An Independent Relationship between Variables

ECON 7921 Price Analysis II Spring Semester 2018 Elliott Fan Economics, NTU The Theory of Games (Chapter 12) Price Analysis, Spring 2018 Elliott Fan Lecture 5 Lets watch a show first -- Golden balls A British TV show http://www.youtube.com/watch?v=p3Uos2fzIJ0 Price Analysis II 2018 Lecture 5 2 Game Matrices Game Matrix is a diagram that:

Shows one players strategy choices across the top. Shows other players strategy choices along left side. Shows corresponding outcomes in appropriate boxes. Example: Pigs in a Box. Price Analysis II 2018 Lecture 5 3 Price Analysis II 2018 Lecture 5 4 Choosing Strategies If a given box was the game matrix outcome, would one or both players want to change their mind? If yes, rule out that outcome. Outcome from which neither player would want to deviate is a Nash Equilibrium outcome. Takes opponents behavior as a given.

Price Analysis II 2018 Lecture 5 5 Nash Equilibrium Price Analysis II 2018 Lecture 5 6 Prisoners Dilemma Revisited Dominant strategy Follow strategy regardless of other players behavior Related to Nash equilibrium Is the outcome a Nash equilibrium? Is a NE always a results from playing DS? Does the outcome reach Pareto optimality? Price Analysis II 2018 Lecture 5 7 Price Analysis II 2018 Lecture 5

8 Dominant Strategies Strategy a player would want to follow regardless of the other players behavior. When both players follow their dominant strategies, Nash equilibrium is reached. Player who wants to know what other player is doing before making a choice has no dominant strategy. Nash equilibria can exist when player(s) have no dominant strategy. Price Analysis II 2018 Lecture 5 Slide 9 Price Analysis II 2018 Lecture 5 10 Price Analysis II 2018 Lecture 5

11 Iterated elimination of strictly dominated strategies Player 2 U Player M 1 D L R 2, 4 6, 3 3, 3 3, 4

5, 4 4, 3 Price Analysis II 2018 Lecture 5 12 Iterated elimination of strictly dominated strategies One way to solve the game above is to use iterated elimination of strictly dominated strategies. So our story may go like, 1 is rational, so 1 will not play M. Given everyone knows this, 2 is rational, 2 will not play R. Given everyone knows this, 1 is rational, 1 will not play U. So (D, L) becomes our prediction. This process uses some common knowledge of rationality. Price Analysis II 2018 Lecture 5 13 Iterated elimination of strictly

dominated strategies But, are you really a person who will play according to this logic of iterated elimination of strictly dominated strategies? Will you get the highest payoff if you do? Despite of this, theorists still feel that iterated elimination does not give sharp predictions. Price Analysis II 2018 Lecture 5 14 Nash Equilibrium as a solution concept Rule for predicting how games will turn out. Popular solution concept but it has some problems: Some games have more than one Nash Equilibrium. Difficult to predict outcome, but once equilibrium is reached it is usually stable. Examples: Car Insurance and Social Status Some games have no Nash Equilibrium. Example: The Copycat Game.

Price Analysis II 2018 Lecture 5 15 Pareto Optima Nash equilibrium is a positive concept: Designed to predict what will happen instead of enabling a discussion of what ought to happen. Pareto improvement or Pareto-preferred: Change to which no one objects. Pareto-optimal: Outcome where no Pareto improvement is possible. Nash equilibrium is not guaranteed to be Pareto optimal. Price Analysis II 2018 Lecture 5 16 Price Analysis II 2018 Lecture 5

17 Price Analysis II 2018 Lecture 5 18 Mixed Strategy A pure strategy is a single choice of a row or column in the game matrix. A mixed strategy involves a random choice among pure strategies. Common in sports. Price Analysis II 2018 Lecture 5 Slide 19 Mixed strategy is hard to comprehend It seems unlikely that the mutual beliefs in the likelihood are met. One way to comprehend a mixed strategy is to consider a game between two groups of individuals, instead of two agents. An alternative way is to consider that choice of an

individual depends on his/her idiosyncratic factors that cannot be determined by opponents. Price Analysis II 2018 Lecture 5 Slide 20 More on Nash Equilibrium The concept of Nash equilibrium is this. A strategy profile (so every player has a part) is a Nash equilibrium, if given your opponent plays his equilibrium strategy, you playing yours is a best response to that. In other words, players are best responding to each other. Price Analysis II 2018 Lecture 5 21 More on Nash Equilibrium The concept of Nash equilibrium is this. A strategy profile (so every player has a part) is a Nash equilibrium, if given your opponent

plays his equilibrium strategy, you playing yours is a best response to that. In other words, players are best responding to each other. 22 More on Nash Equilibrium Equivalently, we can say that given your opponent is playing equilibrium, you have no profitable deviation. That is exactly why we are looking at the reaction function in Cournot competition. A reaction function is simply a function that gives you the best response to every possibly plays of your opponent. We then intersect the two reaction functions to get Nash equilibrium where both are best responding. 23 More on Nash Equilibrium In the battle of sexes game, there are two pure Nash equilibria, (opera, opera) and (boxing,

boxing). Notice that in Nash equilibrium, a players is checking that he is best responding to his opponents play. In other words, implicitly we are assuming that a players belief is correct, or consistent with opponents play. So Nash equilibrium has two important components: best responding and correct beliefs. 24 More on Nash Equilibrium There are some interesting empirical works supporting Nash equilibrium. Yet there are also reasons sometimes people dont feel comfortable playing Nash equilibrium. Or sometimes we can even coordinate better than Nash. Look at the coordination games we played. 25 Do you dare? left

right top 1, 0 1, 1 bottom -1000, 0 2, 1 26 Sequential Games First player chooses a column in the game matrix and then the second player chooses a row. Oligopoly problem: Outcome depends on how the game is played. Stackelberg equilibrium arises when one player announces his strategy before the other.

