Before describing the model, let us examine very shortly the intuition behind the problem. If a firm is wrongfully accused and prosecuted for an offence and imputed with some evidence, it might expect a change in the balance of the expected costs and benefits of its actions. The violation of rules becomes relatively more attractive, and welfare-inducing agreements are concluded either more rarely or interrupted. If this is so, the effects of LPs devoted to reestablishing the oneshot prisoners’ dilemma game between competitors might change compared to the presence of only type II errors. Intuitively, it is quite clear that several types of negative effects can arise, including not only false self-reporting and reporting by counter-agent of agreements but also abstaining from the use of particular clauses in contracts and refraining from concluding these contracts as a whole. That is why we can expect multiple forms of harm related not only to prospective market actors but also to principals of enforcement — tax payers. In our model, we limit ourselves only to direct effects. In any case, the intuition leaves us with some doubts as to what the structure of current and potential strategic interactions between firms will look like.

The presented model is an extension of the model developed by Ghebrihiwet and Motchenkova (2010), which itself builds upon the model by Motta and Polo (2003). Ghebrihiwet and Motchenkova (2010) attempt to fill the void in the study of type I errors and leniency by adding the probability of type I errors to the model of Motta and Polo (2003). They derive some interesting results, e.g., that innocent firms may use plea bargaining as insurance against a type I error. At the same time, this model does not allow us to analyze the self-reporting (including counter-part reporting) of cooperating firms. We extend the model by Ghebrihiwet and Motchenkova (2010) to take into account the effects of LPs on horizontal cooperation agreements that are beneficial to social welfare.

Additionally, the model by Ghebrihiwet and Motchenkova (2010) does not allow innocent firms to apply for leniency because there is no legal uncertainty on particular forms of market behavior. Instead, it gives them the opportunity to plead guilty in a pre-trial settlement. The main reason given for this is that in exchange for leniency, the firm must provide evidence of collusion, whereas an innocent firm can provide none. We assume that firms can enter into agreements that are not aimed at harming competition but can be interpreted as such by an authority that can make errors. That is why the notion of evidence quality is important. In this case, innocent firms — in exchange for leniency — can provide the sort of information that can be used to “prove” the fact of collusion.

Finally, in the model by Ghebrihiwet and Motchenkova (2010), the probabilities of type I and type II errors are the same across all possible behavioral strategies. We propose taking into account that the antimonopoly authority has some experience that allows it to distinguish different types of behavior on a market. In this way, the probability of a colluding firm being found guilty is higher than that for a firm that does not in fact violate the law. This point reflects some particularities of administrative procedures taken into account by the antitrust authority to initialize the case and to make decisions based on the collected and interpreted evidence.

Following Motta and Polo (2003) and Ghebrihiwet and Motchenkova (2010), we analyze a group of perfectly symmetric firms. The firms choose between competing, colluding, deviating from the collusive strategy and cooperating (the corresponding profits are Π N , Π M , Π D and Π COOP) . Because all firms are symmetric, they all choose the same strategy in equilibrium. The antitrust authority chooses an enforcement policy that can include the use of a LP. Firms take into account the policy of the antitrust authority. Thecollusive agreement prescribes both the market behavior and the behavior towards the antitrust authority: whether the firm reveals information about the cartel if monitored.

At period t = 0 the antitrust authority sets the policy parameters: the full fine F (F > 0), the reduced fine R (0 ≤ R < F) 6 and the probabilities of firms being investigated and prosecuted.

We extend the model by Ghebrihiwet and Motchenkova (2010) by assuming that the probabilities of an investigation opening and ending in a conviction are different across different market strategies in the following way. We denote the probability of the antitrust authority starting an investigation against a firm that neither colludes nor cooperates by α 0, and the probability of that investigation ending in a conviction by p 0. For colluding firms, the probabilities are α 1 and p 1; for firms deviating from a cartel agreement, they are α 2 and p 2; for cooperating firms, they are α 3 and p 3, α 0 ≠ α 1 ≠ α 2 ≠ α 3, p 0 ≠ p 1 ≠ p 2 ≠ p 3.

