Economics Letters 95 (2007) 355 – 361 www.elsevier.com/locate/econbase

Can labor markets help resolve collusion? ☆ Jeremy Bertomeu ⁎ Tepper School of Business, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh PA 15213, United States Received 30 May 2006; received in revised form 5 September 2006; accepted 1 November 2006 Available online 22 February 2007

Abstract We show that, in a multi-agent contract setting, the principal can effectively rule out tacit collusion among agents (i.e., “bad” equilibrium) by posting permanent job openings to an external labor market. That is, a simple “market-like” employment mechanism can yield collusion-proof contracts. © 2006 Elsevier B.V. All rights reserved. Keywords: Market; Implementation; Tacit; Collusion; Replacement JEL classification: C7; D2; D8; J4

A well-known problem in the mechanism design literature is the possibility of collusion among agents when the mechanism chosen by the principal has multiple Nash equilibria. Most contracts involving more than one agent, and not explicitly designed to be collusion-proof, may be prone to collusion. Collusion may appear in problems of adverse selection and hidden actions, and will arise in most repeated interactions and in games where agents must truthfully reveal their private information.1 ☆

The author thanks Pierre Liang and Jon Glover for his class on mechanism design and helpful comments. This research was supported by a grant from the William Larimer Mellon fellowship. ⁎ Tel.: +1 412 567 1218; fax: +1 412 268 8163. E-mail address: [email protected]. 1 The possibility of collusion has been documented in repeated interactions (Fudenberg et al., 1994), standard auction settings (Blume and Heidhues, 2004), team production (Demski and Sappington, 1984) and in the industrial organization literature (Ivaldi et al., 2003). There is also an extensive literature, which we do not cite here, that studies collusion in various adverse selection settings. 0165-1765/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.11.007

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Several contributions to implementation theory have shown that one can often complement existing mechanisms in order to rule out collusion. Laffont and Martimort (1997, 2000), among others, consider enriched collusion-proof mechanisms in private information settings when types are uncorrelated or correlated. Several of these results are extended in Che and Kim (2005), for general specifications of the interaction. Interestingly, these mechanisms have not produced yet a clear analogue in existing organizations. There are several plausible reasons for this. First, enriched mechanisms can become more complex than the original contract, and thus may rely on more sophisticated strategies and beliefs. Second, these models usually exclude moral hazard, thus making it easier to redirect the claims to the residual surplus from the game, whereas many interactions may feature hidden actions by the principal (if a firm is sold) or the agents (during production). Third, the design of collusion-proof mechanisms requires a very precise knowledge of the environment, which makes a practical application difficult. In this paper, we proceed in a different direction, asking whether simple labor markets and job offers, can help resolve certain forms of collusion. In contrast to the implementation literature, we do not attempt to design a general mechanism, but consider an employment game that somewhat resembles how organizations operate. Further, the employment game determines which agent will work for the organization, but does not alter the wage contract of an employed agent. We show that permanent openings on all jobs held by agents will rule out a large class of plausible collusive equilibria. The intuition of the model is similar to Palfrey (1992, Theorem 1 p.295) who suggests that the introduction of an uninformed agent can be used strategically by the principal, although the game and the problem are very different. In our model, collusion is inferred in equilibrium by outsiders who rationally apply for the job of a colluding insider. Then, any extra welfare achieved by collusion must be dissipated by market forces, and the equilibrium prescribed by the principal always Pareto-dominates other equilibria such that insiders collude. 1. The model Let us assume that a principal, indexed by i = 0, contracts with n agents, indexed by i = 1, … n; we assume that the number of agents required to operate the firm is fixed. In the basic model (no labor markets), the principal designs a contract and then players choose their best response in the Normal form game induced by the contract. The principal can design a contract based on a signal x, publicly revealed when the game ends. In the standard moral hazard framework, the signal x may include the profit of the firm and, more generally, signals informative on the actions of the agents (Holmström, 1979) as well as different performance measures (Holmström and Milgrom, 1987). In adverse selection settings, the signal x can be a bid (in auction settings), or more generally a type report, but it may also include information acquired by the auctioneer and publicly disclosed (Milgrom and Weber, 1982). A contract is defined as a function ϕ(x), which associates a social choice ϕ(x) ∈ Y to any public signal. This contract may be a vector of wages (in a compensation contracts), an allocation and transfer schedule (in auctions) or the provision of public goods (in public-good problems). It may include private components or common components, which affect all players, depending on the interaction which is considered. For each contract ϕ, the principal and the agents play the following simultaneous-move game. For each i, player i receives a private signal (or type) si ∈ Siϕ and then may choose a (private) action; a vector of signals is denoted s = {s1, …, sn}. A pure strategy in this game is given by a mapping from any signal si to

