Intra-group heterogeneity in collective contests

We can apply a modified version of the two-stage model studied in Katz and Tokatlidu (1996) to describe the case. The effort cost function of an individual in each stage is assumed to be nonlinear (increasing marginal costs), and the contest success functions assigned to each member for the second-stage within-group contest are asymmetric, reflecting the different political power of the members.

The typical assumption in the literature on collective contests has been that the individuals are symmetric within each of the competing groups. The effect of heterogeneity has been mainly confined to differences in group size or effort costs between groups, under alternative assumptions regarding the nature of the prize. Katz et al. (1990) and Riaz et al. (1995) have shown that a group with a larger membership does not attain a lower winning probability, if the contested prize is a group-specific pure public good. Considering a class of impure public-good prizes that includes the pure public-good prize and the pure private-good prize as special cases, Esteban and Ray (2001) derive a sufficient condition ensuring that larger groups have a higher winning probability. For a useful comprehensive survey of the literature, see Konrad (2009).

Sandler (1993) gives a detailed survey of this subject. With the exception of few papers, as Niou and Tan (2005), strategic interactions between the opposed allies are broadly ignored in the relevant literature.

In aggregative games, each player’s payoff is determined by own strategy and the sum of the strategies of all players. A classic example of such games is Cournot oligopoly, and many models of contests by individuals also belong to this class of games. Esteban and Ray (2001) have already adopted this approach (even previous to Cornes and Hartley) to analyse group contests, but their treatment heavily depends on the assumption that the members in each competing group are homogeneous.

We can generalize the model to the case where P and M are different among competing groups, but such an extension does not change the main results.

When we consider the model as a contest for a group specific public-good prize evaluated differently in a group, the stake \(v_{ik}\) is directly interpreted as the valuation of the public-good by member k of group i.

Even if we allow the members in a same group to have different effort cost functions, the results in this section (the existence and uniqueness of equilibrium) are still true.

When no group puts positive effort, we assume that each group attains the winning probability 1/m. Then, every member k in each group has the incentive to deviate by making a small positive effort because \(c_i (0)=0<v_{ik} \). Thus \(A = 0\) cannot be the total effort in equilibrium. The same conclusion is derived even if we assume that the winning probability is zero for each group when \(A = 0\).

Ryvkin (2011) uses another approach to derive existence and uniqueness of equilibrium for a contest model similar to ours. But his proof seems to critically depend on the restriction that the limit of the marginal effort cost at the zero effort level is not positive, i.e. \({c}^{\prime }_i(0)=0\) in terms of our notation. On the other hand, our main arguments are related to the case of \({c}^{\prime }_i (0)>0\). In fact we need the existence and uniqueness result allowing this case.

This is the case where \(c_i(a)=Ka^{\alpha +1}\) with K and \(\alpha \) being positive constants. Then the marginal effort cost function is strictly convex if and only if \(\alpha >1\).

Since \(\sum \nolimits _{k=1}^{N_i } {v_{ik} } =\sum \nolimits _{k=1}^{N_i } {{v}'_{ik} } =P+M\), the last condition is redundantly held in our model.

Notice that the share function of the group itself does not move as long as its properties (effort cost functions of the members and the stake vector) are constant.

Such a case is real. For example, let \(N_{1} = 3,\, P = 11,\, M = 24\), and the private-good prize be equally divided in group 1. If the effort cost function has the form \(c_1 (a)=\frac{a^3}{3}+\frac{3}{2}a\), we can set \(A^M( 1)=12\) (see the proof of Proposition 2 for the details), while the share function has positive values until A reaches to \(\frac{38}{3}\).

In the same framework, we can treat the case where excessive power of rivals is due to higher rival stakes. Let the effort cost function have the same form in group 2 as group 1, but \({\mathbf{v}}_2 =\frac{{\mathbf{v}}_1 }{\theta }\). Then the game is equivalent to that in the text, but now \(\theta \) is the parameter of the stake-size in group 2.

This is analogous to the arguments made by Bergstrom et al. (1986) in the context of voluntary provision of public goods. Under the assumption that the marginal rate of transformation between the private and public goods is unity, they have shown that an income transfer among individuals affects the provision of public goods if it is made between existing contributors and non-contributors. In particular, they have proved that equalizing income redistributions that involve any transfers from contributors to non-contributors will decrease the equilibrium supply of the public good.

In the collective contests with constant marginal costs discussed by Baik (2008), it is not unusual that the expected utilities of the highest-valuation members are the lowest. Due to the assumption of constant marginal costs, only the highest-valuation members in each group are the contributors. We can therefore easily construct such examples. Let member \(h\) have the highest valuation of the prize in group j, and let \(v_{jl} <v_{jh} \) for member l. Denote the constant marginal cost for the members in group j by \(\chi _{j}\). Also, let \(a_{jh}^*>0\) be the equilibrium effort level of member h, and \(p_{j}^{*}\) the winning probability of group j. Member l ends as a non-contributor. If the expected utility of member \(h,\, p_j^*v_{jh} -\chi _j a_{jh}^*\), is higher than that of member \(l,\, p_{j}^{*}v_{jl}\), raise the value of \(v_{jl}\) keeping the inequality \(v_{jl} < v_{jh}\). Since this operation does not change any individual’s effort level, we can eventually reach a situation in which member l attains a higher expected utility than member h.

Notice that this argument does not require interpersonal comparison of utilities.

As in the arguments in the last section, we do not need the actual existence of non-contributors to derive the results here. This is in sharp contrast to the derivation of the results in Baik (2008) and Olson and Zeckhauser (1966).

We could refer to two articles mentioned in the introduction. Also, Ryvkin (2011) uses the property in examining how to sort individuals in competing groups to maximize the total effort. See Esteban and Ray (1999, 2011a) on other important applications of this assumption.

For a comprehensive list of these conjectures, see Sandler (1992).

We can easily imagine the case where the private-part of the prize is thinly distributed among the very large number of group members, which makes the majority of them non-contributors.

Nitzan (1991), Lee (1995), Ueda (2002), Baik and Lee (2007), and Nitzan and Ueda (2011) examine the working of group sharing rules as an incentive scheme in collective contests.