Game Theory: Nash Equilibrium
Strategic interaction, payoff matrices, and the existence of mixed-strategy equilibria.
A finite normal-form game has $n$ players, each with a finite set of pure strategies, and a payoff function $u_i: \prod_j S_j \to \mathbb R$. Players are assumed rational and to know the structure of the game.
A Nash equilibrium is a strategy profile $(s_1^*, \ldots, s_n^*)$ such that no player can gain by unilaterally deviating:
$$u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \quad \text{for all } s_i.$$Pure-strategy NE need not exist (e.g., matching pennies). Nash's theorem (1950): every finite game has at least one mixed-strategy Nash equilibrium (players randomize over pure strategies).
The prisoner's dilemma:
| B: cooperate | B: defect | |
|---|---|---|
| A: cooperate | (−1, −1) | (−3, 0) |
| A: defect | (0, −3) | (−2, −2) |
Defect is dominant for both → unique NE is (Defect, Defect), even though (Coop, Coop) is Pareto-better.