Second player will choose an optimal response. Commitment is important. 27 28 Best Responses and best response curve Think of a 22 game; i.e., a game with two players, A and B, each with two actions. A can choose between actions aA1 and aA2. 29 Best Responses and best response curve B can choose between actions aB1 and aB2. There are 4 possible action pairs; (aA1, aB1), (aA1, aB2), (aA2, aB1), (aA2, aB2). Each action pair will usually cause different payoffs for the players. 30

Best Responses and best response curve Suppose that As and Bs payoffs when the chosen actions are aA1 and aB1 are UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4. Similarly, suppose that UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. 31 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. 32 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4

UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If B chooses action aB1 then As best response is ?? 33 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If B chooses action aB1 then As best response is action aA1 (because 6 > 4). 34 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7.

If B chooses action aB2 then As best response is ?? 35 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If B chooses action aB2 then As best response is action aA2 (because 5 > 3). 36 Best Responses and best response curve If B chooses aB1 then A chooses aA1. If B chooses aB2 then A chooses aA2. As best-response curve is therefore + aA2 As best

response aA1 + a B1 a B2 Bs action 37 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. 38 Best Responses and best response

curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA1 then Bs best response is ?? 39 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA1 then Bs best response is action aB2 (because 5 > 4). 40 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4

UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA2 then Bs best response is ??. 41 Best Responses and best response curve UA(aA1, aB1) = 6 and UB(aA1, aB1) = 4 UA(aA1, aB2) = 3 and UB(aA1, aB2) = 5 UA(aA2, aB1) = 4 and UB(aA2, aB1) = 3 UA(aA2, aB2) = 5 and UB(aA2, aB2) = 7. If A chooses action aA2 then Bs best response is action aB2 (because 7 > 3). 42 Best Responses and best response curve If A chooses aA1 then B chooses aB2. If A chooses aA2 then B chooses aB2. Bs best-response curve is therefore

aA2 As actio n aA1 aB 1 aB 2 Bs best response 43 Best Responses and best response curve If A chooses aA1 then B chooses aB2. If A chooses aA2 then B chooses aB2. Bs best-response curve is therefore Notice that aB2 is a strictly dominant action for B.

a A 2 As action aA1 a B1 a B2 Bs best response 44 Best Responses & Nash Equilibria How can the players best-response curves be used to locate the games Nash equilibria? As response

As choice B aA2 aA2 aA1 aA1 a B1 a B2 Bs choice A + + a B1

a B2 Bs response 45 Best Responses & Nash Equilibria How can the players best-response curves be used to locate the games Nash equilibria? Put one curve on top of the other. As response B A + aA2 aA1 As choice

+ a B1 a B2 Bs choice Bs response 46 Best Responses & Nash Equilibria How can the players best-response curves be used to locate the games Nash equilibria? Put one curve on top of the other. As response + aA2 aA1 Is there a Nash equilibrium?

+ a B1 a B2 Bs response 47 Best Responses & Nash Equilibria How can the players best-response curves be used to locate the games Nash equilibria? Put one curve on top of the other. As response + aA2 aA1 + a B1

a B2 Is there a Nash equilibrium? Yes, (aA2, aB2). Why? Bs response 48 Best Responses & Nash Equilibria How can the players best-response curves be used to locate the games Nash equilibria? Put one curve on top of the other. As response + aA2 aA1 + a B1 a B2

Is there a Nash equilibrium? Yes, (aA2, aB2). Why? aA2 is a best response to aB2. aB2 is a best response to aA2. Bs response 49 Best Responses & Nash Equilibria Player B aB1 a B2 aA1 6,4 3,5 aA2 4,3

5,7 Here is the strategic form of the game. Player A aA2 is the only best response to aB2. aB2 is the only best response to aA2. 50 Best Responses & Nash Equilibria Player B aB1 a B2 aA1 6,4 3,5

aA2 4,3 5,7 Here is the strategic form of the game. Player A aA2 is the only best response to aB2. aB2 is the only best response to aA2. Is there a 2nd Nash eqm.? 51 Best Responses & Nash Equilibria Player B aB1

a B2 aA1 6,4 3,5 aA2 4,3 5,7 Here is the strategic form of the game. Player A aA2 is the only best response to aB2. aB2 is the only best response to aA2. Is there a 2nd Nash eqm.? No, because

aB2 is a strictly dominant action for Player B. 52 Best Responses & Nash Equilibria Player B aB1 a B2 aA1 6,4 3,5 aA2 4,3 5,7

Player A Now allow both players to randomize (i.e., mix) over their actions. 53 Best Responses & Nash Equilibria Player B a B 1 a B 2 aA1 6,4

3,5 aA2 4,3 5,7 Player A A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Now allow both players to randomize (i.e., mix) over their actions. 54 Best Responses & Nash Equilibria Player B a B

1 a B 2 aA1 6,4 3,5 aA2 4,3 5,7 Player A A1 is the prob. A

chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A? 55 Best Responses & Nash Equilibria Player B A 1 is the prob. A a1 a2 chooses action aA1. B1 is the prob. B 6,4 3,5 aA1 chooses action aB1. Player A B

