Eco514-Game Theory Lecture 15: Sequential equilibrium Marciano siniscalchi November 11. 1999 Introduction The theory of extensive games is built upon a key notion, that of sequential rationality, and a key insight, the centrality of off-equilibrium beliefs. The definition of sequential equilibrium brings both to the fore in a straightforward manner, and emphasizes their interrelation From subgame perfection to sequential rationality Let us begin by considering a game I with observed actions(but possibly simultaneous moves). Fix a profile s E S; according to the definition, s is a subgame-perfect equilibrium of r iff, for every history h H\Z, the continuation strategy profile shh is a Nash equilibrium of the subgame r(h) beginning at Equivalently, we can say that s is a SPE iff, for every history h E H\ Z, and every player i E N, there is no strategy si E Si that yields a higher payoff to Player i than si in the subgame r(h) Moreover, we can actually restrict this requirement to subgames starting at histories h E P-(i). Consider a history h' such that i P(h); then either Player i never moves in the subgame r(h,)(in which case si is trivially optimal), or there exists a collection of histories (h" such that (i)h' is a subhistory of every h", and (i)iE P(h) for each such h According to the profile s-i, exactly one of these histories, say h, will be reached starting from h, and by assumption silh is a best reply to s_inin r(h). Therefore, silh, is also a best reply to s-ilh in r(h,) In other words, s is a SPe iff each component strategy si is a best reply against s_i ching any ha are not generated by s, or perhaps explicitly excluded by si itself. This is precisely the notion of sequential rationalityEco514—Game Theory Lecture 15: Sequential Equilibrium Marciano Siniscalchi November 11, 1999 Introduction The theory of extensive games is built upon a key notion, that of sequential rationality, and a key insight, the centrality of off-equilibrium beliefs. The definition of sequential equilibrium brings both to the fore in a straightforward manner, and emphasizes their interrelation. From subgame perfection to sequential rationality . Let us begin by considering a game Γ with observed actions (but possibly simultaneous moves). Fix a profile s ∈ S; according to the definition, s is a subgame-perfect equilibrium of Γ iff, for every history h ∈ H \ Z, the continuation strategy profile s|h is a Nash equilibrium of the subgame Γ(h) beginning at h. Equivalently, we can say that s is a SPE iff, for every history h ∈ H \Z, and every player i ∈ N, there is no strategy s 0 i ∈ Si that yields a higher payoff to Player i than si in the subgame Γ(h). Moreover, we can actually restrict this requirement to subgames starting at histories h ∈ P −1 (i). Consider a history h 0 such that i 6∈ P(h 0 ); then either Player i never moves in the subgame Γ(h 0 ) (in which case si is trivially optimal), or there exists a collection of histories {h 00} such that (i) h 0 is a subhistory of every h 00, and (ii) i ∈ P(h 00) for each such h 00 . According to the profile s−i , exactly one of these histories, say h, will be reached starting from h 0 , and by assumption si |h is a best reply to s−i |h in Γ(h). Therefore, si |h0 is also a best reply to s−i |h0 in Γ(h 0 ). In other words, s is a SPE iff each component strategy si is a best reply against s−i conditional upon reaching any history where Player i moves—including those histories that are not generated by s, or perhaps explicitly excluded by si itself. This is precisely the notion of sequential rationality. 1