Continuous contest game / Positional externality game

Setup

Definition:

Continuous contest game / Positional externality game

• Players: Candidate 1 and Candidate 2.
• Strategies: Candidate 1 chooses campaign spending $x\geq 0$; Candidate 2 chooses campaign spending $y\geq 0$.

Rules

• Start with two candidates competing for a fixed prize.
• Players simultaneously choose non-negative campaign spending, $x$ and $y$.
• The player who chooses the payoff-maximising spending level against the rival’s spending is optimal.
• If at least one candidate spends a positive amount, the prize is shared in proportion to spending.
• If both spend zero, each receives $5$.

Payoff Details

• Payoffs are:

Derivation (Best Response Analysis)

Rule Out Zero Spending

• $(0,0)$ is not a Nash equilibrium because Candidate 1 can deviate to $x=0.1$ when $y=0$ and obtain

Candidate 1 Best Response

• For $x+y>0$, Candidate 1 solves

• FOC for Candidate 1:

• Candidate 1’s interior best response is

• With the non-negativity constraint, for $y>0$:

Candidate 2 Best Response

• By symmetry, for $x>0$:

Solve the System

• At an interior Nash equilibrium, both FOCs hold:

• Hence $x=y$.
• Substituting $x=y$ into either FOC gives

• For the positive solution:

• Therefore:

Diagram (Best Response Functions)

Nash Equilibrium

Result:

$$\{x,y\} \mapsto \{2.5,\ 2.5\}$$

Each candidate receives $\pi_i=2.5$.

Social Optimum

• Total payoff is

• Joint welfare is maximised at

Insights

Insight:

• Best responses are non-linear because each candidate’s marginal return depends on total spending.
• Nash equilibrium has positive spending by both candidates.
• The social optimum has zero spending, so equilibrium campaigning is wasteful rent-seeking.