Real Mathematics – Social Situations #3

Battle of the Couples

Bruce: I have waited for this game since the start of the season. We will be hosting our arch rivals in the final game of the season. And like that is not enough excitement for me, if we win we will crown as champions: It is either us or them.

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Every year when I decide to buy a season ticket, I dream of going to this game. I haven’t missed a game since 2006 and this season’s game will be cherry of top in case we win. We have to win and I must witness it.

Jane: My favorite band is retiring and they are on the road for the last time. Luckily for me they will be visiting my town this time: I will have the opportunity to watch them alive in my city for the first and the last time.

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When I was 11, I bought a random album. I went home, pressed start on my music box and fell in love with them instantly. It was their best-selling album from 1992. I have been laughing with their songs, I have been crying with their songs. I had spent my youth with them. And I will say farewell to them in person.

Bruce and Jane have been together for some time, but they are in a pickle now. Derby game is postponed and both events will be started at the same day, same hour.

Ideal Couple

In this scenario we assume that Jane and Bruce are equally (or at least equivalently) in love with each other.

Since they are ideal couple, spending time together is the most important thing for them. That is why we’ll give them 1 point for being together if they choose to go to the same event.

We’ll give them 1 extra point for their favorite event: For Jane concert is 1 extra point and for Bruce game is 1 extra point.

If they are not together, they get -1 point as they are unhappy for being apart. This is why in case they go to separate events they will get 0 point each.

Then we can construct the game matrix as follows:

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In this game if players choose to be selfish and only consider their happiness, Jane would choose to go to the concert as Bruce would choose to go to the game. We already know that the outcome of concert-game is (0,0). In order to test if this is the Nash equilibrium or not, we must test players one by one. First let Bruce chooses first. He will decide to go to the game.

If Jane knows about it, she will have to choose going to the game also since outcome of the game (1) is larger than the outcome of the concert (0) for her. In this case the equilibrium is game-game.

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Now let Jane do the first selection. She will choose to go to the concert:

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In this case Bruce has to choose to go to the concert as its outcome (1) is larger than game’s outcome (0). Here, equilibrium is concert-concert.

We just found out that there are two Nash equilibriums in this game: Concert-concert or game-game. Both cases could happen if only Jane and Bruce are willing to cooperate. Otherwise if they act selfish, they will get no happiness whatsoever.

Personal Thought: Even with an ideal couple Bruce should compromise since “she” will always win.

Jane Doesn’t Love Bruce

Jane thinks she could do better. And she has a point: Bruce is a 43-year old unsuccessful computer engineer who is slowly going bald. On the other hand Bruce is very much so in love with Jane and deep inside he knows that she is a catch.

For Jane, being in the concert means more.

For Bruce, being with Jane and going to the game are equally important.

Let’s build our game with this information.

If they both decide to go to the concert: Jane will receive 2 points (0 from being with Bruce, 2 from being at the concert). Bruce will get 1 point and that comes from being with Jane.

If they both decide to go to the game: Jane will receive -1 point (0 from being with Bruce, and -1 from not being in the concert). Bruce will get 2 points: 1 from being with Jane, the other from being at the game.

Jane to the concert, Bruce to the game: Jane will get 2 points in total, all coming from being at the concert. Bruce will get 1 point in total and that comes from being at the game.

Jane to the game, Bruce to the concert: Jane will be furious and get -1 points as Bruce will get 0 point.

Now we can construct the matrix of the game as follows:

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If both players are selfish, Jane would choose to go to the concert as Bruce would choose to go to the game. In this scenario concert-game becomes the result of the game. Its outcome is (2,1).

We should check if concert-game is the Nash equilibrium for this game.

If Jane knows that Bruce is going to the game, she would have two options: 2 and -1. Obviously she would choose 2; that is going to the concert. Then in this game result becomes concert-game.

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If Bruce knows that Jane is going to the concert, his choices would get him 1 point in either case. This is why Bruce would have two identical choices. Concert-concert and concert-game will have the same probabilities.

