Game Theory Presentation

Abbas Keramati ( Assistant Professor) University of Tehran – MBA Fall 2008

Game Theory Quantitative Analysis for Decision Making

M.Ghotbi

M.Pajhouh Niya

Title• Lorem ipsum dolor sit amet, consectetuer

adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas.

• Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas.

DGDG

Outline• What is Game Theory?• History of Game Theory• Applications of Game Theory• Key Elements of a game• Types of games• Nash Equilibrium (NE)• Pure Strategies & Mixed Strategies• 2players Zero-Sum games




DGDG

What is Game Theory?What is Game Theory?

• In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.

• The mathematics of human interactions

• In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.

• The mathematics of human interactions




DGDG

History of Game Theory• von Neumann wrote a key paper in 1928• 1944: “Theory of Games and Economic

Behavior” by von Neumann and Morgenstern• 1950: Nash invents concept of Nash

equilibrium• Game theory booms after this…• 1994: Harsanyi, Nash, and Selten win Nobel

Prize in economics for game theory work




DGDG

• Mathematics• Computer Science• Biology• Economics• Political Science• International Relations• Philosophy

• Psychology• Law• Military Strategy• Management• Sports• Game Playing

Applications of Game Theory




DGDG

Key Elements of a game

• Players: Who is interacting?

• Strategies: What are their options?

• Payoffs: What are their incentives?

• Information: What do they know?

• Rationality: How do they think?




DGDG

Types of games

• Cooperative or non-cooperative

• Zero sum and non-zero sum

• Simultaneous and sequential

• Perfect information and imperfect information

• Finite & Infinite Strategies




DGDG

Pure Strategies

•The upper value of the game is equal to the minimum of the maximum values in the columns.

•The lower value of the game is equal to the maximum of the minimum values in the rows.




DGDG

A

BY1 Y2 Minimum

X1 10 6 6

X2 -12 7 -12

Maximum 10 767

An Example:




DGDG

Mixed Strategies

A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy.




DGDG

Nash Equilibrium (NE)

A player’s best strategy is A player’s best strategy is that strategy that maximizes that strategy that maximizes that player’s payoff (utility), that player’s payoff (utility), knowing the strategy's of the knowing the strategy's of the other players.other players.




DGDG

2-players Zero-Sum games

-1 , 1 1 , -1

1 , -1 -1 , 1Player 1

Player 2

Penny Matching: • Each of the two players has a penny.

• Two players must simultaneously choose whether to show the Head or the Tail.

• Both players know the following rules: -If two pennies match (both heads or both tails) then player 2 wins player 1’s penny.

-Otherwise, player 1 wins player 2’s penny.

Tail

Head

Tail

Head




DGDG

Prisoner’s Dilemma

• No communication: - Strategies must be undertaken without the full knowledge of what the other players (prisoners) will do.

• Players (prisoners) develop dominant strategies but are not necessarily the best one.




DGDG

Ted

Confess Not Confess

Confess

Bill

Not Confess

Both get 5 years

1 year for Bill 10 years for

Ted

10 years for Bill

1 year for TedBoth get 3

years

Payoff Matrix for Prisoner’s Dilemma




DGDG

An Example ofMixed Strategy game

A

BB believes B doesn’t

believe

A bluffs 1 0

A doesn’t bluff 0 0.5




DGDG

Now let’s Play This Game

Pirate Game

A

B

D

C

E




DGDG

Nash’s Equilibrium • This equilibrium occurs when each player’s strategy is

optimal, knowing the strategy's of the other players.

• A player’s best strategy is that strategy that maximizes that player’s payoff (utility), knowing the strategy's of the other players.

• So when each player within a game follows their best strategy, a Nash equilibrium will occur.




DGDG

Definition: Nash Equilibrium

In the normal-form game {S1 , S2 , ..., Sn , u1 , u2 , ...,

un}, a combination of strategies ),...,( **1 nss is a Nash

equilibrium if, for every player i,

),...,,,,...,(

),...,,,,...,(

**1

*1

*1

**1

**1

*1

niiii

niiii

sssssu

sssssu

for all ii Ss . That is, *is solves

Maximize ),...,,,,...,( **1

*1

*1 niiii sssssu

Subject to ii Ss

Given others’ choices, player i cannot be better-off if she deviates from si*




DGDG

Nash’s Equilibrium cont.: Bayesian Nash Equilibrium

• The Nash Equilibrium of the imperfect-information game

– A Bayesian Equilibrium is a set of strategies such that each player is playing a best response, given a particular set of beliefs about the move by nature.

– All players have the same prior beliefs about the probability distribution on nature’s moves.

– So for example, all players think the odds of player 1 being of a particular type is p, and the probability of her being the other type is 1-p




DGDG

• Dixit and Nalebuff: Thinking Strategically • Dutta: Strategies and Games: Theory and Practice • www.gametheory.net

Refrences

http://www.amazon.com/exec/obidos/tg/detail/-/0262041693/qid=1076968166/sr=1-1/ref=sr_1_1/103-9367263-5214241?v=glance&s=books

Game Theory Presentation

Business

Transcript of Game Theory Presentation