Gamec Theory

Game Theory

Vikram Singh Slathia

2011MAI025

Dept. of Computer Science

CONTENTS

Introduction History of Game Theory. Elements of games. Basic Concepts of Game Theory Kinds of Strategies. Nash Equilibrium. Types of Games. Applications of Game Theory. Conclusion . References .

INTRODUCTION

Game theory is the mathematical analysis of a conflict of interest to find optimal choices that will lead to a desired outcome under given conditions. To put it simply, it's a study of ways to win in a situation given the conditions of the situation. While seemingly trivial in name, it is actually becoming a field of major interest in fields like economics, sociology, and political and military sciences, where game theory can be used to predict more important trends.

CONT ….

In the broadest terms, game theory analyses how groups of people interact in social and economic situations. An accurate description of game theory is the term used by psychologists ? the theory of social situations. There are two main branches of game theory: co-operative and non-co-operative game theory. Most of the research in game theory is in the field of non-co-operative games, which analyses how intelligent (or rational) people interact with others in order to achieve their own goals

HISTORY OF GAME THEORY.

The ideas underlying game theory have appeared throughout history, apparent in the bible, the Talmud, the works of Descartes and Sun Tzu, and the writings of Chales Darwin. The basis of modern game theory, however, can be considered an outgrowth of a three seminal works: Augustin Cournot’s Researches into the

Mathematical Principles of the Theory of Wealth in 1838, gives an intuitive explanation of what would eventually be formalized as the Nash equilibrium, as well as provides an evolutionary, or dynamic notion of best-responding to the actions of others

CONT…..

Francis Ysidro Edgeworth’s Mathematical Psychics demonstrated the notion of competitive equilibria in a two-person (as well as two-type) economy

Emile Borel, in Algebre et calcul des probabilites, Comptes Rendus Academie des Sciences, Vol. 184, 1927, provided the first insight into mixed strategies - randomization may support a stable outcome.

A modern analysis began with John von Neumann and Oskar Morgenstern's book, Theory of Games and Economic Behavior and was given its modern methodological framework by John Nash building on von Neumann and Morgenstern's results.

ELEMENTS OF GAMES

The essential elements of a game are:a. Players: The individuals who make decisions.b. Rules of the game: Who moves when? What

can they do?c. Outcomes: What do the various combinations of actions produce?d. Payoffs: What are the players’ preferences over the outcomes?e. Information: What do players know when they make decisions?f. Chance: Probability distribution over chance

events, if any.

BASIC CONCEPTS OF GAME THEORY1. Game 2. Move3. Information4. Strategy5. Extensive and Normal Form6. Equilibria

1. GAME

A conflict in interest among n individuals or groups (players). There exists a set of rules that define the terms of exchange of information and pieces, the conditions under which the game begins, and the possible legal exchanges in particular conditions. The entirety of the game is defined by all the moves to that point, leading to an outcome.

2. MOVE

The way in which the game progresses between states through exchange of information and pieces. Moves are defined by the rules of the game and can be made in either alternating fashion, occur simultaneously for all players, or continuously for a single player until he reaches a certain state or declines to move further. Moves may be choice or by chance. For example, choosing a card from a deck or rolling a die is a chance move with known probabilities. On the other hand, asking for cards in blackjack is a choice move.

3. INFORMATION

A state of perfect information is when all moves are known to all players in a game. Games without chance elements like chess are games of perfect information, while games with chance involved like blackjack are games of imperfect information.

4. STRATEGY

A strategy is the set of best choices for a player for an entire game. It is an overlying plan that cannot be upset by occurrences in the game itself.

Strategy combination A strategy profile is a set of strategies for each player which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.

DIFFERENCE BETWEEN

 A Move is a single step a player can take during the game.

A strategy is a complete set of actions, which a player takes into account while playing the game throughout

Move Strategy

CONT …..

Example

KINDS OF STRATEGIES

I. Pure strategy 

II. Mixed Strategy

III. Totally mixed strategy.

I. PURE STRATEGY

A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation he or she could face.

A player‘s strategy set is the set of pure strategies available to that player.select a single action and play itEach row or column of a payoff matrix

represents both an action and a pure strategy

II. MIXED STRATEGY

A strategy consisting of possible moves and a probability distribution (collection of weights) which corresponds to how frequently each move is to be played. A player would only use a mixed strategy when she is indifferent between several pure strategies, and when keeping the opponent guessing is desirable - that is, when the opponent can benefit from knowing the next move.

III. TOTALLY MIXED STRATEGY.

A mixed strategy in which the player assigns strictly positive probability to every pure strategy

 In a non-cooperative game, a totally mixed strategy of a player is a mixed strategy giving positive probability weight to every pure strategy available to the player.

5. PAYOFF

The payoff or outcome is the state of the game at it's conclusion. In games such as chess, payoff is defined as win or a loss. In other situations the payoff may be material (i.e. money) or a ranking as in a game with many players.

5. EXTENSIVE AND NORMAL FORM

Extensive FormThe extensive form of a game is a complete description of:1. The set of players2. Who moves when and what their choices are3. What players know when they move4. The players’ payoffs as a function of the

choices that are made.In simple words we also say it is a graphical representation (tree form) of a sequential game.

