= No dominant positive-value bid in a first-price auction
{title2=$v>0$}
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.
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