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# Static Games with Complete Information ## 1. Basic Elements ### (1) Game A game consists of * Players: $i = 1, 2,..., n$ * Strategies: $s_i \in S_i$, $S_i$ is player i's strategy space * Payoffs: $u_i(s_1, s_2,..., s_n)$ ### (2) Complete Information Each player's payoff function is common...

# Static Games with Complete Information ## 1. Basic Elements ### (1) Game A game consists of * Players: $i = 1, 2,..., n$ * Strategies: $s_i \in S_i$, $S_i$ is player i's strategy space * Payoffs: $u_i(s_1, s_2,..., s_n)$ ### (2) Complete Information Each player's payoff function is common knowledge among all players. ### (3) Static Game Players choose their strategies simultaneously. ### (4) Notation * $s = (s_1, s_2,..., s_n)$: a strategy profile * $s_{-i} = (s_1,..., s_{i-1}, s_{i+1},..., s_n)$: strategies of all players except i * $s = (s_i, s_{-i})$ ## 2. Dominant Strategy Equilibrium ### (1) Definition In the game $G = {I, \{S_i\}, \{u_i(\cdot)\}}$, where $I$ is the set of players, $S_i$ is the strategy space of player $i$, and $u_i$ is the payoff function of player $i$, strategy $s_i^*$ is player $i$'s dominant strategy if for all $s_i \in S_i$ and for all $s_{-i} \in S_{-i}$, $u_i(s_i^*, s_{-i}) \ge u_i(s_i, s_{-i})$ ### (2) Definition If $s_i^*$ is player $i$'s dominant strategy for all $i$, then $(s_1^*, s_2^*,..., s_n^*)$ is a dominant strategy equilibrium. ### (3) Example: Prisoner's Dilemma | Player 2 | Quiet | Fink | | :------- | :---- | :--- | | Player 1 | | | | Quiet | -1, -1 | -9, 0 | | Fink | 0, -9 | -6,-6 | * Fink is the dominant strategy for each player. * (Fink, Fink) is the dominant strategy equilibrium. ## 3. Nash Equilibrium ### (1) Definition In the game $G = {I, \{S_i\}, \{u_i(\cdot)\}}$, strategy profile $s^* = (s_1^*, s_2^*,..., s_n^*)$ is a Nash Equilibrium if for all $i$, $s_i^*$ is player $i$'s best response to the strategies chosen by the other players, $s_{-i}^*$: $u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*)$ for all $s_i \in S_i$ ### (2) How to find Nash Equilibrium * Best response function * $BR_i(s_{-i})$ = { $s_i \in S_i$ | $u_i(s_i, s_{-i}) \ge u_i(s_i', s_{-i})$ for all $s_i' \in S_i$} * A strategy profile $s^* = (s_1^*, s_2^*,..., s_n^*)$ is a Nash Equilibrium if and only if $s_i^* \in BR_i(s_{-i}^*)$ for all i. ### (3) Examples #### (i) Coordination Game | Player 2 | Left | Right | | :------- | :--- | :---- | | Player 1 | | | | Left | 2, 1 | 0, 0 | | Right | 0, 0 | 1, 2 | Nash Equilibria: (Left, Left), (Right, Right) #### (ii) Matching Pennies | Player 2 | Head | Tail | | :------- | :--- | :--- | | Player 1 | | | | Head | 1, -1| -1, 1| | Tail | -1, 1| 1, -1| No pure strategy Nash Equilibrium. ## 4. Existence of Nash Equilibrium ### (1) Theorem If $n$ is finite and $S_i$ is finite for all $i$, then there exists at least one mixed strategy Nash Equilibrium. ### (2) Theorem If $S_i$ is a compact convex subset of a Euclidean space for all $i$, $u_i(s_1,..., s_n)$ is continuous in $s$ and quasi-concave in $s_i$ for all $i$, then there exists at least one Nash Equilibrium (in pure strategies). ## 5. Continuous Strategies ### (1) Example: Cournot Duopoly * Two firms: $i = 1, 2$ * Strategies: $q_i \in [0, \infty)$, $q_i$ is the quantity produced by firm i. * Cost function: $C_i(q_i) = cq_i$ * Inverse demand function: $P(Q) = a - Q$, where $Q = q_1 + q_2$ is the total quantity. * Payoff function: $\pi_i(q_i, q_{-i}) = P(Q)q_i - C_i(q_i) = (a - q_i - q_{-i})q_i - cq_i$ * Best response function: $BR_i(q_{-i}) = \frac{a - q_{-i} - c}{2}$ * Nash Equilibrium: $q_1^* = q_2^* = \frac{a - c}{3}$ ### (2) Example: Bertrand Duopoly * Two firms: $i = 1, 2$ * Strategies: $p_i \in [0, \infty)$, $p_i$ is the price charged by firm i. * Cost function: $C_i(q_i) = cq_i$ * Demand function: $q_i(p_i, p_{-i}) = \begin{cases} D(p_i) & \text{if } p_i < p_{-i} \\ \frac{1}{2}D(p_i) & \text{if } p_i = p_{-i} \\ 0 & \text{if } p_i > p_{-i} \end{cases}$ * Nash Equilibrium: $p_1^* = p_2^* = c$