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; theThus, 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