Given , what A 1 4,3 5,7 a2 value of A1 is best EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1. for A? B B 56 Best Responses & Nash Equilibria Player B A 1 is the prob. A a1 a2

chooses action aA1. B1 is the prob. B 6,4 3,5 aA1 chooses action aB1. Player A B Given , what A 1 4,3 5,7 a2 value of A1 is best EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1. for A? EVA(aA2) = 4B1 + 5(1 - B1) = 5 - B1. B B

57 Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A? EVA(aA1) = 3 + 3B1. EVA(aA2) = 5 - B1. > > = 3 + 3B1< 5 - B1 as B= ?? < 1 58 Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A? EVA(aA1) = 3 + 3B1. EVA(aA2) = 5 - B1.

> > = 3 + 3B1< 5 - B1 as B= . < 1 59 Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A? EVA(aA1) = 3 + 3B1. EVA(aA2) = 5 - B1. > > = 3 + 3B1< 5 - B1 as B= . < 1 As best response is:

aA1 if B1 > aA2 if B1 < aA1 or aA2 if B1 = 60 Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A? EVA(aA1) = 3 + 3B1. EVA(aA2) = 5 - B1. > > = 3 + 3B1< 5 - B1 as B= . < 1 As best response is: aA1 (i.e. A1 = 1) if B1 > aA2 (i.e. A1 = 0) if B1 < aA1 or aA2 (i.e. 0 A1 1) if B1 =

61 Best Responses & Nash Equilibria As best response is: aA1 (i.e. A1 = 1) if B1 > aA2 (i.e. A1 = 0) if B1 < aA1 or aA2 (i.e. 0 A1 1) if As best B1 = response A1 1 0 0 1 B1 62 Best Responses & Nash Equilibria

As best response is: aA1 (i.e. A1 = 1) if B1 > aA2 (i.e. A1 = 0) if B1 < aA1 or aA2 (i.e. 0 A1 1) if As best B1 = response A1 1 0 0 1 B1 63 Best Responses & Nash Equilibria As best response is: aA1 (i.e. A1 = 1) if B1 > aA2 (i.e. A1 = 0) if B1 < aA1 or aA2 (i.e. 0 A1 1) if

As best B1 = response A1 1 0 0 1 B1 64 Best Responses & Nash Equilibria As best response is: aA1 (i.e. A1 = 1) if B1 > aA2 (i.e. A1 = 0) if B1 < aA1 or aA2 (i.e. 0 A1 1) if As best B 1 = response

A1 1 0 This is As best response curve when players are allowed to mix over their actions. 0 1 B1 65 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A

6,4 3,5 aA2 4,3 5,7 A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given A1, what value of B1 is best for B? 66 Best Responses & Nash Equilibria Player B aB1

aB2 A1 is the prob. A chooses action aA1. B1 is the prob. B 6,4 3,5 aA1 chooses action aB1. Player A A Given , what A 1 4,3 5,7 a 2 value of B1 is best A EVB(aB1) = 4A1 + 3(1 - A1) = 3 for + B?

1. 67 Best Responses & Nash Equilibria Player B aB1 aB2 A1 is the prob. A chooses action aA1. B1 is the prob. B 6,4 3,5 aA1 chooses action aB1. Player A A Given , what A 1 4,3

5,7 a 2 value of B1 is best A EVB(aB1) = 4A1 + 3(1 - A1) = 3 for + B? 1. EVB(aB2) = 5A1 + 7(1 - A1) = 7 - 2A1. 68 Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given A1, what value of B1 is best for B? EVB(aB1) = 3 + A1. EVB(aB2) = 7 - 2A1. > > = 3 + A1< 7 - 2A1 as A= ?? < 1

69 Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given A1, what value of B1 is best for B? EVB(aB1) = 3 + A1. EVB(aB2) = 7 - 2A1. 3 + A1 < 7 - 2A1 for all 0 A1 1. 70 Best Responses & Nash Equilibria A1 is the prob. A chooses action aA1. B1 is the prob. B chooses action aB1. Given B1, what value of A1 is best for A? EVB(aB1) = 3 + A1. EVB(aB2) = 7 - 2A1. 3 + A1 < 7 - 2A1 for all 0 A1 1. Bs best response is: aB2 always (i.e. B1 = 0 always). 71 Best Responses & Nash Equilibria

Bs best response is aB2 always (i.e. B1 = 0 always). A1 1 0 This is Bs best response curve when players are allowed to mix over their actions. 0 Bs best 1 B1 72

Best Responses & Nash Equilibria B A As best response A1 A1 1 0 1 0 Bs best 1

B 0 1 0 1 B1 73 Best Responses & Nash Equilibria Is there a Nash equilibrium? B A As best response A1

A1 1 0 1 0 Bs best 1 B 0 1 0

1 B1 74 Best Responses & Nash Equilibria Is there a Nash equilibrium? As best response A1 1 0 0 Bs best 1

B1 75 Best Responses & Nash Equilibria Is there a Nash equilibrium? Yes. Just one. (A1, B1) = (0,0); i.e. A chooses aA2 only As best & B chooses aB2 only. response A1 1 0 0 Bs best 1

B1 76 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,5 aA2 4,3 5,7 Lets change the game. 77

Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 3,5 aA2 4,3 5,7 Here is a new 22 game. 78

Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3 5,7 Here is a new 22 game. Again let A1 be the prob. that A chooses aA1 and let B1 be the prob. that B chooses aB1. What are the NE

of this game? Notice that Player B no longer has a strictly dominant action 79 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3 5,7 A1 is the prob. that A chooses aA1.

