Prisoner's Escape

by Mariano Mosquera

When governments ask companies for bribes, it leads to a prisoner's dilemma situation in public procurement processes. However, companies can manage to escape through collusion.

The Game

The prisoner's dilemma is a strategic game between two actors. Each actor has two choices, to cooperate or to defect: "C." and "D." for player I; "c." and "d." for player II.

Payoffs are defined according to the simultaneous combination of choices made by the players. When both choose cooperation, each player receives a payoff of 2, while when both choose defection, each player receives a payoff of 1. The combination (C., d.) entails that player I cooperates and player II defects, which results in a payoff of 0 for player I and a payoff of 3 for player II. The combination (D., c.) entails that player I defects and player II cooperates, which results in a payoff of 3 for player I and a payoff of 0 for player II. Defection is the dominant strategy in the prisoner's dilemma.

The players’ decisions are the result of the various analyses they carried out. In the first place, a trust or distrust-based analysis (strategic and cultural) can be used to anticipate the other actor’s movement. In the second place, the players can make an analysis (more rational and individual) of the benefit sum of the payoff options. Finally, a Rawlsian analysis. The ban on communication between prisoners acts as the veil of ignorance; that is, the lack of knowledge about one’s relative position with respect to the other actor's decision. This tips the balance in favor of low-risk decisions, since actors would rather not suffer too much damage once the veil is lifted. Therefore, they rule out the option with the worst outcome. According to these three analyses, the best choice is defection.

The prisoner’s dilemma is based on distrust towards the other actor; defection brings a higher benefit sum than cooperation (3+1=4>2=2+0), and the worst possible outcome is that one of the actors cooperates (0) and the other one does not. Mutual defection is a Nash equilibrium, since once both players have stated their decision, neither will have any incentive to change it. Mutual cooperation is a Pareto optimality since it brings the highest collective benefit sum (2+2=4), and it is a situation that cannot be changed without damaging any player.

The Public Procurement Jailer

This game is played in public procurement processes when the government favors institutional corruption. In the original prisoner’s dilemma, the jailer establishes the rules of the game through the payoff system. In this case, there are (formal and informal) rules defined by the government (the jailer) for two companies (prisoners) that want to sell their goods or services to the government. The government offers or promotes, or the companies just know of, the following two possible choices: cooperate with the other actor in a competitive bidding process, or defect from the competitive process and try to win a government contract in return for bribes.

If both actors defect (in this latter sense), the bidding cannot be performed, since another actor's cooperation is necessary to “disguise” a corrupt hiring as a competitive and legal process. The government can make a corrupt agreement with only one actor. If the other actor also decides to offer a bribe but cannot do it, he will not bid (a second decision is not made).

In this game we will not consider the possibility that the government could make a corrupt agreement with both players. For example, the government could make an agreement with one actor so that he wins the contract and also enter into an agreement with the other actor so that he legitimizes the process (which is unlikely in a non-iterated game, since the promise of changing roles in the next bidding process is lost). The combination in which both players defect is spawned by distrust and the logic that another player defects and agrees to pay a bribe to the government. This leads the actor to defect as well, either by offering a bribe or by deciding not to participate in the bidding process. As a result of the combination resulting in the defection of both players, no tenders are submitted. If both players cooperate, a legal and competitive bidding process takes place. If one player cooperates and the other one defects, the cooperating player legitimizes a bidding process that is not actually competitive; it is a charade. Clearly, the cooperating player does not know that the other one has defected. The defecting player wins the contract in return for a bribe.

Complex Payoffs

In this public procurement game, payoffs are defined in connection with two variables: the amount of the contract and the probability of winning the contract. Also, in the case of mutual defection, the variable “time gained” should be considered, since neither player makes any effort to submit tenders.

With regard to the amount, A is defined as the actual value of the contract. It is also stated that A‾ is the value A of the contract less the bribe (A‾=A-bribe). In this regard, A‾ reduces the appeal of the contract for the company. "B" is the complete certainty of winning the contract.

On the other hand, participating in a competitive bidding process involves winning or losing. With two players, the probability is 0.5, assuming that both of them assume an even competition (b.=B/2). If both players defect, neither has a chance of winning the contract, but none of them made an effort to submit tenders (C). If one player cooperates and the other one defects, the cooperating player submits a tender but has no chance of winning the contract. The defecting player will surely win the contract.

This scenario maintains the prisoner's dilemma structure: A‾B0.>Ab.0.>00C.>000.

