Application of Game Theory on an Oligopoly

slides on game theory and its application on an Oligopoly

1. Game Theory & Strategic Behavior

2. Game Theory –The Concept A game is defined as a conflict between two or more opponents, which follows set rules , and involves gains and losses for the players . The mathematical study of games is called Game Theory , and forms an important foundation for mathematics, economics, and computer science. Several fun activities can be easily conducted in a classroom to demonstrate various concepts of game theory Monday, October 24, 2016 2

3. Game Theory: Strategic Decisions Game Theory AKA Multi-person Decision Theory analyzes the decision-making process when there are more than one decision-maker where each agent’s payoff possibly depends on the actions taken by the other agents. Monopolies are a rare occurrence in the free market. Oligopolies and perfect competition firms have to anticipate their competitors’ actions when designing their own strategies. Monday, October 24, 2016 3

4. Game Theory: Strategic Decisions 2 Since an agent’s preferences on his actions depend on which actions the other parties take, his action depends on his beliefs about what the others do. Of course, what the others do depends on their beliefs about what each agent does. Communication between firms in perfect competition and oligopolies is illegal; making it impossible/ unlikely for firms to agree on matters such as price, marketing strategy, sale volume and so forth. This is why most of those firms rely on managerial tools such as Game Theory, in order for them to design the best course of action while taking into consideration the actions of other players in the market. Monday, October 24, 2016 4

5. Game Theory: A Catch 22 A player’s actions depend on: the actions available to each agent, each agent’s preferences on the outcomes, each player’s beliefs about which actions are available to each player and how each player ranks the outcomes, and further his beliefs about each player’s beliefs… Under perfect competition, there are many decision makers => their actions are assumed to be decentralized Monday, October 24, 2016 5

6. Self-interested Agents What does it mean to say that an agent is self-interested? Not that they (the agents) want to harm others or only care about themselves. It means only that the agent has his/her own description of states of the world that it likes, and acts based on this description Each such agent has a utility function It quantifies the degree of preference across alternatives It explains the impact of uncertainty Decision-theoretic rationality: act to maximize expected utility. Monday, October 24, 2016 6

7. Why Utility function? It might seem obvious that preferences can be described by utility functions. But: Why is a single-dimensional function enough? Why should an agent’s response to uncertainty be captured purely by an expected value? Thus, our claim is substantive. There’s a famous theorem (von Neumann & Morgenstern, 1944) that derives the existence of a utility function from a more basic preference ordering and axioms on such orderings. Monday, October 24, 2016 7

8. Defining Games –Key Ingredients Players: who are the decision makers? • People? Governments? Companies? Somebody employed by a Company?... Actions: what can the players do? • Enter a bid in an auction? Decide whether to end a strike? Decide when to sell a stock? Decide how to vote? Payoffs: what motivates players? • Do they care about some profit? Do they care about other players?... Monday, October 24, 2016 8

9. Defining Games - Two Standard Representations • Normal Form (a.k.a. Matrix Form, Strategic Form) Lists what payoffs they get as a function of their actions • It is as if players moved simultaneously • But strategies encode many things... • Extensive Form Includes timing of moves • Players move sequentially, represented as a tree Chess: white player moves, then black player can see white’s move and react... • Keeps track of what each player knows when he or she makes each decision Poker: bet sequentially – what can a given player see when they bet? Monday, October 24, 2016 9

10. Defining Games - The Normal Form • Finite, n-person normal form game: ⟨N; A; u⟩: • Players: N = {1,…,n} is a finite set of n , indexed by i • Action set for player i Ai a = (a 1 ,…,a n ) ∈ A = A 1 × . . . × A n is an action profile • Utility function or Payoff function for player i : u i : A → R u = (u 1 ,…,u n ), is a profile of utility functions Monday, October 24, 2016 10

11. Summary: Defining Games • For Now: Normal Form (Strategic Form, Matrix Representation...) • Players: N • Actions: Ai • Payoffs: ui • Later: Extensive Form • Timing: in what order do things happen? • Information: what do players know when they act • Representation: Tree Games Monday, October 24, 2016 11