B1 is the prob. that B chooses aB1. EVA(aA1) = ?? EVA(aA2) = ?? 80 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3 5,7

A1 is the prob. that A chooses aA1. B1 is the prob. that B chooses aB1. EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1. EVA(aA2) = ?? 81 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3

5,7 A1 is the prob. that A chooses aA1. B1 is the prob. that B chooses aB1. EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1. EVA(aA2) = 4B1 + 5(1 - B1) = 5 - B1. 82 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2

4,3 5,7 A1 is the prob. that A chooses aA1. B1 is the prob. that B chooses aB1. EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1. EVA(aA2) = 4B1 + 5(1 - B1) = 5 - B1. > > B = B B = 3 + 3 1 <5 - 1 as 1 < . 83 Best Responses & Nash Equilibria EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1. EVA(aA2) = 4B1 + 5(1 - B1) = 5 - B1. >

> B = B B = 3 + 3 1 <5 - 1 as 1 < . As best response A1 1 0 0 1 B1 84 Best Responses & Nash Equilibria EVA(aA1) = 6B1 + 3(1 - B1) = 3 + 3B1. EVA(aA2) = 4B1 + 5(1 - B1) = 5 - B1.

> > B = B B = 3 + 3 1 <5 - 1 as 1 < . As best response A1 1 0 0 1 B1 85 Best Responses & Nash Equilibria Player B

aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3 5,7 A1 is the prob. that A chooses aA1. B1 is the prob. that B chooses aB1. EVB(aB1) = ?? EVB(aB2) = ?? 86

Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3 5,7 A1 is the prob. that A chooses aA1. B1 is the prob. that B chooses aB1. EVB(aB1) = 4A1 + 3(1 - A1) = 3 + A1. EVB(aB2) = ??

87 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3 5,7 A1 is the prob. that A chooses aA1. B1 is the prob. that B chooses aB1.

EVB(aB1) = 4A1 + 3(1 - A1) = 4 + A1. EVB(aB2) = A1 + 7(1 - A1) = 7 - 6A1. 88 Best Responses & Nash Equilibria Player B aB1 aB2 aA1 Player A 6,4 3,1 aA2 4,3 5,7 EVB(aB1) EVB(aB2) 3 + A1

A1 is the prob. that A chooses aA1. B1 is the prob. that B chooses aB1. = 4A1 + 3(1 - A1) = 3 + A1. = A1 + 7(1 - A1) = 7 - 6A1. > > 4/ A A = = <7 - 6 1 as 1 < 7 . 89 Best Responses & Nash Equilibria EVB(aB1) EVB(aB2) 3 + A1

= 4A1 + 3(1 - A1) = 3 + A1. = A1 + 7(1 - A1) = 7 - 6A1. > > 4/ A A = = <7 - 6 1 as 1 < 7 . A1 1 4/ 7 0 0 1 Bs best

B1 90 Best Responses & Nash Equilibria EVB(aB1) EVB(aB2) 3 + A1 = 4A1 + 3(1 - A1) = 3 + A1. = A1 + 7(1 - A1) = 7 - 6A1. > > 4/ A A = = <7 - 6 1 as 1 < 7 . A1 1 4/ 7

0 0 1 Bs best B1 91 Best Responses & Nash Equilibria B A As best response A1 A1 1 1

4/ 7 0 0 1 Bs best B 0 1 0 1

B1 92 Best Responses & Nash Equilibria Is there a Nash equilibrium? B A As best response A1 A1 1 1 4/ 7 0 0

1 Bs best B 0 1 0 1 B1 93 Best Responses & Nash Equilibria Is there a Nash equilibrium? As best

response A1 1 4/ 7 0 0 Bs best 1 B1 94 Best Responses & Nash Equilibria Is there a Nash equilibrium? Yes. 3 of them. As best response

A1 1 4/ 7 0 0 Bs best 1 B1 95 Best Responses & Nash Equilibria Is there a Nash equilibrium? Yes. 3 of them. (A1, B1) = (0,0) As best response A1

1 4/ 7 0 0 Bs best 1 B1 96 Best Responses & Nash Equilibria IsIsthere thereaaNash Nashequilibrium? equilibrium? Yes. 3 of them. (A1, B1) = (0,0) (A1, B1) = (1,1)

As best response A1 1 4/ 7 0 0 Bs best 1 B1 97 Best Responses & Nash Equilibria IsIsthere thereaaNash

Nashequilibrium? equilibrium? Yes. 3 of them. (A1, B1) = (0,0) (A1, B1) = (1,1) As best response A1 4/ 7 ( , 1) = ( A 1 B , ) 1 4/ 7

0 0 Bs best 1 B1 98 Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment

Bargaining games 99 Coordination Games Simultaneous play games in which the payoffs to the players are largest when they coordinate their actions. Famous examples are: The Battle of the Sexes Game The Prisoners Dilemma Game Assurance Games Chicken 100 Coordination Games; The Battle of the Sexes Sissy prefers watching ballet to watching mud wrestling. Jock prefers watching mud wrestling to watching ballet. Both prefer watching something together to being apart.

