This is because Player 1 is willing to meet the reserve bid of 1 and Player 2 is not.
One might wonder if the designer can still extract the full surplus with a smaller reserve bid. The answer is yes. In fact, is sufficient for the first best to be attainable under symmetry. That is, the designer need only be able to withhold the prize when both players play zero. To see this, consider the following contest which satisfies both properties except at zero.
This contest has an equilibrium at the first best yet only denies the prize to players when they both exert no effort.
Direct discrimination is similar to a reserve bid. It is possible to mimic any reserve bid through direct discrimination by promising the prize to the weaker player whenever the reserve is not met. In particular, take either example from the previous section and set and . Such a contest is efficient and will be strategically equivalent to the original contest. For example, consider the aforementioned all-pay auction with a reserve bid. With this transformation,
This contest has an equilibrium at the first best.
We now restrict the principal to use an efficient symmetric contest. In equilibrium, each Player must weakly prefer her equilibrium payoff over copying the strategy of her opponent. Therefore, the following weak incentive compatibility condition is necessary for equilibrium:
Together with this implies
where the last line follows from efficiency. Rearranging this equation gives an upper bound on the revenue:
which can be further bounded by
I show by construction that this upper bound is tight. That is, there exist optimal contests which achieve these bounds.
The second bound comes from the fact that because Player 2 cannot win with probability more than one half. To see this, consider that the following two conditions must hold in equilibrium
which imply .
Case 1: . This means that revenue cannot exceed . We can reach this upper bound with an all-pay auction with a bid cap at as in Che and Gale (1998). That is
This has an equilibrium at which achieves the upper bound for .
Case 2: . The above implies that revenue cannot exceed one half. Consider the following difference form contest as in Che and Gale (2000):
The above contest has an equilibrium at and which achieves the upper bound for .
Note that these strategies must be pure because no mixed strategy can have expectation zero and no individually rational mixed strategy can have expectation one. ↩︎