《博弈论》（英文版） lect 10 Extensive Games with(Almost) Perfect Information

Eco514-Game Theory Lecture 10: Extensive Games with(Almost) Perfect Information Marciano siniscalchi October 19. 1999 Introduction Beginning with this lecture, we focus our attention on dynamic games. The majority of games of economic interest feature some dynamic component, and most often payoff uncertainty as The analysis of extensive games is challenging in several ways. At the most basic level describing the possible sequences of events(choices) which define a particular game form is not problematic per se; yet, different formal definitions have been proposed, each with its Representing the players information as the play unfolds is nontrivial: to some extent research on this topic may still be said to be in progress The focus of this course will be on solution concepts; in this area, subtle and unexpected difficulties arise, even in simple games. The very representation of players'beliefs as the play unfolds is problematic, at least in games with three or more players. There has been a fierce debate on the right" notion of rationality for extensive games, but no consensus seems to have emerged among theorists We shall investigate these issues in due course. Today we begin by analyzing a particu- larly simple class of games, characterized by a natural multistage structure. I should point out that, perhaps partly due to its simplicity, this class encompasses the vast majority of extensive games of economic interest, especially if one allows for payoff uncertainty. We shall return to this point in the next lecture Games with Perfect Information Following OR, we begin with the simplest possible extensive-form game. The basic idea is as follows: play proceeds in stages, and at each stage one(and only one) player chooses an

Eco514—Game Theory Lecture 10: Extensive Games with (Almost) Perfect Information Marciano Siniscalchi October 19, 1999 Introduction Beginning with this lecture, we focus our attention on dynamic games. The majority of games of economic interest feature some dynamic component, and most often payoff uncertainty as well. The analysis of extensive games is challenging in several ways. At the most basic level, describing the possible sequences of events (choices) which define a particular game form is not problematic per se; yet, different formal definitions have been proposed, each with its pros and cons. Representing the players’ information as the play unfolds is nontrivial: to some extent, research on this topic may still be said to be in progress. The focus of this course will be on solution concepts; in this area, subtle and unexpected difficulties arise, even in simple games. The very representation of players’ beliefs as the play unfolds is problematic, at least in games with three or more players. There has been a fierce debate on the “right” notion of rationality for extensive games, but no consensus seems to have emerged among theorists. We shall investigate these issues in due course. Today we begin by analyzing a particu￾larly simple class of games, characterized by a natural multistage structure. I should point out that, perhaps partly due to its simplicity, this class encompasses the vast majority of extensive games of economic interest, especially if one allows for payoff uncertainty. We shall return to this point in the next lecture. Games with Perfect Information Following OR, we begin with the simplest possible extensive-form game. The basic idea is as follows: play proceeds in stages, and at each stage one (and only one) player chooses an 1

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 it

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 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

2A1 2,2 0,0 Figure 1: A perfect-information game Strategies and normal form(s) Definition 1 is arguably a "natural "way of describing a dynamic game-and one that is at ast implicit in most applications of the theory According to our formulations, actions are the primitive objects of choice. However, the notion of a strategg, i.e. a history-contingent plan, is also relevant Definition 2 Fix an extensive-form game with perfect information I. For every history h E X, let A(h)=aE A:(h,a)e h be the set of actions available at h. Then, for every player i N, a strategy is a function si: P-(i)-A such that, for every h such that P(h)=i, si(h)E A(h). Denote by Si and S the set of strategies of Player i and the set of all strategy profiles Armed with this definition(to which we shall need to return momentarily) we are ready to extend the notion of Nash equilibrium to extensive games Definition 3 Fix an extensive-form game r with perfect information. The outcome function O is a map o: S-Z defined by h=(a2,……,a)∈z,C<k:a4+h=s The normal form of the game T is G=(N, (Si, ui)ieN), where ui(s)=U1(O(s) The outcome function simply traces out the history generated by a strategy profile. The normal-form payoff function ui is then derived from Ui and O in the natural way. Finally Definition 4 Fix an extensive-form game r with perfect information. A pure-strategy Nash quilibrium of r is a profile of strategies s E which constitutes a Nash equilibrium of its normal form G; a mixed-strategy Nash equilibrium of r is a Nash equilibrium of the mixed extension of G

