Assortative Matching by Lottery Contests

In two-sided matching contests, two contests take place independently within two groups. At the end of these contests, the agents in both groups are assortatively matched according to their efforts and the efforts of the other agents. Then, the prize of each pair who are matched is a function of their types. Two-sided matching contests can be observed, for example, in academic life, in which one of the groups includes universities that invest in hiring the best researchers and teachers as well as in providing the best learning conditions for the students. Such an investment improves its rank and thus will attract better candidates. The other group includes potential student candidates who aspire to be admitted to higher education universities and for this purpose put forth their best efforts in studying for entrance exams, acquiring recommendations, etc. Subsequently, candidates with the best qualities will be admitted to the highest ranked universities. Similar two-sided matching contests can be seen among accounting or law students on the one side and firms on the other, or among models, actors, and artists on the one side and talent agencies on the other.

Two-sided matching contests may involve incentive problems for the designer as well as for the agents. For example, Hoppe, Moldovanu, and Sela studied two-sided matching contests with incomplete information (marginal cost of effort), and compared random matching (without agents’ efforts) to assortative matching (based on agents’ wasteful efforts) in terms of total expected net welfare [1]. They showed that for distribution functions having a decreasing failure rate, assortative matching with wasteful efforts is welfare-superior, while for distribution functions having an increasing failure rate, random matching is superior. Furthermore, they also showed that each agent may be better off under random matching. We, on the other hand, assume that the agents’ efforts are productive for the designer such that these efforts positively affect the net welfare. This means that the benefit from the agents’ efforts is larger than their costs for the agents. For example, in Spence’s model, investment in education represents the agents’ efforts which are obviously not wasteful and have some benefit for the society’s welfare [2]. Then, when the agents’ efforts positively affect welfare, it is clear that assortative matching is welfare superior to random matching in which agents do not exert any effort at all. Therefore a designer’s goal is to activate the agents in order to obtain assortative matching instead of random matching. In this paper, we examine for what reasons agents choose to be non-active in two-sided assortative matching contests and when and how the designer can motivate them to become active.

For this purpose, we study a matching model under complete information where there are two sets of agents, set A with m heterogeneous firms and set B with $n,n\le m,$ heterogeneous workers, each of which has commonly known types. There is only one stage in which the two sets act simultaneously. The firms compete against each other in a Tullock contest [3], and at the same time, the workers compete against each other in another Tullock contest. The agents exert their efforts, and then are assortatively matched, namely, the winner in the contest of set A is matched with the winner in the contest of set B, and so on until all the agents in the set with the smaller number of agents (workers) are matched. The agents have match-value functions that are monotonically increasing in both types of firms and workers. An agent who is matched has a payoff of his match-value minus the cost of their effort. It is worth noting that although the agents’ types are commonly known, since the Tullock contest success function is stochastic, it is possible that a high-type firm from set A is matched with a lower-type worker from set B and vice versa, namely, a low-type firm will be matched with a high-type worker. This reflects real-life situations such as when the best university does not necessarily include all the best students nor all the best researchers. Likewise, students with lower ability may get higher grades than students with higher abilities. In other words, the ability (type) of an agent does not guarantee success in the matching contest.

We first claim that in the $n\times n$ assortative matching contest with any match-value functions, for every $n\ge 2$ there is a corner equilibrium in which the efforts of all the agents in both sets are zero, and therefore the agents are randomly matched and each firm (worker) has the same probability to be matched with each of the workers (firms). When the match-value functions are additive, we prove that the corner equilibrium in which all the agents are non-active is the unique symmetric equilibrium, and also every permutation of the vector of the agents’ equilibrium effort in each set is also in equilibrium. The reason behind this is that the additive function has mixed second order derivatives that are equal to zero, which yields that each agent wins a minimum value that is equal to their own type. As such, the agents in the same set actually have the same prizes based on the agents’ types in the other set, and therefore they are symmetric, and, furthermore, if they exert the same effort in equilibrium it is necessarily equal to zero. However, for different forms of the match-value functions, our matching contests may also have an interior equilibrium in which the agents are active and exert positive efforts, or, at least, a partial interior equilibrium in which some of the agents exert positive efforts. Since the explicit characterization of the agents’ equilibrium efforts on both sides might be very complex, we focus on 2 × 2 assortative matching contests in which there are two agents on each side. We first establish that there is an interior equilibrium by providing necessary and sufficient conditions on the match-value functions.1 In this case, there is no partial interior equilibrium in which some of the agents exert positive efforts. Then, for multiplicative match-value functions of the agents’ types, we have a unique interior equilibrium for which we explicitly characterize the equilibrium efforts. Then, we show that the larger the type of the agent is, the larger is their equilibrium effort.