12. Normal Form Games - The Standard Matrix Representation Writing a 2-player game as a matrix: • “row” player is player 1, “column” player is player 2 • rows correspond to actions a1 ∈ A1, columns correspond to actions a2 ∈ A2 • cells listing utility or payoff values for each player: the row player first, then the column Monday, October 24, 2016 12

13. Game Theory: Example 1 \ 2 L m R T (1.1) (0,2) (2,1) M (2,2) (1,1) (0,0) B (1,0) (0,0) (-1,1) Monday, October 24, 2016 13 Let’s assume that each player knows that these are the strategies and the payoffs , knows that each player knows this, knows that each player knows that each player knows this, etc. In that case, we formally say that the strategies and the payoffs are common knowledge .

14. A Large Collective Action Game • Players: N = {1, . . . ,10.000.000} • Action set for player I Ai = {Revolt; Not} • Utility function for player i : • u i (a) = 1 if #{j : a j = Revolt} ≥ 2.000.000 • u i (a) = −1 if #{j : a j = Revolt} < 2.000.000 and a i = Revolt • u i (a) = 0 if #{j : a j = Revolt} < 2.000.000 and a i = Not Monday, October 24, 2016 14

15. Pure Coordination Game 1 \ 2 Left Right Top (1,1) (0,0) Bottom (0,0) (1,1) Monday, October 24, 2016 15 Let’s assume you want to meet with a friend in one of two places, about which you both are indifferent Player 1 chooses bwn top and bottom rows and Player2 bwn left and right columns: P1prefers Top to Bottom if he knows that P2 plays Left P1 prefers Bottom if he knows that P2 plays Right P1 is indifferent if he thinks that P2 is likely to play either strategy with equal probabilities Similarly for P2

16. Pure Coordination Game Players would look for stable outcomes (strategy profiles) –No player has incentive to deviate if he knows that the other players play prescribed strategies. In the Pure Coordination Game, the Top-Left and Bottom-Right are such outcomes. Bottom-Left and Top-Right are not stable: For example, if Bottom-Left is known to be played, each player would like to deviate (draw figure) Monday, October 24, 2016 16

17. The Battle of the Sexes Game Unlike the Pure Coordination game, most likely players have different preferences on the outcomes, inducing conflict Such game is known as the Battle of the Sexes , in which conflict and the need for coordination are present together Here, once again players would like to coordinate on Top-Left or Bottom-Right The stable outcomes are again Top-Left and Bottom- Right Now P1 prefers to coordinate on Top-Left, while P2 prefers to coordinate on Bottom-Right Monday, October 24, 2016 17

18. The Battle of the Sexes Game 2 1 \ 2 Left Right Top (2,1) (0,0) Bottom (0,0) (1,2) Monday, October 24, 2016 18 We can also show this table in a tree format to show the effect of one decision on the bottom line Draw the tree One solution to this game is called Backward induction (to check what the other player is doing then act) When it is common knowledge that a player has some information or not, the player may prefer not to have that information

19. The Battle of the Sexes Game Monday, October 24, 2016 19 Forward induction (this is current when one of the players has an exit strategy). If P1 has the exit strategy, then playing the game would only be more beneficial than exiting if he ensures to win 2. P2 knows that, and therefore decides for P1, by undertaking the scenario that ensures P1 stays in the game. Therefore, giving P1 the upper-hand of winning 2 is the only positive outcome for P2, thus choosing the second of the stable outcomes Top-Left

20. Consider the following normal form: N={1, 2} Ai ={Movie, Theater} Each player chooses an action of either going to a movie or going to the theater. Player 1 prefers to see a movie with Player 2 over going to the theater with Player 2. Player 2 prefers to go to the theater with Player 1 over seeing a movie with Player 1. Players get a payoff of 0 if they end up at a different place than the other player. a)  a > c ,  b > d  ; c)  a > c ,  b < d  ; b)  a > d ,  b < c  ; d)  a < c ,  b < d  ; Monday, October 24, 2016 20

