Game Theory Fundamentals

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  • Topic: Game theory, Nash equilibrium, Minimax
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APPLICATION OF GAME THEORY

By
Abinaya Naresh
Ananthapadmanabhan P
Dhivya Kannan
Vidyalakshmi P

INTRODUCTION
Game theory is the study of strategic decision making. It is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome. Game theory focuses on how groups of people interact. It focuses on how “players” in economic “games” behave when, to reach their goals, they have to predict how their opponents will react to their moves. It is the study of competitive interaction; it analyzes possible outcomes in situations where people are trying to score points off each other, whether in bridge, politics or war. You do this by trying to anticipate the reaction of your competitor to your next move and then factoring that reaction into your actual decision. It teaches people to think several moves ahead. A famous quote by an anonymous author which goes “it doesn’t matter if you win or lose but how you play the game” missed the point. It matters very much. According to game theory, it’s how you play the game that usually determines whether you win or lose.

A BRIEF HISTORY
0-500 AD : http://www.econ.canterbury.ac.nz/personal_pages/paul_walker/gt/hist.htm

The 1st known recording of an application of game theory.
The Babylonian Talmud is the compilation of ancient law and tradition set down during the first five centuries A.D. which serves as the basis of Jewish religious, criminal and civil law. One problem discussed in the Talmud is the so called marriage contract problem: a man has three wives whose marriage contracts specify that in the case of this death they receive 100, 200 and 300 respectively. The Talmud gives apparently contradictory recommendations. Where the man dies leaving an estate of only 100, the Talmud recommends equal division. However, if the estate is worth 300 it recommends proportional division (50,100,150), while for an estate of 200, its recommendation of (50,75,75) is a complete mystery. This particular Mishna has baffled Talmudic scholars for two millennia. In 1985, it was recognised that the Talmud anticipates the modern theory of cooperative games. Each solution corresponds to the nucleolus of an appropriately defined game.

1713 :
In a letter dated 13 November 1713 James Waldegrave provided the first, known, minimax mixed strategy solution to a two-person game. Waldegrave wrote the letter, about a two-person version of the card game le Her, to Pierre-Remond de Montmort who in turn wrote to Nicolas Bernoulli, including in his letter a discussion of the Waldegrave solution. Waldegrave's solution is a minimax mixed strategy equilibrium, but he made no extension of his result to other games, and expressed concern that a mixed strategy "does not seem to be in the usual rules of play" of games of chance

There were similar instances later in the 1700’s and 1800’s but significant breakthroughs occurred only in the 1900’s

1913 :
E. Zermelo provides the first theorem of game theory; asserts that chess is strictly determined

1928 :
John von Neumann proves the minimax theorem
1944 :
John von Neumann & Oskar Morgenstern write "Theory of Games and Economic Behavior”

1950-1953 :
John Nash describes Nash equilibrium

GAME THEORY BASICS AND ASSUMPTIONS
* Game : A game is any situation in which players (participants) make strategic decision that is decisions that take into account each other’s actions and responses. E.g Firms competing with each other, Group of consumers bidding against each other at an auction * Strategic decisions result in pay offs to the players.

* Game theory is a study of competitive interaction; it analyses possible outcomes in situations where people are trying to score points of each other. * Humans are always rational
* Humans always seek the best alternative in a set of possible choices. The last 2 assumptions help in narrowing down the...
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