All-pay auctions with endogenous asymmetries

We study N-bidders, asymmetric all-pay auctions under incomplete information. First, we solve for the equilibrium of a parametric model. Each bidder’s valuation is independently drawn from an uniform [0, αi] where the parameter αi may vary across bidders. In this game, asymmetries are exogenously given. Next, a two-stage game where asymmetries are endogenously generated is studied. At the first stage, each bidder chooses the level of an observable, costly, value-enhancing action. The second stage is the bidding sub-game, whose equilibrium is simply the equilibrium of the, previously analyzed, game with exogenous asymmetries. Finally, natural applications of the all pay-auction in the context of political lobbying are considered: the effects of excluding bidders, as well as, the impact of caps on bids.

Pure strategy equilibria of multidimensional and non-monotonic auctions

We give necessary and sufficient conditions for the existence of symmetric equilibrium without ties in interdependent values auctions, with multidimensional independent types and no monotonic assumptions. In this case, non-monotonic equilibria might happen. When the necessary and sufficient conditions are not satisfied, there are ties with positive probability. In such case, we are still able to prove the existence of pure strategy equilibrium with an all-pay auction tie-breaking rule. As a direct implication of these results, we obtain a generalization of the Revenue Equivalence Theorem. From the robustness of equilibrium existence for all-pay auctions in multidimensional setting, an interpretation of our results can give a new justification to the use of tournaments in practice.

Asymmetric auctions and risk aversion within independent private valoues

We study tournaments with many ex-ante asymmetric (heterogeneous) contestants as an independent-private-values all-pay auction. The asymmetry is either with respect to the distribution of valuations for the prize or the risk preferences. By characterizing equilibria in tnonotone strategies we show that tournaments \:vith man~y heterogenous contestants are qualitatively distinct. First, with two (or many ex-ante identical) participants, a contestant always exerts some effort with positive probability. In contrast, with many asymmetric participants, one 1night not exert any effort at all, even if there is a positive probability that he has the highest valuation among ali. Second, in tournan1ents with t'wo (o r n1any ex-ante h01nogenous) contestants, equilibrium effort densities are decreasing. This prediction is at odds with experimental evidence that shows the empírica! density might be increasing at high effort levels. V\.lith rnany heterogeneous contestants, however. the increasing bid density is consistent with an equilibrium behavior.