Non-binding agreements and forward induction reasoning Emiliano Catonini May 2013 Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 1 / 20
Motivation In many economic situations players can communicate before the game starts and reach a non-binding agreement on how to play. In dynamic games a violation of the agreement can be observed before the game ends: how is it interpreted? how credible is the remainder of the agreement? Or the agreement can be easily incomplete: agreement on an outcome (path agreement); not willing to discuss the hypothesis of deviation; anticipation of the fact that a violated agreement is not trusted anymore; failure to coordinate in some contingencies. Which agreements are credible? Which outcomes of the game can be implemented through some agreement? Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 2 / 20
Research strategy A non-binding agreement can a ect the behavior of players only through their beliefs. The equilibrium approach is not su cient: deviations are not rationalized. The mere extensive form rationalizability (Pearce, 1984; Battigalli and Siniscalchi, 2002) is not su cient: coordination is not captured. But merging rationalizability and coordination is possible. Under some strategic reasoning assumptions, the tool to do so already existed: strong-delta-rationalizability (Battigalli, 2003). Under a more compelling epistemic priority assumption, a new solution concept is developed and epistemically characterized in a companion paper: selective rationalizability. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 3 / 20
An imperfect Nash can be a "good" agreement P 1 np 2 M R P 1 np 2 N O M (5, 5)! A (2, 0) (1, 4) R (0, 0) (4, 4) S (0, 8) (2, 6) Agree on M and A for player 1; on M for player 2. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 4 / 20
An imperfect Nash can be a "good" agreement P 1 np 2 M R P 1 np 2 N O M (5, 5)! A (2, 0) (1, 4) R (0, 0) (4, 4) S (0, 8) (2, 6) Agree on M and A for player 1; on M for player 2. Suppose that 2 deviates. If 1 tries to rationalize the deviation via forward induction, she must conclude that: Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 4 / 20
An imperfect Nash can be a "good" agreement P 1 np 2 M R P 1 np 2 N O M (5, 5)! A (2, 0) (1, 4) R (0, 0) (4, 4) S (0, 8) (2, 6) Agree on M and A for player 1; on M for player 2. Suppose that 2 deviates. If 1 tries to rationalize the deviation via forward induction, she must conclude that: either 2 believes in the agreement but is not rational (epistemic priority to the agreement); so she could play anything and A is best reply to N; Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 4 / 20
An imperfect Nash can be a "good" agreement P 1 np 2 M R P 1 np 2 N O M (5, 5)! A (2, 0) (1, 4) R (0, 0) (4, 4) S (0, 8) (2, 6) Agree on M and A for player 1; on M for player 2. Suppose that 2 deviates. If 1 tries to rationalize the deviation via forward induction, she must conclude that: either 2 believes in the agreement but is not rational (epistemic priority to the agreement); so she could play anything and A is best reply to N; or 2 does not believe in the agreement but is rational (epistemic priority to rationality); hence she could play R.N and A is best reply to N. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 4 / 20
An imperfect Nash can be a "good" agreement P 1 np 2 M R P 1 np 2 N O M (5, 5)! A (2, 0) (1, 4) R (0, 0) (4, 4) S (0, 8) (2, 6) Agree on M and A for player 1; on M for player 2. Suppose that 2 deviates. If 1 tries to rationalize the deviation via forward induction, she must conclude that: either 2 believes in the agreement but is not rational (epistemic priority to the agreement); so she could play anything and A is best reply to N; or 2 does not believe in the agreement but is rational (epistemic priority to rationality); hence she could play R.N and A is best reply to N. Under both epistemic priority assumptions, 2 can believe that 1 will play A in case of deviation and hence she will not deviate. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 4 / 20
An imperfect Nash can be a "good" agreement P 1 np 2 M R P 1 np 2 N O M (5, 5)! A (2, 0) (1, 4) R (0, 0) (4, 4) S (0, 8) (2, 6) Agree on M and A for player 1; on M for player 2. Suppose that 2 deviates. If 1 tries to rationalize the deviation via forward induction, she must conclude that: either 2 believes in the agreement but is not rational (epistemic priority to the agreement); so she could play anything and A is best reply to N; or 2 does not believe in the agreement but is rational (epistemic priority to rationality); hence she could play R.N and A is best reply to N. Under both epistemic priority assumptions, 2 can believe that 1 will play A in case of deviation and hence she will not deviate. The complete(d) agreement on the imperfect Nash is self-enforcing (credible+complied) and this conclusion is robust to the epistemic priority assumption. This is not true for all SPE! Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 4 / 20
