probability measure ai E A(Ai(note the indices! ) In short, we say"a, is a best reply to ai Note: at first sight, Nash equilibrium seems to say that players' beliefs are require to be" consistent with rationality". However, note that, since the rationality of an act can only be assessed with respect to a given (utility function and)probability measure over states of the world, the definition really says that players' beliefs are required to be "consistent with rationality, given the equilibrium beliefs. We shall be more specific about this point in a subsequent lecture Also, note that this interpretation of Nash equilibrium has no behavioral content it is simply a restriction on beliefs. If we really wish to push this to the extreme we can say that, if players' belief satisfies the restrictions in the definition of a Nash quilibrium, then the players may choose an action that is a best reply to those beliefs Again, we will return to this point later in the course Zero-Sum games and the minmax solution Game theorists were originally interested in a much simpler class of games, namely two-player games in which interests were diametrically opposed Definition 3 A two-player simultaneous game 1, 2, (Ai, Lien) is zero-sum iff, for all a E A, u1(a)+u2(a)=0 war o put this in perspective, think about board games such as chess or checkers,or In the latter case, it appears that doing one's best in the worst-case scenario might be the right"thing to do. I am told this is what they teach you in military academy anyway In a zero-sum game, the worst-case scenario for Player 1 is also the best scenario for Player 2, and vice-versa. Thus, for Player 1 to assume that Player 2 is "out to get her"is not inconceivable-after all, Player 2 has all the right incentives to do so This idea leads to the notion of macmin and minmay strategies. Suppose first that both players can randomize-commit to spinning roulette wheels or something like that, so as to choose strategies according to any pre-specified probability a; E A(Ai) We will get back to this later in the course: for now, let's just assume that this is possible; the reason why this is convenient will become clear momentarily. Again using a conventional notation, let u1(a1, 02)=: 2(anaeA ui(a1, a2)ai(ar)a2(a2)for any(a1,a2)∈△(A1)×△(A2) Consider first a pair of mixed strategies a E A(Ai), i=1, 2probability measure αi ∈ ∆(Ai) (note the indices!). In short, we say “aj is a best reply to αi”. Note: at first sight, Nash equilibrium seems to say that players’ beliefs are required to be “consistent with rationality”. However, note that, since the rationality of an act can only be assessed with respect to a given (utility function and) probability measure over states of the world, the definition really says that players’ beliefs are required to be “consistent with rationality, given the equilibrium beliefs.” We shall be more specific about this point in a subsequent lecture. Also, note that this interpretation of Nash equilibrium has no behavioral content: it is simply a restriction on beliefs. If we really wish to push this to the extreme, we can say that, if players’ belief satisfies the restrictions in the definition of a Nash equilibrium, then the players may choose an action that is a best reply to those beliefs. Again, we will return to this point later in the course. Zero-Sum games and the minmax solution Game theorists were originally interested in a much simpler class of games, namely two-player games in which interests were diametrically opposed: Definition 3 A two-player simultaneous game {{1, 2},(Ai , ui)i∈N } is zero-sum iff, for all a ∈ A, u1(a) + u2(a) = 0. To put this in perspective, think about board games such as chess or checkers, or war. In the latter case, it appears that doing one’s best in the worst-case scenario might be the “right” thing to do. I am told this is what they teach you in military academy, anyway. In a zero-sum game, the worst-case scenario for Player 1 is also the best scenario for Player 2, and vice-versa. Thus, for Player 1 to assume that Player 2 is “out to get her” is not inconceivable—after all, Player 2 has all the right incentives to do so. This idea leads to the notion of maxmin and minmax strategies. Suppose first that both players can randomize—commit to spinning roulette wheels or something like that, so as to choose strategies according to any pre-specified probability αi ∈ ∆(Ai). We will get back to this later in the course: for now, let’s just assume that this is possible; the reason why this is convenient will become clear momentarily. Again using a conventional notation, let u1(α1, α2) =: P (a1,a2)∈A u1(a1, a2)α 1 1 (a1)α 1 2 (a2) for any (α1, α2) ∈ ∆(A1) × ∆(A2). Consider first a pair of mixed strategies α 1 i ∈ ∆(Ai), i = 1, 2, such that u1(α 1 1 , α1 2 ) = min α2∈∆(A2) max α1∈∆(A1) u1(α1, α2) 4