The normal form

The normal form is a matrix representation of a simultaneous game. For two players, one is the "row" player, and the other, the "column" player. Each rows or column represents a strategy and each box represents the payoffs to each player for every combination of strategies. Generally, such games are solved using the concept of a Nash equilibrium. .

6. EQUILIBRIUM

Equilibrium is fundamentally very complex and subtle.  The goal to is to derive the outcome when the agents described in a model complete their process of maximizing behaviour.  Determining when that process is complete, in the short run and in the long run, is an elusive goal as successive generations of economists rethink the strategies that agents might pursue.

NASH EQUILIBRIUM

A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. For games in which players randomize (mixed strategies), the expected or average payoff must be at least as large as that obtainable by any other strategy

CONT ……..

A strategy profile s = (s1, …, sn) is a Nash equilibrium if for every i, si is a best response to S−i , i.e., no agent can do

better by unilaterally changing his/her strategy Theorem (Nash, 1951): Every game with

a finite number of agents andaction profiles has at least one Nash equilibrium

EXAMPLE

BATTLE OF THE SEXES

Two agents need to coordinate their actions, but they have different preferences

Original scenario:• husband prefers football• wife prefers opera

Another scenario:• Two nations must act together to deal

with an international crisis• They prefer different solutions

This game has two pure-strategy Nash equilibria and one mixed-strategy Nash equilibrium

How to find the mixed-strategy Nash equilibrium?

TYPES OF GAMES

A. One-Person Games B. Zero-Sum Games C Non zero sum gameD. Two-Person Games E. Repeated Games

A. ONE-PERSON GAMES A one-person games has no real conflict of

interest. Only the interest of the player in achieving a particular state of the game exists. Single-person games are not interesting from a game-theory perspective because there is no adversary making conscious choices that the player must deal with. However, they can be interesting from a probabilistic point of view in terms of their internal complexity.

B. ZERO-SUM GAMES A zero-sum game is one in which no wealth is

created or destroyed. So, in a two-player zero-sum game, whatever one player wins, the other loses. Therefore, the player share no common interests. There are two general types of zero-sum games: those with perfect information and those without.

If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero

Example a. Rock, Paper, Scissorsb. Poker game

EXAMPLE OF POKER GAME

Let’s there are three players, Rajesh, Suresh and Varun each starting with Rs100, a total of Rs300.

They meet at Rajesh’s house and play for a couple of hours. At the end of the evening Rajesh has Rs200, Suresh has Rs60 and Varun has Rs40. The total amount of money between them is still Rs300.

Rajesh is up Rs100, Suresh is down Rs40 and Varun is down Rs60.

The total of these three numbers is zero (100-40-60), so it is a zero-sum game.

EXAMPLE

In non-zero-sum games, one player's gain needn't be bad news for the other(s). Indeed, in highly non-zero-sum games the players' interests overlap entirely. In 1970, when the three Apollo 13 astronauts were trying to figure out how to get their stranded spaceship back to earth, they were playing an utterly non-zero-sum game, because the outcome would be either equally good for all of them or equally bad.

C. NON ZERO SUM GAME

In game theory, situation where one decision maker's gain (or loss) does not necessarily result in the other decision makers' loss (or gain). In other words, where the winnings and losses of all players do not add up to zero and everyone can gain: a win-win game.

Example  Prisoner's dilemma

C. TWO-PERSON GAMES Two-person games are the largest category

of familiar games. A more complicated game derived from 2-person games is the n-person game. These games are extensively analyzed by game theorists. However, in extending these theories to n-person games a difficulty arises in predicting the interaction possible among players since opportunities arise for cooperation and collusion.

D. REPEATED GAMES In repeated games, some

game G is played multiple times by the same set of agents G is called the stage game Each occurrence of G is

called an iteration or a round

Usually each agent knows what all the agents did in the previous iterations, but not what they’re doing in the current iteration

Usually each agent’s payoff function is additive

Examples 1. Iterated Prisoner’s Dilemma2. Repeated

Ultimatum Game3. Repeated

Matching Pennies4. Repeated Stag

Hunt6. Roshambo

E. SEQUENTIAL GAMES

A sequential game is a game where one player chooses his action before the others choose theirs. Importantly, the later players must have some information of the first's choice, otherwise the difference in time would have no strategic effect. Extensive form representations are usually used for sequential games, since they explicitly illustrate the sequential aspects of a game.

Combinatorial games are usually sequential games.

Sequential games are often solved by backward induction.

F. SIMULTANEOUS GAMES

A simultaneous game is a game where each player chooses his action without knowledge of the actions chosen by other players. Normal form representations are usually used for simultaneous games.

Example Prisoner dilemma .

APPLICATIONS OF GAME THEORY

PhilosophyResource Allocation and

NetworkingBiologyArtificial IntelligenceEconomicsPolitics

CONCLUSION

By using simple methods of game theory, we can solve for what would be a confusing array of outcomes in a real-world situation. Using game theory as a tool for financial analysis can be very helpful in sorting out potentially messy real-world situations, from mergers to product releases.

REFERENCES

Books ; Game theory: analysis of conflict ,Roger B.

Myerson, Harvard University Press Game Theory: A Very Short Introduction, K. G.

Binmore- 2008, Oxford University Press. Links :

http://library.thinkquest.org/26408/math/prisoner.shtml

http://www.gametheory.net

THANK YOU