Proving the existence of a symmetric Bayesian Nash equilibrium

Question

I am currently faced with the following question:

Consider the public goods game. Suppose that there are $I > 2$ players and that the public goods is supplied (with benefit of 1 for all players) only if at least $K$ players contribute. The players' costs of contribution $\theta_1,...,\theta_I$ are private information, and independently and uniformly distributed on $[0.5, 1.5]$.

Let $K=1$. Prove that there is a symmetric Bayesian Nash equilibrium, where each player contributes if his cost is less than or equal to $c^*$ (same for all players) and 0.5 < $c^*$ < 1.

The probability of at least one player other than $i$ contributing = $c^∗−0.5$. If this happens, then the best response of player i is to not contribute since only one player is required to contribute for everyone to benefit

I am not sure how to proceed. I would appreciate your kind help! Thank you!

Answer 1

You may try the following approach: for every strategy $x$, calculate the set of best responses of a player who faces $I-1$ players who all play $x$. Show that the set-valued function just defined satisfies the conditions of your favorite fixed point theorem, and show that the first point is a symmetric equilibrium. You may want to restrict attention to some class of strategies, like monotone strategies, and show that when $x$ is in this class, there is always a best response in this class.

Good luck!