21. Games of Pure Competition Players have exactly opposed interests There must be precisely two players (otherwise they can’t have exactly opposed interests) For all action profiles a ∈ A, u 1 (a) + u 2 (a) = c for some constant c Special case: zero sum Thus, we only need to store a utility function for one player in a sense, we only have to think about one player’s interests Monday, October 24, 2016 21

22. Example: Matching Pennies One player wants to match , the other wants to mismatch Monday, October 24, 2016 22 Heads Tails Heads 1, -1 -1, 1 Tails -1, 1 1, -1

23. Rock-Paper-Scissors Generalized matching pennies. Monday, October 24, 2016 23 Rock Paper Scissors Rock 0,0 -1, 1 1, -1 Paper 1, -1 0,0 -1, 1 Scissors -1, 1 1, -1 0,0 ...Believe it or not, there’s an annual international competition!

24. Games of Cooperation Players have exactly the same interests. no conflict: all players want the same things V a ∈ A; V i; j; u i (a) = u j (a) we often write such games with a single payoff per cell why are such games “non-cooperative”? Monday, October 24, 2016 24

25. Example: Coordination Game Which side of the road should we drive on? Monday, October 24, 2016 25 Left Right Left 1, 1 0, 0 Right 0, 0 1, 1

26. General Games (mixing cooperation and competition) The most interesting games combine elements of coordination and competition The Battle of the Sexes is known to be less conflictual in a sense, that both can win, but one actor in a game would win more than the other if they choose the right action/strategy Monday, October 24, 2016 26 Movies Restaurant Movies 2, 1 0, 0 Restaurant 0, 0 1, 2

27. Prisoner’s Dilemma Prisoner’s dilemma is any game that meets the following conditions: Monday, October 24, 2016 27 Cooperate Defect Cooperate a, a b, c Defect c, b d, d With d > b > a > c.

28. The Prisoner’s Dilemma 1 \ 2 Confess Not Confess confess (-1,-1) (1,-10) Not Confess (-10,1) (2,2) Monday, October 24, 2016 28 This is a well known game, initially about 2 prisoners being caught for the same crime In this game, no matter what the other player does, he would like to confess, yielding a (-1,-1) outcome However, the best outcome occurs when neither one confesses (2,2) This usually comes down to time, because at the end, and with the police tactics, one of the two suspects end up confessing yielding to either a (-10, 1) or (1, -10)

29. Keynes’ Beauty Contest Game You hold a stock and the price is rising... You believe that the price is too high to be justified by the value of the company. You would like to sell it, but would like to wait until the price is almost at its peak. You would like to get out of the market just before other investors do. How will they act? What should you do in response? Monday, October 24, 2016 29

30. Keynes Beauty Contest Game: The Stylized Version Each player names an integer between 1 and 100. The player who names the integer closest to two thirds of the average integer wins a prize, the other players get nothing. Ties are broken uniformly at random. Monday, October 24, 2016 30

31. The Strategic Reasoning What will other players do? What should I do in response? Each player best responds to the others: Nash Equilibrium Monday, October 24, 2016 31

32. Solving the Beauty Contest Game Suppose a player believes the average play will be X (including his or her own integer) That player’s optimal strategy is to say the closest integer to 2/3X. X has to be less than 100, so the optimal strategy of any player has to be no more than 67. If X is no more than 67, then the optimal strategy of any player has to be no more than 2/3 of 67. If X is no more than 2/3 of 67, then the optimal strategy of any player has to be no more than (2/3) 2 of 67. The unique Nash equilibrium of this game is for every player to announce 1! Monday, October 24, 2016 32

