文档介绍：ECN 770, Fall 2004
Professor Lin Zhou
Lecture Note 14: The Existence of Nash Equilibrium
1. Normal Form Game and Nash Equilibrium
A normal form game G consists of ponents:
(1) A set of players: ;
(2) A strategy set Si for each player i;
(3) A payoff function for each player i when player 1 chooses , player 2 chooses ,…, and player I chooses .
Definition 1. Suppose players other than i choose strategies , a strategy by player i is called a best response to if
, for all , or
.
Hence, if player i believes that other players choose , then is (one of) the best strategies he can choose: no other strategies yield higher payoffs.
For a profile of strategies to be a Nash equilibrium, each player i’s strategy must be a best response to other players’ strategies .
Definition 2 A profile of strategies constitute a Nash equilibrium if all of the following inequalities hold,
, for all ,
……………
, for all .
(Remark When all are differentiable and each is an interior point of Si , we may use a set of first order conditions to solve for ,
, for all .
But this is not the primitive concept of a Nash equilibrium!)
Example The Cournot Duopoly Game
Two pete in the market of a homogeneous good. Both firms can produce the good at a constant unit cost of c. Each firm has to decide how many units of the good it will produce and sell. Suppose firm 1 decides to produce Q1 and fir