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an action. Let Aiϕ , with elements ai, denote the set of mixed strategies. A vector of actions is denoted a = (a1, …, an). For a given a and ϕ, Fa,ϕ denotes the joint distribution of (s, x). Finally, let Ui(s; a; ϕ(x); x) denote the utility of player i. U /i ðaÞ

Z ¼

U i ðs; a; /ðxÞ; xÞdFa;/ :

ð1Þ

Denote C/ ¼ fn þ 1; ðAi Þni¼0 ; ðU /i Þni¼0 g the Normal form of the subgame after the contract ϕ has been chosen by the principal. Denote Ωϕ its set of Nash equilibria that are not Pareto-dominated by other Nash equilibria and, for ω ∈ Ωϕ, denote aωϕ , the associated strategy profile. Finally, assume that there exists a unique (ϕ⁎, ω⁎) maximizing U /0 ða/x Þ and denote a⁎, the associated strategy profile. In this paper, we focus only on Nash equilibria that are not Pareto-dominated by another equilibrium. This restriction of the analysis, which is central for our results, is based on the idea that players may always form a grand coalition to move away from outcomes that are dominated. Further, in the case of quasi-linear environments, a Pareto-improving equilibrium can be reached by strictly increasing the utility of all players.2 At ϕ⁎, there may be other equilibria ω ∈ Ωϕ⁎, that are strictly preferred by one or more of the other agents. The principal must enrich the mechanism3 to ensure that the desired equilibrium ω⁎ Paretodominates any collusive equilibrium in the game. 2. Large labor markets We model job openings in order to resolve the problem of collusion. We assume here that, initially, the principal has not found yet the n agents with whom it intends to contract with and posts the contract on a labor market. The following additional assumptions are made. A-1 (large market). There exists a continuum of agents who may be candidate for job assignment i. A-2 (no information). An unemployed agent has no private information when applying for a job offer. A-3 (binding reservation). The reservation utility of an unemployed agent is U /* ða*Þ. i

A-4 (cost of dismissal). An agent hired but subsequently dismissed obtains a utility P U i b U /i * ða*Þ. A-5 (arrival process). There is always a candidate at the first offer, but for subsequent offers, there is a probability pi N 0 that a candidate arrives. These assumptions are intuitive. We assume that there exists a very large pool of potential candidates (A-1.) who do not have private information (A-2.). In an equilibrium with no collusion, all agents receive the reservation wage (A-3.), which corresponds to certain (but not all) competitive labor markets. There is 2

In certain settings involving adverse selection, it may not be plausible to consider such a coalition form before signals are observed. This is an important limitation in our application, although the selection of more efficient equilibria (used here) is a selection concept that is used in many games. 3 It is well-known, in pure adverse selection settings (see previously cited papers) that it is possible to design such mechanisms; further such mechanisms can be made robust to moral hazard. The problem of this paper is more applied, in that we discuss a mechanism that achieves this result, but resembles mechanisms existing in real organizations.

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a cost of being hired, and subsequently fired without having worked (A-4.). Finally, there is always a positive probability that a job offer is considered by an outsider (A-5.). The timeline of the extended game is described. Initially, no agent is employed by the principal. First, the principal chooses a contract ϕ to compensate n agents. We restrict the attention to a contract written conditional on the public signal x but not on signals sent by agents in this extended game (which may be feasible, but requires more sophisticated mechanisms). Second, the principal posts a job opening for assignment 1. This candidate must decide whether to accept the job offer. If in the first round of search, the candidate rejects the offer, the firm is not operated and the principal obtains a reservation utility U 0 bU /0 * ða*Þ. If the first candidate accepts the job offer, the principal posts a new job opening on P assignment 1. If a candidate arrives and accepts the job, the current agent is dismissed (and achieves Ui). The outsider is hired and a new job opening on assignment 1 is posted. A new candidate observes only¯ if the job has already been taken. The process continues until either no candidate arrives or the candidate rejects the job opening. The same procedure is applied for assignment i = 2 up to assignment n. Third, once this process has ended, the n currently employed agents play Γ ϕ described above (abusing the notation by ignoring player identity). Let Γ¯ϕ denote the extended game after ϕ has been chosen. Proposition 1. Suppose that for all i, pi ≥ sup xaX/*

