An Introduction to Game Theory

1. An Introduction to Game Theory Speaker Abhinav Srivastava M.Tech IInd Year School of Information Technology Indian Institute of Technology, Kharagpur 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 An Introduction to Game Theory

3. 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. An Introduction to Game Theory

4. 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. An Introduction to Game Theory

5. 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. An Introduction to Game Theory

6. 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. An Introduction to Game Theory

7. 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 An Introduction to Game Theory

8. 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.) An Introduction to Game Theory

9. Coin Matching Game (Contd…) • Set of Strategies: Sifor 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.) An Introduction to Game Theory

10. Coin Matching Game (Contd…) Player 2 Head Tail Head Player 1 Tail An Introduction to Game Theory

11. 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 An Introduction to Game Theory

12. 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 An Introduction to Game Theory

13. Static games of complete information • The players cooperate? • No only non-cooperative games • Represented as normal-form or strategic form. An Introduction to Game Theory

14. 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. An Introduction to Game Theory

15. The Prisoners’ Dilemma An Introduction to Game Theory

16. 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. An Introduction to Game Theory

17. 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 An Introduction to Game Theory

18. 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. An Introduction to Game Theory

19. Traffic Lights An Introduction to Game Theory

20. 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 An Introduction to Game Theory

21. IT Example An Introduction to Game Theory

22. 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 An Introduction to Game Theory

23. 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. An Introduction to Game Theory

24. 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 An Introduction to Game Theory

25. Bottled Water Game An Introduction to Game Theory

26. 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 An Introduction to Game Theory

27. 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. An Introduction to Game Theory

28. 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. An Introduction to Game Theory

29. Widgets Game An Introduction to Game Theory

30. 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. An Introduction to Game Theory

31. 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. An Introduction to Game Theory

32. Limitations of Nash Equilibrium • Can there be more than one Nash-Equilibrium in the same game? • What if there are more than one? An Introduction to Game Theory

33. 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.  An Introduction to Game Theory

34. The Cola Wars An Introduction to Game Theory

35. Game Over An Introduction to Game Theory

36. Application of GT • Game theory has applications • Economics • International relations • Political science • Military strategy • Operations research An Introduction to Game Theory

37. 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/ An Introduction to Game Theory

38. Questions? An Introduction to Game Theory

39. Thank You An Introduction to Game Theory