Speaker

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An Introduction to Game Theory

• Speaker
• Abhinav Srivastava
• M.Tech IInd Year
• School of Information Technology
• Indian Institute of Technology, Kharagpur

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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

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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.

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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.

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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.

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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.

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What is a Game?

• A game has the following
• Set of players D Pi 1 lt i lt 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

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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 PP1,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.)

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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.)

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Coin Matching Game (Contd)
Player 2
Head
Tail
1, -1 -1, 1
-1, 1 1, -1
Head
Player 1
Tail
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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

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Static games of complete information

• Simultaneous-move
• Each player chooses his/her strategy without
  knowledge of others choices.
• Complete information
• Each players 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

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Static games of complete information

• The players cooperate?
• No only non-cooperative games
• Represented as normal-form or strategic form.

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The Prisoners 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.

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The Prisoners Dilemma
Al Al
Confess Dont Confess
Bob Confess 10 / 10 0 / 20
Bob Dont Confess 20 / 0 1 /1
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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.

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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

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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.

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Traffic Lights
Player II Player II
Obey Disobey
Player I Obey d dD
Player I Disobey 0 D
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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

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IT Example
User
Advanced Proven
Supplier Advanced 20 / 20 0 / 0
Supplier Proven 0 / 0 5 /5
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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

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Zero-Sum Game

• It was discovered by Von Neumann.
• A zero-sum game is a game in which one players
  winnings equal to the other players 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.

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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

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Bottled Water Game
Perrier Perrier
1 2
Evian 1 0, 0 5000, -5000
Evian 2 -5000, 5000 0, 0
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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

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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.

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Widgets Game

• Lets 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.

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Widgets Game
ACME Widgets ACME Widgets ACME Widgets
1 2 3
Widgeon Widgets 1 0, 0 50, -10 40, -20
Widgeon Widgets 2 -10, 50 20, 20 90, 10
Widgeon Widgets 3 -20, 40 10, 90 50, 50
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Nash Equilibrium

• On the name after Nobel Laureate and
  mathematician John Nash.
• Nash, a student of Tuckers 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.

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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.

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Limitations of Nash Equilibrium

• Can there be more than one Nash-Equilibrium in
  the same game?
• What if there are more than one?

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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. 

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The Cola Wars
Coca-Cola Coca-Cola Coca-Cola
Diet Both Classic
Pepsi Diet 25, 25 50, 30 50, 20
Pepsi Both 30, 50 15, 15 30, 20
Pepsi Classic 20, 50 20, 30 10, 10
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• Game Over

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Application of GT

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

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References

• Formal Theory for Political Science by Andrew
  Kydd
• Game Theory An Introductory Sketch by Roger
  McCain
• http//plato.stanford.edu/entries/game-theory/Gam
  es
• http//www.econlib.org/library/Enc/GameTheory.html
  
• http//www.cse.iitd.ernet.in/rahul/cs905/
• http//www.gametheory.net/

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• Questions?

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• Thank You