We proceed by analyzing $m\times n$ assortative matching contests where the number of firms m is larger than the number of workers n. We claim that in a $m\times n$ assortative matching contest where $m>n$, at least n firms exert positive efforts in equilibrium. Furthermore, with additive match-value functions where $m>n$, at least $n+1$ firms exert positive efforts in equilibrium. We focus on 3 × 2 assortative matching contests for which we can explicitly characterize the agents’ equilibrium efforts and show that it is possible that all the agents on both sides are active. On the other hand, we also show that it is possible that all the agents (firms) on the larger side are active while the agents (workers) in the other side are not active. These results establish the existence of an interior equilibrium with positive efforts in $m\times n$ assortative matching contests, and, in particular, the non-existence of a corner equilibrium without any efforts. Hence, by organizing assortative matching contests with different numbers of agents on both sides, the designer can ensure that, independent of the form of the match-value functions, at least k agents, $k=(min(m,n\left)\right)$, will be active.

The rest of the paper is organized as follows: in Section 2, we present our assortative matching contest. In Section 3, we analyze $n\times n$ assortative matching contests, and in Section 4 we analyze $m\times n$ assortative matching contests. Section 5 concludes. Some of the proofs appear in the Appendix A.

We consider a set $A=\left\{1,2,\dots ,m\right\}$ of m firms and a set $B=\left\{1,2,\dots ,n\right\}$ of n workers where $n\le m.$ Firm i’s type is $m_i,$ where $m_i\ge m_{i+1}$, $i=1,\dots ,m-1$. Worker j’s type is $w_j,$ where $w_j\ge w_{j+1}$, $j=1,\dots ,n-1$. All these types are commonly known. As we can see below a firm’s utility function as well as a worker’s utility function increase in their own types. The matching contest proceeds as follows: There is one stage in which both sets act simultaneously. Each firm $i,i=1,2,\dots ,m$ exerts an effort $x_i,$ and each worker $j,j=1,2,\dots ,n$ exerts an effort $y_j$. Efforts are submitted simultaneously in each set. The order of the firms (workers) to be matched is determined according to the method of Clark and Riis (1998a) as follows: The first firm to be matched is determined by the probability success function which takes into account the efforts of all the firms. Formally, firm i, $i=1,\dots ,m$ wins to be the first match with probability $\frac{x_i}{{\sum }_{k=1}^m{x}_k}$, where $x_k$ is firm k’s effort, $k=1,\dots ,m$.3 Then, the second firm to be matched is determined by the probability success function that is based on the efforts of all the firms excluding the effort of the first winner. Thus, firm i, $i=1,\dots ,m$ wins to be the second match with probability ${\displaystyle \sum _{\begin{array}{c}k=1\\ k\ne i\end{array}}^m}\frac{x_k}{{\displaystyle \sum _{j=1}^m}{x}_j}\frac{x_i}{{\displaystyle \sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^m}{x}_j}$, and so on until all the firms are ranked, and similarly, all the workers are ranked. Then, the firm and the worker who win first place in their sets are matched, those who win second place in their sets are matched and so on until all the workers are matched. If firm i is matched with worker j after exerting efforts of $x_i$ and $y_j$, correspondingly, the firm’s utility is $f(m_i,w_j)-x_i$ and, similarly, the worker’s utility is $g(m_i,w_j)-y_j$, where $f,g:R^2\to R^1$ are the match-value functions which are monotonically increasing in the types of the firms and the workers who are matched. We say that a matching contest has an equilibrium if every agent chooses an effort that maximizes their expected utility given the efforts of the other agents in both sets.

In our $n\times n$ assortative matching contests there is always a corner equilibrium in which all the agents do not exert any effort.

The reason behind this corner equilibrium is that if all the n workers exert the same effort, all the workers have the same probability to be in first, second or, any other place. Thus, each firm actually faces n identical prizes since it has the same probability to be matched with each of the workers. As such, the place of each firm is not important, and therefore each of the firms does not have an incentive to exert any effort at all. Likewise, each of the workers exerts an effort of zero, and we have an equilibrium in which all the firms as well all the workers do not exert efforts.

It is worth noting that the existence of a corner equilibrium holds in more general models independent of the preferences of the agents. Furthermore, the existence of the corner equilibrium holds for any contest success function that breaks ties with a fair lottery. For example, consider the assortative all-pay matching contest in which the agent with the highest effort wins for sure, but if there is more than one agent with the highest effort, all these agents win with the same probability [33]. Then, in this assortative all-pay matching contest there is always a corner (trivial) equilibrium in which all the agents in both sets do not exert efforts and the agents are randomly matched.

The corner equilibrium in $n\times n$ assortative matching contests is not necessarily unique and might also be an interior equilibrium in which all or some of the agents exert positive efforts or a partial interior equilibrium in which some of the agents are active. The following result provides sufficient conditions on the match-value functions such that if there is a symmetric equilibrium it is the corner one in which all the agents exert an effort of zero.