To simplify the comparison, we make some additional assumptions about probabilities α and p. This can be done in multiple ways, but the key will be the markers that the antitrust authority uses to identify cartel agreements. A study of cartel behavior and the possible effects that can draw the attention of antitrust authorities can be found in the work of Harrington (2006). We will use two characteristics that can be interpreted by the antitrust authorities as markers of cartels: the existence of an agreement between competitors and the existence of profits that are higher than the competitive level. It seems logical to assume that the lowest probabilities are applicable for firms that originally compete — that is, they neither collude nor cooperate on the market. In this case, not only is there no trace of any agreement, there is also no evidence of excessive profit. By the same logic, the highest probability of investigation and prosecution exists for the case where both a collusive agreement and a collusive profit are present — and this is the case of collusive strategies, so the highest probabilities are α 1 and p 1.

For firms deviating from the agreement, we can assume the following. Although the firm acted competitively in the first period by undercutting its rivals’ price, it has still entered the agreement at some previous point in time — otherwise there would be nothing from which to deviate. Therefore, some proof of the existence of a cartel agreement exists, even though the profits received by the firms do not support the assumption that collusion took place. For these reasons, we maintain that the probability of prosecution in this case, p 2 is higher than in the case of competition, but lower than in the case of collusion: α 0 < α 2 < α 1 and p 0 < p 2 < p 1.

For cooperating firms, the situation is as follows. Because there is a certain agreement between firms, which is difficult to distinguish from a cartel agreement due to the inclusion of ancillary restraints, and because if the cooperation is successful, firms will receive a profit that is higher than the competitive profit (as in the “Uralsib” and “Toyota Bank” example), we assume that the probabilities of prosecution are higher than in the case of competition, but lower than in the case of collusion: α 0 < α 3 < α 1 and p 0 < p 3 < p 1.

A more difficult issue is the correlation between probabilities for deviating firms and cooperating firms. In both cases, some sort of agreement between competitors exists that can be detected by the antitrust authorities (ex post)and interpreted as evidence of collusion. However, in the case of deviating, competition can be observed (as a process): behavior on the market shows that firms actively compete by undercutting each others’ prices. In contrast, in the case of thedeviating strategy, the available evidence that can be used as proof of collusion is only the agreement itself and during a limited period of time. In the case of cooperation, there is both an agreement and a market outcome that can resemblecollusion 7 . Thus, we can assume that a cooperation agreement is more likely to draw attention and end in prosecution than an agreement that has never been executed. Hence, we consider α 0 < α 2 < α 3 < α 1 and p 0 < p 2 < p 3 < p 1.

The timing of the game is as follows. The antitrust authority monitors the behavior of firms in the market, prioritizing the directions and scope of screening. An investigation, once opened, can last one or two periods. In the first phase, an investigation is started with a certain probability. If a firm confesses, the authority ends the investigation and finds a violation with probability 1 (not checking whether the confession is false). The firm that confessed receives a reduced fine and is made to compete in the current period. If none of the firms confess, the investigation continues for a second period and ends in a conviction with a probability that is less than 1. If found guilty, the firm is made to pay the full fine and compete in the second period (it is not assumed that it can exit the market). We assume that any firm that admits to a cartel is granted a reduced fine, independent of whether it was the first to do so. Consequently, the game restarts. We assume infinite repeat.

We now take a closer look at the firms’ strategies and their corresponding values.

To find the subgame perfect equilibria, we compare the values of the strategies listed above. Because from the start we assumed symmetry between the firms, it follows that if one firm finds a certain strategy optimal, so do all other firms.

Following the discussion on the values of α and p presented in section 3.2, we will try to define the conditions for α and p that influence which strategy becomes dominant. To do this, for the purposes of simplification and obtaining an illustration to our conclusions, we assume fixed ratios between probabilities α i and p i and compare the values of the denoted strategies.

We assume that α 0 = 0.2α, α 1= α, α 2 = 0.4α, α 3 = 0.6α, p 0 = 0.2p, p 1= p, p 2 = 0.4p, and p 3 = 0.6p. As mentioned above, these values satisfy the conditions α 0< α 2< α 3< α 1, p 0< p 2< p 3< p 1, and seem feasible in light of the meaning of these parameters. We will also assume that the amount of the reduced fine is zero (R=0), corresponding to a 100% fine discount.

