊆ S i ,m . So, r ih (μhi ) ⊆ S hi,m .

 h



 h





−1 μhi that strongly believes ( S −i,q )q∞=0 and μhi =ζ ( S ∞ ) μhi that strongly believes ( S h−i,q )m q=0 ,

 (( S hi,q )i ∈ I )q≥0 satisfies A0.



 h

As assumed by induction on the length of the game, Lemma 1 holds in ( h ). Hence, ζ ( S h∞ ) ⊇ ζ ( S ∞ ) = ∅. But this

contradicts  h ∈ D l ( S ∞ ). h

2

23

The construction is shown explicitly by Lemma 4 in the Appendix.

24

−1 For i = l, observe that strong belief in ( S −i ,q )nq=0 trivially implies strong belief in ( S −i ,q )nq= 0 , the condition used above.

 h

 h

222

E. Catonini / Games and Economic Behavior 120 (2020) 209–224

8.2.1. Proof of Claim 1 The main challenges for the proof of Claim 1 derive from the following issue. Fix an elimination procedure (( S hi,q )i ∈ I )q≥0 and a player i ∈ I . Consider two sequential best replies  shi , shi to two different CPS’s that strongly believe S h−i ,n , ..., S h−i ,0 . Fix two unordered histories  h, h ∈ H ( shi ) ∩ H (shi ) (that is, h   h and  h  h). Is there always a sequential best reply shi ∈ S i ( h) ∩ S i (h) to a CPS that strongly believes S h−i ,n , ..., S h−i ,0 such that shi | h = shi | h and shi |h = shi |h? No. The reason is the

following: Player i may allow  h and h either because she is confident that  h will be reached and she has optimistic beliefs after  h, or because she is confident that h will be reached and she has optimistic beliefs after h. If  shi is optimal under the first conjecture and shi is optimal under the second conjecture,  shi |h and shi | h may be “emergency plans” for unforeseen

contingencies, where the beliefs do not justify the choice to allow h and  h in the first place. This can be seen already from the set of justifiable strategies of a player. The following is a simplified version of the game of Figure 4 in Battigalli (1997), provided by Gul and Reny. The payoffs are of player 1.

← out − 1 ↓ in ← l− 2 −r →

2 1 ← a− 1

↓b

b

1 −a → 1

↓

3 −c  → 0 d ↓ 3

0 ← c− 3

↓d 3

Player 1 will rationally plan in.a.b if she first expects r and d , and then c once she gets surprised by l. Similarly, player 1 can rationally plan in.b.a . However, player 1 cannot rationally plan in.a.a : the best payoff she can get is lower than the outside option. Actions a and a are emergency plans for unforeseen contingencies, and best respond to beliefs that do not justify playing in in the first place. But in the proof of Claim 1, I will combine a player’s behavior along a set of expected paths with her reactions to opponents’ deviations from those paths. Differently from the example above, all these unforeseen contingencies are allowed by our player under the same rational plan of following those paths. Then, the combination is always possible. To begin, the first lemma formalizes the basic intuition that, after being surprised, a player can come up with any new belief, and thus can combine any possible reaction to the surprise with any plan she had if the surprise had not taken place. −1 Lemma 4. Fix an elimination procedure (( S hi,q )i ∈ I )q≥0 , n ∈ N , i ∈ I ,  h ∈ H h , and μhi that strongly believes ( S h−i ,q )nq= 0 such that     n−1 h h  h h h h  h h h h h    μi ( S −i (h)| p (h)) = 0. Fix si ∈ ri (μi ) ∩ S i (h), μi that strongly believes ( S −i,q (h)|h)q=0 , and si ∈ ri (μi ).  



Consider the unique  shi =h shi such that for every  h∈ / Hh, shi ( h) = shi ( h ).   h i that strongly believes h h r i (i ) ∩ S hi ( h) = ∅).

hi =h There exists μ

S hi ( h)

= ∅ implies

μ μ

−1 h   / Hh , and  h  ( S h−i ,q )nq= shi ∈ r ih ( μhi ) (so, rih (μhi ) ∩ 0 such that μi (·|h ) = μi (·|h ) for all h ∈









 



