Games §

Normal form of a game:

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

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

• $x \succ y$ — consumer strictly prefers $x$ to $y$
• $x \prec y$ — consumer strictly prefers $y$ to $x$
• $x \sim y$ — consumer is indifferent
• $x \succsim y$ — $x$ is at least as good as $y$
• $x \precsim y$ — $y$ is at least as good as $x$

Utility functions provide numerical representations of preferences.

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

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

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

Game exmaples §

Prisoner’s Dilemma §

Symmetric game.

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

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

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

If $s_i$ a strictly dominant strategy for player $i$, they will prefer $s_i$ over all other strategies in $S_i$.

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”