The appendix contains all the necessary calculations.

We find the values of α and p that cause certain strategies to dominate. For our chosen illustrative example (see Appendix), the equilibria are as follows:

1. CNR $0<p<0.75\text{and}0<\alpha <\frac{25}{68p};$
2. CR $0.75<p\le \text{and}0<\alpha <\text{min}\left[\frac{2\text{.5}}{7\text{.5-3.2}p};\frac{5}{9}\right];$
3. COOPNR $\frac{25}{68}<p<\frac{3}{3.2}\text{and}\text{max}\left[\frac{25}{68p};\frac{2\text{.5}}{7\text{.5-3.2}p}\right]\alpha <\frac{2}{3.2p};$
4. COOPR $\frac{3}{3\text{.2}}<p\le 1\text{and}\frac{5}{9}<\alpha <\frac{2}{3};$
5. N —for all other intervals (as long as all the values of α i and p i fall into the segment [0; 1]).

The model by Motta and Polo (2003), which we used as our benchmark model, resulted in three types of subgame perfect equilibria: CR, CNR and N. They are illustrated in Fig. 1.

Fig. 1

The results of Motta and Polo (2003).

One of the main findings of Motta and Polo (2003) was that even when using a very “generous” version of the program —where the applicant can receive full immunity from fines (R = 0) —not all cartels on the market are broken up; there are areas where firms still choose to collude and either reveal or do not reveal (CNR and CR). This happens when the probability of starting an investigation, α, is low. If at the same time the probability of successful prosecution (p) is low, then firms do not have an incentive to confess and we end up in the CNR area, where firms collude and do not reveal information about it. In contrast, if the antitrust authority has sufficient resources and incentive to ensure high probabilities of investigation and prosecution, then cartels are prevented.

For our extended model, we find that the number of possible types of subgame perfect equilibria increases to five:

1. firms collude and do not reveal information about the cartel to antitrust authorities (CNR);
2. firms collude and reveal (CR);
3. firms cooperate and do not confess to colluding (COOPNR);
4. firms cooperate and confess to colluding (COOPR);
5. no collusion or cooperation occurs (N). 10

The results are illustrated in Fig. 2.

Fig. 2

Equilibria of the model in axes (α; p).

The N, COOPNR, COOPR, CNR, and CR areas denote different types of equilibria that depend on the values of α and p.

α COOPNR/DR (p) is a curve above which the firms prefer the strategy DR (resulting in the equilibrium N), and below which the firms prefer COOPNR; thresh-olds α CNR/COOPNR (p), α CR/COOPNR (p), α COOPR/DR and α CR/COOPR have similar interpretation. The line p CNR/CR defines the border between areas of a CNR-type and a CR-type equilibrium; the line p COOPNR/COOPR —the border between COOPNR and COOPR.

Proposition 1. Accounting for the possibility of type I errors and cooperation agreements leads to an increase in the number of types of possible subgame perfect equilibria compared to the benchmark model.

Before attempting to define the role of leniency programs in these results, we will analyze what effect the additional assumption of type I errors has on market behavior.

Proposition 2. Excluding the possibility of type I errors in the model leads to only three types of remaining equilibria: CNR, CR and COOPNR.

Because the probability of being unfairly fined by the antitrust authority is now zero, the value of the COOPNR strategy changes. The value of this strategy is now defined as the following:(1)

COOPNR starts to dominate COOPR, DNR, DR and N for the following reasons.

First, because the antitrust authority now no longer confuses cooperation and collusion, there is no incentive to make a false confession and not only incur an undeserved fine, even if it is reduced, but also to destroy the existing cooperation for one period. Similarly, DNR starts to dominate DR.

Second, because the antitrust authority does not make type I errors, cooperation becomes a better strategy than competition for any given values of parameters of α and p 11 (if П COOP >П N holds). It follows that if a firm has the ability to take part in a cooperation agreement, it will always be profitable for it to do so.

