Vickrey Auction - Proof of Dominance of Truthful Bidding

Proof of Dominance of Truthful Bidding

The dominant strategy in a Vickrey auction with a single, indivisible item is for each bidder to bid their true value of the item.

Let be bidder i's value for the item. Let be bidder i's bid for the item.

The payoff for bidder i is \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 overbidding is dominated by bidding truthfully. Assume that bidder i bids .

If then the bidder would win the item with a truthful bid as well as an overbid. The bid's amount does not change the payoff so the two strategies have equal payoffs in this case.

If then the bidder would lose the item either way so the strategies have equal payoffs in this case.

If then only the strategy of overbidding would win the auction. The payoff would be negative for the strategy of overbidding because they paid more than their value of the item, while the payoff for a truthful bid would be zero. Thus the strategy of bidding higher than one's true valuation is dominated by the strategy of truthfully bidding.

The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids .

If 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 then the bidder would win the item either way so the strategies have equal payoffs in this case.

If then only the strategy of truthfully bidding would win the auction. The payoff for the truthful strategy would be positive as they paid less than their value of the item, while the payoff for an underbid bid would be zero. Thus the strategy of underbidding is dominated by the strategy of truthfully bidding.

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

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