Shreyas’ Notes

Introduction to Game Theory

ECON 205

fall, junior year

Games §

Normal form of a game:

  1. The players in the game: iN={1,,n}i \in \mathbb{N} = \{1, \dots, n\}
  2. For each ii, the set SiS_i of strategies that may be played by player ii.
  3. Payoff function for each player. ui:i=1nSiRu_i : \prod_{i = 1}^{n} S_i \rightarrow \mathbb{R}.

Commodity bundles are elements of Rn\mathbb{R}^n.

Utility functions provide numerical representations of preferences.

The utility function uu represents the preference \succ when for any x,yRnx, y \in \mathbb{R}^n, it holds that u(x)>u(y)    xyu(x) > u(y) \iff x \succ y.

Utility functions represent a consumer’s ordering over commodity bundles, payoff functions represent a player’s ordering over nn-tuples of strategies.

Payoff functions have no “cardinal significance” (but they do have “ordinal significance”).

Game exmaples §

Prisoner’s Dilemma §

Symmetric game.

u1(C,Q)u1(Q,Q)u1(C,C)u1(Q,C)u_1(C, Q) \succ u_1(Q, Q) \succ u_1(C, C) \succ u_1(Q, C)

u2(Q,C)u2(Q,Q)u2(C,C)u2(C,Q)u_2(Q, C) \succ u_2(Q, Q) \succ u_2(C, C) \succ u_2(C, Q)

They do better when they both stay quiet, than when they both confess. Yet, they both have strong incentives to confess.

If sis_i a strictly dominant strategy for player ii, they will prefer sis_i over all other strategies in SiS_i.

iterated elimination of strictly dominated strategies

Battle of the Sexes §

Matching Pennies §

Stag Hunt §

A Nash Equilibrium is an n-tuple of strategies, one for each player, such that each player’s strategy is an optimal response to the other players’ strategies. “steady-state, self-enforcing contract”