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Let $u_i$ denote player $i$'s expected payoff, so the definition also covers mixed strategies. If $p_i^{\ast }$ is a dominant strategy, thenfor every opponents' profile $p_{-i}$ and every alternative $q_i$. In particular this holds at $p_{-i}=p_{-i}^{\ast }$. Thus no player can improve its payoff by a unilateral deviation from $(p_1^{\ast },\dots ,p_n^{\ast })$, which is exactly the definition of a Nash equilibrium. A profile of dominant strategies is a Nash equilibrium.

For the second-price sealed-bid auction, use the usual private-value auction assumptions: a bidder with value $v$ has quasilinear utility $v$ minus its payment if it wins, and zero if it loses. Fix the highest opposing bid $h$. If $h<v$, winning gives $v-h>0$, and bidding $v$ wins. If $h>v$, winning gives $v-h<0$, and bidding $v$ loses. If $h=v$, winning or losing gives zero, irrespective of the tie rule. Any other bid can only change whether the bidder wins, not the payment $h$ conditional on winning. It therefore cannot improve on the truthful bid in any of these cases. Hence bidding the true value is a weakly dominant strategy. Neither symmetry nor independence of the opposing values is needed for this argument.

For part (a), in a first-price sealed-bid auction with $n\ge 2$, continuous nonnegative bids and $v>0$, no fixed bid is a dominant strategy. A positive bid $b$ can be improved against opponents all bidding zero: replace it by $b/2>0$, still win, and increase the payoff by $b/2$. Bid zero can be improved against an opposing highest bid $h=v/2$: bidding $3v/4$ wins and earns $v/4>0$, whereas zero loses. This proves the no dominant positive-value bid in a first-price auction result.

Allowing mixed strategies does not change that conclusion. Against all-zero opponents, a random bid has expected payoff strictly below $v$ unless it is always zero and the tie rule awards this bidder the item with certainty. Any positive bid incurs a strictly positive payment, and any probability of losing at zero also reduces the payoff below $v$. In the strict case choose a sufficiently small positive fixed bid $\delta$ so that $v-\delta$ exceeds the random bid's expected payoff. In the exceptional case the preceding deviation against $h=v/2$ still improves it. The assumptions matter: for value zero, bid zero is weakly dominant; with only one bidder it is also optimal. Consequently there is no dominant strategy for a positive-value bidder in the standard first-price setting.

Part (b)'s printed nonexistence request is false under the stated symmetric independent private values model. Lack of a dominant strategy does not imply lack of a Nash equilibrium. For incomplete private information the appropriate equilibrium is a Bayesian Nash equilibrium, a Nash equilibrium of the game whose strategies are bidding rules as functions of private values. Here is an explicit counterexample with a full deviation check.

Let the $n\ge 2$ risk-neutral bidders have independent valuations with the uniform distribution on $[0,1]$, and put $a=(n-1)/n$. Suppose every opponent bids $\beta (w)=aw$. A bidder with value $v$ who bids $b\in [0,a]$ wins with probability $(b/a{)}^{n-1}$, since the opposing values must all be below $b/a$. Ties have probability zero, including at either endpoint. Its expected quasilinear utility isFor $b>0$,Thus $U_v$ increases up to $b=av$ and decreases thereafter. For $v=0$, bid zero is optimal directly. A bid above $a$ wins with certainty and pays more than bidding $a$, so it cannot improve on the maximum already found in $[0,a]$. A randomized bid is a mixture of these payoffs and also cannot improve on their maximum. The best response for every value is therefore the proposed bidding rule itself. This proves the uniform private-value first-price bidding equilibrium:The value-$v$ expected equilibrium payoff is $v^n/n$. The bidding rule is a pure strategy in the Bayesian game, even though values are random. Therefore the requested general nonexistence claim in (b) has a counterexample, rather than a valid proof. The defect is present in the original PDF, not introduced by the converted text.