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.
I was thinking to use the 'First Price Auction' method to find the symmetric and linear equilibrium for this question. However, I am not sure how to proceed. I would appreciate your kind help! Thank you!
Update:
Player $i$ contributes iff $\theta_i \leq c^*$, then the expected payoff to player $i$ = ($1-\theta_i$)
If $\theta_i \geq c^*$, then the payoff to player $i$ is $1$ (if at least 1 other player contributes), $0$ (no one else contributes)
Expected payoff to player $i$ = $1$ x P(at least one other player contributes) = 1 $-$ P(no one else contributes) = $1 - (1.5 - c^*)^{I-1}$
Hence, player $i$ should contribute iff ($1-\theta_i$) $>$ $1 - (1.5 - c^*)^{I-1}$ and then we solve for $c^*$
I am not sure what you mean by "use the first price auction method." Here's a suggestion on how to proceed: Assume that all $I-1$ opponents of player $i$ use the same cutoff strategy $c$. Show that player $i$'s (Bayesian) best response is also a cutoff strategy and calculate the cutoff (call it, e.g., $c_i$). This should be a function of $c$. Then, to establish a symmetric BNE, assume $c_i=c=c^*$ and solve for $c^*$.