which is exactly what we want In any case, to further simplify things, we introduce the following shorthand notation for best replies to beliefs concentrated on a single action profile Definition 1 Fix a game G=(N, (Ai, Ti, uiieN). For every i E N, Player i's Nash best reply correspondence Pi: A-i= Ai is defined by p(a-)=r(a-)a-a∈A-a That is, Pi(a-i) is the set of Player i's best replies to beliefs concentrated on a_i. Note that this corresponds to OR'sBi (" I wish to reserve the letter B; for something else We are ready for the definition of Nash equilibrium: Definition 2 Fix a game G=(N, (Ai, Ti, lilieN). The profile of actions(ai)ieN is a Nash equilibrium of g iff a;∈p(a-1) for all i∈N The literal interpretation is clear: in a Nash equilibrium, the action of each player is best reply to the(belief concentrated on the) actions of her opponents Existence Note that the above definition applies to both finite and infinite games. Moreover, as stated, it does not guarantee existence in finite games. For example, Matching Pennies(see Lecture 1)does not have an equilibrium according to the above definition At this juncture the theory treats finite and infinite games differently. This might seem odd, and to some extent it so appears to this writer, too. However, the theory has developed as follows: for finite action sets, one employs a"trick"which(in some sense) guarantees existence in arbitrary games: for infinite games, theorists have developed conditions on the action spaces and payoff functions under which Nash equilibria exist. There is no(known) trick "which guarantees existence in arbitrary infinite games I shall focus on existence(with the "trick?")in finite games, although this requires a brief detour on infinite games and some ancillary notions. We shall only need to consider very special, well-behaved infinite games, and I will provide details only for those; however, I will indicate how the basic ideas and results extend to more general settings Upper Hemicontinuity Let us take a step back. Recall that a function f: X-Y, where(X, Tx) and(Y, Iy) are topological spaces, is continuous iff f-(V)=c: f()EV E Tx whenever V Ty 2which is exactly what we want. In any case, to further simplify things, we introduce the following shorthand notation for best replies to beliefs concentrated on a single action profile: Definition 1 Fix a game G = (N,(Ai , Ti , ui)i∈N ). For every i ∈ N, Player i’s Nash best￾reply correspondence ρi : A−i ⇒ Ai is defined by ρi(a−i) = ri(δa−i ) ∀a−i ∈ A−i That is, ρi(a−i) is the set of Player i’s best replies to beliefs concentrated on a−i . Note that this corresponds to OR’s “Bi(·)”: I wish to reserve the letter Bi for something else. We are ready for the definition of Nash equilibrium: Definition 2 Fix a game G = (N,(Ai , Ti , ui)i∈N ). The profile of actions (ai)i∈N is a Nash equilibrium of G iff ai ∈ ρi(a−i) for all i ∈ N. The literal interpretation is clear: in a Nash equilibrium, the action of each player is a best reply to the (belief concentrated on the) actions of her opponents. Existence Note that the above definition applies to both finite and infinite games. Moreover, as stated, it does not guarantee existence in finite games. For example, Matching Pennies (see Lecture 1) does not have an equilibrium according to the above definition. At this juncture, the theory treats finite and infinite games differently. This might seem odd, and to some extent it so appears to this writer, too. However, the theory has developed as follows: for finite action sets, one employs a “trick” which (in some sense) guarantees existence in arbitrary games; for infinite games, theorists have developed conditions on the action spaces and payoff functions under which Nash equilibria exist. There is no (known) “trick” which guarantees existence in arbitrary infinite games. I shall focus on existence (with the “trick”) in finite games, although this requires a brief detour on infinite games and some ancillary notions. We shall only need to consider very special, well-behaved infinite games, and I will provide details only for those; however, I will indicate how the basic ideas and results extend to more general settings. Upper Hemicontinuity Let us take a step back. Recall that a function f : X → Y , where (X, TX) and (Y, TY ) are topological spaces, is continuous iff f −1 (V ) = {x : f(x) ∈ V } ∈ TX whenever V ∈ TY . 2