Third, the ratio of the values of the COOPNR and DNR strategies stops being dependent upon α and p and is now defined by the ratio of the corresponding profits. In our example, the ratio of the profits ensures that COOPNR becomes the dominating strategy.

By comparing values of strategies and using the same parameters as previously, we derive that an analogue of the model of Motta and Polo (2003) in our example would lead to the results illustrated in Fig. 3.

Fig. 3

Results of Motta and Polo (2003) with cooperation.

Finally, we illustrate the comparison of the results derived with and without the assumption of type I errors (Fig. 4). The grey areas are those where in the absence of type I errors, firms used to cooperate (and not make false claims for leniency) in equilibrium —but after taking into consideration type I errors, we find that these are the areas where collusion appears. Not all of this grey area is where firms confess after colluding: if p is low enough, firms collude without confessing.

Fig. 4

Impact of type I errors on the effectiveness of leniency programs.

This result corresponds with the results of Ghebrihiwet and Motchenkova (2010): by taking into account type I errors, we see that for certain policy parameters, firms that in fact never caused damage to social welfare change their behavior and start taking actions that do cause damage. Expecting that even competitive behavior can be prosecuted, firms find it best to start “deserving” their punishment—in this way, they at least compensate by receiving collusive profits.

An effect that was not studied by Ghebrihiwet and Motchenkova (2010) and that has not yet been the object of systematic analysis in the context of leniency programs is the impact on “conscientious” cooperation. Our model shows that in areas where socially beneficial cooperation was possible in equilibrium in the absence of type I errors, “switching on” such errors leads to the appearance of areas where cooperation either never arises (N) or arises only to be terminated if it draws the attention of the antitrust authority (COOPR).

The two latter effects correspond to the findings of Shavell and Polinsky (1989), who argued that an increase in the probability of type I errors can lead to economic agents becoming more inclined towards violating rules, and to the results of Png (1986), who concluded that an increase in the probability of type I errors can lead to an even higher level of compliance. In their own way, our results reconcile these two seemingly contradictory findings: in our model, these effects are not mutually exclusive, but the prevalence of one or the other depends on the deterrence parameters α and p. This leads us to

Proposition 3. The presence of type I errors results in collusion becoming sustainable for a wider set of parameter values and has a detrimental impact on socially beneficial cooperation.

To analyze the effect of leniency on incentives to cooperate in the presence of type I and II errors, we will first look at the case in which a confession is not rewarded by a reduction of fines.

In this case, CNR, DNR and COOPNR become dominant strategies over CR, DR and COOPR, which is intuitively clear. Additionally, the chosen parameters ensure that DNR dominates N.

In this way, three types of equilibria are possible: where all firms collude and do not reveal, where all firms cooperate and do not reveal, and where firms compete. The results are illustrated in Fig. 5.

Fig. 5

Results of the model with and without leniency.

The labeled areas correspond with the equilibria in our main model with leniency. In the dark-grey area, the equilibrium in the absence of leniency is CNR; in the light-grey area, the equilibrium in the absence of leniency is COOPNR. The N equilibrium (white area), where the dominant strategy is DNR, is also possible.

The results make it possible to derive some information about the effect of leniency programs when the antitrust authority can make both type I and type II errors.

First, we confirm the result obtained by Motta and Polo (2003). With the inclusion of leniency, the area where collusion can (in principle) be maintained becomes larger (transition from the dark-grey area to CNR + CR). However, the participants of the newly formed cartels prefer to collude and confess; in addition, some cartels that previously would not have been voluntarily revealed to the authorities are now discovered thanks to confessions exchanged for leniency (dark-grey part of CR). That is why the presented model provides grounds to expect a more complicated picture as could be presupposed intuitively.

It is worth mentioning that in our model the “donor” area for collusion is the locus where in the absence of leniency, cooperation is feasible. One of the most interesting results is that in the appearance of leniency programs in a part of the area where firms used to cooperate they now make false confessions and apply for leniency to insure themselves against possible unfair punishment (locus COOPR). This means that in case an investigation starts, the cooperation will break up. Because we assume the cooperation to be socially beneficial, its destruction due to false self-reporting has a negative impact on social welfare.

