Game Theory: Beauty of it and its Applications

Saptarshi Chowdhury

13th September, 2020

Abstract


This article aims at giving a clear and lucid idea of Game Theory without going deeper into the various mathematical notations and representations. Here, the two primary branches of it, namely Non-Cooperative Game Theory and Cooperative Game Theory have been discussed in detail with appropriate examples corresponding to real-life situations. Also, a few examples mentioned may include the usage of fictional characters but that has been done just to ensure that the readers are kept engaged till the last word. Overall, this article would not have been possible without the contributions of Professor John Forbes Nash Junior who dedicated his entire life to academics.

Introduction 

We are surrounded by the decisions we make, and every single step of the day starts with making these decisions. A human, say X, decides to wake up earlier than usual as she has to attend a meeting that will be organized through Google Meet. Now, X has to make a really quick decision here, and what is that? She has to decide whether to brush her teeth first or to have a quick check at the internet connection just to make sure that she does not have to be in a hassle later. What if X decides to brush her teeth first and worry about the internet issues later as she has much time left? The problem here is that subconsciously X decides to drink water after brushing her teeth, and that leads her to the washroom. What actually happens here? The option that she had, at the beginning of the day, got pushed to the last task before starting the meeting! Luckily, the internet connection was fine, and she had a smooth meeting.

Here, I will ask a simple question. What should be our plan, or rather in light of the above example, what should have been X’s plan, before coming at a conclusion to the decisions we/she make/made? Whatever decision we make has to benefit us at the end of the day! And this is where the concept of “Game Theory” comes into play.

So, what is a game basically? In simple terms, it is the interaction between multiple people (greater than or equal to 2) in which each person’s payoff is influenced by the other person/persons. Here, “game” has been used to describe a “situation” and not a “sport”! They key pioneers of the Game theory were mathematician John von Neumann and economist Oskar Morgenstern in the 1940s.

Non-Cooperative Game Theory

This branch of Game Theory covers all the social interactions or rather competition between individual players where there will be a gainer and a loser, or, to put it simply, a winner and a loser. In terms of Economics, it deals with how rational economic agents basically compete with each other to achieve their own preferred results. A beautiful example to bring more clarity into this sub-topic is “The Prisoner’s Dilemma”, which will be discussed in the below paragraph.

Let us assume that Riddler and Scarcrow were committing a crime together and Batman was able to nab both of them at the same time. After a while, Commissioner Gordon realizes that he does not have enough evidence to put them behind bars, but he has enough to question them. So, Riddler and Scarecrow are kept in two different cells, and no contact is allowed between them. Meanwhile, the prosecutor individually questions them and gives each one of the 3 options and the consequences along with them.

  1. If both of them confess, then both go to jail for a period of 5 years.

  2. If neither confesses, then both go to jail for 1 year (for carrying concealed weapons and plotting).

  3. If one confesses while the other does not, then the confessor goes free (for turning state’s evidence), and the silent one goes to jail for 20 years.

Considering that both Riddler and Scarecrow do not trust each other and have never heard of Prisoner’s Dilemma, the safest option they can choose is confessing the crime that they have committed. This point is called “Nash Equilibrium”. Hence, both of them serve sentences of 5 years each. This is the particular point where apparently the party involved is better off irrespective of what the other has chosen.

Some other real-world applications of the Prisoner’s Dilemma are: 

  1. Rivalry among shopkeepers where both of them end up getting smaller profits.

  2. Football Clubs buying more footballers than necessary and thus ending up having a weak team with no coordination.

  3. Nations competing in an arms race and thus ending up being poorer and morally weaker.

Cooperative Game Theory 

In extremely simple language, we can describe this as a situation where none of the parties completely gain or lose, and in fact, they work together towards a common goal. We can use a very common example here of paying bills in a restaurant. Here, instead of arguing amongst each other, the concerned parties split their bills, preferably using the “Splitwise” app from Google Play Store (no promotions intended).

Some of the important real-world applications of this are stated below:

  1. Marginal Contributions: Here, the contribution of each party is denoted by what is gained or lost when they are removed from the game or the situation. For example, before the coronavirus pandemic started, Raju and his friends gathered every Saturday to eat pizza, drink Coca-Cola, and watch movies. One such week, Sachin decides not to give to attend the gathering. As a result of that, the group orders one pizza and 2 bottles of Coca-Cola less. Now, this is what the marginal contribution of Sachin is to the group!

  2. Equal values of interchangeable players: If two parties should bring the same things to the coalition, then they should have to contribute the same amount and should be rewarded for their contributions equally. Here, we revert to the example of bill splitting.

  3. Zero value of dummy players:  If a person does not contribute anything, then they won’t receive their share of the amount for it. Now, this is debatable in the sense that a particular person may not have anything to contribute at that certain instant of time, and may have even lost the medium through which he/she was about to contribute.

So, what is exactly Shapley Value? Suppose, Martha and Jonas love to bake cookies. However, Jonas can bake just 10 cookies per hour whereas Martha can bake almost 20 cookies per hour. Now, they decide to collectively work together (yes, in the same world) and end up baking 40 cookies an hour. After selling it for 1€ per cookie, they earn a total of 40€. However, there is confusion here as to how will they distribute the money amongst themselves! According to Shapley Value, we can distribute the situation this way:

  1. Jonas can bake 10 cookies per hour, and hence Martha’s marginal contribution to him is 30 cookies, since, 40-10=30.

  2. Martha can bake 20 cookies per hour, and hence Jonas’ marginal contribution to her is 20 cookies, since 40-20=20.

In the first situation, Jonas’ value to the coalition is only 10 cookies, while in the 2nd case, it is 20 cookies! Under the Shapley Value Equation, we calculate the arithmetic mean of 10 and 20, which gives us 15.

Hence, Jonas gets 15€ while Martha gets 25€.

Conclusion

Now, this article was not Mathematical, instead it was aimed at readers who want to have an idea about Game Theory and not go much deeper into it. However, it is highly advisable to go through several books and videos related to this concept. I have merely touched a droplet of this beautiful theory, and there is much more to it!

 

Reference:

  1. “Cooperative Game Theory”:http://www.coalitiontheory.net/research-areas/cooperative-game-theory

  2. “How Game Theory affects your Everyday Life” by “The London Globalist”: http://www.thelondonglobalist.org/how-game-theory-affects-your-everyday-life

  3. “What is Game Theory and What are Some of its Applications?” (2003, June 02). Retrieved 31st August from:  https://www.scientificamerican.com/article/what-is-game-theory-and-w/

  4. Ecotalker,-.(2015, January 15). Game On: A Game Theory Quiz! Arthashastra. https://ecotalker.wordpress.com/2015/01/06/game-on-a-game-theory-quiz/.

About the Author

IMG_20200422_132540 - Saptarshi Chowdhur

Saptarshi, currently majoring in Statistics has worked with a lot of topics related to his subject. Along with SiC, he is also an author for his university departmental magazine 'Prokarsyo'.