Forward Induction Reasoning versus Equilibrium Reasoning Andrés Perea EPICENTER & Dept. of Quantitative Economics Maastricht University Toulouse, November 20, 2015 Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 1 / 18 Introduction In game theory, we distinguish between equilibrium concepts and rationalizability concepts. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 2 / 18 Introduction In game theory, we distinguish between equilibrium concepts and rationalizability concepts. What is the di¤erence between the two? Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 2 / 18 Introduction In game theory, we distinguish between equilibrium concepts and rationalizability concepts. What is the di¤erence between the two? Precise answers have been given by Epistemic Game Theory, since the late 80’s. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 2 / 18 Introduction Correct beliefs assumption In two-player games, the condition that separates equilibrium concepts from rationalizability concepts is the correct beliefs assumption : Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 3 / 18 Introduction Correct beliefs assumption In two-player games, the condition that separates equilibrium concepts from rationalizability concepts is the correct beliefs assumption : Player i believes that player j is correct about i’s beliefs, and Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 3 / 18 Introduction Correct beliefs assumption In two-player games, the condition that separates equilibrium concepts from rationalizability concepts is the correct beliefs assumption : Player i believes that player j is correct about i’s beliefs, and player i believes that j believes that i is correct about j’s beliefs. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 3 / 18 For two-player static games: common belief in rationality + # (induced choices) rationalizability correct beliefs assumption # (induced …rst-order beliefs) ! Nash equilibrium See Brandenburger and Dekel (1987, 1989), Tan and Werlang (1988), Aumann and Brandenburger (1995), Asheim (2006) and Perea (2007). Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 4 / 18 For two-player dynamic games: common belief in future rationality + # (induced choices) backwards dominance procedure correct beliefs assumption # (induced …rst-order beliefs) ! sequential equilibrium See Perea and Predtetchinski (2015). Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 5 / 18 For two-player games: epistemic conditions + # (induced choices or …rst-order beliefs) rationalizability concept correct beliefs assumption # (induced choices or …rst-order beliefs) ! equilibrium counterpart In this talk, I show that there is no equilibrium counterpart to common strong belief in rationality (Battigalli and Siniscalchi (2002)). Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 6 / 18 Common Strong Belief in Rationality Informal description Common strong belief in rationality (Battigalli and Siniscalchi (2002)) is a forward induction concept. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 7 / 18 Common Strong Belief in Rationality Informal description Common strong belief in rationality (Battigalli and Siniscalchi (2002)) is a forward induction concept. Forward induction reasoning: Whenever possible, try to …nd a plausible explanation for the choices your opponent made in the past. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 7 / 18 Common Strong Belief in Rationality Informal description Common strong belief in rationality (Battigalli and Siniscalchi (2002)) is a forward induction concept. Forward induction reasoning: Whenever possible, try to …nd a plausible explanation for the choices your opponent made in the past. Key condition in common strong belief in rationality: At every instance of the game, you must believe that your opponent chooses optimally given his beliefs, whenever believing so is possible. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 7 / 18 Common Strong Belief in Rationality Why it contradicts the correct beliefs assumption e c 2, 2 2, 1 0, 0 d 1, 1 1, 2 4, 0 3 a u Q g f Stage 2 1 Stage 1 Q Q b QQ Q s 3, 0 Q At stage 1, player 1 believes that player 2 will not choose g . Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 8 / 18 Common Strong Belief in Rationality Why it contradicts the correct beliefs assumption e c 2, 2 2, 1 0, 0 d 1, 1 1, 2 4, 0 3 a u Q g f Stage 2 1 Stage 1 Q Q b QQ Q s 3, 0 Q At stage 1, player 1 believes that player 2 will not choose g . At stage 2, player 2 believes that player 1 chose a because he believes that player 2 will choose g with high probability. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 8 / 18 Common Strong Belief in Rationality Why it contradicts the correct beliefs assumption e c 2, 2 2, 1 0, 0 d 1, 1 1, 2 4, 0 3 a u Q g f Stage 2 1 Stage 1 Q Q b QQ Q s 3, 0 Q At stage 1, player 1 believes that player 2 will not choose g . At stage 2, player 2 believes that player 1 chose a because he believes that player 2 will choose g with high probability. Hence, player 1 believes that player 2 is incorrect about his belief. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 8 / 18 Common Strong Belief in Rationality Why it contradicts the correct beliefs assumption e c 2, 2 2, 1 0, 0 d 1, 1 1, 2 4, 0 3 a u Q g f Stage 2 1 Stage 1 Q Q b QQ Q s 3, 0 Q At stage 1, player 1 believes that player 2 will not choose g . At stage 2, player 2 believes that player 1 chose a because he believes that player 2 will choose g with high probability. Hence, player 1 believes that player 2 is incorrect about his belief. Therefore, CSBR is inconsistent with the correct beliefs assumption. