An Introduction to Game Theory

04/22/23 An Introduction to Game Theory

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An Introduction to Game Theory

SpeakerAbhinav SrivastavaM.Tech IInd Year

School of Information TechnologyIndian Institute of Technology, Kharagpur

04/22/23 An Introduction to Game Theory 2

Contents

History Introduction Definitions related to game theory What is a game? Types of Games Static games of complete information Dominant Strategy Zero-Sum Game Nash Equilibrium References


History

Game Theory is an interdisciplinary approach to the study of human behavior.

It was founded in the 1920s by John von Neumann. In 1994 Nobel prize in Economics awarded to three

researchers. HARSANYI, JOHN C., U.S.A., University of California, Berkeley, CA, b. 1920 (in

Budapest, Hungary); NASH, JOHN F., U.S.A., Princeton University, NJ, b. 1928; and SELTEN, REINHARD, Germany, Rheinische Friedrich-Wilhelms-Universit,t, Bonn,

Germany, b. 1930:

“Games” are a metaphor for wide range of human interactions.


Introduction

Game Theory (GT) can be regarded as A multi-agent decision problem.

Many people contending for limited rewards/payoffs.

Moves on which payoffs depends.

Follow certain rules while making the moves.

Each player is supposed to behave rationally.


Definitions Related to GT

Game Theory: Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) who behave strategically.

Player: Each participant (interested party) is called a player.

Strategy: A strategy of a player is the predetermined rule by which a player decides his course of action from his own list of actions during the game.

Rationality: It implies that each player tries to maximize his/her payoff irrespective to what other players are doing.


Definitions of GT (Contd…)

Rule: These are instructions that each player follow. Each player can safely assume that others are following these instructions also.

Outcome: It is the result of the game.

Payoff: This is the amount of benefit a player derives if a particular outcome happens.


What is a Game?

A game has the following

Set of players D = { Pi | 1 <= i <= n} Set of rules R

Set of Strategies Si for each player Pi Set of Outcomes O Pay off ui (o) for each player i and for each

outcome o e O


Coin Matching Game

Coin Matching Game : Two players choose independently either Head or Tail and report it to a central authority. If both choose the same side of the coin , player 1 wins, otherwise 2 wins.

This game has the following :- Set of Players: P={P1,P2} (The two players who are choosing

either Head or Tail.) Set of Rules: R (Each player can choose either Head or Tail. Player

1 wins if both selections are the same otherwise player 2 wins.)


Coin Matching Game (Contd…)

Set of Strategies: Si for each player Pi (For example S1 = { H, T} and S2 = {H,T} are the strategies of the two players.)

Set of Outcomes: O {Loss, Win} for both players (This is a function of the strategy profile selected. In our example S1 x S2 = {H,H),(H,T),(T,H),(T,T)} is the strategy profile.)

Pay off: ui (o) for each player i and for each outcome o e O. (This is the amount of benefit a player derives if a particular outcome happens.)


Coin Matching Game (Contd…)

1, -1 -1, 1

-1, 1 1, -1

Player 2

Head Tail

Head

Tail

Player 1


Types of Games

There are four types of Games: Static Games of Complete Information Dynamic Games of Complete Information Static Games of Incomplete Information Dynamic Games of Incomplete Information


Static games of complete information

Simultaneous-move Each player chooses his/her strategy without knowledge of

others’ choices. Complete information

Each player’s strategies and payoff function are common knowledge among all the players.

Assumptions on the players Rationality

Players aim to maximize their payoffs Players are perfect calculators

Each player knows that other players are rational


Static games of complete information

The players cooperate? No only non-cooperative games

Represented as normal-form or strategic form.


The Prisoner’s Dilemma

Two burglars, Bob and Al, are captured and separated by the police.

Each has to choose whether or not to confess and implicate the other.

If neither confesses, they both serve one year for carrying a concealed weapon.

If each confesses and implicates the other, they both get 10 years.

If one confesses and the other does not, the confessor goes free, and the other gets 20 years.


The Prisoners’ Dilemma

Al

Confess Don’t Confess

Bob Confess 10 / 10 0 / 20

Don’t Confess

20 / 0 1 /1


Dominant Strategies

The prisoners have fallen into a “dominant strategy equilibrium”

DEFINITION: Dominant Strategy Evaluate the strategies. For each combination, choose the one that gives the best payoff. If the same strategy is chosen for each different combination, that strategy is

called a “dominant strategy” for that player in that game DEFINITION: Dominant Strategy Equilibrium

If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs constitute the dominant strategy equilibrium for that game.


Traffic Lights

There are two players in this game: Player I and Player II.

Player I is the commuter and All other people at the intersection (signal) can be considered as the second player in the game.

