Progressive Education Society’s Modern College Of Engineering Department Of Computer Engineering

Game TheoryBy

Soumyashree Bilwar Department Of Computer

Engineering

Overview What is Game Theory ( layman’s language) Game Theory (Formal Definition) Major Assumptions Types Of Games Representation Of Games Nash Equilibrium Popular Games Prisoner’s Dilemma Chicken Game General and applied uses Conclusion References

Game Theory?????Lets us first understand what is a Game??Game, in the mathematical sense, is defined as strategic situation in which there are multiple participants.

Is Sudoku a "game" ? No. Is Chess a "game" ? Yes.

What is Game Theory?(layman’s language)

Game Theory is one way of studying how an individual or a group makes a strategic choice.Practical applications in everyday life: Friends choosing where to go for dinner Gamblers betting in a card game Diplomats negotiating a treaty Commuters deciding how to go to work

What is Game Theory?(Formal Definition )

Game Theory is a set of tools and techniques for decisions under uncertainty involving two or more intelligent opponents in which each opponent aspires to optimize his own decision at the expense of the other opponents.

Major Assumptions

Players – the number of participants may betwo or more. A player can be a singleindividual or a group with the same objective. Timing – the conflicting parties decidesimultaneously. Conflicting Goals – each party is interestedin maximizing his or her goal at the expenseof the other.

Major Assumptions(cont)

Repetition – most instances involverepetitive solution. Payoff – the payoffs for each

combination ofdecisions are known by all parties. Information Availability – all parties areaware of all pertinent information. Eachplayer knows all possible courses of actionopen to the opponent as well as anticipated payoffs

Types of Games:Cooperative or non-cooperativeSymmetric and asymmetricZero-sum and non-zero-sumSimultaneous and sequentialPerfect information and imperfect

informationCombinatorial gamesInfinitely long gamesDiscrete and continuous gamesMany-player and population gamesMetagames

Representation of gamesType 1:Extensive form-> Tree

Point of choice for a player. The player is specified by a number listed by the vertex.

Possible action for that player.

Payoffs

Representation of gamesType 2: Normal form-> Matrix

4,3 -1,-1

0,0

3,4

Player 1chooses Up

Player 1chooses Down

Player 2chooses Left

Player 2chooses Right

Normal form or payoff matrix of a 2-player, 2-strategy game

Payoffs

Player1 chooses RowsPlayer2 chooses Columns

Nash Equilibrium John Nash John Nash was a mathematician

and an economist. He developed several theories in

economics . He was a Princeton and CMU

graduate. His most important contribution

was the theory of Nash equilibrium

He is the person portrayed in the movie “A beautiful mind”.

What is Nash Equilibrium ?For any two groups that do not co-operate

there will be a point at which neither group can benefit from unilateral action , and that the groups will hold their strategies constant at this point.

The Nash equilibrium is not usually the most effective strategy; it is only the best one without co-operation.

Through co-operation it is only that both parties will be able to increase their utility.

Some of the popular Games of Game TheoryPrisoner's dilemmaBattle of the sexesDeadlockRock, Paper, ScissorsTrust gameCake cuttingChicken (aka hawk-dove)Traveller's dilemma

Prisoner’s Dilemma

Prisoner B stays silent Prisoner B confesses

Prisoner A stays silent Each serves 1 month Prisoner A: 1 yearPrisoner B: goes free

Prisoner A confesses Prisoner A: goes freePrisoner B: 1 year Each serves 3 months

http://pespmc1.vub.ac.be/PRISDIL.html

Cooperation is usually analysed in game theory by means of a non-zero-sum game called the "Prisoner's Dilemma“. The prisoner's dilemma is meant to study short term decision-making .

Analysis of Prisoner’s DilemmaEach player gains when both stay silent.

(one month) One player stays silent and other confesses then one

who confesses will gain more. (confess- freed, silent-1 year) If both confess , both lose (or gain very little) but not

as much as the "cheated" silent prisoner whose cooperation is not returned.(3 months)

Prisoner’s Dilemma has single Nash equilibrium.

Friend or Foe? is a game show that aired from 2002 to 2005 on the Game Show Network in the United States. It is an example of the prisoner's dilemma game tested by real people

Chicken Game

Driver BSwerve

Driver B Straight

Driver ASwerve Tie , Tie Lose, Win

Driver A Straight Win , Lose Crash

http://pespmc1.vub.ac.be/PRISDIL.html

Chicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash.

Analysis of Chicken Game

Both lose when both swerve. One player wins when one swerves and

other goes straight. If both go straight, both lose(lose more

than what they would have lost when both swerve.

Because if both go straight they CRASH) Chicken Game has 2 Nash Equilibrium.

General and applied uses Economics and business E.g. modelling competing behaviours of interacting agents , auctions, bargaining, social network formation. Political Science E.g. public choice, social choice, players are voters, politicians , states. Biology E.g. evolution , mobbing, animal communication Computer Science and logic E.g. game semantics, online algorithms , equilibrium in games and peer to peer systems, time complexity Philosophy E.g. co ordination games , convention , common knowledge

ConclusionGame theory is exciting because although the

principles are simple, the applications are far-reaching.

Game theory is the study of cooperative and non cooperative approaches to games and social situations in which participants must choose between individual benefits and collective benefits.

Game theory can be used to design credible commitments, threats, or promises, or to assess propositions and statements offered by others.

References Research papers and books Game Theory at Work by James D. Miller Thinking Strategically: Competitive Edge in

Business, Politics and Everyday Life by Avinash Dixit Existence of Equilibrium in Discrete Market Games by Somdeb Lahiri

URL http://en.wikipedia.org/wiki/Game_theory http://faculty.lebow.drexel.edu/mccainr/top/eco/

game/game-toc.html http://www2.owen.vanderbilt.edu/mike.shor/

courses/game-theory/quiz/problems2.html http://en.wikipedia.org/wiki/Nash_equilibrium

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