33. Summary Nash Equilibrium A consistent list of actions: Each player’s action maximizes his or her payoff given the actions of the others. A self-consistent or stable profile Each player’s action maximizes his or her payoff given the actions of the others. Nobody has an incentive to deviate from their action if an equilibrium profile is played. Someone has an incentive to deviate from a profile of actions that do not form an equilibrium. Monday, October 24, 2016 33

34. Nash Equilibrium Should we expect an equilibrium to be played? That depends on whether we know for a fact that people understand the reasoning behind the game played; in which case we expect every rational player to play towards the equilibrium Should we expect a non-equilibrium to be played? We actually expect non-equilibrium to move away and toward equilibrium as time goes and as people understand better the structure of the game Monday, October 24, 2016 34

35. Simultaneous vs. Sequential Games A distinction between ways of representing games comes with the order of the play. So far we have studied games where moves were done simultaneously ( simultaneous-move games ). Another type of games, represented mostly by trees rather than matrices, is the sequential move games , where you see what the other party has done before you choose your own action Chess for example is a sequential-move game Monday, October 24, 2016 35

36. Simultaneous vs. Sequential Games2 Games of perfect information (sequential games) are the simplest sorts of games, because in such games (as long as the games are finite) players and analysts can use a straightforward procedure for predicting outcomes: A player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her. She then asks herself which of the available final outcomes brings her the highest utility, and chooses the action that starts the chain leading to this outcome. This process is called  backward induction  (because the reasoning works backwards from eventual outcomes to present choice problems). Monday, October 24, 2016 36

37. Sequential Games: Game Trees A game tree is an example of what mathematicians call a directed graph. It is a set of connected nodes  in which the overall graph has a direction. We can draw trees from the top of the page to the bottom, or from left to right. In the first case, nodes at the top of the page are interpreted as coming earlier in the sequence of actions. In the case of a tree drawn from left to right, leftward nodes are prior in the sequence to rightward ones. Monday, October 24, 2016 37 An unlabeled tree has a structure of the following sort:

38. More General Form: Prisoner’s Dilemma We use the prisoner’s dilemma game as an extended example, only because it's particularly helpful for illustrating the  relationship  between strategic-form and extensive-form Our first step in modeling the two prisoners' situation as a game is to represent it in terms of utility functions. Go free => payoff is 4 2 years => payoff is 3 5 years => payoff is 2 10 years => payoff is 0 Monday, October 24, 2016 38

39. Prisoner’s Dilemma The PD game is a combination of Competition and Coordination : a Normal-Form game Wherever one action for a player is superior to his other actions for each possible action by the opponent, we say that the first action  strictly dominates   the second one (it is strictly dominant ). Monday, October 24, 2016 39

40. The Sequential-move of the PD game Let’s assume that the prisoners do  not  move simultaneously. Let’s assume they move sequentially; that is, suppose that Player II can choose after observing Player I's action. This would enable us to represent the game in extensive form, so to introduce game-trees and the method of analysis appropriate to them. Monday, October 24, 2016 40

41. The Sequential-move of the PD game First, here are definitions of some concepts that will be helpful in analyzing game-trees: Node : A point at which a player chooses an action. Initial node : The point at which the first action in the game occurs. Terminal node : Any node which, if reached, ends the game. Each terminal node corresponds to an  outcome . Monday, October 24, 2016 41

42. The Sequential-move of the PD game Subgame : Any connected set of nodes and branches descending uniquely from one node. Payoff : an ordinal utility number assigned to a player at an outcome. Outcome : an assignment of a set of payoffs, one to each player in the game. Strategy : a program instructing a player which action to take at every node in the tree where she could possibly be called on to make a choice. Monday, October 24, 2016 42

43. Prisoner’s Dilemma in Game-Tree At the end of each branch, we have possible outcomes . Each outcome is identified with an assignment of payoffs Each of the structures descending from the nodes 1, 2 and 3 respectively is a subgame . In PD, the outcome: (2,2), indicating mutual defection, is said to be the ‘solution ’ to the game. Following the general practice in economics, game theorists refer to the solutions of games as  equilibria . Monday, October 24, 2016 43