3,3 r 1 2,2 d1 a1 r 2 D A 1,1 r 1 d2 a2 0,0 Figure 1: A perfect-information game Strategies and normal form(s) Definition 1 is arguably a “natural” way of describing a dynamic game—and one that is at least implicit in most applications of the theory. According to our formulations, actions are the primitive objects of choice. However, the notion of a strategy, i.e. a history-contingent plan, is also relevant: Definition 2 Fix an extensive-form game with perfect information Γ. For every history h ∈ X, let A(h) = {a ∈ A : (h, a) ∈ H} be the set of actions available at h. Then, for every player i ∈ N, a strategy is a function si : P −1 (i) → A such that, for every h such that P(h) = i, si(h) ∈ A(h). Denote by Si and S the set of strategies of Player i and the set of all strategy profiles. Armed with this definition (to which we shall need to return momentarily) we are ready to extend the notion of Nash equilibrium to extensive games. Definition 3 Fix an extensive-form game Γ with perfect information. The outcome function O is a map O : S → Z defined by ∀h = (a 1 , . . . , ak ) ∈ Z, ` < k : a `+1 = sP((a 1,...,a`))((a 1 , . . . , a` )) The normal form of the game Γ is GΓ = (N,(Si , ui)i∈N ), where ui(s) = Ui(O(s)). The outcome function simply traces out the history generated by a strategy profile. The normal-form payoff function ui is then derived from Ui and O in the natural way. Finally: Definition 4 Fix an extensive-form game Γ with perfect information. A pure-strategy Nash equilibrium of Γ is a profile of strategies s ∈ S which constitutes a Nash equilibrium of its normal form GΓ ; a mixed-strategy Nash equilibrium of Γ is a Nash equilibrium of the mixed extension of GΓ . 3

Thus, in the game of Figure 1, both(a1a2, A)and(d1d2, D)are Nash equilibria Observe that a strategy indicates choices even at histories which previous choices dictated y the same strategy prevent from obtaining. In the game of Figure 1, for instance, dian is a strategy of Player 1, although the history(a1, A)cannot obtain if Player 1 chooses di at 0 It stands to reason that d2 in the strategy did cannot really be a description of pla 1s action--she will never really play d2! We shall return to this point in the next lecture. For the time being, let us provisionally say that d2 in the context of the equilibrium(did2, D)represents only Player 2's beliefs about Player 1s action in the counterfactual event that she chooses an at o, and Player 2 follows it with A The key observation here is that this belief is crucial in sustaining(d1d2, D)as a Nash equilibrium Games with observable actions and chance moves The beauty of the OR notation becomes manifest once one adds the possibility that more than one player might choose an action simultaneously at a given history. The resulting game is no longer one of perfect information, because there is some degree of strategic uncertainty Yet, we maintain the assumption that histories are observable: that is, every player on the move at a history h observes all previous actions and action profiles which comprise h The Or definition is a bit vague, so let me provide a rigorous, inductive one. i als the possibility of chance moves, i.e. erogenous uncertainty Definition 5 An extensive-form game th observable actions and chance moves is a tuple T=(N, A,H, P, Z, U, fc) where N is a set of players: Chance, denoted by c, is regarded as an additional player, socE N a is a set of actions H is a set of sequences whose elements are points in Ilie, A for some ACNuic Z and x are as in Definition 1 P is the player correspondence P: X= NUCh U: Z-R as in definition H satisfies the conditions in Definition 1. Moreover, for every k>1,( )∈H implies that(a2,…,a4-1)∈ h and a∈IiP(a,ak-1)A For everyiE NU{e},letA()={an∈A:a-∈1l/eP( O\A s..(h,(a,a-)∈H Then fc:h:cE P(h)- A(A) indicates the probability of each chance move, and f∈(h)(A(h)=1 for all h such that c∈P(h) The definition is apparently complicated, but the underlying construction is rather nat ural: at each stage, we allow more than one player(including Chance)to pick an action; the

