action. Sequences of actions are called histories; some histories are terminal, i. e. no further actions are taken, and players receive their payoffs. moreover, at each stage every player gets to observe all previous actions Definition 1 An extensive-form game with perfect information is a tupleT=(N, A, H, P,Z,U) N is a set of players a is a set of actions h is a collection of finite and countable sequences of elements from A, such that (i)0∈H; (i)(a2,,a)∈ H implies(a1,,a)∈ h for all e<k i)Ifh=(a1,,a.,…)and(a1,……,a^)∈ h for all k≥1, then h∈H PAA . a, a)E H for all a E A. Also let X=H\Z. All infinite histories are terming nd Z is the set of terminal histories: that is,(a, .,a')E Z iff(a, .,a)EH -N is the player function, associating with each non-terminal history h E X the er P(h) on the move after history h U =(UDien: Z-R is the payoff function, associating a vector is to terminal history I differ from OR in two respects: first, I find it useful to specify the set of the definition of an extensive-form game. Second, at the expense of some(but not much!) generality, I represent preferences among terminal nodes by means of a vN-M utility function Interpreting Definition 1 A few comments on formal aspects are in order. First, actions are best thought of as move labels; what really defines the game is the set H of sequences. If one wishes, one can think of A as a product set(i.e. every player gets her own set of move labels), but this is inessential Histories encode all possible partial and complete plays of the game T. Indeed, it is precisely by spelling out what the possible plays are that we fully describe the game under consideration Thus, consider the following game: N=(1, 2; A=a1, d1, a2, d2, A, D;H=[0, (d1),(a1), (al, D), (ar hus,z={(d1),(a1,D),(a1,A,d2),(a1,A,a2)}andX={0,(a1),(a1,A),};fnll,P(0) P(a1,A))=1,P(a1)=2,andU(d4)=(2,2),U(a1,D)=(1,1),U(a1,A,d1)=(0,0) U((a1, A, a2))=(3, 3). Then T=(N, A, H, Z, P, U)is the game in Figure 1 The empty history is always an element of H, and denotes the initial point of the game Part(ii) in the definition of H says that every sub-history of a history h is itself a history in its own right. Part (ii) is a"limit"definition of infinite histories. Note that infinite histories are logically required to be terminal A key assumption is that, whenever a history h occurs, all players(in particular, Pla P(h)) get to observe itaction. Sequences of actions are called histories; some histories are terminal, i.e. no further actions are taken, and players receive their payoffs. Moreover, at each stage every player gets to observe all previous actions. Definition 1 An extensive-form game with perfect information is a tuple Γ = (N, A, H, P, Z, U) where: N is a set of players; A is a set of actions; H is a collection of finite and countable sequences of elements from A, such that: (i) ∅ ∈ H; (ii) (a 1 , . . . , ak ) ∈ H implies (a 1 , . . . , a` ) ∈ H for all ` < k; (iii) If h = (a 1 , . . . , ak , . . .) and (a 1 , . . . , ak ) ∈ H for all k ≥ 1, then h ∈ H. Z is the set of terminal histories: that is, (a 1 , . . . , ak ) ∈ Z iff (a 1 , . . . , ak ) ∈ H and (a 1 , . . . , ak , a) 6∈ H for all a ∈ A. Also let X = H \ Z. All infinite histories are terminal. P : X → N is the player function, associating with each non-terminal history h ∈ X the player P(h) on the move after history h. U = (Ui)i∈N : Z → R is the payoff function, associating a vector of payoffs to every terminal history. I differ from OR in two respects: first, I find it useful to specify the set of actions in the definition of an extensive-form game. Second, at the expense of some (but not much!) generality, I represent preferences among terminal nodes by means of a vN-M utility function. Interpreting Definition 1 A few comments on formal aspects are in order. First, actions are best thought of as move labels; what really defines the game is the set H of sequences. If one wishes, one can think of A as a product set (i.e. every player gets her own set of move labels), but this is inessential. Histories encode all possible partial and complete plays of the game Γ. Indeed, it is precisely by spelling out what the possible plays are that we fully describe the game under consideration! Thus, consider the following game: N = {1, 2}; A = {a1, d1, a2, d2, A, D}; H = {∅,(d1),(a1),(a1, D),(a1, A),(a1, A, d2),(a1, A, a2)}; thus, Z = {(d1),(a1, D),(a1, A, d2),(a1, A, a2)} and X = {∅,(a1),(a1, A), }; finally, P(∅) = P((a1, A)) = 1, P(a1) = 2, and U((d1)) = (2, 2), U((a1, D)) = (1, 1), U((a1, A, d1)) = (0, 0), U((a1, A, a2)) = (3, 3). Then Γ = (N, A, H, Z, P, U) is the game in Figure 1. The empty history is always an element of H, and denotes the initial point of the game. Part (ii) in the definition of H says that every sub-history of a history h is itself a history in its own right. Part (iii) is a “limit” definition of infinite histories. Note that infinite histories are logically required to be terminal. A key assumption is that, whenever a history h occurs, all players (in particular, Player P(h)) get to observe it. 2