For every i∈NuU{e},letA(h)={a∈A:(h,a)∈H}. Then fc:{h:c∈P(h)}→△(4) indicates the probability of each chance move, and f(h)(Ai(h))=1 for all h such that c∈P(h Note that we are not allowing for simultaneous moves at a given history. Th easily fixed, but it is not necessary because simultaneous moves can be modelled exploiting the fact that moves may be unobservable The cells of each partition Ii are called information sets Perfect recall Note that, in general, it may be the case that, for some sequences of actions h, h', both h and(h, h)belong to the same information set. This represents a situation in which a player forgets her own previous choice More generally, a player may forget that her opponents have chosen certain actions in the past Definition 2 allows for both possibilities. However, it is conventional(and expedient) to assume that players have perfect recall: they forget neither their own actions, nor their previous observations o formulate this assumption rigorously, we first define, for each player i E N, a collection of sequences(Xi(h))hex such that, for every h E X, Xi (h) includes all the actions Player has chosen and all the information sets in Ii she has visited in the partial history h Formally, let X;(0)=0; next, assuming that Xi(h) has been defined for all histories of ength e, for every a E A(h), let Xi(,a)=Xi(hyufauIi(h, a) if P(h)=P(h, a)=i; Xi(h, a)=Xi(h)ua if P(h)=it p(h, a); Xi(h, a)=Xi(h)UIi(h, a) if P(h,a)=it P(h);and ·X(h,a)=X(h)ifP(h)≠ i and f(h,a)≠i Then we say that an extensive game r has perfect recall iff, for every player i N information set Ii E Ti, and histories h, h'E Ii, Xi(h)=Xi(h ). Observe that Xi is defined at every history, but the notion of perfect recall imposes direct restrictions only at histories where Player i moves Note that, in particular, perfect recall implies that it is never the case that, for some history h and sequence of actions h, both h and(h, h) belong to the same information setFor every i ∈ N∪{c}, let Ai(h) = {a ∈ A : (h, a) ∈ H}. Then fc : {h : c ∈ P(h)} → ∆(A) indicates the probability of each chance move, and fc(h)(Ai(h)) = 1 for all h such that c ∈ P(h). Note that we are not allowing for simultaneous moves at a given history. This can be easily fixed, but it is not necessary because simultaneous moves can be modelled exploiting the fact that moves may be unobservable. The cells of each partition Ii are called information sets. Perfect Recall Note that, in general, it may be the case that, for some sequences of actions h, h0 , both h and (h, h0 ) belong to the same information set. This represents a situation in which a player forgets her own previous choices. More generally, a player may forget that her opponents have chosen certain actions in the past. Definition 2 allows for both possibilities. However, it is conventional (and expedient) to assume that players have perfect recall: they forget neither their own actions, nor their previous observations. To formulate this assumption rigorously, we first define, for each player i ∈ N, a collection of sequences (Xi(h))h∈X such that, for every h ∈ X, Xi(h) includes all the actions Player i has chosen and all the information sets in Ii she has visited in the partial history h. Formally, let Xi(∅) = ∅; next, assuming that Xi(h) has been defined for all histories of length `, for every a ∈ A(h), let: • Xi(h, a) = Xi(h) ∪ {a} ∪ Ii(h, a) if P(h) = P(h, a) = i; • Xi(h, a) = Xi(h) ∪ {a} if P(h) = i 6= P(h, a); • Xi(h, a) = Xi(h) ∪ Ii(h, a) if P(h, a) = i 6= P(h); and • Xi(h, a) = Xi(h) if P(h) 6= i and P(h, a) 6= i. Then we say that an extensive game Γ has perfect recall iff, for every player i ∈ N, information set Ii ∈ Ii , and histories h, h0 ∈ Ii , Xi(h) = Xi(h 0 ). Observe that Xi is defined at every history, but the notion of perfect recall imposes direct restrictions only at histories where Player i moves. Note that, in particular, perfect recall implies that it is never the case that, for some history h and sequence of actions h 0 , both h and (h, h0 ) belong to the same information set. 4