101 Coordination Games; The Battle of the Sexes Jock B MW B 8,4 1,2 MW 2,1 4,8 SB is the prob. that Sissy chooses ballet. JB is the prob. that

Jock chooses ballet. Sissy 102 Coordination Games; The Battle of the Sexes Jock B MW B 8,4 1,2 MW 2,1 4,8

Sissy SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. What are the players best-response functions? 103 Coordination Games; The Battle of the Sexes Jock B MW B 8,4

1,2 MW 2,1 4,8 Sissy SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. What are the players best-response functions? EVS(B) = 8JB + (1 - JB) = 1 + 7JB. 104 Coordination Games; The Battle of the Sexes

Jock B MW B 8,4 1,2 MW 2,1 4,8 Sissy SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. What are the players

best-response functions? EVS(B) = 8JB + (1 - JB) = 1 + 7JB. EVS(MW) = 2JB + 4(1 - JB) = 4 - 2JB. 105 Coordination Games; The Battle of the Sexes Jock B MW B 8,4 1,2 MW 2,1

4,8 Sissy SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. What are the players best-response functions? EVS(B) = 8JB + (1 - JB) = 1 + 7JB. EVS(MW) = 2JB + 4(1 - JB) = 4 - 2JB. > > 1/3. J J J = 1 + 7 B = 4 2

as B B < < 106 Coordination Games; The Battle of the Sexes Jock B B MW 8,4 1,2 Sissy MW

2,1 4,8 SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. SB 1 EVS(B) = 8JB + (1 - JB) = 1 + 7JB. EVS(MW) = 2JB + 4(1 - JB) = 4 - 2JB. > > 1/3. J J J = 1 + 7 B = 4

2 as B B < 0 < 0 1/3 1 J 107 B Coordination Games; The Battle of the Sexes Jock B B

MW 8,4 1,2 Sissy MW 2,1 4,8 SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. SB Sissy 1 EVS(B) = 8JB + (1 - JB) = 1 + 7JB. EVS(MW) = 2JB + 4(1 - JB) = 4 - 2JB.

> > 1/3. J J J = 1 + 7 B = 4 2 as B B < 0 < 0 1/3 1 J 108

B Coordination Games; The Battle of the Sexes SB SB Sissy 1 Jock 1 1/3 0 0 1/3 1

JB 0 0 1 JB 109 Coordination Games; The Battle of the Sexes The games NE are ?? SB SB Sissy 1 Jock

1 1/3 0 0 1/3 1 JB 0 0 1 JB 110 Coordination Games; The Battle of the Sexes The games NE are ??

SB SB Sissy 1 Jock 1 1/3 0 0 1/3 1 JB 0

0 1 JB 111 Coordination Games; The Battle of the Sexes The games NE are ?? Sissy SB 1 1/3 0 0 1/3 Jock

1 JB 112 Coordination Games; The Battle of the Sexes The games NE are: (JB, SB) = (0, 0); i.e., (MW, MW) Sissy SB 1 1/3 0 0 1/3 Jock 1

JB 113 Coordination Games; The Battle of the Sexes The games NE are: (JB, SB) = (0, 0); i.e., (MW, MW) (JB, SB) = (1, 1); i.e., (B, B) Sissy SB 1 1/3 0 0 1/3 Jock 1

JB 114 Coordination Games; The Battle of the Sexes The games NE are: (JB, SB) = (0, 0); i.e., (MW, MW) (JB, SB) = (1, 1); i.e., (B, B) 1/,3 1/); J S 3 i.e., both ( , ) = ( Sissy B B SB watch the ballet with prob. 1/9, both watch the mud wrestling with prob. 4/9, and with prob. 4/9 they

1 watch different events. 1/3 0 0 1/3 Jock 1 JB 115 Coordination Games; The Battle of the Sexes Jock B MW B

8,4 1,2 MW 2,1 4,8 SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. Sissy or Sissy the expected value of the NE (JB, SB) =1/(3 1/3 , 2+ 2/+ / 1 / 2

8 1+ 9 9 9 4 4/9= 10/3< 4 and 8. ) is 116 Coordination Games; The Battle of the Sexes Jock B MW B 8,4 1,2 MW

2,1 4,8 SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. Sissy or Sissy the expected value of the NE (JB, SB) =1/(3 1/3 , ) is 2/+ / 1 8 1+ 9 9 2 2/+ 9 4 4/9= 10/3< 4 and 8. For Jock the expected value of the NE (JB, SB) =1/(3 1/,3 ) is 2/+ / 2 4 1+

+ 8 4/9= 14/3; 4 < 14/3< 8. 117 9 9 1 2/9 Coordination Games; The Battle of the Sexes Jock B MW B 8,4 1,2 MW 2,1 4,8

Sissy SB is the prob. that Sissy chooses ballet. JB is the prob. that Jock chooses ballet. So, is the mixed strategy NE a focal point for the game? or Sissy the expected value of the NE (JB, SB) =1/(3 1/3 , ) is 2+ 2/+ / 1 / 2 8 1+ 9 9 9 4 4/9= 10/3< 4 and 8. For Jock the expected value of the NE (JB, SB) =1/(3 1/,3 ) is 2+ / 2

/ 1 2/+ 8 4/ = 14/ ; 4 < 14/ < 8. 4 1+ 118 9 9 9 9 3 3 Coordination Games; Assurance Games A simultaneous play game with two coordinated NE, one of which gives strictly greater payoffs to each player than does the other. The question is: How can each player give the other an assurance that will cause the better NE to be the outcome of the game? 119 Coordination Games; Assurance Games

