Game Theory: The Mathematics of Decision Making

Today I am going to introduce you to Game Theory, as I feel it is a subject to which people can see immediate and considerable effect in their life. In simple words, game theory is the math behind the decisions you take on a daily basis.

What is a game?

downloadA game is any interactive situation between multiple people in which every persons pay off is affected by decisions made by others, keeping certain rules in mind. Pay off reflects the desirability of an outcome to a player.

What is game theory?

Game theory is the branch of mathematics concerned with the analysis of strategies for dealing with situations where the outcome of a participant’s choice of action depends on the actions of other participants. It looks at relationships and tries to predict the optimal decisions people will make. It is very wide ranging and is used by economists, psychologists, military tacticians, political scientists, biologists, etc.

It has two major segments:

  1. Cooperative Game – In a cooperative game, the players make their choices towards getting mutual benefit.
  2. Non-cooperative/competitive Game – In a non-cooperative game, there are some winners and some losers. Each player makes his/her choice keeping his own pay off in mind.

We are now going to discuss The Prisoner’s Dillema, one the most famous thought experiments of competitive game theory to understand the foundations of game theory.

Prisoner’s Dilemma

Given

a) PLAYERS: Prisioner 1 and Prisoner 2

b) OUTCOME: Years in jail- (Prisoner 1/Prisoner 2)

2 prisoners have been arrested for fleeing a scene of crime and based on the evidence already collected, they are each going to have to spend 2 years in jail.

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c) The police wants more so they have placed both of them in two different rooms, with no way to contact each other and have offered each one of them a deal:

1. If one prisoner confesses about his partner’s crime while the other doesn’t-  0 years of punishment for the one who confesses, while the partner will be jailed for 10 years                               

2. If both confess about each other’s crimes- Both will be jailed for 5 years 

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3. If neither of them confess- Both will be jailed for 2 years

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They have to make their decisions independently as they have been split up. 

d) Emotions do not play a role in the game. Both the prisoners are concerned with their own jail time.

Tabular Presentation

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  • If both the players do not confess, they will have to spend as little time as possible ie 2 years each in jail.

If they could trust each other, that would be their best bet but they have no reason to trust each other. Therefore, both of them will have an incentive to confess and they will also consider the fact that the other prisoner might confess for his own benefit.

  • If one prisoner confesses and the other doesn’t, the snitch gets to walk out ie one prisoner will be jailed for 0 years while the other will get 10 years of jail time.
  • Compared to the risk of getting 10 years of jail time, 5 years each of jail time doesn’t sound so bad to both of them. Therefore, according to game theory’s solution both should choose to confess.

This outcome is known as Nash Equilibrium which one of the foundational concepts of game theory. When a player makes a choice that leaves him better off compared to his other choices, no matter what the other player’s choice is, it is called his dominant strategy. The combination of both the players dominant strategies coming together would be called Nash Equilibrium.

If Prisoner 1 confesses and Prisoner 2 doesn’t, Prisoner 1 is better off and gets no time in jail compared to 2 years in jail. If Prisoner 2 does confess ie 5 years of jail time each, Prisoner 1 is still better off as if he hadn’t confessed he would have had to serve 10 years of jail time. Therefore, his dominant strategy will be to confess. Likewise, for Prisoner 2.

If both do not confess they will each get 2 years of jail time which is better for both of them however, it’s an unstable state as in a non-cooperative game all the players stand to gain from stabbing each other in the back.

dilemma

We can use this concept in various situations in our daily life,

  1. A person is in love and has to make the decision about whether he is going to convey it to his partner or not.
    Checkout out this link to see how he goes about the decision using game theory:  https://www.quora.com/What-are-some-real-world-examples-of-game-theory/answer/Srishti-Bhatia-4?srid=vIGr
    This is a very basic example for you to understand how you could use the concept in daily life. Do keep in mind that it gives the most logical solution, not the best possible outcome.
  2. A person has been assigned a group project and he is going to make the decision whether he is going to work on it or not.
    Checkout out this link to understand the situation and then try finding the solution using game theory yourself:
     http://lesswrong.com/lw/del/real_world_solutions_to_prisoners_dilemmas/
  3. Checkout the above link for many more examples and try solving the ones you find interesting using game theory.

I hope this helped you get an introduction to Game Theory. Please feel free to contact me if you do not understand any part or feel that some part of this post is unclear. Happy Learning!

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