An agreement on a SPE outcome may be not credible Even if o -the-path punishments are not ruled out a priori, they can be ruled out via forward induction. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 5 / 20
An agreement on a SPE outcome may be not credible Even if o -the-path punishments are not ruled out a priori, they can be ruled out via forward induction. Twice repeated prisoner dilemma with a punishment action. AnB C D P C 5, 5 2, 6 0, 2 D 6, 2 3, 3 0, 2 P 2, 0 2, 0 1, 1 Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 5 / 20
An agreement on a SPE outcome may be not credible Even if o -the-path punishments are not ruled out a priori, they can be ruled out via forward induction. Twice repeated prisoner dilemma with a punishment action. AnB C D P C 5, 5 2, 6 0, 2 D 6, 2 3, 3 0, 2 P 2, 0 2, 0 1, 1 Agree on colluding in the rst stage and, if the agreement has gone through, defecting in the second. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 5 / 20
An agreement on a SPE outcome may be not credible Even if o -the-path punishments are not ruled out a priori, they can be ruled out via forward induction. Twice repeated prisoner dilemma with a punishment action. AnB C D P C 5, 5 2, 6 0, 2 D 6, 2 3, 3 0, 2 P 2, 0 2, 0 1, 1 Agree on colluding in the rst stage and, if the agreement has gone through, defecting in the second. This is not credible because by deviating in the rst stage the deviator univoquely signals the intention to defect in the second. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 5 / 20
An agreement on a SPE outcome may be not credible Even if o -the-path punishments are not ruled out a priori, they can be ruled out via forward induction. Twice repeated prisoner dilemma with a punishment action. AnB C D P C 5, 5 2, 6 0, 2 D 6, 2 3, 3 0, 2 P 2, 0 2, 0 1, 1 Agree on colluding in the rst stage and, if the agreement has gone through, defecting in the second. This is not credible because by deviating in the rst stage the deviator univoquely signals the intention to defect in the second. I show that the same applies to a "path that can be upset by a convincing deviation" (Osborne, 1990). Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 5 / 20
Notation For simplicity, I refer to nite, complete information, dynamic games with observable actions, i.e. games where information sets are singletons. Endowing each player with a "dummy action" at histories where she is not active, non-terminal histories and information sets coincide. I set of players A i set of actions potentially available to pl. i (A := i2i A i ) H A <N set of non-terminal histories, including h 0 := Z A <N set of terminal histories (outcomes, paths) u i : Z! R payo function of player i A i (h) A i set of actions of player i available at history h 2 H S i A H i set of strategies of player i (S i := j6=i S i, S := i2i S i ) S i (h) S i set of strategies of player i that are compatible with h 2 H H(S) H set of non-terminal hist. that are compatible with S S ζ(s) Z set of terminal histories induced by S S Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 6 / 20
Agreements De nition (Agreement) An agreement is a pro le of correspondences e = (e i : H A i ) i2i such that for every i 2 I and h 2 H, 6= e i (h) A i (h). De nition (Path Agreement) A path agreement on z = (ea 1,..., ea 8 t ) 2 Z is an agreement e = (e i ) i2i < ea i 1 for h = h 0 such that for every i 2 I, e i (h) = ea l+1 : i for h = (ea 1,..., ea l ), l < t A i (h) else De nition (Complete Agreement) A complete agreement is an agreement e = (e i ) i2i such that for every i 2 I and h 2 H, je i (h)j = 1. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 7 / 20
Beliefs (possibly) induced by an agreement De nition (Conditional Probability System) A CPS on (S i, (S i (h)) h2h ) is a mapping µ(j) : 2 S i (S i (h)) h2h! [0, 1] satisfying the following axioms: 1 for every C 2 (S i (h)) h2h, µ(c jc ) = 1; 2 for every C 2 (S i (h)) h2h, µ(jc ) is a probability measure on S i ; 3 for every E 2 2 S i and C, D 2 (S i (h)) h2h, if E D C, then µ(e jd)µ(djc ) = µ(e jc ). I denote the set of all CPS on (S i, (S i (h)) h2h ) by H (S i ). De nition (First-order-belief restrictions corr. to the agreement) Consider an agreement e = (e i ) i2i. For every i 2 I, let µ i = (µ i (jh)) h2h 2 e i H (S i ) if for every h 2 H: suppµ i (jh) n o s i 2 S i (h) : 8bh % h, 8j 6= i, s j (bh) 2 e j (bh). Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 8 / 20
Rationality De nition A strategy s i 2 S i is a sequential best reply to a CPS µ i 2 H (S i ) if for every h 2 H(s i ) and es i 2 S i (h), u i (ζ(s i, s i ))µ i (s i jh) u i (ζ(es i, s i ))µ i (s i jh). s i 2suppµ i (jh) s i 2suppµ i (jh) The set of sequential best replies to a CPS µ i is denoted by ρ i (µ i ). Players are rational when they play a sequential best reply to some CPS on co-players strategies. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 9 / 20