A general assumption in this payoff system is that actors view the risk of participating in illegal activities as very low. That is why the risk variable will not be considered. In real life, the equilibrium in a game with more than two players is achieved with the combination of defection (D.) and legitimization (c.). This is the combination desired by the corrupt government. With more than two players, the probabilities of finding a legitimizing actor increase.

Expanded Cooperation

Collusion is the means by which the prisoners can escape the dilemma. Collusion refers to the expansion of cooperation between prisoners against the jailer; in this case, the government. That is to say, the cooperation between prisoners goes beyond the pre-established rules of the game and its payoff system.

This requires an option of expanded cooperation for the actors. If they make this decision, the actors increase the payoffs in two variables: an increase in price (due to an agreement between suppliers) and an increase in the probability of winning (due to a subcontracting agreement with the loser.1) Only by means of expanded cooperation can the actors change the rules and the payoff system imposed by the jailer.

Acting on their own, the players are condemned to the jailer's payoffs. If only one of them cooperates, the cooperator does not get any payoff. If only one of them defects, the defector will be conditioned to the bribe demanded by the government.

Collusion can increase significantly the contract value (A+). In addition, with a subcontracting agreement between two actors, the probability of winning the contract is B. In this scenario, the payoff structure is changed: A+B0.(collusion)>A‾B0.(solitary defector)>Ab.0.(simple cooperation)>00C.(mutual defection)>000.(solitary cooperator). It is assumed that even with subcontracting, the increased amount is very attractive to the actors.

Rawls Obstacle

In this new scenario, two of the three analyses carried out by the actors are modified. This enables the prisoner's escape. The culture of distrust is not eliminated in this scheme; however, it is transferred to the jailer. Companies trust more in themselves than in the government, since the latter is the source of the initial corruption.

The benefit sum is modified as well. Cooperating now entails: A+B0.=A+B0.+000. Defecting entails: A‾BC.=A‾B0.+00C. Being: A+B0.>A‾BC. This is the case as long as the amount of the increase is higher than the time-money relationship assigned to a tender submission process. These two criteria outweigh the Rawlsian analysis of preventing the worst of all evils. The actor may consider that the other actor will also add benefits and choose expanded cooperation. Any actor who does not cooperate in an expanded way and defects is not doing it due to the lack of good payoffs, but merely because he thinks the other actor will defect or try to hurt the other actor.

The Rawlsian analysis continues to be the worst obstacle, since cooperating could still be the worst outcome if the other actor defects. However, in this public procurement scenario, it is the jailer who is distrusted.

The original prisoner's dilemma has two players but also has rules based on the ban on communication, the payoff system, and a jailer who imposes its authority. Therefore, the prisoner's escape entails:
1. A set of rules and a payoff system that do not favor cooperation.
2. A jailer without the legitimacy to demand obedience, who is favored by the defection/ legitimization combination.
3. A payoff system that can be increased with cooperation between prisoners.
These conditions will lead to expanded cooperation.


By examining how to cooperate beyond pre-established rules, we can rethink institutions and their incentives.2 Expanded cooperation shows better payoffs, which are useful for the players' individual analyses. Through the increase of cooperation payoffs, it is possible to switch from the prisoner's dilemma rules to Rousseau's stag hunt incentives. Also, a new payoff structure could enable a Nash equilibrium with a dominant reward (Pareto optimality) in expanded mutual cooperation.

Offering bribes is possible in a prisoner's dilemma scheme. However, the offer of bribes is reduced by the improvement of incentives for cooperation (stag hunt), even with a corrupt government. In this case, legally favoring the subcontracting of the losing companies that cooperated in the competitive process could be considered.

Distrust in government makes it necessary to look more deeply into corporative transparency methodologies and integrity agreements under the coordination of third parties, such as universities and civil organizations. On the other hand, in an iterated game, additional scores could be assigned during the bidding process to losing companies from previous processes, whenever they face the same competitors. This entails generating advantages in other games; incentives for cooperation and not only applying sanctions in new games, as the classic way out of the iterated game.

*The author would like to thank Dr. José María Rodriguez, Economist, for reviewing this article.

1. The iterated game usually includes the promise of changing roles in the next bidding process. In this game, repetition is not considered.
2. This framework can be used for the purpose of investigating other expanded cooperation processes, such as social uprisings triggered by inefficient institutions and governments that promote the combination of no participation/legitimization.