ς : S h−i → S h−i such that for each sh−i ∈ S h−i , ς (sh−i ) =h sh−i and ς (sh−i ) ∈ S h−i,m ( h) for all m ≥ 0 with  h h h hi = ( s−i ∈ S −i ,m ( h)| h. Since ς is injective, I can construct an array of probability measures μ μhi (·| h)) h∈ H h on S −i as        hi (·| hi (ς (sh−i )| μ h) = μhi (·| h) for all  h∈ / H h and μ h) = μhi (sh−i | h) for all  h ∈ H h and sh−i ∈ S h−i . From the definition of ς , it   −1 h h h i ( S −i ( i strongly believes ( S −i ,q )nq= h h h immediately follows that μ h)| h) = 1 for all  h ∈ H h , that μ 0 , and that μi = μi . Finally, hi ( S −i ( hi satisfies the chain rule. Thus, μ hi is a CPS. since μ h)| p ( h)) = 0, μ   h h h h  h h     Fix h ∈ H ( si )\ H = H (si )\ H . If h ≺ h, by μi ( S −i (h)| p (h)) = 0 we have μhi ( S h−i ( h)| h) = 0, and for every sh−i ∈ / S h−i ( h ),

Proof. Fix a map

ζ (shi , sh−i ) = ζ ( shi , sh−i ). If  h ⊀ h, for every sh−i ∈ S h−i ( h ),  h∈ / H (shi , sh−i ), so ζ (shi , sh−i ) = ζ ( shi , sh−i ). Hence, in both cases, shi ∈         h (ς (sh )| r i (μh ,  h) implies  sh ∈ r i (μh ,  h) = r i ( μh , h). Fix  h ∈ H ( sh ) ∩ H h = H (sh ). For every sh ∈ S h , μ h) = μh (sh | h). For i

i

i

i



 

i



i



−i



−i

i

−i

i

−i

every  shi ∈ S hi ( h), letting  shi := shi | h, we have ζ ( shi , sh−i ) = ζ ( shi , ς (sh−i )). So,  shi | h = shi ∈ r i (μhi ,  h) implies  shi ∈ r i ( μhi , h). Thus,  shi

hi . is a sequential best reply to μ

2

The same combination argument is now formalized from the point of view of the deviator and her beliefs, for different deviations from the same set of paths. I will refer directly to the context of Claim 1. h Lemma 5. Let (( S hi,q )i ∈ I )q≥0 denote (( S hi,q )i ∈ I )q≥0 (resp., (( S i ,q )i ∈ I )q≥0 ). Fix m ≤ n (resp., m ∈ N ) and  D ⊆ D l . For every  h∈ D, fix  h h m    l that strongly believes ( S −l,q (h)|h)q=0 . μ  h





  lh = Z λlh (slh ) (resp., μ lh = Z \∪h∈D Z λlh (slh )) that strongly believes ( h h h There exists μ S h−l,q )m q=0 such that μl = μl for all h ∈ D.

E. Catonini / Games and Economic Behavior 120 (2020) 209–224 h

Proof. I am going to show that for each i = l and shi ∈ S i ,∞ , and for each map  h

223

 ς : h∈ D → shi ∈  S hi,m ( h)| h, there exists

  shi ∈  S hi,m such that  shi = Z shi (resp.,  shi = Z \∪h∈D Z shi ) and  shi =h ς ( h) for all  h∈ D.25 Redistributing probability from each shi

lh . to the corresponding  shi ’s in the supports of λlh (slh ), it is easy to obtain the desired μ ih (shi ) (resp., λhi (shi ) and shi ) as λhi (shi ) and ς Relabel 

Fix  h∈ D ∩ H (shi ). Since

−1 h h h μhi and shi .26 Thus, μhi strongly believes ( S h−i ,q )m q=0 , and s i ∈ r i (μi ).

μhi = Z λhi (shi ) and λhi (shi ) strongly believes S −i,∞ , μhi ( S h−i ( h)| p ( h)) = 0. Since ς ( h) ∈  S hi,m ( h)| h, there    m−1 27 h h h h     exists μi that strongly believes ( S −i ,q (h)|h)q=0 ( ) such that ς (h) ∈ r i (μi ). Hence, by Lemma 4, there exist: 

hi =h i) μ

h





−1 h   / Hh; h  μhi that strongly believes ( S h−i ,q )m q=0 with μi (·|h ) = μi (·|h ) for all h ∈ 

ii)  shi ∈ r ih ( μhi ) such that  shi =h



ς ( h) and  shi ( h) = shi ( h) for all  h∈ / Hh.

Iterating for each  h∈ D, we obtain:

hi = Z \∪h∈D Z i) μ

 h





−1   h h h μhi that strongly believes ( S h−i ,q )m q=0 with μi = μi for all h ∈ D;  h



shi ∈ r ih ( μhi ) such that  shi = Z \∪h∈D Z shi and  shi =h ii)  shi ∈  S hi,m . By A0, 

ς ( h) for all  h∈ D.

