1. Games, Strategies, and Decision Making By Joseph E. Harrington First Edition Chapter 2: Building a Model of a Strategic Situation Prepared by Debashis Pal Copyright © 2009 by Worth Publishers Lecture Notes on

2. Building a model of strategic interaction First, we need to develop manageable models, so that we can apply game theory to analyze complex interactions. Second, we need to learn methods to “solve” the models that we develop. This chapter focuses on using game theory to model strategic situations. The next chapter begins solving game theory models.

3. How to model a scenario involving kidnapping Involves three individuals: Guy (kidnapper), Orlando (victim), and Vivica (victim’s wife). First, Guy needs to decide whether to kidnap Orlando. Second, if Guy kidnaps Orlando, Vivica needs to decide whether to pay ransom. Third, after knowing whether Vivica has paid ransom or not, Guy needs to decide whether to kill Orlando, or release him. The (inverted) decision tree diagram in the next slide summarizes the interactions.

4. Decision tree for the kidnapping game:

5. Extensive form representation: When the interactions are represented using a tree diagram, it is called the extensive form representation of a game. A decision tree is read from top to bottom. Each of the dots is called a decision node, which represents a point in the game at which someone has a decision to make. Coming out of a decision node is a series of branches, where each branch represents a different action available to the decision maker.

6. Payoffs in the kidnapping game: There are five possible outcomes in the kidnapping game. Each outcome should represent a payoff for Guy and a payoff for Vivica. How do we rank Guy’s payoffs over the outcomes? How do we rank Vivica’s payoffs over the outcomes? The next slide represents Guy and Vivica’s (possible) payoffs for the outcomes. Payoffs are determined from their preferences. At the end of a game, we need to have the payoffs representing the well being of the players.

7. Payoffs in the kidnapping game:

8. Baseball game: The Orioles and the Yankees Right handed batters generally perform better against left handed pitchers and left handed batters typically perform better against right handed pitchers. (See the next slide for the statistics.) It is the bottom of the ninth inning and the game is tied between the Orioles and the Yankees. The pitcher on the mound for the Yankees is Mariano Rivera, who is a right hander, and the batter due up for the Orioles is Javy Lopez, who is also a right hander. The Orioles can send Jay Gibbons, who is a left hander, but then the Yankees can send in left handed pitcher Randy Johnson.

9. Do right handed batters do better against left handed pitchers?

10. Baseball game: The Orioles and the Yankees.

11. Galileo and the Inquisition In 1633, the great astronomer and scientist Galileo was under consideration for interrogation by the Inquisition. The church contends that in 1616 Galileo was ordered not to teach and support the Copernican theory that the earth revolves around the sun, but Galileo violated the order in his new book “The Dialogue Concerning the Two Chief World Systems.” The players in this game are Pope Urban VIII, Galileo, and the Inquisitors.

12. Galileo and the Inquisition

13. Haggling at an auto dealership Donna wants to buy a Lexus and shows up at a Lexus dealership. Marcus is the salesman. The car can be sold at three possible prices: high, moderate, and low. Marcus quotes a price. Donna may accept the price or reject the price. If she rejects the price, she may leave or make a counter offer. Marcus can accept the price or make a counter offer. The haggling continues until they agree on a price or Donna leaves the dealership.

14. Haggling at an auto dealership

15. Simplifying the extensive form of the haggling game:

16. Perfect information games: In the preceding examples, when it is one player’s turn to make a decision, he/she knows what decisions are made by the other players until that point. For example, in the earlier kidnapping game, Guy learns whether ransom has been paid prior to deciding what to do with Orlando. When each player knows at the time of his/her decision making what decisions are made by other players, the game is called a game of perfect information. All preceding examples are perfect information games.

17. Games of imperfect information When at least one player does not know what the other(s) has done, when it is his/her turn to make a decision, it is called a game of imperfect information. Suppose in the kidnapping game, after Guy kidnaps Orlando, Guy and Vivica may have to make decisions without knowing what the other has done. Then, it will be a game of imperfect information. We use a concept, information set, to formally distinguish between games of perfect and imperfect information.

18. Information set An information set is made up of all of the decision nodes that a player is incapable of distinguishing among. Every decision node belongs to one and only one information set. If the information set has more than one node, then the player is uncertain as to where exactly he/she is in the game. In a perfect information game, each information set has exactly one node. In an imperfect information game, at least one information set has more than one node.

19. Kidnapping game when the exchange is simultaneous:

20. Kidnapping game when the exchange is simultaneous: an alternative representation

21. A mugging game The players: Simon (good guy) and a mugger (Jessica). The mugger may or may not have a gun. If she has a gun, she may or may not want to show the gun. If the mugger shows her gun, Simon knows for sure that she has a gun. Otherwise, Simon does not know whether or not the mugger has a gun. Simon can resist or may not resist. This is a game of imperfect information: since Jessica does not show her gun, Simon does not know whether she has a gun or not.

22. The mugging game:

23. U.S. Court of Appeals

24. The Iraq War and weapons of mass destruction Three players: Iraq, the United Nations, and the United States. Iraq may or may not have acquired WMD. UN may decide to inspect or not. Iraq may decide whether to allow inspection or not. Also, if it has acquired WMD, it may decide to disclose or not. U.S. may decide to take military actions against Iraq. This is a game of imperfect information.

