Abstract
Many important games are aggregative games which allows for robust comparative statics analysis even when the games do not exhibit strategic complements or substitutes (Acemoglu and Jensen, Games Econ Behav 81:27–49, 2013). This paper establishes such comparative statics results for contests improving upon existing results by (i) allowing payoff functions to be discontinuous at the origin, and (ii) allowing for asymmetric rent-seeking contests and patent races. A leading example where (i) is relevant is the classical Tullock contest (Tullock G (1980) Efficient rent-seeking. In: Buchanan JM, Tollison RD, Tullock G (eds) Toward a theory of the rent-seeking society. Texas A&M Press, College Station, pp 97–112). The paper also studies existence and uniqueness of equilibria extending the results of Szidarovszky and Okuguchi (Games Econ Behav 18:135–140, 1997) and Cornes and Hartley (Econ Theory 26:923–946, 2005) to patent races.
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Notes
- 1.
- 2.
For a general discussion of aggregative games see Jensen (2010).
- 3.
Note that strategy sets are assumed to be compact. This is for technical convenience. If one insists on having \(S_{i} = \mathbb{R}_{+}\), one can instead assume, for example, that for all \(s_{-i} \in \mathbb{R}_{+}^{I-1}\), π i (⋅ , s −i ) strictly decreases for \(s_{i} \geq \bar{ s}_{i}\).
- 4.
- 5.
Use that \(\frac{\partial ^{2}c_{i}\circ h_{i}^{-1}(s_{i})} {\partial s_{i}^{2}} = \frac{c_{i}^{{\prime\prime}}(h_{i}^{-1}(s_{i}))(h_{i}^{{\prime}}(h_{i}^{-1}(s_{i})))^{-1}h_{i}^{{\prime}}(h_{i}^{-1}(s_{i}))-h_{i}^{{\prime\prime}}(h_{i}^{-1}(s_{i}))(h_{i}^{{\prime}}(h_{i}^{-1}(s_{i})))^{-1}c_{i}^{{\prime}}(h_{i}^{-1}(s_{i}))} {(h_{i}^{{\prime}}(h_{i}^{-1}(s_{i})))^{2}}\).
- 6.
See statement 3 of Theorem 2 where conditions for z to be locally increasing/decreasing play an important role.
- 7.
- 8.
Note that the payoff of an agent in Loury is in fact \(\frac{V } {R} \frac{h_{i}(s_{i})} {R+\sum _{j}h_{j}(s_{j})} - s_{i}\) (see equation (7) on page 399 of Loury (1979)). Multiplying with R—which does not alter the optimal decision—leads to the functional form considered here.
- 9.
Some Cournot models of course fit directly into the set-up of the current paper. Thus with inverse demand \(P(Q) = \frac{1} {R+Q}\) and cost c i , firm i’s profit function is of the form (1) with h i (s i ) = s i and V i = 1. Note, however, that this Cournot model is not a game of strategic substitutes (as one may show, firm i’s optimal output is decreasing in the opponents’ aggregate output if and only if \(s_{i} \leq R +\sum _{j\neq i}s_{j}\)).
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Acknowledgements
I would like to thank Pierre von Mouche, Federico Quartieri, and an anonymous referee for helpful comments. All remaining errors are my responsibility.
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Jensen, M.K. (2016). Existence, Uniqueness, and Comparative Statics in Contests. In: von Mouche, P., Quartieri, F. (eds) Equilibrium Theory for Cournot Oligopolies and Related Games. Springer Series in Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-29254-0_14
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