Strategic choice of sharing rules in collective contests

Starting with Nitzan (1991), the literature has considered both exogenous and endogenous sharing rules, while it has assumed that the choice of such rules may occur under either public or private information. For a recent survey on prize-sharing rules in collective rent seeking, see Flamand and Troumpounis (2015).

These transfers are analogous to the ones proposed in individual contests by Hillman and Riley (1989). Recent literature has considered cost-sharing in collective contests for purely public prizes, which can also be interpreted as within-group transfer schemes (Nitzan and Ueda 2014b; Vazquez 2014). Nitzan and Ueda (2014b) provide several examples of situations involving transfers among members of a group, in contexts such as labor unions, ethnic conflicts, or academic institutions.

While the two previous works provide further results on the influence of a convex cost of effort on the elimination of the GSP, the introduction of a strategic choice of sharing rules obliges us to restrict ourselves to the linear cost case. A convex cost of effort penalizes higher levels of individual effort, hence it works against the occurrence of the GSP. Corchón (2007) provides an intuition for the latter: “Intuitively, it is clear that Olson’s conjecture cannot hold if costs rise very quickly with effort: for instance if costs are zero up to a point, say \({\bar{G}}\) where they jump to infinity, all agents will make effort \({\bar{G}}\) and smaller groups will exert less effort than large ones.” Our conjecture is then that monopolization should also hold with convex costs, while the GSP should be less likely.

In fact, in the literature on collective rent seeking and sharing rules the assumption of perfect information as in this paper is the usual one. To the best of our knowledge, the only papers analyzing the case of private information are Baik and Lee (2007, 2012), Nitzan and Ueda (2011), Baik (2014) and Nitzan and Ueda (2014b). Monopolization never arises in that context, even when considering intermediate degrees of privateness of the contested prize.

Relaxing the property of non-rivalry by assuming that the public part of the prize is congestionable does not qualitatively alter our results (see footnotes 16 and 17).

We abstract from the possibility of intra-group heterogeneity regarding lobbying effectivity, which can be reflected in different valuations of the prize by the members. One can ask whether a group whose members have highly unequal interests in the collective action will be more or less active. The literature has provided contrasted answers to the latter question, due to the difference in the assumed form of the effort cost function [see the discussion in Nitzan and Ueda (2014a) and the references they cite]. Assuming very weak and plausible restrictions on the form of the effort cost functions, Nitzan and Ueda (2014a) show that if a group competes for a prize with sufficiently many rivals or with a very superior rival, unequal stakes among the members can enhance its performance.

Observe that \(1/n_i\) substitutes \(e_{ki}/E_i\) in (1) to avoid an indeterminacy for \(E_i=0\).

Cost-sharing in collective contests for purely public prizes can also be interpreted in terms of within-group transfers (Nitzan and Ueda 2014b; Vazquez 2014).

Pecorino and Temimi (2008) modify the model of Esteban and Ray (2001) to a standard public good setting, and show that their results are robust to the presence of small fixed costs of participation in the case of a pure public good, but not in the case of a fully rival good. Furthermore, when the degree of rivalry is sufficiently high, the introduction of small fixed costs of participation implies that collective action must break down in large groups.

This case is also equivalent to the collective contest with pure public goods studied by Baik (2008).

Notice that although Proposition 2 considers the cases of \(p=0\) and \(p=(0,1]\) separately, there is no discontinuity between the two cases. One can easily verify that independently of the occurrence of the GSP, the winning probability of both groups converges to one half as p approaches zero. The reason we chose to separate between the two cases is that when the prize is a pure public good (i.e., \(p=0\)), expression (2) is not necessary since the sharing rules do not apply.

As the aggregate welfare of group i only depends on aggregate effort, and as we know from Proposition 1 that in equilibrium aggregate effort is unique for any \(\alpha _i\), it follows that the sharing rule \(\alpha _i\) that maximizes the expected utility of the representative individual in a within-group symmetric equilibrium also maximizes \(\sum _{k\in i} EU_{ki}\).

The results for \(p=0\) are not analyzed here. They can be obtained from Proposition 1.

On pure public goods, see also Riaz et al. (1995), Ursprung (1990) and Katz et al. (1990), among others. They have shown that with a pure public good, a group with larger membership attains a winning probability larger than or equal to that of a smaller group. Neither monopolization nor the GSP arise in such cases.

This discontinuity might strike some readers as puzzling as it cannot arise in a continuous game, by upper hemicontinuity property of the equilibrium correspondence. However, it is worth emphasizing that two different games are played for \(p=0\) or \(p>0\). Indeed, for \(p>0\) we are solving a two-stage game where sharing rules are chosen prior to the effort stage (with an action space \({\mathbb {R}}_+^{2+n_A+n_B}\)). In contrast, for \(p=0\) sharing rules do not apply so that one can focus only on the single-stage effort game (with action space \({\mathbb {R}}_+^{n_A+n_B}\)). This difference in the definition of the game and the absence of the strategic choice of sharing rules in the case of a pure public good allow for the discontinuity.

Notice that the numerical value of \(p_1\) is large and very close to one as the size of the groups increases, and/or when there is a large difference between group sizes. For instance, if \(n_A=15\) and \(n_B=7\) then \(p_1=0.98\). However, this critical value of the degree of privateness can be rescaled by assuming that the public good is congestible. This could be captured by a parameter \(c\in [0,1)\) such that the utility attributed to the public good by any member of group i is \((1-p)V/n_i^c\). In that case the critical value \(p_1\) is given by \(p_1^c=\frac{n_B \left[ n_A^c (1 + n_B)-n_A n_B^c\right] }{n_A^c n_B (1 + n_B) - n_B^c (n_A^c + n_A n_B)}\), which for the previous example rescales \(p_1\) to \(p_1^c=0.76\) for \(c=0.6\). In order to avoid unnecessary complications in the main text, we consider the case of \(c=0\).

The numerical value of \(p_{GSP}\) is large and very close to one as the size of the groups increases, and/or when there is a large difference between group sizes. For instance, if \(n_A=15\) and \(n_B=7\) then \(p_{GSP}=0.99\). Again, this value can be rescaled if we assume that the public good is congestible. In that case \(p_{GSP}^c=\frac{n_A n_B [n_B^c+2 n_A n_B^c-n_A^c (1 + 2 n_B)]}{n_B^c[n_A^c (n_A - n_B) + n_A (1 + 2 n_A) n_B]-n_A^{1+c} n_B (1 + 2 n_B)}\), which for the previous example yields \(p_{GSP}^c=0.95\) for \(c=0.6\).

To avoid introducing more notation, we do not define formally the thresholds \(p_2\) and \(p_3\) presented in the interpretation of our results. They can be easily calculated and one can show that \(p_1<p_2<p_3<p_{GSP}\).

The equilibrium (restricted) sharing rules for the case of n groups are provided by Ueda (2002).

The analytical expression of the threshold \(\mathring{p}\) can be found in the Appendix.

Two more possible equilibria characterized by the equilibrium sharing rules presented in Propositions 3 and 5 are the ones such that (i) only the large group decides to restrict the choice of its sharing rule (if p is large enough) and (ii) only the small group decides to restrict the choice of its sharing rule (if \(p<p_1\)).