Highest Bidder Collection

Product Information

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 > 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 < v i {\displaystyle \max _{j\neq i}b_{j}

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}} . Truthful bidding dominates the other possible strategies (underbidding and overbidding) so it is an optimal strategy. 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 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. 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}}}