When a commuter arrives and faces a red light he/she has two options:

Wait for light to turn Green Jump the Red light


Traffic Lights

If the commuter obeys and others also obey he will have to suffer delay of 'd' that is the time required for the red light to turn green.

If he disobeys but others obey his delay is 0.

If he obeys but others disobey let additional delay is D ( due to congestion ) over 'd' .

If all disobey total delay is D.


Traffic Lights

Player II

Obey Disobey

Player I Obey d d+D

Disobey 0 D


Information Technology Example

Players Company considering a new internal computer system A supplier who is considering producing it

Choices To install an advanced system with more features To install a proven system with less functionality

Payoffs Net payment of the user to the supplier

Assumptions A more advanced system really does supply more functionality


IT Example

User

Advanced Proven

Supplier Advanced 20 / 20 0 / 0

Proven 0 / 0 5 /5


Complications

There are no dominant strategies. The best strategy depends on what the other player chooses! Need a new concept of game-equilibrium Nash Equilibrium

Occurs when each participant chooses the best strategy given the strategy chosen by the other participant

Advanced/Advanced Proven/Proven Can be more than one Nash equilibrium

This is considered a cooperative game


Zero-Sum Game

It was discovered by Von Neumann. A zero-sum game is a game in which one player’s

winnings equal to the other player’s losses. If there is even one strategy set for which the sum

differs from zero, then the game is not zero sum. In a zero-sum game, the interest of the players are

directly opposed, with no common interest at all.


Bottled Water Game

Players: Evian, Perrier Each company has a fixed cost of $5000 per period, regardless

of sales They are competing for the same market, and each must chose a high

price ($2/bottle), and a low price ($1/bottle) At $2, 5000 bottles can be sold for $10,000 At $1, 10000 bottles can be sold for $10,000 If both companies charge the same price, they split the sales evenly

between them If one company charges a higher price, the company with the lower

price sells the whole amount and the higher price sells nothing Payoffs are profits – revenue minus the $5000 fixed cost


Bottled Water Game

Perrier

$1 $2

Evian $1 $0, $0 $5000, -$5000

$2 -$5000, $5000 $0, $0


The Maximin Criterion

There is a clear concept of a solution for the zero-sum game.

Each player chooses the strategy that will maximize the minimum payoff.

The pair of strategies and payoffs such that each player maximizes her minimum payoffs is the “solution of the game”.

Both choose $1 prices


Non-constant Sum Games

Games that are not zero-sum or constant sum are called Non-constant sum games.

These are in some sense natural games.


Widgets Game

Let’s sell widgets Set pricing to $1, $2, or $3 per widget Payoffs are profits, allowing for costs General idea is that company with lower

price gets more customers, and more profits, within limits.


Widgets Game

ACME Widgets

$1 $2 $3

Widgeon Widgets

$1 0, 0 50, -10 40, -20

$2 -10, 50 20, 20 90, 10

$3 -20, 40 10, 90 50, 50


Nash Equilibrium

On the name after Nobel Laureate and mathematician John Nash.

Nash, a student of Tucker’s contributed several key concepts to game theory around 1950.

Nash Equilibrium is the most widely used “solution concept” in game theory.

If there is a set of strategies with the property that no player can benefit by changing their strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.


Nash Equilibrium

Maximin is a Nash Equilibrium. Dominant Strategy is also a Nash

Equilibrium. Nash Equilibrium is an extension of the

concepts of dominant strategy and the maximin solution for zero-sum games.


Limitations of Nash Equilibrium

Can there be more than one Nash-Equilibrium in the same game?

What if there are more than one?


Multiple Nash Equilibrium

There are two cola companies, Pepsi and Coke.

Each own a vending machine in the dormitory.

Each must decide how to stock its machine. They can fill the machine with diet soda,

regular soda, or a combination of the two.


The Cola Wars

Coca-Cola

Diet Both Classic

Pepsi Diet 25, 25 50, 30 50, 20

Both 30, 50 15, 15 30, 20

Classic 20, 50 20, 30 10, 10


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Game Over


Application of GT

Game theory has applications Economics International relations Political science Military strategy Operations research


References

Formal Theory for Political Science by Andrew Kydd

Game Theory: An Introductory Sketch by Roger McCain

http://plato.stanford.edu/entries/game-theory/#Games

http://www.econlib.org/library/Enc/GameTheory.html

http://www.cse.iitd.ernet.in/~rahul/cs905/

http://www.gametheory.net/

http://plato.stanford.edu/entries/game-theory/#Games

http://www.econlib.org/library/Enc/GameTheory.html

http://www.cse.iitd.ernet.in/~rahul/cs905/


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An Introduction to Game Theory

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