25. The Iraq War and weapons of mass destruction

26. Extensive form games : So far, we have learned about extensive form games. An extensive form game describes: (a) a concrete sequence with which the players act. (b) what actions they have available and what each player knows at the time of his/her action. (c) payoffs for each player at each outcome. There is an alternative way of describing a game. It is called strategic form game, which we study next.

27. Games in strategic form: what is a strategy? A strategy is a fully specified decision rule for how to play a game that incorporates every possible contingency. Suppose a bad guy first moves to the left or right. James Bond watches the bad guy’s move, and then moves to the left or right. Bad guy has two strategies, either move to the left or move to the right. (Say, B1 and B2.) Since James Bond moves second, however, his decisions are contingent on bad guy’s decisions.

28. James Bond and the bad guy James Bond has four strategies. They are: Strategy J1: Move to the left if the bad guy moves to the left, and move to the left if the bad guy moves to the right. Strategy J2: Move to the right if the bad guy moves to the left, and move to the right if the bad guy moves to the right. Strategy J3: Move to the left if the bad guy moves to the left, and move to the right if the bad guy moves to the right. Strategy J4: Please complete.

29. Strategy set The strategy set for a player is defined as the collection of all feasible strategies for that player. The strategy sets for bad guy and James Bond are: Strategy set of bad guy: {B1, B2} Strategy set of James Bond: {J1, J2, J3, J4}

30. Try it: for the extensive form game below, write down the strategy sets for Guy and Vivica.

31. Solution: Vivica has two strategies: If a kidnapping occurs, then pay a ransom. If a kidnapping occurs, then do not pay a ransom. Guy has four strategies: Kidnap and if a kidnapping occurs, then kill. Kidnap and if a kidnapping occurs, then release. Do not kidnap and if a kidnapping occurs, then kill. Do not kidnap and if a kidnapping occurs, then release.

32. Strategic form games: A strategic form game describes: (1) who is making decisions? (2) over what they are making decisions? (3) payoffs at different outcomes. So, a strategic form game describes: (1) a set of players. (2) strategy set for each player. (3) payoffs for each player at different outcomes. Strategic form games are also called normal form games.

33. Strategic form game: Tosca Tosca is a popular opera composed by famous opera composer Giacomo Puccini. Two players: a corrupt chief of police, Scarpia, and an attractive woman, Tosca. Scarpia arrests Tosca’s lover and tells Tosca that her lover will be shot by a firing squad. Scarpia may order real bullets or blanks. Tosca’s lover will survive if real blanks are used, otherwise he will die. Scarpia meets Tosca and makes sexual demands. Tosca can either stab Scarpia or concede to his demands.

34. Tosca: A game in a strategic form Set of Players: {Tosca, Scarpia} Tosca’s strategies: {Stab, Consent} Scarpia’s strategies: {Real Bullets, Blanks} Four outcomes: A = (Stab, Real), B = (Stab, Blanks), C = (Consent, Real), and D = (Consent, Blanks). Tosca’s preference ordering: B > D > A > C. Scarpia’s preference ordering: C > D > A > B. Payoffs are presented in a tabular form.

35. Tosca: A game in a strategic form

36. Example: competition for elected office The Republican and Democratic candidates (for presidency) simultaneously choose their respective political platform. Positions available for the Democratic candidate are Moderate, Moderately Liberal, and Liberal. Positions available for the Republican candidate are Moderate, Moderately Conservative, and Conservative. Each candidate’s goal is to win the election. See the next slide for their payoffs and a strategic form representation of the game.

37. Competition for elected office

38. Switching between extensive and strategic forms : For every extensive form game, there is a unique strategic form representation. Consider the earlier baseball game example. Players: {Orioles, Yankees} Orioles strategies: {(1) Substitute Gibbons for Lopez, (2) Retain Lopez} Yankees strategies: {(1) Substitute Johnson for Rivera, (2) Retain Rivera} Payoffs are as specified in the earlier example. The next slide represents the strategic form.

39. Strategic form of the baseball game

40. Strategic form of the Galileo, the Pope, and the Inquisition game

41. Going from the strategic to extensive form: From a strategic form representation of a game, we can draw the corresponding extensive form representation. Although every extensive form game has a unique strategic form game associated with it, the same strategic form game can be associated with more than one extensive form game. Consider the game between Tosca and Scarpia. The next slide shows two extensive form representations of the game.

42. Two extensive form representations of the Tosca game

43. Lack of common knowledge : Jack and Kate are to meet at the French restaurant Per Se in New York City. Jack has since learned that the restaurant is closed for the day. He emails Kate suggesting they meet at Artie’s at 7.00 P.M. Kate emails back a confirmation to Jack. Kate arrives at Artie’s at 7.00 P.M., but Jack does not show up. Kate thinks that Jack has not received her confirmation she and leaves. Jack shows up late. By then Kate has left.

44. What do we need for common knowledge? Jack knows the plan. Kate knows the plan. Jack knows that Kate knows the plan. Kate knows that Jack knows the plan. Jack knows that Kate knows that Jack knows the plan. Kate knows that Jack knows that Kate knows the plan. And so on, and so on.

45. Common Knowledge Common knowledge is much more than players knowing something: it involves them knowing what the others’ know, and knowing what the others’ know about what the others’ know, and so forth.

46. A few more issues in modeling games: Can a player forget? Can a player change the game? Does the game has to be factually accurate?

47. Can a player change the game? Extensive form of the film “Ransom”

48. Summary: Concepts Learned Extensive form game Information set Perfect information game Imperfect information game Payoffs Strategic form game Set of players Strategies Common knowledge