(i)There exists sequences a"- a and en-0 such that, for each n, an is an E-perfect equilibrium of G i) There exists a sequence an→ a such that:(a) for every r,i∈ n and a;∈A a2(a)>0;(b) for every n,t∈ N and a∈A; such that ail(a)>0.,a∈r(am) You will recognize that(iii)is the familiar characterization of perfect equilibria. Condition (b)states formally that ai is a best reply to each an I advise you to try and reconstruct the proof of this result from your notes Pr properness In an e-perfect equilibrium, informally speaking, " right choices"are infinitely more likely than mistakes. However, mistakes can be more or less costly--some mistakes entail a larger loss of utility compared with a best reply. Hence Myerson's idea: let us assume that more costly mistakes are infinitely more likely We are led to Definition 4 Fix a game G=(N, (Ai, uiieN and E>0. An e-proper equilibrium of G a profile a such that, for all iE N,(i) for every a; E Ai, ai(ai>0, and(ii)for every pair ai,a E Ai, ui(ai, a-i)<ui(a,a_i)=ai(ai)<Eai(a?. A profile a is a proper equilibrium of G iff there exist sequences a-a and e-0 such that, for each n, a" is an e"-proper equilibrium of G Clearly, every proper equilibrium is perfect, but not vice-versa The key result about proper equilibria is stated below Proposition 0. 2 Let i be an extensive-form game and let g be its normal form. Then every proper equilibrium a of G can be extended to a sequential equilibrium ofr Again, you should try to reconstruct the proof of this result from your class notes Observe that, by construction, proper equilibria are invariant to the addition or deletion of actions which yield payoff vectors which can be duplicated by existing actions. Thus, proper equilibria of a normal-form game are also proper equilibria of its reduced normal Thus, here is the tie-in with our preceding discussion of invariance: every proper equilib rium of a reduced normal-form game g induces payoff-equivalent sequential equilibria in every extensive game having g as its(reduced) normal form. We have identified a normal-form solution concept which exhibits "nice"properties in every "extensive-form presentation"of given game. Not For those of you who are(still! )interested, let me point out that, once we start eliminating duplicate actions from a game, it comes relatively natural to think about eliminating actions(ii) There exists sequences α n → α and  n → 0 such that, for each n, α n is an -perfect equilibrium of G. (iii) There exists a sequence α n → α such that: (a) for every n, i ∈ N and ai ∈ Ai , α n i (ai) > 0; (b) for every n, i ∈ N and ai ∈ Ai such that αi(ai) > 0, ai ∈ ri(α n −i ). You will recognize that (iii) is the familiar characterization of perfect equilibria. Condition (b) states formally that αi is a best reply to each α n −i . I advise you to try and reconstruct the proof of this result from your notes. Properness In an -perfect equilibrium, informally speaking, “right choices” are infinitely more likely than mistakes. However, mistakes can be more or less costly—some mistakes entail a larger loss of utility compared with a best reply. Hence Myerson’s idea: let us assume that more costly mistakes are infinitely more likely. We are led to Definition 4 Fix a game G = (N,(Ai , ui)i∈N ) and  > 0. An -proper equilibrium of G is a profile α such that, for all i ∈ N, (i) for every ai ∈ Ai , αi(ai) > 0, and (ii) for every pair ai , a0 i ∈ Ai , ui(ai , α−i) < ui(a 0 i , α−i) ⇒ αi(ai) ≤ αi(a 0 i ). A profile α is a proper equilibrium of G iff there exist sequences α n → α and  n → 0 such that, for each n, α n is an  n -proper equilibrium of G. Clearly, every proper equilibrium is perfect, but not vice-versa. The key result about proper equilibria is stated below: Proposition 0.2 Let Γ be an extensive-form game and let G be its normal form. Then every proper equilibrium α of G can be extended to a sequential equilibrium of Γ. Again, you should try to reconstruct the proof of this result from your class notes. Observe that, by construction, proper equilibria are invariant to the addition or deletion of actions which yield payoff vectors which can be duplicated by existing actions. Thus, proper equilibria of a normal-form game are also proper equilibria of its reduced normal form. Thus, here is the tie-in with our preceding discussion of invariance: every proper equilib￾rium of a reduced normal-form game G induces payoff-equivalent sequential equilibria in every extensive game having G as its (reduced) normal form. We have identified a normal-form solution concept which exhibits “nice” properties in every “extensive-form presentation” of a given game. Not bad! For those of you who are (still!) interested, let me point out that, once we start eliminating duplicate actions from a game, it comes relatively natural to think about eliminating actions 6