Real life disputes, negotiations and competitive situations involve multi-issue considerations in which the final outcome depends on the aggregated effort over several dimensions. We consider two allocation systems, the I-system, in which each issue is disputed and award independently, and the A-system, in which all issues are aggregate in a single prize award. In the A-system, we propose a contest success function that aggregates the individuals’ multi-issue efforts in a single outcome. Among other results, we found that the A-system tends to induce higher total effort than the I-system. The model is also able to reproduce a large set of strategic behaviors. For instance, under decreasing returns to effort, individuals maximize their payoffs by distributing effort over all issues, while under increasing returns to effort, individuals focus on a single issue. Hybrid equilibria, in which one individual focus in a single issue while the other individual diversifies effort over all issues, may also emerge when individuals hold different returns to effort. Strategic behavior is simultaneously influenced by the weight of each issue on the final outcome and by comparative advantages. Throughout the manuscript, we link our results with strategic behavior observed in electoral competition, i.e., “issue ownership”,“issue divergence/convergence”and“common value issues”. We expect that our findings will help researchers and practitioners to better understand the process of endogenous selection of issues in competitive contexts and to provide guidance in the implementation of the optimal allocation mechanism.
In the main text of this paper, in order to simplify the analysis and to provide better intuition, we have focused in the two individuals and two issues case. In this Appendix, we briefly consider the general model with an arbitrary number of issues. In this context, we should introduce extra notation to distinguish between individuals and issues.
A.1 The general model
Let denotes the effort of individual in issue under the system Let denote the ability of individual in issue Recall that denotes the prize of issue and denotes the aggregated prize.
In this context, under the multi-issue -system, the probability that individual wins by providing effort in all (or some) issues is given by:
Under the multi-issue -system, each individual simultaneously chooses a profile of efforts, for that maximizes the expected payoff net of the cost of effort, which is given by:
In order to have a sufficiently tractable -system model, we do the following simplifying assumptions: and with for all and and
A.2 General results and proofs
In the case the solution to the general problem eq. (16) is framed in the following result.
Forthe individualsequilibrium efforts in each issueare given by:
[Proof of Proposition 10 (and Propositions 3 and 4)] The proof of Proposition 3 is just a particular case of this proof. From the problem 16, the associated set of first order conditions is given by:
for and Under for all and and , after some algebra on the system of first order conditions we obtain that:
for which corresponds to independent equations. Consequently, we need two additional equations in order to solve the system. One such equation is obtained by noticing that since individuals are not budget constrained, and the total prize and the unit cost of effort are the same, in equilibrium we must have: This equality relation can be easily shown by rewriting the system of first order conditions eq. (18) in the form:
and, then summing over all issues for each The last independent equation is obtained by any of the first order conditions in the system eq. (18). After some algebra, on this system of equations, we obtain the unique solution given by expression eq. (17), from where expression eq. (8) is a particular case. Now, we need to verify under which conditions such solution corresponds to a Nash equilibrium (Pérez-Castrillo and Verdier 1992; Szidarovszky and Okuguchi 1997). Since the second derivative of each first order condition is strictly negative, i.e.:
then, is strictly concave for In addition, since the vector is defined on a convex space, then the first order condition is simultaneously necessary and sufficient for a maximum. Finally, since the effort in expression eq. (17) is positive we are left to show that in equilibrium Note that the winning probability of individual is given by: and the individual total effort is given by: (i.e., ). Consequently, after replacing these expressions into we obtain the expected payoff:
for Since effort does not create value, the minimum payoff is obtained when effort is maximal, i.e., at (see Proposition 4). Then, it is easy to show that participation is guaranteed for which is always true for
The proof of Proposition 4 is just a particular case of the following more general proof. Since in equilibrium each individual provides the same aggregate effort, i.e., and Then, after some algebra we obtain which is equivalent to expression eq. (9). In order to study whether the aggregate effort increases with the ability simply differentiate with respect to to obtain that:
is strictly positive if because for all and and the opposite otherwise. Since is twice continuously differentiable, the maximal effort occurs at At this point, the Hermitian matrix of is negative-semidefinite, i.e., with non-positive eigenvalues, which are either or
In the case individuals place all effort in a single issue - the one in which they have highest ability. Let denotes the issue in which individual has highest ability.
the individuals equilibrium efforts in each issueare given by:
Proof of Proposition 11 (and Proposition 5). The proof of Proposition 5 is just a particular case of this proof. Note that for in order for the sequence of equalities eq. (19) obtained from the first order condition eq. (18) to be satisfied, the issues with larger ability must receive lower effort. However, such behavior cannot be an equilibrium because since the effort output is convex in and the marginal utility from effort increases. Consequently, optimal behavior requires that each individual places effort in a single issue, i.e., the one with largest abilities (see the discussion in Section 5.1). Let denotes the issue in which individual has the highest ability. Under for all and and individual chooses to maximize eq. (16) and set for all The associated system of two first order conditions is given by:
for The solution returns the unique, symmetric and strictly positive equilibrium effort given in eq. (21), from where expression eq. (11) is a particular case. The expected payoff is non-negative if for  The individual second order condition for a maximum is given by:
which is negative if This condition is implied by the more restrictive non-negative expected payoff condition for This inequality corresponds to the existence condition eq. (20), from where the condition eq. (10) is a particular case.
Financial support from the Barcelona Graduate School of Economics and Univ. Rovira i Virgili are gratefully acknowledged. I would like to thank Matthias Dahm, Sabine Flamand, David Pérez-Castrillo, Ricardo Ribeiro and Santiago Sanchez-Pages as well as several seminars and congresses participants for helpful comments and discussions. All remaining errors are mine.
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