44. Problem 1 Consider the following: Each player chooses an action of either going to a movie or going to the theater. P1 prefers to see a movie with P2 over going to the theater with P2. P2 prefers to go to the theater with P1 over seeing a movie with P1. Players get a payoff of 0 if they end up at a different place than the other player. Monday, October 24, 2016 44

45. Problem 1 (cont.) Player 1\ Player 2 Movie Theater Movie a,b 0,0 Theater 0,0 c,d Monday, October 24, 2016 45 Based on what we saw in class, what Game Theory is this? Which restrictions should  a ,  b ,  c  and  d  satisfy, and why?

46. Problem 2 What should be filled in ? Choose the right answer: Monday, October 24, 2016 46 H T H 1,-1 T 0, ? Consider the following constant-sum game: c) -1; d) 2; 1; b) 0;

47. Problem 3 n people guess an integer between 1 and 100, and the winner is the player whose guess is closest to the mean of the guesses + 1 (ties broken randomly). Guess an integer, and which of the following is an equilibrium? –and make sure you explain why Monday, October 24, 2016 47 All announce 50 b) All announce 100 c) All announce 75 d) All announce 51

48. How to Solve Strategic Games? There are three main concepts to solve strategic games: Dominant Strategies & Dominant Strategy Equilibrium 2. Dominated Strategies & Iterative Elimination of Dominated Strategies 3. Nash Equilibrium Monday, October 24, 2016 48

49. 1. Dominant Strategies A strategy is a dominant strategy for a player if it yields the best payoff (for that player) no matter what strategies the other players choose . If all players have a dominant strategy, then it is natural for them to choose the dominant strategies and we reach a dominant strategy equilibrium . Monday, October 24, 2016 49

50. Dominance Let’s s i and s i ’ be two strategies for player i , and let S - i be the set of all possible strategy profiles for the other players What’s a strategy? For now, just choosing an action (“pure strategy”) Monday, October 24, 2016 50 Definition : s i strictly dominates s’ i if √ s - i є S -1 , u i ( s i , s - i ) > u i ( s’ i , s - i ) Definition : s i weakly dominates s’ i if √ s - i є S -1 , u i ( s i , s - i ) >= u i ( s’ i , s - i )

51. Example of Dominance Determine which strategies are dominated in following normal game: Monday, October 24, 2016 51 W X Y Z U 3,6 4,10 5,0 0,8 M 2,6 3,3 4,10 1,1 D 1,5 2,9 3,0 4,6 For Player 2, strategy X dominates strategy Z

52. Dominance & Equilibrium If one strategy dominates all others, we say it’s dominant A strategy profile consisting of dominant strategies for every player must be a Nash Equilibrium An equilibrium in strictly dominant strategies must be unique Monday, October 24, 2016 52 C D C -1, -1 -4, 0 D 0, -4 -3, -3

53. Example 1 (Prisoner’s Dilemma): Prisoner 2 Confess Deny Prisoner 1 Confess -10, -10 -1, -25 Deny -25, -1 -3, -3 Confess is a dominant strategy for both players and therefore (Confess, Confess) is a dominant strategy equilibrium yielding the payoff vector (-10,-10). Monday, October 24, 2016 53

54. Example 2 (Time vs. Newsweek): Newsweek AIDS BUDGET Time AIDS 35,35 70,30 BUDGET 30,70 15,15 The AIDS story is a dominant strategy for both Time and Newsweek. Therefore (AIDS,AIDS) is a dominant strategy equilibrium yielding both magazines a market share of 35 percent. Monday, October 24, 2016 54

55. Example 3: Player 2 X Y A 5,2 4,2 Player 1 B 3,1 3,2 C 2,1 4,1 D 4,3 5,4 Here Player 1 does not have a single strategy that “beats” every other strategy. Therefore she does not have a dominant strategy. On the other hand Y is a dominant strategy for Player 2. Monday, October 24, 2016 55