2

Now the proof of Claim 1 can be formalized. Proof of Claim 1. Suppose by contraposition that there is a partition ( D , D ) of D l such that for every  h ∈ D, there exist m( h) ≤ n and







 h

h μlh ∈ Mm that violate C1, and for every  h ∈ D there exist m( h) ∈ N and μlh ∈ M m( h) that violate C2. (Note: ( h)

 lh that strongly believes ( S h−l,q ( D or D may be empty.) For each  h ∈ D l , fix μ h)| h)nq=0 under which player l expects a non  h h h h h Z l = μlh that strongly believes ( S h−l,q )nq=0 such higher payoff than under μl . Let μl := λl (sl ). By Lemma 5, there exists μ 



lh =h μ lh . I want to show that there exists slh ∈ rlh ( that for every  h ∈ Dl , μ μlh ) such that slh = Z slh , contradicting NIH.  h

l with Fix  h ∈ D. Substitute μ









(h) lh and obtain a new μlh =h μlh that strongly believes ( S h−l,q )m μlh in the construction of μ q=0 





lh ( S −l (z)| μlh ( S −l (z)| h) = μ h) for all  h∈ / H h and z ∈ / Z h . Since player l expects a non lower payoff against μlh than against l , rlh (μlh ) ∩ S lh ( μ h) = ∅ (which holds by the contrapositive hypothesis) implies rlh ( μlh ) ∩ S lh ( h) = ∅. So, H (rlh ( μlh )) ∩ D = ∅. with  h

Write D = {h1 , ..., hk } where m(h1 ) ≥ ... ≥ m(hk ). By Lemma 5, for each j = 1, ..., k, there exists t m(h j ) strongly believes ( S −l,q )q=0 such that lh, j =h shown that rlh ( lh, j −1 ) = rlh ( lh ). Then, rlh ( lh, j −1 ) j j well. For all  h∈ / H h and z ∈ / Z h , lh, j ( S −l (z)| h) =

h

μ

μ

μ

μ

μ

So, we have:

μlh, j = Z

h

j

\∪t =1 Z h

t

μlh that

t

μlh for all 1 ≤ t ≤ j. Let μlh,0 := μlh . Fix j = 1, ..., k and suppose to have ∩ S lh (h j ) = ∅. By the contrapositive hypothesis, rlh (μlh, j ) ∩ S lh (h j ) = ∅ as

μlh, j−1 ( S −l (z)| h). Then, rlh (μlh, j ) = rlh (μlh, j −1 ). Inductively, rlh (μlh,k ) = rlh (μlh ).

i) rlh (μlh,k ) ∩ D l = ∅; ii) slh ∈ rlh (μlh,k ); iii) H (rlh ( μlh )) ∩ D = ∅;

iv) for each  h ∈ D, player l initially expects a non lower payoff under v)

  h =h lh ; l,k lh = Z lh = Z

μ

μ

μ

μ









lh , and recall that μ lh =h μ lh and μlh than under μ

h

μlh,k , and μlh strongly believes S −l,∞ .

lh and μlh,k she expects the same By (v), we obtain the following two facts: (a) if player l deviates out of Z , under μ distribution over histories in D l and terminal histories that immediately follow the deviation; (b) if she does not deviate lh and μlh,k she expects the same outcome distribution. By (i), we have that (c) player l has no incentive out of Z , under μ to deviate out of Z under

25 26 27

μlh,k . By (iv) and (a), deviations that can only lead to histories in D (or terminal histories) bring

For (( S hi,q )i ∈ I )q≥0 , the map ς is well defined because by the induction hypothesis of the proof of Lemma 1, S hi,m ( h) = ∅ for all  h ∈ Dl .  ih (shi ) are taken from the induction hypothesis of the proof of Lemma 1. λhi (shi ) and ς h Here is where the convention that every CPS strongly believes the empty set comes in handy:  S h−i ,q ( h) can be empty (S −i ,q ( h) certainly is for sufficiently

high q, because S l,∞ ( h) = ∅). h

224

E. Catonini / Games and Economic Behavior 120 (2020) 209–224

lh than under a lower expected payoff under μ

lh as well. μlh,k . Hence, by (b) and (c), such deviations are suboptimal under μ

lh ,

μ deviations that may lead to a history in D are suboptimal too. Hence, (d) player l has no incentive to h lh as well. By (b), (c), and (d), we have that for each  deviate out of Z under μ h ∈ H ( S ∞ ),  rl ( μlh , h) = rl (μlh,k ,  h). Then, by (ii), By (iii), under

there exists slh ∈ rlh ( μlh ) such that slh ( h) = slh ( h) for all  h ∈ H ( S ∞ ). h

2

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