By Sean Ferris

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What is game theory? Who created game theory? The Prisoners’ Dilemma A Beautiful Mind Real life application to game theory References

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Game theory is the study of the ways in which strategic interactions among rational players produce outcomes with respect to the preferences (or utilities) of those players, none of which might have been intended by any of them

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Click here to view Game Theory video

http://www.youtube.com/watch?v=HCinK2PUfyk&feature=related

This video is an example of how game theory is used in a practical environment

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The first two initial people who organized this economic theory were John van Neumann and Oskar Morgenstern in the 1944 classic The Theory of Games and Economic Behavior.

John Nash then claimed that one should be able to reduce all co-operative games into some non-cooperative form. This position is what is known as the "Nash Programme". Within the non-cooperative literature, a distinction can also be made between "normal" form games (static) and "extensive" form games (dynamic).

In 1950, John Nash introduced the concept of a "Nash Equilibrium"

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Game theory then developed into evolutionary game theory by Thomas C. Schelling who apply the concepts of non-cooperative game theory to explain such phenomena which are often thought to be the result of cooperation or human design - i.e. "institutions" and "conventions“

Several Nobel Prizes have been awarded to some of major figures of Game Theory: the Nobel was shared by John Nash, J.C. Harsanyi and R. Selten in 1994 and William Vickrey and James Mirrlees in 1996

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Game theory attempts to look at the relationships between participants in a particular model and predict their optimal decisions. One frequently cited example of game theory is the prisoner's dilemma. Suppose there are two brokers accused of fraudulent trading activities: Dave and Henry. Both Dave and Henry are being interrogated separately and do not know what the other is saying. Both brokers want to minimize the amount of time spent in jail and here lies the dilemma. The sentences vary as follows:

1) If Dave pleads not guilty and Henry confesses, Henry will receive the minimum sentence of one year, and Dave will have to stay in jail for the maximum sentence of five years. 2) If nobody makes any implications they will both receive a sentence of two years. 3) If both decide to plead guilty and implicate their partner, they will both receive a sentence of three years. 4) If Henry pleads not guilty and Dave confesses, Dave will receive the minimum sentence of one year, and Henry will have to stay in jail for the maximum five years.

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Obviously, pleading guilty is the most attractive should the other plead not guilty since the sentence is only one year. However, if the other party also chooses to plead guilty, both will have to serve three years. On the other hand, if both parties plead not guilty, they'd have to serve two years in jail. Consequently, the risk of pleading not guilty is a five-year sentence, should the other choose to confess.

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At Princeton University, John Nash struggles to make a worthwhile contribution to serve as his legacy to the world of mathematics. He finally makes a revolutionary breakthrough that will eventually earn him the Nobel Prize. After graduate school he turns to teaching, becoming romantically involved with his student Alicia. Meanwhile the government asks his help with breaking Soviet codes, which soon gets him involved in a terrifying conspiracy plot. Nash grows more and more paranoid until a discovery that turns his entire world upside down. Now it is only with Alicia's help that he will be able to recover his mental strength and regain his status as the great mathematician we know him as today

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Click here to view a Beautiful mind clip

http://www.youtube.com/watch?v=0JDoQb6A2YI

In this video clip you will see how John Nash applies Game Theory to an

everyday situation

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An excerpt of a real life problem solved through game theory.

We see a lot of seats/berths vacant on the running trains whilst having a lot of people wait‐listed. This is because of lot of last minute cancellations, some even after the preparation of the reservation charts. This may lead to more cancellations by wait listed people not knowing that berths are available. Even if they carry on their journey it causes them tremendous amount of inconvenience to get the berths confirmed while on travel etc and also gives a scope for corruption. Thus last minute cancellations have many undesired implications resulting in losses to both the Railways and also defeating the purpose of Social Welfare which is very important. But then again most of the people prefer to book tickets well advance to make sure they are on board on the day. There are also intermediaries who buy tickets in advance and then cancel unsold tickets in the last moment to get their money back. There is no way to differentiate these from the end users.

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We feel that most of these problems stem from the pricing and penalty model followed by the Indian Railways. Currently there are two kinds of tickets. A fixed percentage of Tatkal tickets are sold at a slightly higher price only for a period of 5 days before the day of journey and have refund in case of cancellation. For the remaining tickets, the window is open 90 days before the actual date of journey and these can be cancelled till the departure time with varying penalty.

This model has a scope for getting away with minimal penalty for a cancellation as late as 48 hrs before the departure. This can be exploited by a rational intermediary who works only on profit motive and is bounded by limited budget. In some situations, the intermediary can even afford to gamble by not cancelling tickets at all and still managed to reap profits by selling the remaining tickets at a higher price. This is possible as the tickets are highly underpriced. This can be understood by the fact that people are willing to buy tickets even at a higher price. On the other hand, the pricing and penalty models followed in countries like China for railways and also the world airlines industry seem to work efficiently and eliminate any scope of such deliberate blockages by intermediaries.

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The whole scenario can be looked at as a two player game where the rational intermediary is the adversary and tries to beat the system and maximize his profit. We should be able to show that there exists equilibrium for a sound model under certain assumptions.

In view of the above, we model the reservation schema into a mathematical model, keeping in mind the social tendencies for booking tickets well in advance and cancelling the tickets once booked etc while facilitating any genuine changes in plans. Then we propose a new price and penalty function and show that they do better than the existing set of functions in minimizing the number of unused seats on the running trains. We can also show that there exists equilibrium for the same.

Game theory takes complex situations and allows individuals to help come up with solutions by figuring out their opponents strategy.

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"A Beautiful Mind (Adam Smith was wrong)." Youtube. 2008. Youtube. 4 Nov 2008 <http://www.youtube.com/watch?v=0JDoQb6A2YI>.

"Game Theory." Getty images. 2008. Getty Images Inc.. 3 Nov 2008 www.gettyimages.com

"Game Theory." Investopedia. 2008. Investopedia ULC. 3 Nov 2008 <http://www.investopedia.com/terms/g/gametheory.asp>.

"Game Theory." Stanford Encyclopedia . 1997. Don Ross. 3 Nov 2008 <http://plato.stanford.edu/entries/game-theory/#Mot>.

Game Theory. Youtube. 2008. Youtube. 3 Nov 2008 <http://www.youtube.com/watch?v=HCinK2PUfyk&feature=related>

Mlohner, "A Beautiful Mind." IMDB. 2008. The Internet Movie Database. 4 Nov 2008 <http://www.imdb.com/title/tt0268978/plotsummary>.

Venkatesh Krishna, Kinthali, Duvvuri Vamsi. "Application of Game Theory to Real Life Situations." 2-2. 4 Nov 2008 <http://www.cse.iitk.ac.in/users/vamsi/BTPsubmit/interim.pdf>.

Walker, Paul. "History of Game Theory." A Chronology of Game Theory. 2005. University of Canterbury. 3 Nov 2008 <http://www.econ.canterbury.ac.nz/personal_pages/paul_walker/gt/hist.htm>.

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