No dominant positive-value bid in a first-price auction ($v>0$)

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.