A common example is the arms race problem. India and Pakistan can both increase their stockpiles of nuclear weapons. This is very costly. 120 Coordination Games; Assurance Games Having nuclear superiority over the other gives a higher payoff, but the worst payoff to the other. Not increasing the stockpile is best for both. 121 Coordination Games; Assurance Games Pakistan Dont Stockpile Dont 5,5 1,4

4,1 3,3 India Stockpile 122 Coordination Games; Assurance Games Pakistan Dont Stockpile Dont 5,5 1,4 4,1 3,3

India Stockpile The games NE are ?? 123 Coordination Games; Assurance Games Pakistan Dont Stockpile Dont 5,5 1,4 4,1 3,3 India Stockpile

he games NE are (Dont, Dont) and (Stockpile, Stockpile). Which is the likely NE? 124 Coordination Games; Assurance Games Pakistan Dont Stockpile Dont 5,5 1,4 4,1 3,3 India Stockpile he games NE are (Dont, Dont) and (Stockpile, Stockpile). Which is the likely NE? What if India moved first? What ction would it choose? Wouldnt Dont be best?

125 Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment Bargaining games 126 Games of Competition Simultaneous play games in which any increase in the payoff to one player is exactly the decrease in the payoff to the other player. These games are thus often called constant (payoff) sum games.

127 Games of Competition An example is the game below. What NE can such a game possess? 2 L R U 0,0 2,-2 D x,-x 1,-1 1

128 Games of Competition An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down. 2 If x < 1 then Left dominates L R Right. Therefore, if x < 0 the NE is (Up, Left) and if 0 < x < 1 0,0 2,-2 U the NE is (Down, Left). 1 If x > 1 then there is no NE in pure strategies. Is there 1,-1

x,-x D a mixed-strategy NE? 129 Games of Competition The probability that 2 chooses Left is L. The probability that 1 chooses Up is U. x > 1. 2 L R U 0,0 2,-2 D

x,-x 1,-1 1 130 Games of Competition The probability that 2 chooses Left is L. The probability that 1 chooses Up is U. x > 1. EV1(U) = 2(1 - L). EV1(D) = xL + 1 - L. 2 L R U

0,0 2,-2 D x,-x 1,-1 1 131 Games of Competition The probability that 2 chooses Left is L. The probability that 1 chooses Up is U. x > 1. 2 L R

U 0,0 2,-2 D x,-x 1,-1 EV1(U) = 2(1 - L). EV1(D) = xL + 1 - L. > 2 - 2l = < 1 + (x - 1)L > as L = <1/(1 + x). 1

132 Games of Competition The probability that 2 chooses Left is L. The probability that 1 chooses Up is U. x > 1. 2 L R U 0,0 2,-2 D x,-x

1,-1 1 EV1(U) = 2(1 - L). EV1(D) = xL + 1 - L. > 2 - 2l = < 1 + (x - 1)L > as L = <1/(1 + x). EV2(L) = - x(1 - U). EV2(R) = - 2U - (1 - U). 133 Games of Competition The probability that 2 chooses Left is L. The probability that 1 chooses Up is U. x > 1.

2 L U 0,0 R 2,-2 1 D x,-x 1,-1 EV1(U) = 2(1 - L). EV1(D) = xL + 1 - L. > 2 - 2l = < 1 + (x - 1)L >

as L = <1/(1 + x). EV2(L) = - x(1 - U). EV2(R) = - 2U - (1 - U). > - x + xU = < - 1 - U > as (x 1)/(1 + x) = < U. 134 Games of Competition 1 chooses Up if L > 1/(1 + x) and Down if L < 1/(1 + x). 2 chooses Left if U < (x 1)/(1 + x) and Right if U > (x 1)/(1 + x). 2 L

R U 0,0 2,-2 D x,-x 1,-1 1 135 Games of Competition 1: Up if L > 1/(1 + x); Down if L < 1/(1 + x). 2: Left if U < (x 1)/(1 + x); Right if U > (x 1)/(1 + x). U

L 1 2 1 1 0 0 L 0 0 1 U 136 Games of Competition 1: Up if L > 1/(1 + x); Down if L < 1/(1 + x). 2: Left if U < (x 1)/(1 + x); Right if U > (x 1)/(1 + x). U L

1 2 1 1 0 0 1/(1+x) L 0 0 (x-1)/(1+x) 1 U 137 Games of Competition 1: Up if L > 1/(1 + x); Down if L < 1/(1 + x). 2: Left if U < (x 1)/(1 + x); Right if U > (x 1)/(1 + x). U

U 1 2 1 1 (x-1)/(1+x) 0 0 1/(1+x) L 0 0 1 L 138 Games of Competition 1: Up if L > 1/(1 + x); Down if L < 1/(1 + x). 2: Left if U < (x 1)/(1 + x); Right if U > (x 1)/(1 + x). U

1 When x > 1 there is only a mixed-strategy NE in which 1 plays Up with (x-1)/(1+x) probability (x 1)/(x + 1) and 2 plays Left with probability 1/(1 + x). 0 0 1/(1+x) 1 L 139 Some Important Types of Games Games of coordination Games of competition Games of coexistence

Games of commitment Bargaining games 140 Coexistence Games Simultaneous play games that can be used to model how members of a species act towards each other. An important example is the hawk-dove game. 141 Coexistence Games; The HawkDove Game Hawk means be aggressive. Dove means dont be aggressive. Two bears come to a fishing spot. Either bear can fight the other to try to drive it away to get more fish for itself but suffer battle injuries, or it can tolerate the presence of the other, share the fishing, and avoid injury. 142

