Highest Bidder Collection

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We now argue that in the sealed first price auction the equilibrium bid of a buyer with valuation v {\displaystyle v} is Truthful bidding dominates the other possible strategies (underbidding and overbidding) so it is an optimal strategy. The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids b i < v i {\displaystyle b_{i}

The payoff for bidder i is { v i − max j ≠ i b j if b i > max j ≠ i b j 0 otherwise {\displaystyle {\begin{cases}v_{i}-\max _{j\neq i}b_{j}&{\text{if }}b_{i}>\max _{j\neq i}b_{j}\\0&{\text{otherwise}}\end{cases}}} If b i < max j ≠ i b j < v i {\displaystyle b_{i}<\max _{j\neq i}b_{j} v i {\displaystyle b_{i}>v_{i}} . If v i < max j ≠ i b j < b i {\displaystyle v_{i}<\max _{j\neq i}b_{j}

Let v i {\displaystyle v_{i}} be bidder i's value for the item. Let b i {\displaystyle b_{i}} be bidder i's bid for the item. If max j ≠ i b j > b i {\displaystyle \max _{j\neq i}b_{j}>b_{i}} then the bidder would lose the item either way so the strategies have equal payoffs in this case. If max j ≠ i b j < v i {\displaystyle \max _{j\neq i}b_{j}
If max j ≠ i b j > v i {\displaystyle \max _{j\neq i}b_{j}>v_{i}} then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies have equal payoffs for this case. If max j ≠ i b j < b i {\displaystyle \max _{j\neq i}b_{j}