Equilibrium Aggregates in Common-Agency Settings

This note outlines the approach I am taking to characterizing the set of equilibrium aggregates in our common-agency setting. The analysis proceeds in several steps. First, I follow Martimort and Stole (Games and Economic Behavior, 2012) and derive a self-generating maximization (SGM) program, the solutions of which coincide with the set of equilibrium aggregate contracts. I then derive some intermediate results that simplify the constraint set in the SGM program and derive some of its properties, and then I provide some sufficient conditions under which the operator defined by the SGM program is monotone.

The Model

An agent $$A$$ chooses an effort level $$e$$ from some set $$E \subset \mathbb{R}$$ at cost $$c\left( e \right)$$ that induces a measure $$\phi \left( e \right)$$ over a set of outcomes $$Y$$. There are $$N$$ principals $$\left( {{P_1}, \ldots ,{P_N}} \right)$$ who receive benefits $${B_1}\left( y \right), \ldots ,{B_N}\left( y \right)$$, the distribution of which depends on $$e$$. The principals simultaneously offer contracts $${w_i}$$ from some set of contracts $$W$$. Throughout, we will focus on environments in which $$W$$ is a convex cone. That is, if two contracts $$w$$ and $$w'$$ are in $$W$$, then for any $$\alpha ,\beta \ge 0$$, the contract $$\alpha w + \beta w'$$ is also in $$W$$. $$A$$ can either accept all contracts or reject all in favor of outside option yielding utility $$0$$. If he accepts, then his incentives are given by the aggregate contract $$w = {w_1} + \ldots + {w_N}$$. Notice that $$w \in W$$. The agent then chooses an effort level to solve ${e^*}\left( w \right) = \max \mathop {\arg \max }\limits_{e \in E} w \cdot \phi \left( e \right) - c\left( e \right),$where $$w \cdot \phi \left( e \right) \equiv \int\limits_Y {w\left( y \right)d\phi \left( {y,e} \right)} {\rm{ }}$$. I am assuming here that if the agent is indifferent, he chooses the highest effort level that he is indifferent among.

An equilibrium of this game is a set of equilibrium contracts $${w_1}, \ldots ,{w_N}$$ and an equilibrium effort level $$e$$ such that $$e = {e^*}\left( {{w_1} + \ldots + {w_N}} \right)$$ and given $${w_{ - i}}$$, $${w_i}$$ is optimal for principal $$i$$. An equilibrium aggregate contract is a contract $$w \in W$$ such that there are equilibrium contracts $${w_1}, \ldots ,{w_N}$$ with $${w_1} + \ldots + {w_N} = w$$.

For most of the results, I will make the following assumption of symmetric benefits across principals.

Assumption S: Principals' benefits are symmetric. That is, $${B_1} = \ldots = {B_N}$$.

Step 1: Self-Generating Maximization (SGM) Program

This section shows that when Assumption S holds, $$\bar w$$ is an equilibrium aggregate contract if and only if it solves a self-generating maximization (SGM) program. This result is not directly implied by the Martimort and Stole (2012) characterization, because I allow for a class of restrictions on the set of contracts. Define the objective $\Lambda \left( {w,\bar w} \right) = \left( {B - Nw + \left( {N - 1} \right)\bar w} \right) \cdot \phi \left( {{e^*}\left( w \right)} \right).$Lemma 4 provides necessary and sufficient conditions for $$\bar w$$ to be an equilibrium aggregate contract.

Lemma 4: Suppose Assumption S holds. Then $$\bar w \in W$$ is an equilibrium aggregate contract if and only if $\bar w \in \mathop {\arg \max }\limits_{w \in W + \left( {1 - 1/N} \right)\bar w} \Lambda \left( {w,\bar w} \right).$

Step 2: Properties of SGM Program

We can now establish properties of the SGM program. Define a cost-minimizing contract that implements effort $$e$$ by $$w_e^*$$, which solves $w_e^* \in \mathop {\arg \min }\limits_{w \in W} w \cdot \phi \left( e \right)$subject to incentive compatibility $e \in \mathop {\arg \max }\limits_{\tilde e \in E} w \cdot \phi \left( {\tilde e} \right) - c\left( {\tilde e} \right)$and individual rationality $w \cdot \phi \left( e \right) - c\left( e \right) \ge 0.$The next lemma shows that all equilibrium aggregate contracts are cost-minimizing contracts.

