Dynamic Games with Complete Information

Dynamic Games with Complete Information

• Dynamic games of complete information involve sequential moves by players.
• Multistage games with observed actions involve players observing previous actions at each stage.
• These games can be represented using game trees (extensive form).

Example Dynamic Game

• Firm 1 chooses a quantity $q_1 \ge 0$.
• Firm 2 observes $q_1$ and then chooses a quantity $q_2 \ge 0$.
• Payoff to firm $i$ is $\pi_i(q_1, q_2) = q_i[P(Q) - c]$, where $Q = q_1 + q_2$ and $P(Q)$ is the market-clearing price.

Strategies in Dynamic Games

• A strategy is a complete plan of action for every possible contingency.
• In multistage games, strategies must specify actions at each stage, even those off the equilibrium path.
• In the example, Firm 1's strategy is a quantity $q_1 \ge 0$, and Firm 2's strategy is a function $q_2(q_1)$ that specifies $q_2 \ge 0$ for each $q_1 \ge 0$.

Nash Equilibrium in Dynamic Games

• Defined the same as in static games: each player's strategy must be a best response to others' strategies.
• The Nash equilibrium concept may allow incredible threats.
• With Firm 1 choosing $q_1^* = Q^m$ (Cournot equilibrium output) and Firm 2 choosing $q_2(q_1) = 0$ if $q_1 = Q^m$ and $\infty$ if $q_1 \ne Q^m$, this is a Nash equilibrium, but Firm 2's threat to choose $\infty$ if $q_1 \ne Q^m$ is not credible.

Subgame-Perfect Nash Equilibrium

• This is a refinement of Nash equilibrium that eliminates equilibria based on non-credible threats.
• A subgame is the game remaining to be played from any point where the history is common knowledge.
• A subgame-perfect Nash equilibrium is a strategy profile that is a Nash equilibrium in every subgame.
• Any finite multistage game with observed actions possesses at least one subgame-perfect Nash equilibrium.
• It can be found using backward induction: determine the optimal actions of the last player, then the second to last, and so on.

Example (cont.)

• Using backward induction, Firm 2 chooses $q_2$ to maximize its profit given $q_1$:

$$ \max_{q_2} \pi_2(q_1, q_2) = q_2[P(q_1 + q_2) - c] $$

• The first-order condition is:

$$ P(q_1 + q_2) + q_2 P'(q_1 + q_2) = c $$

• Let $q_2^*(q_1)$ denote Firm 2's best response to $q_1$.
• In the first stage, Firm 1 chooses $q_1$ to maximize profit, knowing Firm 2 will choose $q_2^*(q_1)$:

$$ \max_{q_1} \pi_1(q_1, q_2^(q_1)) = q_1[P(q_1 + q_2^(q_1)) - c] $$

• The first-order condition is:

$$ P(q_1 + q_2^(q_1)) + q_1 P'(q_1 + q_2^(q_1))[1 + \frac{\partial q_2^*(q_1)}{\partial q_1}] = c $$

• Solving the two first-order conditions simultaneously yields the subgame-perfect Nash equilibrium $\langle q_1^, q_2^(q_1)\rangle$.