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 8 / 18 Modelling Belief Hierarchies In a belief hierarchy, player i has a belief, at each of his information sets, about the possible strategy choice of player j. First-order belief. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 9 / 18 Modelling Belief Hierarchies In a belief hierarchy, player i has a belief, at each of his information sets, about the possible strategy choice of player j. First-order belief. Also, player i has a belief, at each of his information sets, about the possible belief that j has, at each of his information sets, about i’s strategy choice. Second-order belief. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 9 / 18 Modelling Belief Hierarchies In a belief hierarchy, player i has a belief, at each of his information sets, about the possible strategy choice of player j. First-order belief. Also, player i has a belief, at each of his information sets, about the possible belief that j has, at each of his information sets, about i’s strategy choice. Second-order belief. And so on. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 9 / 18 Modelling Belief Hierarchies In a belief hierarchy, player i has a belief, at each of his information sets, about the possible strategy choice of player j. First-order belief. Also, player i has a belief, at each of his information sets, about the possible belief that j has, at each of his information sets, about i’s strategy choice. Second-order belief. And so on. How can we encode such in…nite belief hierarchies in an easy way? Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 9 / 18 Modelling Belief Hierarchies De…nition (Epistemic model) Consider a dynamic game G with two players. An epistemic model for G is a tuple M = (T1 , T2 , b1 , b2 ) where (a) Ti is a set of types for player i, (b) bi assigns to every type ti 2 Ti and every information set h 2 Hi some conditional belief bi (ti , h) 2 ∆(Sj (h) Tj ). M is belief-complete if for every conditional belief vector βi = ( βi (h))h 2H i on Sj Tj there is some ti 2 Ti with bi (ti ) = βi . Here, Sj (h) is the set of strategies for player j that are consistent with reaching information set h. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 10 / 18 Modelling Belief Hierarchies De…nition (Epistemic model) Consider a dynamic game G with two players. An epistemic model for G is a tuple M = (T1 , T2 , b1 , b2 ) where (a) Ti is a set of types for player i, (b) bi assigns to every type ti 2 Ti and every information set h 2 Hi some conditional belief bi (ti , h) 2 ∆(Sj (h) Tj ). M is belief-complete if for every conditional belief vector βi = ( βi (h))h 2H i on Sj Tj there is some ti 2 Ti with bi (ti ) = βi . Here, Sj (h) is the set of strategies for player j that are consistent with reaching information set h. For every type ti 2 Ti , we can derive an in…nite belief hierarchy. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 10 / 18 Correct Beliefs Assumption De…nition (Correct beliefs assumption) Consider a two-player dynamic game G , and an epistemic model M = (T1 , T2 , b1 , b2 ) for G . (a) Type ti 2 Ti believes that j is correct about his beliefs, if at every information set h 2 Hi , the belief bi (ti , h) only assigns positive probability to types tj 2 Tj which, at every information set h0 2 Hj , assign probability 1 to his true type ti . (b) Type ti 2 Ti satis…es the correct beliefs assumption if ti believes that j is correct about his beliefs, and if at every information set h 2 Hi , the belief bi (ti , h) only assigns positive probability to types tj that believe that i is correct about his beliefs. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 11 / 18 Common Strong Belief in Rationality Strong belief Key condition: If at information set h 2 Hi it is possible for player i to believe that j chooses rationally, player i must believe at h that j chooses rationally. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 12 / 18 Common Strong Belief in Rationality Strong belief Key condition: If at information set h 2 Hi it is possible for player i to believe that j chooses rationally, player i must believe at h that j chooses rationally. How to de…ne this condition formally? Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 12 / 18 Common Strong Belief in Rationality Strong belief Key condition: If at information set h 2 Hi it is possible for player i to believe that j chooses rationally, player i must believe at h that j chooses rationally. How to de…ne this condition formally? A strategy si is rational for a type ti if at every information set h 2 Hi where si 2 Si (h), ui (si , bi (ti , h)) ui (si0 , bi (ti , h)) for all si0 2 Si (h). Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 12 / 18 Common Strong Belief in Rationality Strong belief Key condition: If at information set h 2 Hi it is possible for player i to believe that j chooses rationally, player i must believe at h that j chooses rationally. How to de…ne this condition formally? A strategy si is rational for a type ti if at every information set h 2 Hi where si 2 Si (h), ui (si , bi (ti , h)) for all si0 2 Si (h). Consider an event E Sj ui (si0 , bi (ti , h)) Tj . Type ti strongly believes the event E if bi (ti , h)(E ) = 1 at all information sets h 2 Hi where (Sj (h) Perea (Maastricht University) Forward Induction vs. Equilibrium Tj ) \ E 6= ∅. Toulouse, November 20, 2015 12 / 18 Common Strong Belief in Rationality Formal de…nition De…nition (Common Strong Belief in Rationality) Consider a two-player dynamic game G , and a belief-complete epistemic model M = (T1 , T2 , b1 , b2 ) for G . Induction start. Let Ti0 := Ti and Ri0 := f(si , ti ) 2 Si Ti0 j si rational for ti g. Induction step. Let k 1, and suppose Tik for both players i. Then, for both players i, Tik Rik : = fti 2 Tik 1 1 and Rik 1 j ti strongly believes Rjk : = f(si , ti ) 2 Si Tik have been de…ned 1 g, and j si rational for ti g. Type ti expresses common strong belief in rationality if ti 2 Tik for all k. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 13 / 18 CSBR is Inconsistent with Correct Beliefs Assumption Theorem (CSBR is Inconsistent with Correct Beliefs Assumption) There are two-player dynamic games G such that, for every belief-complete epistemic model M = (T1 , T2 , b1 , b2 ) for G , there is no type ti 2 Ti that (a) expresses common strong belief in rationality, and (b) satis…es the correct beliefs assumption. Hence, there is no equilibrium counterpart to common strong belief in rationality. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 14 / 18 CSBR is Inconsistent with Correct Beliefs Assumption Theorem (CSBR is Inconsistent with Correct Beliefs Assumption) There are two-player dynamic games G such that, for every belief-complete epistemic model M = (T1 , T2 , b1 , b2 ) for G , there is no type ti 2 Ti that (a) expresses common strong belief in rationality, and (b) satis…es the correct beliefs assumption. Hence, there is no equilibrium counterpart to common strong belief in rationality. In the paper, I characterize the class of dynamic games for which CSBR is consistent with equilibrium reasoning, and show that this class is very small. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 14 / 18 Games with Perfect Information A dynamic game is with perfect information if at every stage only one player is active, and this player always observes all past choices by his opponent. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 15 / 18 Games with Perfect Information A dynamic game is with perfect information if at every stage only one player is active, and this player always observes all past choices by his opponent. A game with perfect information is without relevant ties if every player i, at each of his information sets h 2 Hi , is never indi¤erent between any two di¤erent outcomes that follow h. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 15 / 18 Games with Perfect Information A dynamic game is with perfect information if at every stage only one player is active, and this player always observes all past choices by his opponent. A game with perfect information is without relevant ties if every player i, at each of his information sets h 2 Hi , is never indi¤erent between any two di¤erent outcomes that follow h. A strategy si for player i is rational if there is conditional belief vector bi such that si is optimal for bi (h) at every information set h 2 Hi where si 2 Si (h). Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 15 / 18 Games with Perfect Information A dynamic game is with perfect information if at every stage only one player is active, and this player always observes all past choices by his opponent. A game with perfect information is without relevant ties if every player i, at each of his information sets h 2 Hi , is never indi¤erent between any two di¤erent outcomes that follow h. A strategy si for player i is rational if there is conditional belief vector bi such that si is optimal for bi (h) at every information set h 2 Hi where si 2 Si (h). An information set h is consistent with both players’rationality is h is reached by a strategy-pair (s1 , s2 ) where both s1 and s2 are rational. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 15 / 18 Games with Perfect Information Theorem (Games with Perfect Information) Consider a two-player dynamic game with perfect information and without relevant ties. If common strong belief in rationality is consistent with the correct beliefs assumption in this game, then the backward induction path must reach all information sets that are consistent with both players’rationality. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 16 / 18 Reny’s Game w 1 ∅ a 2 b - w h1 1 d - w h2 c f - w 2 h - h3 4, 0 g e ? ? ? ? 3, 3 2, 2 1, 1 0, 4 Backward induction path is a. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 17 / 18 Reny’s Game w 1 ∅ a 2 b - w h1 1 d - w h2 c f - w 2 h - h3 4, 0 g e ? ? ? ? 3, 3 2, 2 1, 1 0, 4 Backward induction path is a. Yet all information sets are consistent with both players’rationality. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 17 / 18 Reny’s Game w 1 ∅ a 2 b - w h1 1 d - w h2 c f - w 2 h - h3 4, 0 g e ? ? ? ? 3, 3 2, 2 1, 1 0, 4 Backward induction path is a. Yet all information sets are consistent with both players’rationality. Hence, common strong belief in rationality is not consistent with the correct beliefs assumption. Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 17 / 18 Thanks for your attention. Any questions? Perea (Maastricht University) Forward Induction vs. Equilibrium Toulouse, November 20, 2015 18 / 18