A wide variety of social and economic problems can be analyzed with Game Theory. It is interesting to see how Game Theory is applied in our daily lives without even noticing it. Whatever exists in our society is because it is in equilibrium achieved by people. Having said that, we will try to explain corruption under the lens of Game Theory.

## Game Theory in High Level

Game theory constructs mathematical models to examine how people behave in strategic situations whereby “strategic situation” means that **whatever is best for you** **depends on what others do**.

We can use the following notation for any problem which can be considered as a Game

- The number of
**players i**of the game, where i=1,2,…,N - The strategy
**a**of each player**i**notated as \(a_i\) - The payoff of each strategy
**a**for each player**i**notated as \(g_i(a_1,a_2,…,a_N)\). Note that the payoff for each player depends on the strategy of other players.

**Nash Equilibrium**

In Nash Equilibrium a common assumption is that players act **rationally**. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in a Nash equilibrium, **they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing**.

A nice example to explain the Nash Equilibrium by taking into account only two players is the Prisoner’s Dilemma.

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:

- If A and B each betray the other, each of them serves two years in prison
- If A betrays B but B remains silent, A will be set free and B will serve three years in prison
- If A remains silent but B betrays A, A will serve three years in prison and B will be set free
- If A and B both remain silent, both of them will serve only one year in prison (on the lesser charge).

Here the **group equilibrium** is for both prisoners to co-operate by serving 1 year in prison each, but if we look carefully, both prisoners have an incentive to deviate their strategies.

**Prisoner A Payoff**

For example, let’s have a closer look at prisoner A. **Given that prisoner B betrays**, Prisoner A can either remain silent which means -3 years or betray which means -2 years. Since -2>-3 Prisoner A chooses to betray. Now, given that prisoner B remains silent, Prisoner A can either remain silent which means -1 years, or betray which means 0 years. Since 0>-1 Prisoner A again chooses to betray. Thus, in any case, **the best strategy for Prisoner A is to betray **

The same logic applies for prisoner B and we ended up that the Nash Equilibrium is both prisoners to betray and both of them to serve two years in prison.

## How to Explain Corruption with Game Theory

Corruption can be explained with Game Theory. I would like to share an example from Greece and the process to get a driving license **back in** **2000**. Let me try to describe the process briefly.

In order to be eligible to take the exams for the driving license, you needed to do **at least 20 lessons**. The exam fees were around **100€** and every driving lesson cost around **18€**. If you failed in the exams, you were obliged to take at least another 10 lessons in order to be able to sit the exams again. This means, that if you failed then you had to pay another 18 x 10 + 100 = **280€** just to be able to sit the exams again. The process above can be considered as a social game with the following parameters.

**Players**

There are three players. The **candidate**, the **examiner **and the **driving teacher**. All of them want the best for themself.

**The Strategy**

The game has to do with bribery. If the examiner will ask for money, if the driving teacher will be a mediator in the process and finally if the candidate will bribe the examiner or not.

Let’s see how Nash Equilibrium was created. The goal of the corrupted system was to enforce the candidates to bribe. For that reason, it made obligatory to take another **10 driving lessons** in case of exam failure.

**Candidate’s Dilemma**

The candidate’s dilemma is the following:

- The first option is to bride and receive the driving license.
- The second option is not to bribe. If the examiners are corrupted, then no matter how well you drive during the exams, they will find a way to make you fail. This means that the candidate will have to pay at least another
**280€**. If the examiners are not corrupted, then there are two possibilities. Either to pass or fail the exams fairly. Thus the candidate should think about the probability of the examiners to be corrupted, the probability to pass the exams fairly and then if he/she fails will pay at least another**280€**and will have to face again the same situation, which means that again he/she will have to take a decision to bribe or not. - The third option is to report the examiners. This means that he/she has to pay at least
**50€**to sue them and then to start the trial process which takes around 10 years. In the meanwhile, he/she will have to pay the lawyers and if he/she failed to prove the bribery, then he/she will have to pay compensation for non-pecuniary damage.

Clearly, the best strategy for the candidate is to bribe. The question is how much s the premium of the Nash Equilibrium. The price was around **180€**. From this amount, the majority was going to the examiners and the rest to the driving teacher, and all the players were happy. Let’s have a closer look at the “utility” of each player.

**Benefit for the Driving Teacher**

The driving teacher was receiving an amount from the bribery for acting as a middleman. If for any reason, the driving teacher didn’t want to be part of the game, then his/her students will have very low success rates which implies a damage to his reputation. So for the driving teacher the best strategy was to be part of the bribery.

**Benefit for the Examiners**

The examiners were employees in the public sector receiving a decent salary. Because these positions were allowing them to make a bunch of black money, they were filled by corrupted people from the government. It was like a promotion to be a driving examiner. Thus, the government was paying indirectly their faithful watchdogs. From the bribery, they were receiving the lion’s share.

**Benefit for the Candidate**

The candidate without stress and without taking any risk, was making sure to get the driving license.

## Final Thoughts

I used the past tense because the situation was around 2000. I do not know how is it right now. The situation was so bad that they were even sending the driving licenses at home. So, many people without even know how to switch on the car were able to receive a driving license. The question is how we can disturb this bad Nash Equilibrium. This is a long discussion and it out of the scope of this post. I decided to choose the example with the driving license because I faced. I could have used other examples like cases from hospitals where the patient is willing to bribe the doctors of the public sector etc.