Highest Bidder Collection

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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.

If max j ≠ i b j < b i {\displaystyle \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} v i {\displaystyle b_{i}>v_{i}} .If max j ≠ i b j < v i {\displaystyle \max _{j\neq 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. U ( b ) = w ( b ) ( v − b ) = 2 b ( v − b ) = 1 2 [ v 2 − ( v − 2 b ) 2 ] {\displaystyle U(b)=w(b)(v-b)=2b(v-b)={\tfrac {1}{2}}[{{v}

If b i < max j ≠ i b j < v i {\displaystyle b_{i}<\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. 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.

Seller Update: September 2023