The intuitive explanation to the above result is that in this assortative matching contest, the agents from each set face the same list of prizes which are the types in the other set. In other words, the agents in each set are actually symmetric with the same strategy. We prove that this symmetric strategy has to be an effort of zero. We conjecture that this result will hold for all match-value functions that have mixed second order derivatives that are equal to zero.

In the following, we demonstrate that there is an interior equilibrium in which firms and workers exert positive efforts or at least a partial interior equilibrium in which some of the firms and/or some of the workers exert positive efforts. A characterization of the equilibrium efforts in an assortative matching contest with a large number of agents on both sides is very complex, and therefore, for simplicity, we first focus on the smaller $n\times n$ matching contest with two firms and two workers.

Consider now that the two sets do not necessarily have the same size such that there is a set $A=\left\{1,2,\dots ,m\right\}$ of $m\ge 2$ firms and a set $B=\left\{1,2,\dots ,n\right\}$ of $n\ge 2$ workers where $n<m.$ The firms’ types are $m_i,$ where $m_i\ge m_{i+1}$, $i=1,\dots ,m-1$. The workers’ types are $w_j,$ where $w_j>w_{j+1}$, $j=1,\dots ,n-1$. We showed that in an equilibrium of the $n\times n$ assortative matching contest the agents from both sets may not exert efforts in equilibrium. However, this does not occur in the $m\times n$ matching contests if $m>n$ or vice versa. However, $m\times n$ matching contests might have a partial interior equilibrium in which some of the agents exert positive efforts but others do not exert any effort at all.

The intuition behind Proposition 5 is as follows: if less than n firms exert positive efforts, a firm that does not exert any effort will not be matched with a positive probability. Then, if such a non-active firm will choose to exert a sufficiently small effort, it will be matched for sure with one of the workers and then their expected payoff significantly increases.

We showed that in an equilibrium of the 2 × 2 assortative matching contest with an additive match-value function the agents from both sets do not exert efforts. By Proposition 5, this does not occur in the $m\times n$ matching contests if $m>n$ or vice versa. Furthermore, the minimum number of agents who exert positive efforts is even larger when the match-value function is additive.

By Proposition 5 at least n firms exert positive efforts. If exactly n firms exert positive efforts we actually have a $n\times n$ assortative matching contest, and by Proposition 2, all the firms do not exert any effort. Then, similarly to the case in Proposition 5, any non-active firm has an incentive to exert a sufficiently small positive effort such that there will be at least $n+1$ firms that exert positive efforts.

Consider now that $n=2$ such that the firms’ types are $m_i,$ where $m_i\ge m_{i+1}$, $i=1,\dots ,m-1$, and the workers’ types are $w_h$ and $w_l$, where $w_h\ge w_l$. Then, if the firms and the workers have a multiplicative match-value function, we have the following result:

Furthermore, if the firms and the workers have additive match-value functions we have

Propositions 7 and 8 indicate the relation between the two workers’ efforts, but they do not exclude the option that these efforts are equal to zero. An explicit characterization of the equilibrium efforts in these matching contests with large numbers of agents on both sides is very complex and therefore, for simplicity, in the next subsection we focus on the smaller $m\times n$ matching contest with three firms and two workers.

We studied assortative matching contests in which there are two sets of agents. In each set, the agents compete against each other in a Tullock contest, and then, according to the results of both Tullock contests, if the agents exert positive efforts, the agents from both sets are assortatively matched such that the first agents from both sets are matched, the second agents are matched, and so on until all the agents from the smaller set are matched. Every two agents who are matched win a reward according to match-value functions that depend on both agents’ types. Our findings for this simultaneous competition are summarized in the following Table 1:

We can see that in symmetric assortative matching contests in which the number of agents in both sets is the same ($n\times n$), there is always an equilibrium in which all the agents in both sets do not exert efforts and as such the agents are randomly matched. However, these matching contests may also have an interior equilibrium in which the equilibrium strategies depend on the form of the match-value function. It is important to note that when the number of agents in both sets is not the same ($m\times n$), independent of the form of the match-value function, there is no equilibrium in which all the agents do not exert efforts.

In sum, the reason for non-activity of the agents in assortative matching contests is that the match-value functions have the same property of the additive function according to which the mixed second derivatives are equal to zero. Then, if the number of agents in both sets are the same, agents from the same set face the same differences between their possible prizes (values of winning) and so that they have the same strategy to be non-active. Thus, a contest designer has two ways to activate the agents: the first is to choose the correct match-value functions and the second is to organize two sets with different sizes. Since in many situations the forms of the match-value function are exogenous such that a designer does not necessarily have control over their forms, by choosing the correct number of agents on each side of the matching contest, independent of the form of the match-value functions, he can ensure that the results of the contest will not be random and the agents will exert positive efforts in order to find their best match.