Abstract
We prove the existence of pure strategy Bayesian equilibria in group contests under individual-level and group-level private information. For the latter type, we develop a novel approach reducing group contests to contests between individuals with multi-dimensional types, with far-reaching implications for the existence of equilibrium in various group contest settings, including the case of complete information.
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Notes
The linear costs of effort are without loss of generality. Any costs of effort \(c_{ij}g_{ij}(x_{ij})\), where \(g_{ij}\) are strictly increasing continuous functions satisfying \(g_{ij}(0)=0\), can be accommodated by redefining efforts as \(y_{ij}=g_{ij}(x_{ij})\) and modifying the group production functions \(h_i\), see the following paragraph.
The modified game is thus von Neumann–Morgenstern equivalent to the original contest (Morris and Ui 2004).
The CES production function in Example 1 with \(\rho _i\le 1\) leads to a cost function of this form.
Ewerhart (2014) also shows that the equilibrium in the contest between individuals is unique. However, in group contests equilibrium uniqueness is not a generic property. For example, it is easy to see that if in all groups there are at least two players and production functions \(h_i\) have a Cobb–Douglas form, then all players exerting zero effort constitutes an equilibrium. Moreover, for \(n>2\), all players exerting zero effort in any number of groups less than \(n-1\) is also an equilibrium.
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Brookins, P., Ryvkin, D. Equilibrium existence in group contests. Econ Theory Bull 4, 265–276 (2016). https://doi.org/10.1007/s40505-015-0085-5
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DOI: https://doi.org/10.1007/s40505-015-0085-5