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Dynamic contests with resource constraints

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Abstract

We study a dynamic contest between two players who compete against each other in \(n\) different stages. The players have winning values for each stage of the contest that may vary across the stages as well as heterogeneous resource budgets that decrease from a given stage to the next proportionally to the resources allocated in that stage. We characterize a subgame perfect equilibrium of this dynamic contest and show that when the winning value is equal between the stages, the players’ resource allocations are weakly decreasing over the stages. We also study the effect of several distributions of winning values on the players’ resource allocations. We show both the distribution of winning values that balances the players’ resource allocations and the distribution of winning values that maximizes the players’ total resource allocations.

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Notes

  1. Matros (2006) studied elimination tournaments where, similarly to our model, players have budget resources instead of cost functions. However, an elimination tournament is different than our dynamic model because players in elimination tournaments do not have an incentive to deplete the budgets of their rivals in the first stages since the losers in each stage are eliminated before the following stages.

  2. Ryvkin (2011) studied a different multi-stage contest known as the best-of \( k\) contest in which the players’ probabilities of winning in each stage depend on the players’ efforts in that stage as well as their efforts in the previous stages. He found that agents are more likely to exert higher efforts in the later stages of the contest which is the exact opposite of our findings.

  3. Additional works on allocation of resources in sequential contests include, among others, Warneryd (1998), Konrad (2004) and Kovenock and Roberson (2009).

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Acknowledgments

We would like to thank Dan Kovenock for for his helpful comments.

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Authors and Affiliations

  1. Department of Economics, Ben-Gurion University of the Negev, 84105 , Beer-Sheva, Israel

    Aner Sela & Eyal Erez

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  1. Aner Sela
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  2. Eyal Erez
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Correspondence to Aner Sela.

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Sela, A., Erez, E. Dynamic contests with resource constraints. Soc Choice Welf 41, 863–882 (2013). https://doi.org/10.1007/s00355-012-0711-1

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