Collusion enforcement with private information and private monitoring☆
Introduction
Despite an extensive literature on repeated games, there has been little work that seeks to provide a theoretical justification of observed cartel practice. In an important exception, Harrington and Skrzypacz [24] note that many cartels used transfer schemes to enforce collusion. Under these schemes, colluding firms agreed to a set of sales quotas, and firms that reported sales above quota compensated firms that reported sales below through inter-firm purchases. In one case, Haarmann & Reimer purchased 7000 tons of citric acid from Archer Daniel Midlands. Inspired by Harrington and Skrzypacz [24], we propose a new collusion enforcement scheme in a repeated game with both private monitoring and private information. We show that despite observing neither costs, prices, nor sales, a cartel can still enforce a collusive agreement that maximizes joint profit. When the firms are patient and the demand shocks small, the equilibrium cartel profit could be close to the monopoly level.
We call the static version of our scheme an output–target mechanism. Under this mechanism firms first agree to a set of output targets. At the end of each period the cartel can calculate the “reported” profit of each firm based on the reported output and cost of each firm. The cost reports and profit shortfalls—the difference between the profit targets (i.e., the firm's profit if it makes the output target) and the reported profits—jointly determine the side-payments between firms and the probability of switching to a non-collusive continuation path (i.e., having a price war) in the next period. A firm tends to pay less when it reports lower sales, but the gain is offset by a higher probability of a price war. Taking into account its payments to other firms and the loss in case of a price war, each firm's continuation profit in equilibrium is linear in the reported output shortfalls of the other firms.
One can view the mechanism as a truncated Clarke–Groves mechanism (Vickrey [38], Clarke [15] and Groves [23]). If the output targets were set higher than the maximum outputs, then the continuation profit of each firm (including side-payments) would be equal to the reported profits of the other firms (minus a constant), and our scheme would resemble a classic Clarke–Groves mechanism. But since our cartel problem involves hidden actions, we cannot balance the budget by transferring the penalty of one firm to another firm. Instead, all penalties are destroyed through a price war. If the output targets were set above the maximum outputs, the amount to be destroyed could be very large. The first contribution of this paper is to show that we can reduce the efficiency loss by lowering the output targets. When the demand shocks are small, the efficiency loss would be small when the output targets are set near the expected output levels. However, when output targets are set below the maximum outputs, the transfer scheme would no longer fully capture the external effects of the firms' actions. A firm that has secretly cut price may escape punishment when a positive demand shock masks the effect of the price cut. A second contribution of the paper is to show that despite this “truncation” problem, the mechanism can still enforce an efficient collusive agreement under a fairly weak condition.
The main feature of our mechanism—that a firm is punished when the reported outputs of the other firms fall below certain targets—is consistent with the transfer schemes described by Harrington and Skrzypacz [24]. Since these schemes are common, it is important to understand how they work. Harrington and Skrzypacz [24] show that the main features of these schemes are consistent with equilibrium behavior in a model of repeated Bertrand competition with private monitoring. Our model is more general in that it allows private information and applies to both Bertrand and Cournot competition. In Harrington and Skrzypacz [24], inter-firm transfers serve as a linear output tax. By choosing the right tax rate, a cartel can discourage the firms from over-production. However, since a firm is required to pay more when it reports higher sales, it has an incentive to under-report. Harrington and Skrzypacz [24] show that it is possible to induce truth-telling when the industry demand is completely inelastic. Under our mechanism, firms have no incentive to mis-report as the continuation profit of a firm does not depend on its own report.
The standard approach to enforce an efficient outcome in repeated games with private monitoring is to punish a player whenever he fails a statistical test (Kandori and Matsushima [28], Compte [16], Fudenberg and Levine [20], Obara [32], and Zheng [39]). Our approach is to make each firm internalize the external effects of its action. The main advantage of this approach is that it is very robust. The existing literature on collusion enforcement can be divided into two strands. One strand assumes that prices are publicly observable and focuses on the issue of private cost information (Athey and Bagwell [6], [7]).1 Another strand assumes costs are public and focuses on the issue of secret price cutting (Green and Porter [22], Aoyagi [3], Harrington and Skrzypacz [25], [24]). There is, however, no economic reason why a cartel must be able to observe either the costs, prices, or sales of the firms. Because a firm can lie and cheat at the same time, a cartel in general cannot use one scheme to elicit the private information and apply a separate scheme to induce the collusive prices. Our approach provides a natural solution to the twin problems of inducing the firms to reveal their costs and set the right prices.
