Elements of a game#

Players (N): set of decision makers
Actions ( $A_i$ $A_{i}$ ): possible actions for each player $i \in N$ $i \in N$
- $A = A_1 \times A_2 \times ... \times A_N$ : combination of actions
Payoffs ( $u_i(a)$ ): reward for layer $i$ based on action $a$
Information ( $I_i$ ): what player $i$ knows about the game
Strategy ( $s_i$ ): plan of action for player $i$ based on its known information
Objective: to choose optimal strategy $S_i^*$ that maximizes payodd, given strategy of others $s_{-i}$ . $S_i^* = argmax_{s_i} u_i(s_i, s_{-i})$
Equilibrum ( $S_1^*, S_2^*, ..., S_N^*$ ): stable outcome where no player can improve payoff by changing strategy

Type of game#

players have opposing interests
involve 2 players
the sum of payoffs is a constant: $u_1(a) + u_2(a) = c$ $u_{1} (a) + u_{2} (a) = c$
- payoff sum of zero-sum game is $0$

Players choose simultaneously, with immediate outcome
example: Prisoner’s dilemma
Solution: a strategy profile where each player adopts a rational strategy.
- Pareto Optimal: No player can be made better off without making anotherplayer worse off
- Nash Equilibrium: No player can improve their payoff by unilaterally changing their strategy: $u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*)$ $u_{i} (s_{i}^{*}, s_{- i}^{*}) \geq u_{i} (s_{i}, s_{- i}^{*})$ for all $i$ $i$ and $s_i \in S_i$ $s_{i} \in S_{i}$
  - every game has at least one Nash equilibrium

Dominant Strategy: strategy dominates all others if it always yoelds a better or equal outcome regardless of other playes’ strategies
Strongly dominant strategy: $u_i(s_i, s_{-i}) > u_i(s_i', s_{-i})$
Weakly dominant strategy: $u_i(s_i, s_{-i}) \geq u_i(s_i', s_{-i})$

Probabilistic action selection with randomized policy

Minimax Theorem
- in a finite, two-player zero-sum game, the maximin value (the maximum of a player’s minimum payoffs) is equal to the minimax value (the minimum of the opponent’s maximum payoffs) 0 both players have optimal mixed strategies that ensure the same game value, regardless of the order of play