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In conclusion, there are two Nash equilibriums for this game: Concert-concert and concert-game. Both cases have the same outcome that is (2,1).

One wonders…

Find the matrix of the game and its Nash equilibrium when Bruce doesn’t love Jane while Jane does love him.

Real Mathematics – Social Situations #2

There is only one rule: No rules!

Istanbul is a great city. But some days I feel overwhelmed with its traffic jam. I am aware of the fact that almost every metropolis on earth has the same problems. But there is something you don’t know if you have never driven a car in Istanbul: Everything could happen in Istanbul’s traffic.

In this city if there is a pedestrian and a car on a pedestrian crossing it is the pedestrian who gives way, not the car. In this city it is absolutely normal to see a car on the far right lane trying to make a left turn. In this city if multiple cars meet at a crossing, nobody knows who will be the first car to pass.

This poor reporter is trying to use the pedestrian lane. You can see what kind of situation she got into. Don’t worry; she is fine. She’s made it safe and sound.

These are just a few examples of how irregular traffic is in Istanbul. You might think that I am exaggerating. Okay, you can continue to think like that, just beware even if it is green light for you.

What strikes me most about Istanbul’s traffic is that everybody learns about these rules when they study for the driving license exam. However most of the drivers forget about who has priority at a crossroads.

Who will lose?

Assume that you are the driver 1 and you are coming closer to an intersection where you have the priority to pass. Unfortunately for you, this intersection is in Istanbul and you have no idea how the driver 2 will behave in the next few seconds: Will he/she stop or not?

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Situation 1: You both decided to stop. Both cars are stopping in the middle of the road and no one is crossing the intersection. There are no winners.

Situation 2: You both decided not to stop and eventually crashed. In this situation not only there are no winners, you two are both losers.

Situation 3: You decided to stop, and driver 2 didn’t. Driver 2 won the game as he/she is the first one the pass the intersection.

Situation 4: You decided not to stop as driver 2 did. You are the crowned winners: You will be home 3 seconds before driver 2.

Nash Equilibrium

Let’s use game theory to analyze these four situations. If a car stops it will get 0, if they crashed they’ll each get -1 and if one passes first he/she will get 1 point.

Then, matrix of the game will be as follows:

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If drivers are selfish, meaning that everyone is looking out for their own interest, both drivers would decide to get 1 point. This means they would both decide not to stop and crash.

Yet, Nash equilibrium of this game demonstrates interesting results.

If either one of them stops as the other one knows about it, then the other driver would choose not to stop.

If either one of them doesn’t stop as the other one knows about it, then the other driver would choose to stop. This way he/she can prevent the crash.

This means that the situation has two Nash equilibriums: (1,0) or (0,1).

In these types of games every player needs one another. As seen above in certain situations drivers must think of preventing an accident instead of passing the intersection first.

A person needing one another is a necessity if we all want a just and organized social life. If every individual in a society keeps thinking only about his/her own interests and makes decisions which would only help themselves, then every aspect of social life would get worse than it should be. This is the case why traffic of Istanbul is in such conditions: Drivers don’t respect others and disobey social contracts just because they have something personal to gain.

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“Thou shalt not steal!” Oh you are driving… Sure, you can steal my time!

An example from Istanbul: A driver has to wait in traffic for 50 minutes even though that distance can be taken in 10 minutes with normal speed limit. After having spent 50 frustrating minutes, this driver would double or even triple a single lane so that he/she would gain 5 minutes. Driver’s decision to double a single lane makes traffic even denser; it would affect thousands of people. And craziest of all is that this driver’s behavior is not any different from the ones who initiated the traffic jam in the first place…

M. Serkan Kalaycıoğlu

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-dilemma

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:

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Possibilities

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

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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.

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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.

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John Nash

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.

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If second student stays loyal, first would choose 0 over 1.

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.

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If second student betrays, first student would choose 10 days over 15 days.

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