U /i * ða/x* Þ−U /i * ða*Þ U /* ða/* Þ− U i

x

ð2Þ

Pi

Then, there exists a Nash equilibrium of Γ¯ϕ⁎ that Pareto-dominates all other equilibria. It is such that agents: (i) accept the job offer if no other agent accepted, (ii) reject the offer if an agent already accepted, (iii) follow the equilibrium strategy ω⁎ if they are employed (i.e. no collusion). Proof. Since Nash equilibrium is immediate, let us prove that it is more efficient than the other equilibria. Note that a candidate must always accept the offer conditional on no other offer (or else, there would be shutdown, which is less efficient). Next, conditional on not being fired, agents must choose a Nash equilibrium in Γ ϕ⁎ . / Suppose there is ĩ, an agent who achieves strictly more than Ui˜ * ða*Þ conditional on not being dismissed. If agents choose a mixed strategy when an agent has already accepted an offer, this cannot hold / since conditional on mixing, agents must achieve an expected utility Ui˜ * ða*Þ. Suppose then that agents always accept the proposal even when an agent has already accepted the offer. But it follows from Eq. (2) that if all agents accept a job offer, they must all obtain less than their reservation, a contradiction. □ The principal posts a job opening even though there is a currently employed agent. There are two cases. If there is no collusion, the job opening will not be attractive4 to an outsider and thus, the first agent to accept the job will never be fired. If there is collusion, market forces will attract a job candidate and the colluding agent will be fired. Entry by outsiders is sufficiently intense so that all rents from collusion are eliminated. The assumptions required for this result to hold are reasonably strong, and thus there may be cases such that simple job offers will not fully prevent collusion. First, the gains from collusion must be low or the arrival of new candidates and the cost of dismissal must be large. When these assumptions are 4

However, there are equilibria where the outsider takes the job of the insider; however, these equilibria are inefficient.

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violated, there will be equilibria such that employed agents collude and all candidates accept the offer. The principal will need to increase the personal cost of dismissal, which may be achieved by, for example, limitations to future employment opportunities such as reputation or competition clauses, or a nonrefundable sunk payment when the agent is hired. Second, if agents receive more than their reservation wage in equilibrium, acceptance by an outsider may only indicate that the offer is attractive. We do not have a simple solution in this case since such problems are usually due to limited liability constraints. For similar reasons, if job candidates have private information, acceptance may be due to private information and may not be a signal of collusion. Third, the information held by agents about past dismissals is also very important. If agents know whether dismissals have taken place, collusive equilibria can be sustained as follows: agents choose the collusive equilibrium if no agent has been dismissed but choose the equilibrium ω⁎ if there was at least one dismissal. However, such strategies will be impossible when the original contracting problem is among agent with no personal ties, and thus who do not observe prior dismissals. To a certain extent, then, unions in large organizations can be used to record dismissal history and thus enforce collusive equilibria more desirable to the agents. 3. Small labor markets We revisit (A-1.), relaxing the assumption that the pool of potential candidates is very large. Assume that all potential candidates who may work in assignment i are ranked by a parameter j = 1, …, ∞ which corresponds to the order in which they may be candidates. For notational simplicity, we focus on the case where j is known to the agent, since the case where j is imperfectly known follows easily. The effective maximum number of candidates is given by the realization of a random variable Ñi drawn from a distribution such that: λi,N ≡ P(Ñi = N|Ñi ≥ N). As before, there is a probability pi that the principal fails to find a new candidate although less than Ñi have appeared. Let us assume that the realization of Ñi is not observable. We restrict the analysis to equilibria such that actions conditional on employment do not depend on the order of the candidates. 5 Corollary 1. Suppose that for all i, pi ≥ sup sup Ni xaX/*

U /i * ða/x* Þ−U /i * ða*Þ =ki;Ni U /* ða/* Þ− U i

x

ð3Þ

Pi

Then, there exists a Nash equilibrium in Γ¯ϕ⁎ that Pareto-dominates all other equilibria. It is such that: (i) for each assignment, the first agent accepts the offer, (ii) all subsequent agents reject the offer, (iii) employed agents follow the equilibrium strategy ω⁎ (i.e. no collusion). We show that collusion rents may still vanish in the case of a small market. An important difference with our previous setting is that unemployed agents know the order in which they may be called by the 5

In this respect, the behavior conditional on employment is symmetric. We could not find an example such that collusion may be enforced with more asymmetric strategies; and further, the degree of coordination that this may require seems very contrived in practical settings.