Thus, in the game of Figure 1, both (a1a2, A) and (d1d2, D) are Nash equilibria. Observe that a strategy indicates choices even at histories which previous choices dictated by the same strategy prevent from obtaining. In the game of Figure 1, for instance, d1a1 is a strategy of Player 1, although the history (a1, A) cannot obtain if Player 1 chooses d1 at ∅. It stands to reason that d2 in the strategy d1d2 cannot really be a description of Player 1’s action—she will never really play d2! We shall return to this point in the next lecture. For the time being, let us provisionally say that d2 in the context of the equilibrium (d1d2, D) represents only Player 2’s beliefs about Player 1’s action in the counterfactual event that she chooses a1 at ∅, and Player 2 follows it with A. The key observation here is that this belief is crucial in sustaining (d1d2, D) as a Nash equilibrium. Games with observable actions and chance moves The beauty of the OR notation becomes manifest once one adds the possibility that more than one player might choose an action simultaneously at a given history. The resulting game is no longer one of perfect information, because there is some degree of strategic uncertainty. Yet, we maintain the assumption that histories are observable: that is, every player on the move at a history h observes all previous actions and action profiles which comprise h. The OR definition is a bit vague, so let me provide a rigorous, inductive one. I also add the possibility of chance moves, i.e. exogenous uncertainty. Definition 5 An extensive-form game with observable actions and chance moves is a tuple Γ = (N, A, H, P, Z, U, fc) where: N is a set of players; Chance, denoted by c, is regarded as an additional player, so c 6∈ N. A is a set of actions H is a set of sequences whose elements are points in Q i∈J A for some A ⊂ N ∪ {c}; Z and X are as in Definition 1; P is the player correspondence P : X ⇒ N ∪ {c} U : Z → R N as in Definition 1; H satisfies the conditions in Definition 1. Moreover, for every k ≥ 1, (a 1 , . . . , ak ) ∈ H implies that (a 1 , . . . , ak−1 ) ∈ H and a k ∈ Q i∈P((a 1,...,ak−1)) A. For every i ∈ N ∪ {c}, let Ai(h) = {ai ∈ A : ∃a−i ∈ Q j∈P(h)\{i} A s.t. (h,(ai , a−i)) ∈ 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). The definition is apparently complicated, but the underlying construction is rather nat￾ural: at each stage, we allow more than one player (including Chance) to pick an action; the 4

chosen profile then becomes publicly observable. We quite simply replace individual actions with action profiles in the definition of a history, and adapt the notation accordingly Remark 0.1 Let A(h)=ae llieP(h) A: (h,a)EH. Then A(h)=IliEP(h) Ai(h) The definition of a strategy needs minimal modifications Definition 6 Fix an extensive-form game r with observable actions and chance moves Then, for every player i E NUc, a strategy is a function s;: h: i E P(h))-A such that, for every h such that iE P(h),si(h)E Ai(h). Denote by Si and s the set of strategies of Player i and the set of all strategy profiles In the absence of chance moves, Definition 4 applies verbatim to the new setting. You can think about how to generalize it with chance moves(we do not really wish to treat Chance as an additional player in a normal-form game, so we need to redefine the payoff functions in the natural way ). Finally, the definition of Nash equilibrium requires no change For those of you who are used to the traditional, tree-based definition of an extensive game, note that you need to use information sets in order to describe games without perfect information, but with observable actions. That is, you need to use the full expressive power of the tree-based notation in order to describe what is a slight and rather natural extension of perfect-information games Most games of economic interest are games with observable actions, albeit possibly with payoff uncertainty; hence, the OR notation is sufficient to deal with most applied problems (payoff uncertainty is easily added to the basic framework, as we shall see vr. l On the other hand, the OR notation is equivalent to the standard one for games with perfect information call histories“ 'nodes”, actions“arcs”, terminal histories "leaves”andW“root

chosen profile then becomes publicly observable. We quite simply replace individual actions with action profiles in the definition of a history, and adapt the notation accordingly. Remark 0.1 Let A(h) = {a ∈ Q i∈P(h) A : (h, a) ∈ H}. Then A(h) = Q i∈P(h) Ai(h). The definition of a strategy needs minimal modifications: Definition 6 Fix an extensive-form game Γ with observable actions and chance moves. Then, for every player i ∈ N ∪ {c}, a strategy is a function si : {h : i ∈ P(h)} → A such that, for every h such that i ∈ P(h), si(h) ∈ Ai(h). Denote by Si and S the set of strategies of Player i and the set of all strategy profiles. In the absence of chance moves, Definition 4 applies verbatim to the new setting. You can think about how to generalize it with chance moves (we do not really wish to treat Chance as an additional player in a normal-form game, so we need to redefine the payoff functions in the natural way). Finally, the definition of Nash equilibrium requires no change. For those of you who are used to the traditional, tree-based definition of an extensive game, note that you need to use information sets in order to describe games without perfect information, but with observable actions. That is, you need to use the full expressive power of the tree-based notation in order to describe what is a slight and rather natural extension of perfect-information games.1 Most games of economic interest are games with observable actions, albeit possibly with payoff uncertainty; hence, the OR notation is sufficient to deal with most applied problems (payoff uncertainty is easily added to the basic framework, as we shall see). 1On the other hand, the OR notation is equivalent to the standard one for games with perfect information: just call histories “nodes”, actions “arcs”, terminal histories “leaves” and ∅ “root”. 5