Lemma 5: Suppose $$\bar w$$ solves $\mathop {\max }\limits_{w \in W + \left( {1 - 1/N} \right)\bar w} \Lambda \left( {w,\bar w} \right).$Then $$\bar w = w_e^*$$ for some $$e \in E$$.

Next, define the operator $\hat w\left( {\bar w} \right) = \mathop {\arg \max }\limits_{w \in W + \left( {1 - 1/N} \right)\bar w} \Lambda \left( {w,\bar w} \right).$We want to show that, under some conditions, $$\hat w\left( {\bar w} \right)$$ is a monotone operator. In order to define monotonicity in this setting, define the supplement partial ordering on contracts $${ \ge _W}$$ by $$w{ \ge _W}w'$$ if $$w = w' + \tilde w$$ for some $$\tilde w \in W$$. That is, $$w$$ supplements $$w'$$ if $$w$$ is the sum of $$w'$$ and a feasible contract. We ultimately would like to show that if $$\bar w{ \ge _W}\bar w'$$, and $$w \in \hat w\left( {\bar w} \right)$$ and $$w' \in \hat w\left( {\bar w'} \right)$$, then $$w = w' + \tilde w$$ for some $$\tilde w \in W$$.

If this operator is indeed monotone, then we can define a compact subset $$\bar W$$ of $$W$$ such that $$\hat w\left( {\bar w} \right)$$ is a monotone operator on a compact space, and therefore by an extension of Topkis's fixed point theorem, the set of fixed points $${W^*}$$ of this operator forms a complete lattice. Further, we know that if Assumption S is satisfied, then $${W^*}$$ coincides with the set of equilibrium aggregate contracts, and from Lemma 5, all equilibrium aggregate contracts are cost-minimizing contracts.

Finally, if the set of cost-minimizing contracts is ordered by the supplement partial ordering (i.e., if $$e \ge e'$$ implies that $$w_e^* = w_{e'}^* + \tilde w$$ for some $$\tilde w \in W$$), then we can define the operator $\hat e\left( {\bar e} \right) = \mathop {\arg \max }\limits_{e \in E} \Lambda \left( {w_e^*,w_{\bar e}^*} \right).$I conjecture that the contracts corresponding to the fixed points of this operator coincide with the fixed points of the operator $$\hat w\left( {\bar w} \right)$$. If this is the case, the problem can be simplified greatly.

A Class of Examples

In this section, I develop an example in which cost-minimizing contracts are ordered by the supplement partial ordering, and the operator $$\hat w\left( {\bar w} \right)$$ is monotone. Let $$W = \left\{ {w \ge 0:y \ge y' \Rightarrow w\left( y \right) \ge w\left( {y'} \right)} \right\}$$ be the set of non-negative, weakly increasing contracts on $$Y$$. Let $$Y = \left\{ {0,1} \right\}$$, $$E = \left[ {0,1} \right]$$, and $$\Pr \left[ {y = 1|e} \right] = e$$. I will take no stance on the functional form of $$c\left( e \right)$$ except that it is (weakly) increasing.

Define $${E^{feas}} \subset E$$ to be the set of $$e$$ for which there is some $$w \in W$$ such that $w \cdot \phi \left( e \right) - c\left( e \right) \ge w \cdot \phi \left( {e'} \right) - c\left( {e'} \right)\,\,for\,all\,e' \in E.$Define $$\tilde c\left( e \right)$$ to be the largest convex function such that $$\tilde c\left( e \right) \le c\left( e \right)$$ for all $$e \in E$$. Then $${E^{feas}} = \left\{ {e \in E:\tilde c\left( e \right) = c\left( e \right)} \right\}$$.

When there are only two possible output levels, it is straightforward to characterize the set of cost-minimizing contracts. For all $$e \in {E^{feas}},$$ $$w_e^*\left( 0 \right) = 0$$, and $$\tilde c{'^ - }\left( e \right) \le w_e^*\left( 1 \right) \le \tilde c{'^ + }\left( e \right)$$, where $$\tilde c{'^ - }\left( e \right)$$ and $$\tilde c{'^ + }\left( e \right)$$ are the left and right derivatives of $$\tilde c$$ at $$e$$.

