Continuous Contribution Public Good Game

Setup

Definition:

Continuous Contribution Public Good Game

• Players: $N>2$ players.
• Strategies: Each player $i$ chooses a contribution $x_i \in [0,5]$.

Rules

• Start with each player endowed with $5$ units; Players simultaneously choose contributions $x_i\in[0,5]$ to a common pool.
• The player who chooses the payoff-maximising contribution against others’ contributions is optimal.
• Each player is endowed with $5$ units of a private asset; Contributions are added to a common pool.
• Payoffs are realised simultaneously after all contributions are chosen.

Payoff Matrix

• Player $i$‘s payoff is

• Rewriting:

Derivation (Best Response Analysis)

• Holding other contributions fixed, player $i$‘s problem is

• The derivative with respect to own contribution is

• Therefore payoff falls as $x_i$ increases, so the best response is the boundary choice

Derivation (Nash Equilibrium)

• Since every player’s best response is $0$, the unique Nash equilibrium is

• To find the social optimum, maximise total welfare:

• Because $N>2$, the coefficient on each $x_i$ is positive.
• Therefore welfare is maximised by setting every contribution at the upper bound:

Nash Equilibrium

Result:

The unique Nash equilibrium is zero contribution by every player:

$$(0,0,\dots,0).$$

Social Optimum

• The welfare-maximising outcome is full contribution by every player:

• The gap between equilibrium and optimum is the free-rider problem.

Diagram (Best Response Functions)

• Diagram shown for the symmetric projection with $N=3$, so $X_{-i}=2x_i$ on symmetric profiles.

Insights

Insight:

• Each player bears the full private cost of contribution but captures only part of the social benefit.
• This is the continuous-strategy analogue of a Prisoner’s Dilemma.