Coexistence Games; The HawkDove Game Bear 2 Hawk Dove Hawk Bear 1 Dove -5,-5 8,0 0,8 4,4 Are there NE in pure strategies? 143 Coexistence Games; The HawkDove Game Bear 2 Hawk Dove

Hawk Bear 1 Dove -5,-5 8,0 0,8 4,4 Are there NE in pure strategies? Yes (Hawk, Dove) and (Dove, Hawk). Notice that purely peaceful coexistence is not a NE. 144 Coexistence Games; The HawkDove Game Bear 2 Hawk Dove Hawk Bear 1 Dove

-5,-5 8,0 0,8 4,4 Is there a NE in mixed strategies? 145 Coexistence Games; The HawkDove Game Bear 2 Hawk Dove Hawk Bear 1 Dove -5,-5 8,0

0,8 4,4 1H is the prob. that 1 chooses Hawk. 2H is the prob. that 2 chooses Hawk. What are the players best-response functions? Is there a NE in mixed strategies? 146 Coexistence Games; The HawkDove Game Bear 2 Hawk Dove Hawk Bear 1 Dove

-5,-5 8,0 0,8 4,4 1H is the prob. that 1 chooses Hawk. 2H is the prob. that 2 chooses Hawk. What are the players best-response functions? EV1(H) = -52H + 8(1 - 2H) = 8 - 132H. EV1(D) = 4 - 42H. 147 Coexistence Games; The HawkDove Game Bear 2 Hawk

Dove Hawk Bear 1 Dove EV1(H) = EV1(D) = 8 - 132H -5,-5 8,0 0,8 4,4 1H is the prob. that 1 chooses Hawk. 2H is the prob. that 2 chooses Hawk. What are the players best-response functions?

-52H + 8(1 - 2H) = 8 - 132H. 4 - 42H. > < 2 2 = <4 - 4 H as H = > 4/9. 148 Coexistence Games; The HawkDove Game Bear 2 Hawk Dove Hawk Bear 1 Dove EV1(H) = EV1(D) = 8 - 132H -5,-5 0,8

8,0 4,4 1H is the prob. that 1 chooses Hawk. 2H is the prob. that 2 chooses Hawk. 1 H Bear 1 1 -52H + 8(1 - 2H) = 8 - 132H. 4 - 42H. > < 2 2 = 0 <4 - 4 H as H = > 4/9.

0 4/9 1 2 149 H Coexistence Games; The HawkDove Game 1H Bear 1 1H 1 Bear 2 1

4/9 0 0 4/9 1 2H 0 0 1 2H 150 Coexistence Games; The HawkDove Game The game has a NE in mixed-strategies in which each bear plays Hawk with probability 4/9. 1H 1

0 0 4/9 1 2H 151 Coexistence Games; The HawkDove Game Bear 2 Hawk Dove Hawk Bear 1 Dove -5,-5 8,0 0,8

4,4 or each bear, the expected value of the mixed-strategy NE is / 8 25/81 /, a value between 20+ / 4 5) 16+ = 18081 81 81 5 and +4. 152 Some Important Types of Games Games of coordination

Games of competition Games of coexistence Games of commitment Bargaining games 153 Commitment Games Sequential play games in which One player chooses an action before the other player chooses an action. The first players action is both irreversible and observable by the second player. The first player knows that his action is seen by the second player. Price Analysis II 2018 Lecture 5 Slide 154 Commitment Games c 5,9

2 Game Tree a d 1 5,5 Direction of play e b 7,6 2 f 5,4

Price Analysis II 2018 Lecture 5 Player 1 has two actions, a and b. Player 2 has two actions, c and d, following a, and two actions e and f following b. Player 1 chooses his action before Player 2 chooses 155 Commitment Games c 2 a d 5,9 Is a claim by Player 2 that she will commit to choosing action c if

Player 1 chooses a credible to Player 1? 5,5 Yes. 1 b 2 7,6 Is a claim by Player 2 e that she will commit to choosing action e if Player 1 chooses b credible to Player 1? f 156 Price Analysis II 20185,4 Lecture 5 Yes. Commitment Games c 5,9 So Player 1 should choose action b.

d 5,5 2 a 1 e b 7,6 2 f 5,4 Price Analysis II 2018 Lecture 5 157 Commitment Games

c 5,3 Change the game. 2 a d 5,5 1 e b 7,6 2 f 5,4 Price Analysis II 2018 Lecture 5

158 Commitment Games 2 a 1 b 2 5,3 Is a claim by Player 2 c that she will commit to choosing action c if Player 1 chooses a credible to Player 1? d 5,5 No. If Player 1 chooses 7,6 action a then Player 2 e does best by choosing action d. What should Player 1

do? Still choose b. f 159 Price Analysis II 20185,4 Lecture 5 Commitment Games c 5,9 Change the game. d 15,5 2 a 1 e b 7,3