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principal (if they do not, the likelihood of a candidate will be averaged, thus strengthening the result). These agents will use this information to estimate the likelihood of an outsider obtaining their job. However, this sufficient condition will be violated in the realistic case where the number of potential candidates is a bounded random variable. In Proposition 2, we extend the result in cases where Eq. (3) is violated, which includes the case of a finite maximum number of applicants. Let ω ≠ ω⁎ be a collusive equilibrium that is weakly preferred by all agents to ω⁎ and strictly for i ∈ I. Then, denote Ni⁎, the smallest Ni such that Eq. (3) is violated. Finally, assume that Eq. (3) is not met at equality prior to Ni⁎. Proposition 2. Suppose N⁎i is finite for all i ∈ I. There is no equilibrium such that agents are employed and collude on ω if (and only if) there exists i ∈ I such that N⁎i is even. Proof. Consider the last employable agent N⁎. Under collusion, this agent will always accept the job offer if i

he is reached, since he cannot be fired. The penultimate agent then, will anticipate being fired with probability pi and thus will never accept. By backward induction, it must then hold that if Ni⁎ is odd, the first agent will accept the job offer, knowing that the second agent will always reject. Then, if this condition is met for all i, the equilibrium will be such that the first agents achieve collusion. If Ni⁎ is even for one i, then the first candidate will reject and thus the contract will not be signed, which contradicts efficiency. □ We develop the intuition for this result in the case of a known finite number of applicants. It will be common knowledge that the last employable agent will always accept if there is collusion, thus implying that the penultimate agent rejects, until either the first agent accepts or rejects. In the former case, markets will fail to prevent collusion since the threat of dismissal will effectively refrain any outsider from accepting a job offer. In the latter case, collusion will always imply rejection by the first agent and thus (endogenous) shutdown of the firm. 4. Concluding remarks To conclude, it is important to note that posting a job opening, and searching for a new candidate will, in general not be costless and there may be a gain to production with (n + 1) agents rather than only n agents. Then, when pi is endogenous, the cost of posting job openings and forfeiting more agents will depend on the potential gain from collusive behavior. This employment mechanism can only rule out equilibria that are not Pareto-dominated, and more sophisticated mechanisms (discussed in previously cited papers) are necessary to rule out other equilibria. In the case of very tense labor negotiations between agents and the principal, agents may not be interested in being part of a coalition that strictly increases the welfare of the principal but leave theirs unchanged, thus leading to inefficient equilibria rather than collusion breakdown. References Blume, A., Heidhues, P., 2004. All equilibria of the Vickrey auction. Journal of Economic Theory 114 (1), 170–177. Che, Y.-K., Kim, J., 2005. Robustly Collusion-Proof Implementation. Demski, J.S., Sappington, D.E.M., 1984. Optimal incentive contracts with multiple agents. Journal of Economic Theory 33 (1), 152–171. Fudenberg, D., Levine, D., Maskin, E., 1994. The folk theorem with imperfect public information. Econometrica 62 (5), 997–1039. Holmström, B., 1979. Moral hazard and observability. Bell Journal of Economics 10 (1), 74–91. Holmström, B., Milgrom, P.R., 1987. Aggregation and linearity in the provision of intertemporal incentives. Econometrica 55 (2), 303–328.

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Ivaldi, M., Jullien, B., Seabright, P., Tirole, J., 2003. The economics of tacit collusion. IDEI, Toulouse - Final Report for DG Competition, European Commission. Laffont, J.-J., Martimort, D., 1997. Collusion under asymmetric information. Econometrica 65 (4), 875–911. Laffont, J.-J., Martimort, D., 2000. Mechanism design with collusion and correlation. Econometrica 68 (2), 309–342. Milgrom, P.R., Weber, R.J., 1982. A theory of auctions and competitive bidding. Econometrica 50 (5), 1089–1122. Palfrey, T.R., 1992. In: Laffont, Jean-Jacques (Ed.), Bayesian Implementation, Vol. 1 of Advances in Economic Theory Sixth World Congress. Econometric Society Monographs, vol. 20. Cambridge University Press.