56. Example 4 (with 3 players): P3 A B P2 P2 L R L R U 3,2,1 2,1,1 U 1,1,2 2,0,1 P1 M 2,2,0 1,2,1 M 1,2,0 1,0,2 D 3,1,2 1,0,2 D 0,2,3 1,2,2 Here U is a dominant strategy for Player 1, L is a dominant strategy for Player 2, B is a dominant strategy for Player 3, and therefore (U;L;B) is a dominant strategy equilibrium yielding a payoff of (1,1,2). Monday, October 24, 2016 56

57. Dominated Strategies A strategy is dominated for a player if she has another strategy that performs at least as good no matter what other players choose. Of course if a player has a dominant strategy then this player’s all other strategies are dominated. But there may be cases where a player does not have a dominant strategy and yet has dominated strategies. Monday, October 24, 2016 57

58. Example 5: Player 2 X Y A 5,2 4,2 Player 1 B 3,1 3,2 C 2,1 4,1 D 4,3 5,4 Here B & C are dominated strategies for Player 1 and X is a dominated strategy for Player 2. Therefore it is natural for Player 1 to assume that Player 2 will not choose X, and Player 2 to assume that Player 1 will not choose B or C. Monday, October 24, 2016 58

59. Therefore the game reduces to Player 2 Y Player 1 A 4,2 D 5,4 In this reduced game D dominates A for Player 1. Therefore we expect players to choose (D;Y) yielding a payoff of (5,4). This procedure is called iterated elimination of dominated strategies . Monday, October 24, 2016 59

60. Example 6: Player 2 L R U 10,5 10,10 Player 1 M 20,10 30,5 D 30,10 5,5 U is dominated for Player 1 = ⇒ Eliminate. Monday, October 24, 2016 60

61. Player 2 L R Player 1 M 20,10 30,5 D 30,10 5,5 R is dominated for Player 2 = ⇒ Eliminate . Player 2 L Player 1 M 20,10 D 30,10 M is dominated for Player 1 = ⇒ Eliminate and (D;L) survives. Monday, October 24, 2016 61

62. Example 7: Player 2 V W X Y Z A 4,-1 3,0 -3,1 -1,4 -2,0 B -1,1 2,2 2,3 -1,0 2,5 Player 1 C 2,1 -1,-1 0,4 4,-1 0,2 D 1,6 -3,0 -1,4 1,1 -1,4 E 0,0 1,4 -3,1 -2,3 -1,-1 Here the order of elimination is: D-V-E-W-A-Y-C-X and hence, (B;Z) survives the elimination yielding a payoff of (2,5). Monday, October 24, 2016 62

63. Example 8: Each of two players announces an integer between 0 and 100. Let a 1 be the announcement of Player 1 and a 2 be the announcement of Player 2. The payoffs are determined as follows: If a 1 + a 2 ≤ 100: Player 1 receives a 1 and Player 2 receives a 2 ; If a 1 + a 2 > 100 and a 1 > a 2 : Player 1 receives 100 − a 2 and Player 2 receives a 2 ; If a 1 + a 2 > 100 and a 1 < a 2 : Player 1 receives a 1 and Player 2 receives 100 − a 1 ; If a 1 + a 2 > 100 and a 1 = a 2 : Both players receive 50. Solve this game with iterated elimination of dominated strategies. Monday, October 24, 2016 63

64. Observation: If Player 1 announces 51 her payoff is 50 if Player 2 announces 50 or 51, and 51 if Player 2 announces anything else. Likewise for Player 2 . Round 1: If Player 1 announces a 1 < 51 she’ll get a 1 no matter what Player 2 announces. Therefore any strategy smaller than 51 is dominated by 51. Likewise for Player 2. We can delete all strategies between 0 and 50 for both players. Round 2: If Player 1 announces 100 she can get at most 50. This is because Player 2 announces a number between 51-100. Therefore 100 is dominated by 51 in this reduced game. Likewise for Player 2. We can delete 100 for both players. Round 3: If Player 1 announces 99 she can get at most 50. This is because Player 2 announces a number between 51-99. Therefore 99 is dominated by 51 in this further reduced game. Likewise for player 2. We can delete 99 for both players. Monday, October 24, 2016 64

