Equilibrium contingent compensation in contests with delegation.

1. Introduction

Consider a lawsuit between a plaintiff and a defendant. Each litigant first hires an attorney and writes a contract with him. Then, each attorney expends his effort to win the lawsuit on behalf of his client. Because the outcome of the lawsuit depends on the attorney's effort, which in turn depends on the contract, the litigant must take into account the strategic aspects of contracts when designing her contract.

The purpose of this paper is to consider contests with delegation, like the illustrative example above, focusing on equilibrium contracts. (1,2) Specifically, we consider contests in which two players each want to win a prize, and each player hires a delegate who expends his effort to win the prize on the player's behalf. We endogenize delegation contracts between the players and their delegates while explicitly taking into account the delegates' participation constraints based on their reservation wages.

Contests with delegation abound. Examples include litigation in which litigants hire lawyers to win lawsuits; rent-seeking contests in which firms, organizations, or individuals hire lobbyists to acquire government favors and business; and research and development contests in which firms hire research groups or university professors to obtain patents.

We consider two-player contests with bilateral delegation. The players are risk-neutral, and Player 1 values the prize more highly than Player 2. The players design and provide compensation schemes for their delegates. The delegates are risk-neutral. They have the same nonnegative reservation wage, and have equal ability for the contest. The delegates' effort is not verifiable to a third party, which implies that contracts contingent on the delegates' effort are precluded. We assume that each delegate's compensation is contingent on the outcome of the contest--it depends on whether he wins or loses the prize.

We formally consider the following two-stage game. In the first stage, each player hires a delegate and writes a contract with him. The contract specifies how much the delegate will be paid if he wins the prize and how much if he loses it. Then the players simultaneously announce the contracts written independently. In the second stage, after knowing both contracts, the delegates choose their effort levels simultaneously and independently. At the end of the second stage, the winner is determined and each player pays compensation to her delegate according to the contract written in the first stage.

Fershtman and Kalai (1997) distinguish between two types of delegation: incentive delegation and instructive delegation. In the case of incentive delegation, a player provides an incentive scheme for her delegate, and the delegate chooses an effort level that maximizes his own payoff, given the incentive scheme. In the case of instructive delegation, a player designs a set of instructions and requires her delegate to follow the instructions. According to this classification, then, this paper adopts incentive delegation. The players in this paper provide compensation schemes for their delegates that are based on the observables, and the delegates choose their effort levels given the compensation schemes.

Solving for the subgame-perfect equilibrium of the two-stage game, we obtain the equilibrium contracts between the players and their delegates, and show that each player's equilibrium contract is a no-win-no-pay contract--a contract that specifies zero compensation for a delegate if he loses the prize. Then, we examine the delegates' equilibrium compensation spreads, effort levels, probabilities of winning, expected payoffs, and the players' equilibrium expected payoffs. We define a delegate's compensation spread as the difference between what he earns if he wins the prize and what he earns if he loses it.

We obtain the result of no-win-no-pay contracts because of the constraint that a delegate's compensation should not be negative if he loses the prize, and the assumption that the delegates are risk-neutral. The result of no-win-no-pay contracts makes intuitive sense. By choosing such a contract, each player makes her delegate's compensation spread as wide as possible so that she can most strongly motivate her delegate to win the prize.

Another interesting result is that when a delegate's participation constraint is not binding in equilibrium, his equilibrium expected payoff is greater than his reservation wage. Recall that economic rent is defined as that part of the compensation received by the owner of a resource that exceeds the resource's opportunity cost. Then we may say that the gap between the delegate's equilibrium expected payoff and his reservation wage constitutes the economic rent for the delegate. This economic rent is not created because of restrictions on entry into the "delegate industry," but created because of both the inability to write contracts based on a delegate's effort and the players' strategic decisions on their delegates' compensation. Indeed, competition among potential delegates to become this particular delegate, if any, cannot reduce the delegate's equilibrium expected payoff to his reservation wage.

We also obtain: (i) Delegate 1's compensation spread is greater than Delegate 2's, and (ii) the equilibrium expected payoff for Delegate 1 is greater than that for Delegate 2. These occur unless both delegates' participation constraints are binding in the subgame-perfect equilibrium. Part (i) implies that the player with a higher valuation--the hungrier player--offers her delegate better contingent compensation than her opponent does. Part (ii) is very interesting because the delegates are identical before signing up for their players: They have equal ability for the contest and have the same reservation wage. The difference in the delegates' expected payoffs arises because of the inability to write contracts based on the delegates' effort and because Player 1 motivates her delegate more strongly than Player 2--that is, Delegate l's compensation spread is greater than Delegate 2's. In this case, even though there exists competition among potential delegates to be employed by Player 1, it cannot lead to the same expected payoff for the delegates.

The assumption that the delegates' effort is not verifiable to a third party--which implies the inability to write contracts based on the delegates' effort--is crucial in obtaining the result that the economic rents for the delegates exist. Indeed, the economic rent for each delegate exists because the delegate's effort is his private information. In this respect, the economic rent for each delegate can be interpreted as an informational rent, which is a well-known concept in the principal-agent literature. (3)

There are two main motives of delegation. The first is that a player wants to use superior ability by hiring a delegate who has more ability than herself; the second is that a player wants to achieve strategic commitments through delegation. Baik and Kim (1997) first introduced delegation into the literature on the theory of contests. They present a model that involves both motives of delegation. Considering two-player contests in which each player has the option of hiring a delegate, they first establish that buying superior ability is an important motive of delegation. They then show that, as compared with the model without delegation, a total effort level is less when unilateral delegation by the player with a higher valuation or bilateral delegation arises, but it is greater when unilateral delegation by the player with a lower valuation arises. However, they assume that the delegation contracts are exogenously given, and assume implicitly that each delegate's reservation wage is zero. Warneryd (2000) considers two-player contests with bilateral delegation. He shows that compulsory delegation with moral hazard--that is, where the delegates' effort is unobservable--may be beneficial to the players. He also shows that this result holds even when secret renegotiation opportunities are given to the players and delegates. Schoonbeek (2002) considers a two-player contest in which only one player, say Player 1, has the option of hiring a delegate. He compares the equilibrium expected utility of Player 1 in the unilateral-delegation case with that in the no-delegation case, focusing on the impact of the risk aversion of Player 1 with respect to her money income. Konrad, Peters, and Warneryd (2004) consider a first-price all-pay auction with two buyers in which each buyer has the option of hiring an agent. They show that in equilibrium each buyer delegates the bidding to her agent; and both buyers are better off. They also show that the buyers provide their agents with incentives to make bids that differ from the bids the buyers would like to make, and the delegation contracts are asymmetric even if the buyers and the auction are perfectly symmetric.

The paper proceeds as follows. In section 2, we develop the model and set up the two-stage game. We then obtain a unique Nash equilibrium of a second-stage subgame. In section 3, we analyze the first stage of the two-stage game. We first show that each player writes a no-win-no-pay contract with her delegate. Then we obtain the equilibrium contracts chosen by the players. Section 4 examines the delegates' equilibrium compensation spreads, effort levels, probabilities of winning, their equilibrium expected payoffs, and the players' equilibrium expected payoffs. Finally, section 5 offers our conclusions.

2. The Model