Suppose a town is building a bridge, and it costs $B$. There are $n$ villagers.
Each village's valuation of the bridge is private information, $v_i$.
It is common knowledge that this valuation is drawn from a uniform distribution $[0,1]$. $B\in[0,1]$.
Villager can only submit $0$ or $B$.
If one villager submits $B$, then the bridge is built and every other villager pays their submission.
If no bridge is built, everyone gets $0$.
How do I construct an expected payoff a village $i$?
What I got down to is having 2 scenarios: $v_i>B$ and $v_i\leq B$.
But in each case, I have two possible payoffs. For the former case,
if every other player submits $0$, $i$ should submit B, because $v_i-B>0$.
if someone pays $B$, $i$ should submit 0, because she gets $v_i$.
You get a similar type of for the other scenario.
But how would I incorporate this into expected payoff of $i$ and how should I go about constructing a social welfare function?
I feel like this is just a variation of all-payout auction with discrete action space for each $i$.