Strong-delta-rationalizability De nition (strong-delta-rationalizability) Consider the following procedure. (Step 0) For every i 2 I, let S 0 i, e = S i. (Step n > 0) For every i 2 I and s i 2 S i, let s i 2 S n exists a CPS µ i 2 e i such that: 1 s i 2 ρ i (µ i ) i, e if and only if there 2 8p = 0,..., n 1, 8h 2 H, S p T i, S e i (h) 6= ) µ i (S p i, jh) = 1 e (i.e. µ i strongly believes S p i, ); e Finally let Si, = T S n e i,. The pro les in S e are called e n0 strongly-delta-rationalizable. For e i := H (S i ), it is strong rationalizability (extensive-form rat.). Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 10 / 20
Strong-delta-rationalizability Step by step, strong-delta-rationalizability captures the following assumptions: 1 Players are rational and believe in the agreement (at every order) 2 1 holds and players believe that 1 holds as long as not contradicted by observation* 3 2 holds and...... * players believe in 1 at every information set that can be reached if 1 truly holds. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 11 / 20
Selective rationalizability De nition (selective rationalizability) Denote by (S m ) m0 the strong rationalizability procedure. Consider the following procedure. (Step 0) For every i 2 I, let S 0 i,r e = S i. (Step n>0) For every i 2 I and s i 2 S i, let s i 2 S n exists µ i 2 e i such that: 1 s i 2 ρ i (µ i ); 2 8p = 0,..., n 1, 8h 2 H, S p i,r e T S i (h) 6= =) µ i (S p i,r e jh) = 1; i,r e if and only if there 3 8q = 0,..., 8h 2 H, S q T i S i (h) 6= =) µ i (S q i jh) = 1; Finally, let Si,R = T S n e i,r. The pro les in S e R are called e n0 selectively-rationalizable. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 12 / 20
Selective rationalizability Step by step, selective rationalizability captures the following assumptions: 1 Players are rational, believe in the agreement, and hold common strong belief in rationality* 2 1 holds and players believe that 1 holds as long as not contradicted by observation 3 2 holds and...... * at every information set, players believe in rationality up to the highest order that is not contradicted by observation. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 13 / 20
Self-enforceability An agreement is credible when believing in it is compatible with the strategic reasoning hypotheses. De nition (Credibility) An agreement e = (e i ) i2i is credible under priority to rationality (resp. to the agreement) if S R e 6= (resp. S e 6= ). A credible agreement is self-enforcing when all the behavioral implications of the agreement comply with the agreement itself. De nition (Self-enforceability) An agreement e = (e i ) i2i is self-enforcing under priority to rationality (resp. to the agreement) if it is credible and for every i 2 I, h 2 H(SR ) e (resp. h 2 H(S )) and s e i 2 Si,R (h) (resp. s e i 2 Si, (h)), s e i (h) 2 e i (h). Players comply with the agreement at reached information sets and believe in the agreement at unreached information sets. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 14 / 20
Results - path agreements Theorem (Robustness to epistemic priority assumptions) A path agreement is self-enforcing/credible under priority to rationality if and only if it is self-enforcing/credible under priority to the agreement. Proposition (Strong rationalizability conditions) Take a path z 2 Z. If the corresponding path agreement is credible, then z 2 ζ(s ). If ζ(s ) = fzg, then the path agreement is self-enforcing. Proposition (Equilibrium conditions) Take a path z 2 Z. If the path agreement is self-enforcing, then there exists a SPE inducing z with probability 1. If there exists a SPE inducing z with probability 1 such that for every SPE of every subgame following a unilater deviation from z the deviator is worse o than under z, the path agreement is credible*. * credibility def. in Gossner (2012) for incomplete codes with inf. horizon. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 15 / 20
Results - complete agreements Proposition Consider a SPE s 2 S of a game with observable actions and no relevant ties. The agreement e = (e i ) i2i such that for every i 2 I and h 2 H, e i (h) = s i (h) is self-enforcing under priority to the agreement. What about self-enforceability of pure SPE under priority to rationality? Some SPE outcomes are not strongly rationalizable ) no hope for credibility. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 16 / 20
Results - complete agreements Proposition Consider a SPE s 2 S of a game with observable actions and no relevant ties. The agreement e = (e i ) i2i such that for every i 2 I and h 2 H, e i (h) = s i (h) is self-enforcing under priority to the agreement. What about self-enforceability of pure SPE under priority to rationality? Some SPE outcomes are not strongly rationalizable ) no hope for credibility. Some SPE outcomes are but even where a SPE outcome is the sole strongly rationalizable one, the inducing strongly rationalizable strategies may not be the SPE ones ) no hope for credibility. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 16 / 20
