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 \(0.\)