# Two-Group Contests with Communication Within and Between Groups

## Abstract

We consider a two-group contest game with weakest-link social composition functions and convex cost functions and prove the existence of a unique group-proof Nash equilibrium. Such a refinement of the Nash equilibrium prescribes the same communication possibilities as those required by a coalition-proof Nash equilibrium—in the precise sense of Bernheim et al. (J Econ Theory 42:1–12, 1987)—only among the contenders of the same group and between the two groups. We show how a fictitious game with “most inefficient fictitious contenders” can be constructed to prove the existence of a unique group-proof Nash equilibrium of the original two-group contest game. An example evidences that cautious arguments on the (twice) differentiability of cost functions must be used in such a construction.

## Keywords

Nash Equilibrium Payoff Function Contest Success Function General Nash Equilibrium Nash Equilibrium Uniqueness## Notes

### Acknowledgements

The first author gratefully acknowledges financial support: part of his work was carried out in the frame of Programme STAR, financially supported by UniNA and Compagnia di San Paolo. The second author gratefully acknowledges JSPS KAKENHI Grant Number 15K03361.

## References

- Baik KH (1993) Effort levels in contests: the public-good prize case. Econ Lett 41:363–367CrossRefGoogle Scholar
- Baik KH (2008) Contests with group-specific public-good prizes. Soc Choice Welf 30:103–117CrossRefGoogle Scholar
- Bernheim BD, Peleg B, Whinston MD (1987) Coalition-proof Nash equilibria. I. Concepts. J Econ Theory 42:1–12CrossRefGoogle Scholar
- Choi JP, Chowdhury SM, Kim J (2015) Group contests with internal conflict and power asymmetry. Scand J Econ (Forthcoming). doi:10.1111/sjoe.12152Google Scholar
- Chowdhury SM, Topolyan I (2015) The attack-and-defense group contests: best-shot versus weakest-link. Econ Inq (Forthcoming). doi:10.1111/ecin.12246Google Scholar
- Chowdhury SM, Lee D, Sheremeta RM (2013) Top guns may not fire: best-shot group contests with group-specific public good prizes. J Econ Behav Org 92:94–103CrossRefGoogle Scholar
- Corchón LC (2007) The theory of contests: a survey. Rev Econ Des 11:69–100Google Scholar
- Cornes R (1993) Dyke maintenance and other stories: some neglected types of public goods. Q J Econ 108:259–271CrossRefGoogle Scholar
- Cornes R, Hartley R (2005) Asymmetric contests with general technologies. Econ Theory 26:923–946CrossRefGoogle Scholar
- Hirshleifer J (1983) From weakest-link to best-shot: the voluntary provision of public goods. Public Choice 41:371–386CrossRefGoogle Scholar
- Katz E, Nitzan S, Rosenberg J (1990) Rent seeking for pure public goods. Public Choice 65:49–60CrossRefGoogle Scholar
- Kolmar M, Rommeswinkel H (2013) Contests with group-specific public goods and complementarities in efforts. J Econ Behav Org 89:9–22CrossRefGoogle Scholar
- Konishi H, Le Breton M, Weber S (1999) On coalition-proof Nash equilibria in common agency games. J Econ Theory 85:122–139CrossRefGoogle Scholar
- Lee D (2012) Weakest-link contests with group-specific public good prizes. Eur J Pol Econ 28:238–248CrossRefGoogle Scholar
- Milgrom P, Roberts J (1996) Coalition-proofness and correlation with arbitrary communication possibilities. Games Econ Behav 17:113–128CrossRefGoogle Scholar
- Münster J (2007) Simultaneous inter- and intra-group conflicts. Econ Theory 32:333–352CrossRefGoogle Scholar
- Quartieri F (2013) Coalition-proofness under weak and strong Pareto dominance. Soc Choice Welf 40:553–579CrossRefGoogle Scholar
- Quartieri F, Shinohara R (2015) Coalition-proofness in a class of games with strategic substitutes. Int J Game Theory 44:785–813CrossRefGoogle Scholar
- Shinohara R (2005) Coalition-proofness and dominance relations. Econ Lett 89:174–179CrossRefGoogle Scholar
- Szidarovszky F, Okuguchi K (1997) On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games Econ Behav 18:135–140CrossRefGoogle Scholar
- Yamazaki T (2008) On the existence and uniqueness of pure-strategy Nash equilibrium in asymmetric rent-seeking contests. J Public Econ Theory 10:317–327CrossRefGoogle Scholar