An elimination contest with non-sunk bids

In this paper we study a multi-stage elimination contest with non-sunk bids: differently from existing literature, we realize that when players are budget-constrained, they do not regard past bids as strategically irrelevant in their decision of how much to bid in following stages. This happens because they face a basic trade-off when allocating scarce resources over stages. We believe that although non-sunk bids make the analysis more complex, they allow to improve the quality of the modelization for many real scenarios, like R&D contests and sport tournaments. In our simple two-stage framework with complete information and asymmetric players, we find that: (i) there is a unique SPNE where in the first stage only the strongest player bids positive, while forcing the others to bid zero; in the second stage shortlisted bidders play mixed strategies, and the strongest player wins the game on average; (ii) relative ex-ante strengths of players are relatively more important, in determining the outcome of the game, than their relative abilities of allocating limited resources over the stages; (iii) the two-stage contest yields a lower expected revenue than the one-stage one, due to the fact that the first stage yields basically no revenue and that shortlisting to the second stage is inefficient. On the basis of these results, our elimination contest does not seem to be a very advantageous allocation mechanism for the contest sponsor.