Endogenous Detection of Collaborative Crime: The Case of Corruption

Graetz et al. (1986) developed a similar model already before Tsebelis. This will be discussed later.

Bianco et al. (1990) argued that if the game was iterated or had more players, it might lead to different results. Graetz et al. (1986), Cox (1994), Friehe (2008) and Pradiptyo (2007) all offered different results in slight variations of Tsebelis’ model.

The inspector is assumed to be a rational self-interested utility maximiser with regard to inspection effort, but does not engage in accepting bribery herself.

See also Rose-Ackerman (2006) for a more extensive handbook on the economics of corruption.

Lambsdorff (2007) elaborated on this idea and developed what he refers to as the invisible foot principle. Basu et al. (1992) also use asymmetric penalties. However, they do so only for bribing and accepting bribery, but not for reciprocating.

The upper bound of prison sentences and fines is equal for bribing and accepting bribes under 331–335 of the German criminal code (Strafgesetzbuch). This is argued to be justified, since the misappropriation of public funds or goods happens in a mutual act. Thus, as people are deemed equal before the law as per the constitution, they deserve equal punishment.

If we were to convert our game into a cooperative one, this would require a bargaining game between all three parties, where briber, official and inspector have to negotiate on the payoff to the inspector. This might however lead to issues of crime control as shown in Marjit and Shi (1998).

Note that in this game the official automatically accepts, if she is offered a bribe. She cannot reject. This is plausible if the size of the bribe is always high enough to achieve acceptance, but not necessarily reciprocation. Alternatively, one might think that an official would first decide whether to accept a bribe or not and then reciprocate in case of acceptance. However, since we are focussed on detection there is no need to make alterations of this kind.

Lambsdorff and Nell’s original model also included additional nodes, where players could defect and report on each other. We exclude these for simplicity.

The bonus (r) can then either be seen as an investment in the future (for instance as a positive discount factor) or one could replace it with a threat in form of a cost of punishment for not reciprocating.

As shown later, r needs to outbalance αpO2. The official gains r by reciprocating, but she also risks an additional penalty (pO2) with probability α.

For completeness, note that aˉI>cˉI represents the intuition that it is worse not to inspect if there is reciprocated bribery, i.e. full-blown corruption, than if there is a mere unreciprocated bribe.

A negative bribe could be seen as a bribe from official to briber, but this case shall not concern us here.

This assumption can be changed to p≥0, depending on what we believe to be more realistic or more suitable with a view to determining the most effective way of deterring corruption. We might for example believe that not all four penalties are distinguished by some legislations. Moreover, in Lambsdorff and Nell’s original game, players could not only report on each other but reporting was rewarded. They called this leniency. An investigation of this case might be worthwhile in a variation of our game. In yet another variation it might also be useful to think of the penalty for bribing as a positive incentive that cancels out the cost of the bribe. This makes sense, if bribing occurred in order to gain access to a public good, to which one was entitled to anyway (cf. Basu 2011).

We will see later that changing either penalty on the briber has the same qualitative effect on the equilibrium and penalty pO2 on the official has the opposite effect on the equilibrium. Penalty pO1 has no effect, because it features in the payoff for both inspected reciprocation and inspected non-reciprocation.

At this point and from now onwards α, β1 and β2 denote equilibrium values of these variables.

As noted in Footnote 15, changing pO1 has no effect on any of the probabilities.

Eq. [1] is simplified.