Given a proposed aggregate contract $$\bar w \in W$$, the minimal effort level that can be implemented is defined as the $${e_{\min }}\left( {\bar w} \right)$$ that satisfies $$\tilde c{'^ - }\left( {{e_{\min }}\left( {\bar w} \right)} \right) \le \left( {1 - 1/N} \right)\left( {\bar w\left( 1 \right) - \bar w\left( 0 \right)} \right) \le \tilde c{'^ + }\left( {{e_{\min }}\left( {\bar w} \right)} \right).$$ Given $$\bar w$$, define $$E_{\bar w}^{feas} = {E^{feas}} \cap \left[ {{e_{\min }}\left( {\bar w} \right),1} \right]$$ to be the set of feasible effort levels relative to $$\bar w$$. Next, define $${C_{\bar w}}\left( e \right) = \left( {1 - 1/N} \right)\left( {\bar w\left( 1 \right) - \bar w\left( 0 \right)} \right)e + \left( {1 - 1/N} \right)\bar w\left( 1 \right)$$ if $$e = {e_{\min }}\left( {\bar w} \right)$$ and $${C_{\bar w}}\left( e \right) = C\left( e \right) + \left( {1 - 1/N} \right)\bar w\left( 1 \right)$$ if $$e > {e_{\min }}\left( {\bar w} \right).$$ Finally, define the operator$\hat e\left( {\bar e} \right) = \mathop {\arg \max }\limits_{e \ge {e_{\min }}\left( {w_{\bar e}^*} \right)} \left( {B + \left( {N - 1} \right)w_{\bar e}^*} \right) \cdot \phi \left( e \right) - N{C_{w_{\bar e}^*}}\left( e \right).$

Theorem 1: $$\bar e$$ is an equilibrium effort level if and only if $$\bar e \in \hat e\left( {\bar e} \right)$$.

We can therefore focus on the much simpler problem of solving for the fixed points of $$\hat e\left( {\bar e} \right).$$  Further, $$\hat e\left( {\bar e} \right)$$ has some nice properties, which the following proposition shows.

Proposition 1: $$\left( {B + \left( {N - 1} \right)w_{\bar e}^*} \right) \cdot \phi \left( e \right) - N{C_{w_{\bar e}^*}}\left( e \right)$$ satisfies increasing differences in $$\left( {e,\bar e} \right)$$, and $${e_{\min }}\left( {w_{\bar e}^*} \right)$$ is increasing in $$\bar e$$.

By Topkis's theorem, we therefore have that $$\hat e\left( {\bar e} \right)$$ defines a monotone operator on a compact space. The intuition behind this proposition is that, given any cost-minimizing target contract $$w_{\bar e}^*$$, each principal either wants to leave $$\left( {1 - 1/N} \right)w_{\bar e}^*$$ in place by contributing $$w = 0$$, or they want to top up $$\left( {1 - 1/N} \right)w_{\bar e}^*.$$ If they choose to top it up, they will top it up to a cost-minimizing contract, which is feasible, because $$w_{\bar e}^*$$ is increasing in $$e$$. As a result of the monotonicity of $$\hat e\left( {\bar e} \right)$$, by an extension of Tarski's fixed-point theorem, there is at least one fixed point, and the set of fixed points forms a complete lattice.

Theorem 2: The set of equilibrium effort levels $${E^*}$$ is a complete lattice with least and greatest elements $$e_L^*$$ and $$e_H^*$$.

Further, the program described by the operator $$\hat e\left( {\bar e} \right)$$ can be solved explicitly given any (weakly increasing) cost function $$c\left( e \right)$$ and aggregate benefits $$B$$. We can therefore ask when it is the case that $$e_L^* < e_H^* < {e^{SB}}$$ so that the worst equilibrium effort level is strictly worse than the second-best equilibrium effort level. In particular, given $$B$$, I think we will be able to say that for any $$c$$ such that $$\tilde c$$ has points $$e < e_H^*$$ at which it is non-differentiable, for sufficiently large $$N$$, then the smallest such point is an equilibrium effort level. This is satisfied, for instance, by $$c$$ such that $$c\left( 0 \right) = 0$$ and $$c\left( e \right) = F + \frac{c}{2}{e^2},$$ in which case $$\tilde c\left( e \right)$$ is not differentiable at $$0$$. If this conjecture is correct, it is actually nondifferentiability of $$\tilde c$$ , rather than nonconvexities in $$c$$, that leads to coordination failures. In some cases, such as the example with fixed costs, nonconvexities in $$c$$ lead to nondifferentiabilities in $$\tilde c$$, but they are neither necessary nor sufficient. They are not necessary, because our initial example with $$E = \left\{ {0,1} \right\}$$, $$\tilde c$$ is linear (and therefore convex), but it is not differentiable at $$0.$$