• Content map: SMU H3 Game Theory Map

Chapter 5: Simultaneous Games with Continuous Strategies

Continuous-Strategy Games

Definition:

A continuous-strategy game is a simultaneous-move game in which each player’s strategy set is a subset of the real numbers.

• Typical choices are prices, quantities, effort, and spending.
• Matrix comparison is replaced by optimisation.
• Calculus is the main tool whenever the payoff function is differentiable.

Payoff Maximisation

Definition:

For a differentiable payoff function, an interior optimum is characterised by a first-order condition that sets the derivative with respect to the player’s own choice equal to zero.

• Fix the opponent’s choice.
• Differentiate the player’s payoff with respect to the player’s own strategy.
• Solve the first-order condition to obtain the payoff-maximising candidate.
• This plays the same role as row-by-row or column-by-column comparison in finite games.

Best-Response Functions

Definition:

A best-response function maps each opponent choice into the player’s payoff-maximising choice.

• Solve the first-order condition for the player’s own variable.
• Express that solution as a function of the opponent’s choice.
• In continuous games, equilibrium is often visualised by intersecting best-response curves.

Continuous-Strategy Nash Equilibrium

Definition:

A continuous-strategy Nash equilibrium is a pair of strategies that satisfy both players’ best-response functions at the same time.

• Derive one best-response function for each player.
• Solve the resulting system simultaneously.
• Algebra and diagram should identify the same equilibrium point.

Result:

In continuous games, Nash equilibrium is the intersection point of the best-response curves.

Competitive and Collusive Benchmarks

Definition:

The competitive benchmark solves each player’s own optimisation problem, the collusive benchmark maximises joint payoff across players.

• Competition treats each player’s objective separately.
• Collusion changes the objective from individual profit to total profit.
• Collusion can raise joint payoffs relative to Nash equilibrium.
• In a one-shot game, collusion is typically unstable because unilateral deviation can be profitable.

Insight:

Competition and collusion solve different optimisation problems, so they generally produce different prices, quantities, and profits.