Another effect is the dramatic decrease in the area where cooperation can be maintained at all. Previously, with our chosen parameters and without leniency, all the firms that did not collude preferred to cooperate, if given the possibility — but after introducing leniency, the area where COOPNR and even COOPR are feasible decreased noticeably, whereas the area where no cooperation arises increased in size.

The effects described above are summarized in

Proposition 4. Leniency in the presence of type I errors can lead to the destruction of welfare-enhancing cooperation in the market and can also depress incentives to enter into new cooperation agreements.

It is difficult to say whether the total effect on welfare will be negative or positive. With the introduction of leniency, the less harmful CR strategy partially replaces CNR, but the overall collusive area expands by reducing the potential for cooperation. In addition, incentives for choosing to compete grow, which, on its own, may be beneficial for welfare. Still, the possibilities of welfare-reducing effects should be enough to make regulators consider the importance of raising the standards of evidence, including access to relevant information and adequate interpretation by means of economic analysis while looking at horizontal agreements.

The topic of leniency has strong ties with that of corruption that are obviously underdiscussed. The link is two-fold. First, antitrust violations — primarily cartels and bid-rigging — are known to have strong correlations with corrupt practices. In public procurement, for example, collusion is often facilitated through government agents, and antitrust investigations often lead to uncovering cases of bribery and other types of corruption. For this reason, effective leniency programs contribute to the fight against corruption by helping in the acquisition of information about potential violations at a lower cost for the regulator. However, some complications can arise from the fact that when a firm applies for leniency, it can factor into its decision the risk of becoming subject to an anticorruption investigation. If corruption indeed took place, then in terms of modeling it could mean that the reduced fine would be greater than zero, and the effectiveness of leniency policies could be critically reduced. A potential solution would be to ensure that whoever is exempt from liability for an antitrust violation should also be guaranteed protection from sanctions for corruption. The legal mechanism for such a construct requires further discussion, but one point is that the whistle-blower should also be required to collaborate with authorities in investigating the corruption case to receive additional leniency. The discussion of this link between corruption and leniency programs is outside the scope of our paper.

Another aspect linking leniency for cartel participants and the fight against corruption is the fact that leniency programs are also widely used to uncover other types of violations, namely corruption schemes. The effectiveness of such programs can be unclear (for a review, see Berlin and Spagnolo, 2015), but an aspect that has not been previously studied is the effect of leniency on corruption, which in fact facilitates welfare-maximizing transactions. The fact that corruption can, under certain circumstances, promote efficiency has been widely debated, with proponents appealing to arguments ranging from the familiar “greasing the wheels” metaphor to more complicated ones, such as in Huntington (1968), and opponents drawing attention to disastrous long-term effects. A relevant concept would be one put forward by Basu (2011) — the category of “harassment bribes”, or bribes that are given by actors to receive benefits to which they are already legally entitled. Assuming that the entitlement is derived from a social welfare-maximizing strategy, corruption becomes a less costly way to attain an efficient outcome in a rigid system, with no tendency towards positive change in the foreseeable future. If, by this logic, some forms of corruption can indeed promote efficiency, then treating all acts of corruption as per se illegal creates the risk of type I errors, interpreted not as wrongfully prohibiting a practice that is in fact legal but wrongfully prohibiting an efficiency-enhancing practice.

In that case, the introduction of leniency programs for acts of corruption can be modeled in a way similar to that which we use above, where collusion can be replaced, for example, by bribery with an inefficient outcome, and cooperation would be an act of efficiency-promoting corruption. We can easily envision a system of undiscriminating corruption, where bribes are part of the universally accepted rules of the game and even the most efficient companies participate in corrupt schemes, while the enforcement of anticorruption laws is selective (whether due to a lack of resources or due to political concerns). The “deviating” strategy also has a meaning because often corrupt relationships are shown to be susceptible to risks of opportunistic behavior on both sides (and consequently to the “hold up” problem —see Buccirossi and Spagnolo, 2006). Differing probabilities of investigation and conviction would also still be rationalized in a system with only limited resources that can be devoted to fight corruption: while it is impossible to monitor all transactions for potential corruption all the time, it makes sense that transactions that lead to socially unbeneficial outcomes (e.g., receiving a bribe to appoint an inefficient company to be the supplier of certain goods for the government, resulting in disruptions of contracts) will draw more attention from the authorities than transactions that eventually lead to beneficial outcomes (e.g., receiving a bribe for appointing that same contract to the most efficient company on the market, following which the contract is fulfilled without failure and for a low price).