Our paper is also related to the recent works that extend the Clarke–Groves approach to a dynamic environment. Bergemann and Välimäki [12] introduce a dynamic pivot mechanism that generalizes the pivot mechanism to a dynamic setting. Athey and Segal [11] introduce a dynamic generalization of the Arrow–d'Aspremont–Gerard-Varet mechanism (d'Aspremont and Gerard-Varet [18] and Arrow [5]). Both Bergemann and Välimäki [12] and Athey and Segal [11] involve only private information, but in any period the distribution of an agent's types may depend on his previous types and past collective decisions.2 In our model the cost distributions are stationary, but each firm can choose a hidden action that directly affects the profits of the other firms. Mezzetti [30] considers a two-period mechanism-design problem. He shows that even when payoffs are interdependent, it is possible to enforce the efficient outcome by setting the transfer of each player equal to the reported realized payoffs of the other players. Our output–target mechanism is similar in that it uses the reported realized profits of the firms to enforce the efficient action profile. The difference is that in our model it is the action, rather than the type of a firm, that directly affects the profits of the other firms. Lastly, Hörner et al. [27] consider a general dynamic Bayesian game that incorporates both hidden actions and Markovian types. In the case of private independent types, they solve the private-information and imperfect-monitoring problems simultaneously by combining the Clarke–Groves approach of Athey and Segal [11] and Mezzetti [30] with standard techniques in repeated games (Kandori and Matsushima [28]). We show that under a fairly weak condition a truncated Clarke–Groves mechanism is still incentive compatible.
The rest of the paper is organized as follows. The next section introduces the model. Section 3 describes the standard Clarke–Groves approach and explains why it may lead to a large efficiency loss in a model with hidden actions. Section 4 introduces the output–target mechanism in a static setting and shows that it implements the efficient collusive outcome when the stochastic demand functions satisfy a monotone condition. Section 5 uses the mechanism to construct a perfect public equilibrium in a repeated game. Section 6 concludes.
Section snippets
Demand
We consider an infinitely repeated oligopoly game. Let denote a set of n firms, each with constant marginal cost. In each period , each firm i chooses an “action” from a compact interval . Let denote an action profile and the set of action profiles. The “output” for each firm i is denoted by an output function where is the output function of firm i, and a random shock that is distributed according to a smooth distribution function
The Clarke–Groves approach
Although a standard Clarke–Groves mechanism involves only private information, it can be easily extended to allow hidden actions. Consider a one-shot collusion game with transfers defined by steps 1–5 of the stage game introduced in the last section. But instead of making side-payments each firm i directly receives a transfer that depends on and .11 The total payoff of firm i in this
Output–target mechanism
Inspired by the real-life cartel practice of requiring firms selling above quota to compensate firms selling below, we show that, for many stochastic demand functions, it is possible to reduce the efficiency loss of the Clarke–Groves approach by setting the profit caps strictly below the maximum profits.
Let denote firm j's expected output truncated from above at . For any i, with and any such that , define
Implementation
In this section we apply the results in the last section to construct a perfect public equilibrium in the original repeated game. The equilibrium trigger-strategy profile, denoted by , is characterized by a probability function and a matrix of side-payment functions , where . There are two states: collusive and non-collusive. The equilibrium starts off in the collusive state in period 1. In the collusive state each firm i reports its cost
Concluding remarks
In this paper we show that for a wide class of stochastic demand functions, it is possible to enforce the cartel-profit-maximizing outcome by tying each firm's continuation profit to the truncated profits of the other firms. We show that when the firms are patient and the demand shocks are small there is a perfect public equilibrium with cartel profit close to the monopoly level. One lesson we can draw from the analysis is that without active antitrust enforcement, a cartel can operate
Acknowledgments
We thank Joseph Harrington, Andrzej Skrzypacz, an associate editor, two anonymous referees, and participants at various seminars and conferences for helpful comments. This research is funded by the Chinese National Science Foundation (Project No. 71171125). Chan is supported by the Shanghai Dongfang Xuezhe Program.
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An earlier version of this paper has been circulated under the title “Price Wars, Side-Payments, and Collusion Enforcement.”