How Game Theory Models Continuous Strategies

While basic game theory models decisions as discrete choices—such as cooperating or defecting—real-world economic and social decisions usually involve a continuous spectrum of options, such as setting a price, choosing a quantity to produce, or allocating a budget. Game theory models these continuous strategies by defining strategy spaces as intervals of real numbers and utilizing calculus to determine optimal decisions. This article explores how mathematical payoff functions, first-order conditions, and best response curves are used to identify Nash equilibria in continuous games.

Defining Continuous Strategy Spaces

In a continuous game, a player’s strategy space is not a finite list of actions, but rather an infinite set represented by an interval of real numbers. For example, a firm’s strategy space for choosing a production quantity might be represented as \(S_i = [0, \infty)\), while a politician’s strategy space for choosing a policy position on a spectrum might be \(S_i = [0, 1]\).

Because players have an infinite number of possible actions, standard payoff matrices cannot be used to find the equilibrium. Instead, payoffs are represented as continuous, differentiable mathematical functions:

\[u_i(s_i, s_{-i})\]

where \(s_i\) is the strategy chosen by player \(i\), and \(s_{-i}\) represents the strategies chosen by all other players.

Finding Equilibrium Using Calculus

To find the Nash equilibrium in a continuous strategy game, players maximize their payoff functions relative to their opponents’ actions. Because the payoff functions are continuous and differentiable, calculus is used to solve the optimization problem.

1. First-Order Conditions (FOC)

Assuming the payoff function is concave (meaning it has a clear peak representing maximum payoff), each player finds their optimal action by taking the partial derivative of their payoff function with respect to their own strategy, and setting it to zero:

\[\frac{\partial u_i(s_i, s_{-i})}{\partial s_i} = 0\]

This step identifies the point where marginal payoff is zero, indicating that the player cannot increase their payoff by slightly increasing or decreasing their strategy.

2. Best Response Functions

Solving the first-order condition for \(s_i\) yields player \(i\)’s best response function:

\[s_i^* = g_i(s_{-i})\]

This equation defines the optimal strategy for player \(i\) as a direct function of the strategies chosen by the other players.

3. Solving the System of Equations

The Nash equilibrium occurs when all players are simultaneously playing their best responses to one another. Mathematically, this is the point where the best response functions intersect. By solving the system of equations generated by the players’ best response functions, economists find the specific strategy profile \((s_1^*, s_2^*, ..., s_n^*)\) where no player has an incentive to unilaterally deviate.

Classic Examples of Continuous Models

Continuous strategy models are foundational to microeconomics and political science. Two of the most common applications include:

• Cournot Competition: In this economic model, competing firms choose the continuous quantity of a homogeneous good to produce. Each firm’s profit depends on its own output and the total market output, which dictates the market price. The intersection of the firms’ quantity best-response curves yields the Cournot-Nash equilibrium.
• Bertrand Competition: Rather than choosing quantities, firms in a Bertrand model compete by choosing a continuous price. In a market with homogeneous goods, consumers buy exclusively from the cheapest provider, leading to a Nash equilibrium where firms continuously undercut each other until price equals marginal cost.