65. . . . . . . Round 49: If Player 1 announces 53 she can get at most 50. This is because Player 2 announces a number between 51-53. Therefore 53 is dominated by 51 in this further, further, . . . , further reduced game. Likewise for Player 2. We can delete 53 for both players. Round 50: If Player 1 announces 52 she can get at most 50. This is because Player 2 announces a number between 51-52. Therefore 52 is dominated by 51 in this further, further, . . . , further reduced game. Likewise for Player 2. We can delete 52 for both players. Hence only 51 survives the iterated elimination of strategies for both players. As a result the payoff of each player is 50. Monday, October 24, 2016 65

66. Nash Equilibrium In many games there will be no dominant and/or dominated strategies. Even if there is, iterative elimination of dominated strategies will usually not result in a single strategy profile. Consider a strategic game. A strategy profile is a Nash equilibrium if no player wants to unilaterally deviate to another strategy, given other players’ strategies. Monday, October 24, 2016 66

67. Example 9: Player 2 L R Player 1 U 5, 5 2, 1 D 4, 7 3, 6 Consider the strategy pair (U;L). If Player 1 deviates to D then his payoff reduces to 4. If Player 2 deviates to R then her payoff reduces to 1. Hence neither player can benefit by a unilateral deviation. Therefore (U;L) is a Nash equilibrium yielding the payoff (5,5). Monday, October 24, 2016 67

68. Example 10: Consider the following 3-person simultaneous game. Here Player 1 chooses between the rows U and D, Player 2 chooses between the columns L and R, and Player 3 chooses between the matrices A and B. P3 A B P2 P2 L R L R P1 U 5,5,1 2,1,3 U 0,2,2 4,4,4 D 4,7,6 1,8,5 D 1,1,1 3,7,1 In this game (U;R;B) is the only Nash equilibrium Monday, October 24, 2016 68

69. Example 11 (Battle of the Sexes): The following game has two Nash equilibria (U;L) and (D;R). Player 2 L R Player 1 U 3, 1 0, 0 D 0, 0 1, 3 Monday, October 24, 2016 69

70. Example 12 (Matching Pennies): The following game has no Nash equilibrium. Player 2 L R Player 1 U 1, -1 -1, 1 D -1, 1 1, -1 Monday, October 24, 2016 70

71. Tricks for Finding Nash Equilibrium in Complicated Games Example 13: P2 V W X Y Z A 4,-1 4,2 -3,1 -1,2 -2,0 B -1,1 2,2 2,3 -1,0 2,5 P1 C 2,3 -1,-1 0,4 4,-1 0,2 D 1,3 4,4 -1,4 1,1 -1,2 E 0,0 1,4 -3,1 -2,3 -1,-1 Monday, October 24, 2016 71

72. Example 13 (cont.) In column V , if there is a Nash equilibrium at all it should be (A;V); otherwise P1 deviates. But it is not a Nash equilibrium since P2 deviates. In column W , if there is a Nash equilibrium at all it should be (A;W) or (D;W); otherwise P1 deviates. Since P2 does not deviate in either, both strategy profiles are Nash equilibrium. In column X , if there is a Nash eq. at all it should be (B;X); otherwise P1 deviates. But it is not a Nash eq. since P2 deviates. Monday, October 24, 2016 72

73. In column Y , if there is a Nash equilibrium at all it should be (C;Y); otherwise P1 deviates. But it is not a Nash equilibrium since P2 deviates. In column Z , if there is a Nash equilibrium at all it should either be (B;Z); otherwise P1 deviates. Since P2 does not deviate here it is a Nash eq uilibrium Monday, October 24, 2016 73 Example 13 (cont.)

74. Monday, October 24, 2016 74