First-price sealed-bid auction

Each bidder submits one private bid; the highest bidder wins and pays its own bid, while losers pay nothing. A multi-unit version can award one item to each of the highest bidders, each paying its own bid. In a monotone symmetric equilibrium, a bidder can imitate another type's bid when checking incentive constraints.

In a first-price sealed-bid auction with at least two bidders, continuous nonnegative bids, quasilinear utility and positive private value $v$, no bid is a dominant strategy. Against opponents all bidding zero, any positive bid can be lowered while still winning, increasing the payoff. Bid zero fails against an opposing bid $v/2$, since bidding $3v/4$ earns $v/4>0$. Nor does randomizing create a dominant strategy. Against all-zero opponents, a random bid has expected payoff strictly below $v$ unless it is identically zero and receives the item with certainty under the tie rule. In the strict case, a sufficiently small positive fixed bid earns more. In the exceptional case, the deviation against $v/2$ again improves the payoff. Finite expected utility is implicit; an expected payoff of negative infinity cannot dominate finite-payoff bids. The positive-value condition matters: a zero-value bidder weakly prefers bid zero.

For $n\ge 2$ risk-neutral bidders with independent uniform valuations on $[0,1]$, this rule is a symmetric Bayesian Nash equilibrium. If opponents use $\beta (v)=av$, a bid $b\in [0,a]$ has expected utility $(v-b)(b/a{)}^{n-1}$. Its derivative has the sign of $(n-1)v-nb$, so with $a=(n-1)/n$ the best bid is $av$. Bids above $a$ guarantee winning but only increase payment. The type-$v$ equilibrium utility is $v^n/n$.