# Real Mathematics – Social Situations #1

Lazy Students

Mathematics test is the next day. You and your best friend have two options: Study for it or play video games. Your decision was made easily and you ended up playing Fifa for 6 hours. Obviously you choked during the exam and now you two are just waiting for an F to be announced. Although, you were prepared for such situation: Your plan was to memorize the questions, solve them at night and try to change your papers the next day.

Next day, early in the morning you went directly to the floor where teacher’s lounge is at. You hid inside the toilets and waited for first lessons to start. When you heard the bell, you let yourselves out of the toilet and managed to change your exam papers. You were cocky as you were walking down the corridors which caused principal to catch you.

Principal was wondering what these two were doing at this floor while they should have been in class. Principal was sure of one thing: They did something bad. He (let’s assume principal is a he) decided to separate students and interrogate them one by one.

As you were waiting in principal’s room alone, you were thinking that principal has no evidence whatsoever. You were aware that you will get suspended for 1 day as you were caught ditching class. Nevertheless if they understand that you changed your exam papers, punishment would be much harder.

Now principal is back. He claims that your friend betrayed you and told everything, and if you come clean you will be getting a minor penalty. In case you reject confessing you will get a much harder punishment.

Is he right? Did your friend really give you away?

What should you do? Confess or deny?

Prisoner’s Dilemma

This story is actually a different version of an example known as prisoner’s dilemma. Prisoner’s dilemma is the core example to explain an important branch of mathematics called game theory. To sum it up briefly prisoner’s dilemma examines whether you would sell your friend to avoid punishment or be loyal to him and make the best possible choice for both of you.

In game theory’s cooperative games this question is vital: When you are in a situation where everyone’s choices affects the result, would you be selfish and only consider your outcome or would you choose for the benefit of the group?

Punishment Matrix

• List all the players.
• Show alternative choices for each player. These choices are also called “strategies”.
• If there are two players, choices for the first player would be represented in the rows. For the second player choices are represented in the columns.
• Every entry inside matrices represents the utility or payoff to the first and second player respectively.

Then the punishment matrix for the lazy students become as follows:

Possibilities

If first student decides to betray, he/she will get either 10 or 0 days of suspension.

In this case, if second student betrays as well, both students will get 10 days of suspension each. If second student stays loyal and decides not to confess, he/she will get 15 days of suspension as the other gets 0.

If first student sticks loyal to his/her friend, then he/she will get either 15 or 1 day of suspension.

In this case, if second student betrays, he/she will get no punishment. If second student also stays loyal, then they will get 1 day suspension each.

What would you choose if you were in this situation?

Selfish

Best scenario for both students is them to get 1 day suspension each. This could happen only if they both stay loyal to each other. But there is also another scenario to consider: Worst one.

The worst scenario in this game is to get 15 days of suspension which could only happen if one student stays loyal and the other betrays. In this case, the student who was loyal would get 15 days.

Hence there is the dilemma: Staying loyal gives the best possible outcome for both students. But individually it can cause the worst scenario too.

If students are selfish, they would consider the choice where they can get the minimum damage. For this game, it is getting 0 days of suspension. For both students this could only happen if they choose to betray. Here comes the second dilemma: If they both betray, they would get 10 days of suspension each.

If students are up for teamwork, they would consider the choice where there is least punishment in total for both of them. For this game it is getting 1 day of suspension each. This could only happen if they choose to stay loyal.

Nash Equilibrium: Each player’s strategy is optimal when considering the decisions of other players. In Nash equilibrium every player wins because everyone gets the outcome they desire. To test if the Nash equilibrium exists in a game, reveal each player’s strategy to the other players. If no one changes his/her strategy, then there is a Nash equilibrium.

Nash equilibrium was discovered by famous mathematician John Nash whom you might have heard from the popular Hollywood movie called “A Beautiful Mind”.

What is so special about prisoner’s dilemma? And also what is the Nash equilibrium for our lazy students?

Prisoner’s dilemma is a very special example as it gives out an incredible result. In our lazy student game Nash equilibrium is when both students choose to betray. This is fascinating as the best possible outcome for the students to stay loyal and get 1 day of suspension each.

How?

Loyalty

Let’s assume that both students stay loyal and also assume that we told them what decisions they have made.

If first student knows that the second student stayed loyal, he/she would change his move and decide to betray so that he/she will get 0 days of punishment.

Same goes for the second student!

In the end both students would betray each other.

Betrayal

Now let’s assume that they both betrayed each other and we told them what decision they have made.

If first student knows that the second betrayed him/her, he/she wouldn’t change his/her first decision.

Same goes for the second student!

In the end both students wouldn’t change their decisions and betrays each other.

This is why betraying and getting 10 days of punishment instead of 1 day is the Nash equilibrium.

M. Serkan Kalaycıoğlu