Bribery – a game theoretic approach

Mr. Kaushik Basu, Chief Economic Adviser, Ministry of Finance, Government of India has published a working paper on why bribing should be made legal to stop bribery. The paper can be found here: http://finmin.nic.in/WorkingPaper/Act_Giving_Bribe_Legal.pdf

For general people, attached is my take on it, from a game theoretic perspective, as he also used game theory to prove his point. Mine is purely non mathematical argument, and I wish to work with the mathematics behind it soon.

In the working paper, Mr. Kaushik Basu wrote that “In other words, what is being argued is that this entire punishment should be heaped on the bribe taker and the bribe giver should not be penalized at all, at least not for the act of offering or giving the bribe. We may in fact go further and say that, in the event of a case of bribery being established in the court of law, the bribe taker is required to give the bribe, to the extent that its size can be uncovered, back to the giver. Let us, for now, go with this assumption.

The main argument of this paper is that such a change in the law will cause a dramatic drop in the incidence of bribery. The reasoning is simple. Under the current law, discussed in some detail in the next section, once a bribe is given, the bribe giver and the bribe taker become partners in crime. It is in their joint interest to keep this fact hidden from the authorities and to be fugitives from the law, because, if caught, both expect to be punished. Under the kind of revised law that I am proposing here, once a bribe is given and the bribe giver collects whatever she is trying to acquire by giving the money, the interests of the bribe taker and bribe giver become completely orthogonal to each other. If caught, the bribe giver will go scot free and will be able to collect his bribe money back. The bribe taker, on the other hand, loses the booty of bribe and faces a hefty punishment.

Hence, in the post-bribe situation it is in the interest of the bribe giver to have the bribe taker caught. Since the bribe giver will cooperate with the law, the chances are much higher of the bribe taker getting caught. In fact, it will be in the interest of the bribe giver to have the taker get caught, since that way the bribe giver can get back the money she gave as bribe. Since the bribe taker knows this, he will be much less inclined to take the bribe in the first place. This establishes that there will be a drop in the incidence of bribery.

Indeed, under the new law, when a person gives a bribe, she will try to keep evidence of the act of bribery—a secret photo or jotting of the numbers on the currency notes handed over and so on—so that immediately after the bribery she can turn informer and get the bribe taker caught. The upshot of this is not that the bribe taker will get caught but he will not take the bribe in the first place”

Let us take the situation a bit more technically, in the game theoretic terms.

This situation can be modeled as the classical problem of prisoner’s dilemma. The prisoner’s dilemma problem deals with two convicted persons. Now the situations are, both can tell about other’s crime to the police, both can keep quite about other’s crime, and one can keep quiet while other can tell. If the punishment for the crime is same, then we can show that though both keeping quiet is the best option for both the convicted, they end up telling about the other’s crime, and in consequence both gets the punishment. Now, the basis of this game theoretic calculation in the amount of mistrust between two offenders. If we play this game for long time, essentially, the mistrust comes down to a great extent, and we can again show that in the limiting scenario, the game converges to the strategy pair where both convicted keep quiet about other’s crime.

Now let us take Mr. Basu’s suggestion into consideration and form the game matrix. We take the giver’s strategies as “tell” and “not tell”, while the taker’s strategy is “do (the job)” or “not do”. We are working with some fictitious numbers for our reader’s ease of understanding. Let the amount of bribe is 10, and the amount of punishment, when converted in the same unit of the bribe is also 10. Let the cost of the job being not done in time is 3, and the gain of the job being done in time is 15 (normally the bribe amount is less than the importance of the job).

Then the game matrix looks like:

	Tell	Not tell
Do	-10, 15	10,5
Not do	-10, -3	10, -13

The row strategies are played by the bribe taker and the column strategies are played by the bribe giver.

Evidently, the strategy “not tell” always gets dominated by the strategy “tell”, and hence, it seems that the taker will eventually get caught. So far, the argument of Mr. Basu works fine. But, the question arises beyond the scope of the standard non-realistic assumption of non-cooperative settings between the players. Furthermore, the game is not an one shot game, it is to be played over a period of time and over various combination of players. So, the dynamics of learning through repetitive games is a crucial part of the analysis of this type of game. The evolutionary character of this game is also to be considered as the character of the society and the people will surely change over the time.

So, what happens if we play a repetitive game?

Whatever the bribe giver decides, the bribe taker can act independent of her action as both the “do” and “not do” will give her the same payoff, according to the payoff matrix showed above. Obviously, for the case of “harassment bribes”, the urge or goal for the client is to get the work done, that is why she is giving the money, otherwise she always have a choice to walk away. So, we can eliminate the “not do” strategy from the matrix as that is not going to help any of the players. Now comes the most important aspect of the game, whether its cooperative or non-cooperative. If the game had been of the non cooperative nature, the game would have been perfectly followed the course of Mr. Basu, but unfortunately the thing is not so rosy. In most, if not all of the cases, there is substantial amount of preplay communication between the two players, and in any such cases, the standard idea of non-cooperation fails, and eventually which leads to the strategy pair (Do, Not tell) in the long run, because that is the only case where both the players are having positive payoff!

This is a very superficial level of analysis to show that the strategy of punishing the taker and no punishment for the giver will not work in the long run as long as the preplay communication between the two players are there. For more detail analysis and concrete result, a detail mathematical argument is necessary which is beyond the scope of this writing.

Now, this game can be seen from a different angle. Till now, we haven’t included any option to the players of not to give the bribe, or not to take the bribe. If we include that option to the strategies of the players, and if we add a substantial amount of reward to those who are not going to take the bribe, being offered, and add the punitive measure to the giver also, the things may completely be changed. Let us have a look at the changed payoff matrix. Let us assume that the reward for not taking bribe is 10 and the punishment for offering bribe is 10. So, the new payoff matrix will look like:

	Not give bribe	Give and tell	Give and quite
Honest + tell + do work	10,15	10,-10	10,-10
Take and do work	0,3	-10,5	10,-5
Take and doesn’t do	0,-3	-10,-13	10,-13

Evidently, in this game there is only one pure strategy strict Nash Equilibrium, which is “being honest”, both for the giver and the taker! Being a strict equilibrium, it is evolutionarily stable, and stable against any perturbation also, i.e., in technical terms, this is a proper and perfect equilibrium point.

So, in the long run, as long as the preplay communication is there between the bribe giver and the bribe taker, no law which differentiates between the bribe giver and bribe taker can reduce bribery or corruption, instead, if the law starts rewarding the good and punishing the bad, from the very conservative sense of terms, then only in the long run, the good sense will prevail!