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Social Choice and Welfare

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12.08.2020 | Original Paper

Effort complementarity and sharing rules in group contests

verfasst von: Katsuya Kobayashi, Hideo Konishi

Erschienen in: Social Choice and Welfare | Ausgabe 2/2021

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Abstract

In this paper, we consider a prize-sharing rule design problem in a group contest with effort complementarities within groups by employing a CES effort aggregator function. We derive the conditions for a monopolization rule that dominates an egalitarian rule if the objective of the rule design is to maximize the group’s winning probability. We find conditions under which the monopolization rule maximizes the group’s winning probability, while the egalitarian rule is strictly preferred by all members of the group. Without effort complementarity, there cannot be such a conflict of interest.

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This is called PNC (potential non-contributor condition) in Nitzan and Ueda (2014). In addition, this result needs a mild technical condition called RC (regularity condition) for the cost function around zero effort.

They also show that the egalitarian sharing rule maximizes the winning probability if the marginal cost starts from zero and is convex (their anti-Olson theorem).

Esteban and Ray (2001) clarify that Olson’s (1965) results do not depend on whether the prize is public or private. They explain that after fixing the equal allocation of the private good among all members in a group, whether its winning probability increases or not depends on the elasticity of the marginal effort cost when the population increases.

Choi et al. (2016) analyze a two-group contest when the prize is allocated by an intra-group contest between the group members with asymmetric powers. Cheikbossian and Fayat (2018) analyze a contest between two groups by introducing a rivalry parameter of an impure public good prize.

Kolmar and Rommeswinkel (2013) are the first in the literature to introduce group members’ effort complementarity in group contests by using a CES production function. They call this CES function a group impact function.

We employ the Tullock-form contest success function (Tullock 1980).

Nitzan and Ueda (2011) assume that individual effort levels are observable by the group leader and analyze the case of the convex combination between the egalitarian rule and a relative-effort sharing rule, which allocates the winning prize proportionally to their effort levels.

Baik (2008) calls this Nash equilibrium a group-i-specific equilibrium.

In a group contest for multiple (homogenous) indivisible prizes, Crutzen et al. (2020) obtain a closely related result. When $2r<\beta$, allocating prizes according to a fair lottery maximizes group success, while when $2r>\beta$, allocating prizes according to a predetermined (priority) list is better than the fair lottery allocation rule.

See also the “anti-Olson theorem” in Nitzan and Ueda (2014).

In Proposition 3, condition $1-\frac{n_{i}}{n_{i}+1}\beta >P_{iM}$ appears to be a condition for an endogenous variable $P_{iM}$ because $P_{iM}$ is explicitly unsolvable. However, $P_{iM}$ is uniquely led by

$$\begin{aligned} P_{iM}=\frac{P_{iM}^{\frac{1}{\beta }}(1-P_{iM})^{\frac{1}{\beta }}}{P_{iM}^{ \frac{1}{\beta }}(1-P_{iM})^{\frac{1}{\beta }}+X_{-i}}. \end{aligned}$$

This is determined by the exogenous variables $\beta$ and $X_{-i}$ only. Thus, we can confirm if condition $1-\frac{n_{i}}{n_{i}+1}\beta >P_{iM}$ is satisfied with economic data $\beta$, $n_{i}$, and $X_{-i}$.

Readers may think that it is unrealistic to assume that groups can observe other groups’ sharing rules. Nitzan and Ueda (2011) assume that sharing rules are the private information of each group and use perfect Bayesian equilibrium with the same beliefs for other groups’ sharing rules at every information set. Since the model does not involve a real asymmetric information problem, their perfect Bayesian equilibrium coincides with our subgame perfect equilibrium under complete information.

We thank Kaoru Ueda for suggesting that we use the share function approach.

When the group leader chooses the monopolization rule at Stage 1, effort complementarity is irrelevant on the equilibrium path. Effort complementarity is in effect only off the equilibrium path.

Baik KH (2008) Contests with group-specific public-good prizes. Soc Choice Welf 30:103–117CrossRef

Cheikbossian G, Fayat R (2018) Group size, collective action and complementarities in efforts. Econ Lett 168:77–81CrossRef

Choi JP, Chowdhury SM, Kim J (2016) Group contests with internal conflict and power asymmetry. Scand J Econ 118:816–840CrossRef

Cornes R, Hartley R (2005) Asymmetric contests with general technologies. Econ Theor 26–4:923–946CrossRef

Crutzen BSY, Flamand S, Sahuguet N (2020) A model of a team contest, with an application to incentives under list proportional representation. J Public Econ 182:104109CrossRef

Epstein GS, Mealem Y (2009) Group specific public goods, orchestration of interest groups with free riding. Public Choice 139(3–4):357–369CrossRef

Esteban J, Ray D (2001) Collective action and the group size paradox. Am Polit Sci Rev 95–3:663–672CrossRef

Kolmar M, Rommeswinkel H (2013) Contests with group-specific public goods and complementarities in efforts. J Econ Behav Organ 89:9–22CrossRef

Nitzan S, Ueda K (2011) Prize sharing in collective contests. Eur Econ Rev 55:678–687CrossRef

Nitzan S, Ueda K (2014) Intra-group heterogeneity in collective contests. Soc Choice Welf 43:219–238CrossRef

Olson M (1965) The logic of collective action. Harvard University Press, Cambridge

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 University Press, College Station, pp 97–112

Ueda K (2002) Oligopolization in collective rent-seeking. Soc Choice Welf 19–3:613–626CrossRef

Titel: Effort complementarity and sharing rules in group contests
verfasst von: Katsuya Kobayashi
Hideo Konishi
Publikationsdatum: 12.08.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 2/2021
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-020-01277-9

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