OR covers perfection and proves a key result: equilibria that are perfect in the ectensive form can be extended to sequential equilibria. I will add alternative characterizations of erects We begin by defining perturbations Definition 1 Fix a game G=(, (Ai, wiieN). A perturbation vector is a point n (na(ai))ieN, a eA, E Ra:EN A such that, for all iE N, 2o.eA ni(ai)<1. The r-perturbation of g is the game G(n=(N, (Ai, UiieN), where A={a;∈△(A):a1∈A,a(a1)≥n(a)} and each payoff function U; is the usual extension of ui from I A; to I A(Ai) That is, in the game G(n), the minimum probability with which a; E A; is played is ni(ai) Note that n-perturbations have convex and compact strategy sets, and payoff functions are continuous; therefore, every n-perturbation of g has a Nash equilibrium I use the notation A"=Lien A:, A-i=ll+i A Remark 0.1 An action profile(alieN E A" is a Nash equilibrium of G(n) iff, for every i∈ n and a;∈Al Sgr(a-)→a(a1)=mn(a) (where ri ( is the best-reply correspondence of the original game) Definition 2 Fix a game G=(N, (Ai, ui)ieN). A(mixed-action) equilibrium a of G perfect iff there exist sequences a"n-a and n'-0 such that, for each n, a" is a Nash quilibrium of G(n") The intuition is that a is perfect iff it is the limit of equilibria of perturbed games in which every action gets played with positive(but vanishing) probability We now provide two alternative characterizations, one of which is the version you are most familiar with Definition 3 Fix a game G=(N,(Ai, ui)ieN). A mixed action profile a is an E-perfect quilibrium of G iff for all i∈ N and a1∈A1,(i)a(a1)>0and(i)agra(a-)→a(a)≤E That is, in an e-equilibrium, actions that are not best replies receive"vanishingly small probability. Note that an E-perfect equilibrium need not be a Nash equilibrium I conclude with the main characterization result of this subsection Proposition 0.1 Fix a game G=(N, (Ai, uiieN). Then the following statements are (i) a is a perfect equilibrium of GOR covers perfection and proves a key result: equilibria that are perfect in the extensive form can be extended to sequential equilibria. I will add alternative characterizations of perfection. We begin by defining perturbations: Definition 1 Fix a game G = (N,(Ai , ui)i∈N ). A perturbation vector is a point η = (ηi(ai))i∈N,ai∈Ai ∈ R P i∈N |Ai| ++ such that, for all i ∈ N, P ai∈Ai ηi(ai) < 1. The η-perturbation of G is the game G(η) = (N,(A η i , Ui)i∈N ), where A η i = {αi ∈ ∆(Ai) : ∀ai ∈ Ai , αi(ai) ≥ ηi(ai)} and each payoff function Ui is the usual extension of ui from Q i Ai to Q i ∆(Ai). That is, in the game G(η), the minimum probability with which ai ∈ Ai is played is ηi(ai). Note that η-perturbations have convex and compact strategy sets, and payoff functions are continuous; therefore, every η-perturbation of G has a Nash equilibrium. I use the notation Aη = Q i∈N A η i , A η −i = Q j6=i A η j . Remark 0.1 An action profile (αi)i∈N ∈ Aη is a Nash equilibrium of G(η) iff, for every i ∈ N and ai ∈ Ai , si 6∈ ri(α−i) ⇒ αi(ai) = ηi(ai) (where ri(·) is the best-reply correspondence of the original game). Definition 2 Fix a game G = (N,(Ai , ui)i∈N ). A (mixed-action) equilibrium α of G is perfect iff there exist sequences α n → α and η n → 0 such that, for each n, α n is a Nash equilibrium of G(η n ). The intuition is that α is perfect iff it is the limit of equilibria of perturbed games in which every action gets played with positive (but vanishing) probability. We now provide two alternative characterizations, one of which is the version you are most familiar with. Definition 3 Fix a game G = (N,(Ai , ui)i∈N ). A mixed action profile α is an -perfect equilibrium of G iff for all i ∈ N and ai ∈ Ai , (i) αi(ai) > 0 and (ii) ai 6∈ ri(α−i) ⇒ αi(ai) ≤ . That is, in an -equilibrium, actions that are not best replies receive “vanishingly small” probability. Note that an -perfect equilibrium need not be a Nash equilibrium. I conclude with the main characterization result of this subsection. Proposition 0.1 Fix a game G = (N,(Ai , ui)i∈N ). Then the following statements are equivalent. (i) α is a perfect equilibrium of G. 5