Highest Bidder Collection

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If v i < max j ≠ i b j < b i {\displaystyle v_{i}<\max _{j\neq i}b_{j} 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}}} The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids b i < v i {\displaystyle b_{i}

It does not allow for price discovery, that is, discovery of the market price if the buyers are unsure of their own valuations, without sequential auctions. We now argue that in the sealed first price auction the equilibrium bid of a buyer with valuation v {\displaystyle v} is Suppose that buyer 2 bids according to the strategy B ( v ) = v / 2 {\displaystyle B(v)=v/2} , where B ( v ) {\displaystyle B(v)} is the buyer's bid for a valuation v {\displaystyle v} . We need to show that buyer 1's best response is to use the same strategy. The strategy of overbidding is dominated by bidding truthfully. Assume that bidder i bids b i > v i {\displaystyle b_{i}>v_{i}} .If max j ≠ i b j < v i {\displaystyle \max _{j\neq 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}>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.