With these assumptions, the mechanics of the model will remain the same, while the results can gain an additional interpretation. Assuming corruption can be welfare maximizing (even if only in the short term), introducing a leniency program could not only induce firms to employ bribery to gain benefits to which they have no right in the first place (moving from welfare-beneficial to welfare-unbeneficial corruption as an analogue of the “deserved punishment” effect) but also destabilize existing efficient, but illegally established schemes (as in the “disrupted cooperation effect”) and preclude firms from engaging in welfare-maximizing activities if the only way to access them is by corruption (as in the “prevented cooperation” effect).

As the idea of encouraging companies to implement antitrust compliance policies, possibly by way of providing a reduction of fines, becomes more and more popular in Russia, it is interesting to look at the possible consequences of such a measure, given what we know about the effects of leniency programs and how these two instruments can enforce or hinder each other.

On the first point, it is worth noting that our results highlight the risks of importing institutions without sufficiently taking into account the nuances of the local institutional environment, including working mechanisms of rules enforcement. Antitrust norms rely heavily upon economic analysis and expert judgment, and countries that have relatively less experience in applying the specific methods of economic analysis used in antitrust, as well as countries with insufficient resources dedicated to antitrust enforcement, run a higher risk of type I errors. Knowing this, firms tend to use all the instruments available to them to minimize the risk of wrongful conviction — and sometimes it becomes economically feasible to use as the means of insurance some instruments that were initially not developed for these purposes. Consequently, while antitrust fines in Russia remain large and an asymmetry of information persists between firms and the antitrust authority as to what constitutes a violation (taking into account the general assumption that norms do not necessarily promote the most efficient of all possible outcomes), it is quite possible to expect that compliance programs might be used by firms for which they were not initially meant (i.e., firms with low risks of antitrust violations). All in all, this might result in an additional cost for business (the cost of devising and implementing an unnecessary compliance program) and, eventually, a devaluation of the whole concept of compliance programs —that is, unless specific measures are taken to curb these possible effects.

The second point might be better understood if we relax the assumption of a firm functioning as a “black box” with the single purpose of maximizing profit and revert to an approach more in line with methodological individualism. Assuming that an asymmetry of information is present between the owners of a firm and its managers, as well as between top-managers and managers and so on, and a discrepancy exists between the goals of the principals and the agents on different levels, adopting an antitrust compliance program that identifies not only external antitrust risks (such as the types of violations that are most likely to occur based on the market structure and market position of the firm) but also internal ones and develops the necessary corporate procedures to minimize those risks seems both individually and socially beneficial. Firms benefit from an individual reduction of the probability of conviction, society benefits from a reduced probability of violations, and corporate procedures provide the necessary sources of evidence to keep the costs of possible investigations down, including the costs of identifying the individuals responsible, which can be critical for criminal sanctions. In practice, the effects of such policies can be ambiguous depending on, among other things, the design of the liability rules (Buccirossi and Spagnolo, 2008; Shastitko, 2016).

If a compliance policy is in place, it may complicate the matter of leniency. First, in countries where both corporate and individual leniency programs exist, if an internal investigation follows a certain procedure and takes time, then a company faces an additional risk of individual whistle-blowers applying for individual leniency throughout the time of such an investigation. If an individual self-reports and the antitrust authority finds out that the company is still investigating his behavior, which is why it has not applied for corporate leniency, the question arises of whether the company should be punished as severely as if an internal investigation was not underway or if their compliance policy was inefficient in preventing and uncovering the violation.

The intuition behind possible type I errors in the case of compliance policies—the company mistakenly self-diagnosing a violation —and their potential effects on owners, managers at different levels and employees, as well as the actions of the company as a whole, remains an issue for further discussion. We hope that these questions will be expanded upon in future research.