Results - complete agreements Proposition Consider a SPE s 2 S of a game with observable actions and no relevant ties. The agreement e = (e i ) i2i such that for every i 2 I and h 2 H, e i (h) = s i (h) is self-enforcing under priority to the agreement. What about self-enforceability of pure SPE under priority to rationality? Some SPE outcomes are not strongly rationalizable ) no hope for credibility. Some SPE outcomes are but even where a SPE outcome is the sole strongly rationalizable one, the inducing strongly rationalizable strategies may not be the SPE ones ) no hope for credibility. Hence, it is more appopriate to tackle the issue through the opposite perspective: which SPE outcomes can be enforced through some agreement? Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 16 / 20
Enforceability De nition (Enforceability) An outcome z 2 Z. is enforceable under priority to rationality (resp. to the agreement) if there exists an agreement e = (e i ) i2i such that ζ(s R e ) = z (resp. ζ(s e ) = z). But how? Natural attempt is to reach the corresponding path agreement. Theorem Consider an outcome z 2 Z and the corresponding path agreement e = (e i ) i2i. If z is enforced by some agreement e 0 = (e 0 i ) i2i such that for every i 2 I and h 2 H, e i (h) e 0 i (h), then e = (e i ) i2i is self-enforcing. Kind of "revelation principle" for agreements: if players want to achieve a given outcome and are not willing to put o -the-path restrictions, they cannot do any better than just declaring it. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 17 / 20
Conditions for enforceability Proposition If an outcome z 2 Z is enforceable under priority to rationality, it is enforceable also under priority to the agreement.* Proposition If an outcome z 2 Z is enforceable, then there exists an Extensive Form Best Response Set (Battigalli and Friedenberg, 2012) Q S i such that i2i ζ(q) = fzg. Proposition If an outcome z 2 Z is enforceable, then there exists a Nash equilibrium s 2 S such that ζ(s) = z. * this does not hold for self-enforceability! Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 18 / 20
Is there always an enforceable SPE outcome? Pure SPE may not exist and agreeing on the support of a SPE in behavioral strategies does not ensure credibility. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 19 / 20
Is there always an enforceable SPE outcome? Pure SPE may not exist and agreeing on the support of a SPE in behavioral strategies does not ensure credibility. A pure SPE is self-enforcing under priority to the agreement, so its outcome is enforceable. Under priority to rationality, I must rst check that the outcome is induced by some strongly rationalizable strategy pro le. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 19 / 20
Is there always an enforceable SPE outcome? Pure SPE may not exist and agreeing on the support of a SPE in behavioral strategies does not ensure credibility. A pure SPE is self-enforcing under priority to the agreement, so its outcome is enforceable. Under priority to rationality, I must rst check that the outcome is induced by some strongly rationalizable strategy pro le. This may be false for all pure SPE, but as a by-product I found the following: Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 19 / 20
Is there always an enforceable SPE outcome? Pure SPE may not exist and agreeing on the support of a SPE in behavioral strategies does not ensure credibility. A pure SPE is self-enforcing under priority to the agreement, so its outcome is enforceable. Under priority to rationality, I must rst check that the outcome is induced by some strongly rationalizable strategy pro le. This may be false for all pure SPE, but as a by-product I found the following: Theorem In every game with observable actions, there always exists a set of outcomes P ζ(s ) such that there exists a (not necessarily pure) SPE inducing P with probability 1. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 19 / 20
Is there always an enforceable SPE outcome? Pure SPE may not exist and agreeing on the support of a SPE in behavioral strategies does not ensure credibility. A pure SPE is self-enforcing under priority to the agreement, so its outcome is enforceable. Under priority to rationality, I must rst check that the outcome is induced by some strongly rationalizable strategy pro le. This may be false for all pure SPE, but as a by-product I found the following: Theorem In every game with observable actions, there always exists a set of outcomes P ζ(s ) such that there exists a (not necessarily pure) SPE inducing P with probability 1. Reconciling result: backward induction and forward induction never give disjoint predictions. Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning 05/13 19 / 20