Least unique bid auction

Each bidder chooses a permitted amount. The bidder with the smallest amount submitted by exactly one bidder wins and pays that amount; if no amount is unique, nobody wins. With object value $V$, the winner's payoff is $V$ minus their bid, and losers pay nothing. Unlike a first-price sealed-bid auction, repeated lowest bids can be disqualified.

For bids $1,2$ and $V>2$, write $p_i$ for player $i$'s probability of bidding $1$. The two expected payoffs are $(V-1)(1-p_j)(1-p_k)$ and $(V-2)p_j{p}_k$. Their unique symmetric equilibrium uses the displayed probability. Every permutation of $(1,0,t)$, $0\le t\le 1$, is also a Nash equilibrium: the mixed bidder is indifferent, while the two pure bidders weakly prefer their prescribed bids. There are no further equilibria. Three interior probabilities must all coincide, since subtracting two indifference equations gives a nonzero positive factor times their difference. Two interior probabilities with a pure opponent are impossible, and a single interior probability requires the other two bids to differ. Thus the nonsymmetric equilibria form six line segments with six pure endpoints, and their number is uncountable.