# Game of Variable Contributions to the Common Good under Uncertainty

## TL;DR

This paper analyzes a stochastic game of contribution to the common good, revealing how gradual concession strategies emerge and differ from no-delay equilibria, with implications for efficiency and asymmetry effects.

## Contribution

It introduces a variable contribution game extending the Nerlove-Arrow model, demonstrating regular control equilibria that generalize mixed strategies in wars of attrition.

## Key findings

- Equilibrium characterized by gradual, regular control strategies.
- Stochasticity accentuates differences between singular and regular control solutions.
- Asymmetry reduces inefficiency caused by gradual concession strategies.

## Abstract

We consider a stochastic game of contribution to the common good in which the players have continuous control over the degree of contribution, and we examine the gradualism arising from the free rider effect. This game belongs to the class of variable concession games which generalize wars of attrition. Previously known examples of variable concession games in the literature yield equilibria characterized by singular control strategies without any delay of concession. However, these no-delay equilibria are in contrast to mixed strategy equilibria of canonical wars of attrition in which each player delays concession by a randomized time. We find that a variable contribution game with a single state variable, which extends the Nerlove-Arrow model, possesses an equilibrium characterized by regular control strategies that result in a gradual concession. This equilibrium naturally generalizes the mixed strategy equilibria from the canonical wars of attrition. Stochasticity of the problem accentuates the qualitative difference between a singular control solution and a regular control equilibrium solution. We also find that asymmetry between the players can mitigate the inefficiency caused by the gradualism.