By S. Nicholls
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28 1. STATIC GAMES AND SOLUTION CONCEPTS We next relate the −equilibria of close games. 35 If G is an α-approximation to G and σ is an -equilibrium of G , then σ is an ( + 2α)-equilibrium of G. Proof. For all i ∈ I and all si ∈ Si , we have ui (si , σ−i ) − ui (σ ) = ui (si , σ−i ) − ui (si , σ−i ) + ui (si , σ−i ) − ui (σ ) + ui (σ ) − ui (σ ) ≤ + 2α. ✷ The next proposition shows that we can approximate a continuous game with an essentially finite game to an arbitrary degree of accuracy. 36 For any continuous game G and any α > 0, there exists an “essentially finite" game which is an α-approximation to G.
For games with multiple Nash equilibria, one may want to allow for randomizations between Nash equilibria by some form of communication prior to the play of the game. , they randomize between two pure strategy Nash equilibria, resulting in a payoff of (3/2, 3/2). The coin flip is one way of communication prior to the play. A more general form of communication is to find a trusted mediator who can perform general randomizations, as illustrated in the next example. 19 Traffic Intersection Game Consider a game where two cars arrive at an intersection simultaneously.
All demand has to be supplied. The payoff functions of each firm is the profit they make (we assume for simplicity that cost of supplying the good is equal to 0 for both firms). (a) We first study the pure strategy Nash equilibria of this game. For this, we consider all possible candidate strategy profiles and check if there is any profitable unilateral deviation: – p1 = p2 > 0: each of the firms has an incentive to reduce their price to capture the whole demand and increase profits. – p1 < p2 : Firm 1 has an incentive to slightly increase his price.