2 f 5,12 Price Analysis II 2018 Lecture 5 160 Commitment Games c 2 a 1 b 2 5,9 Can Player 1 get 15 points? If Player 1 chooses a then Player 2 will d 15,5 choose c and Player 1

will get only 5 points. 7,3 e If Player 1 chooses b then Player 2 will choose f and again Player 1 will get only f 161 Price Analysis II 20185,12 Lecture 5 5 points. Commitment Games c 2 a d 1 e b 5,9 If Player 1 can change payoffs so that a

commitment by Player 2 to choose d after a is credible then Player 15,5 1s payoff rises from 5 to 15, a gain of 10. 7,3 2 f 5,12 Price Analysis II 2018 Lecture 5 162 Commitment Games 2 a 1 b 2

5,9 If Player 1 can change c payoffs so that a commitment by Player 2 to choose d after a is credible then Player d 10,101s payoff rises from 5 to 15, a gain of 10. 7,3 If Player 1 gives 5 of e these points to Player 2 then Player 2s commitment is credible. Player 1 f 163 Price Analysis II 20185,12 Lecture 5 cannot get 15 points. Commitment Games c 2 a

d 1 e b 5,9 Credible NE of this type are called subgame perfect. What exactly is this games SPE? It 10,10insists that every action chosen is 7,3 rational for the player who chooses it. 2 f 5,12 Price Analysis II 2018 Lecture 5 164

Subgame perfect equilibrium In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that if (1) the players played any smaller game that consisted of only one part of the larger game and (2) their behavior represents a Nash equilibrium of that smaller game, then their behavior is a subgame perfect equilibrium of the larger game. Price Analysis II 2018 Lecture 5 165 Commitment Games e 5,9 Credible NE of this type are called subgame perfect.

What exactly is this games SPE? It 10,10insists that every action chosen is 7,3 rational for the player who chooses it. f 5,12 c 2 a d 1 b 2 Price Analysis II 2018 Lecture 5 166 Commitment Games

c 2 a d 1 e b 5,9 Credible NE of this type are called subgame perfect. What exactly is this games SPE? It 10,10insists that every action chosen is 7,3 rational for the player who chooses it. 2 f 5,12

Price Analysis II 2018 Lecture 5 167 Commitment Games c 2 a d 1 e b 5,9 Credible NE of this type are called subgame perfect. What exactly is this games SPE? It 10,10insists that every action chosen is 7,3 rational for the player who chooses it.

2 f 5,12 Price Analysis II 2018 Lecture 5 168 More example 1 Price Analysis II 2018 Lecture 5 169 More example 1 Price Analysis II 2018 Lecture 5 170 More example 2 Price Analysis II 2018 Lecture 5

171 More example 3 Price Analysis II 2018 Lecture 5 172 Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment Bargaining games Price Analysis II 2018 Lecture 5

Slide 173 Bargaining Games Two players bargain over the division of a pie of size 1. What will be the outcome? Two approaches: Nashs axiomatic bargaining. Rubinsteins strategic bargaining. Price Analysis II 2018 Lecture 5 Slide 174 Strategic Bargaining The players have 3 periods in which to decide how to divide the pie; else both get nothing. Player A discounts next periods payoffs by . Player B discounts next periods payoffs by . The players alternate in making offers, with Player A starting in period 1. If the player who receives an offer accepts it then the game ends immediately. Else the game continues to the next period.

Price Analysis II 2018 Lecture 5 Slide 175 Strategic Bargaining Period 2: B offers x2. A responds. 1 (x1,1-x1) 1 Y (x3,1-x3) 1 x3 x1 A B

N 0 Period 1: A offers x1. B responds. B x2 A 0 N Y (x2,1-x2) Price Analysis II 2018 Lecture 5 A B Y

N (0,0) 0 Period 3: A offers x3. B responds. 176 Strategic Bargaining How should B respond to x3? (x3,1-x3) 1 x3 A Price Analysis II 2018 Lecture 5 B Y N

(0,0) 0 Period 3: A offers x3. B responds. 177 Strategic Bargaining How should B respond to x3? Accept if 1 x3 0; i.e., accept any x3 1. (x3,1-x3) 1 x3 A Price Analysis II 2018 Lecture 5 B

Y N (0,0) 0 Period 3: A offers x3. B responds. 178 Strategic Bargaining How should B respond to x3? Accept if 1 x3 0; i.e., accept any x3 1. (x3,1-x3) 1 Knowing this, what should A offer? x3

A Price Analysis II 2018 Lecture 5 B Y N (0,0) 0 Period 3: A offers x3. B responds. 179 Strategic Bargaining How should B respond to x3? Accept if 1 x3 0; i.e., accept any x3 1. (1,0)

x3=1 Knowing this, what should A offer? x3 = 1. B N A Price Analysis II 2018 Lecture 5 Y (0,0) 0 Period 3: A offers x3 = 1. B accepts. 180 Strategic Bargaining Period 2: B offers x2.

A responds. 1 (x1,1-x1) 1 Y (1,0) x3=1 x1 A B Y B N N 0 Period 1:

A offers x1. B responds. B x2 A 0 N Y (x2,1-x2) Price Analysis II 2018 Lecture 5 A (0,0) 0 Period 3: A offers x3 = 1. B accepts. 181

Strategic Bargaining In Period 3 A gets a payoff of 1. In period 2, when replying to Bs offer of x2, the present-value to A of N is thus ?? Period 2: B offers x2. A responds. 1 (1,0) x3=1 Y B N B x2 A

A (0,0) N 0 Y (x2,1-x2) Price Analysis II 2018 Lecture 5 0 Period 3: A offers x3 = 1. B accepts. 182 Strategic Bargaining In Period 3 A gets a payoff of 1. In period 2, when replying to Bs offer of x2, the

present-value to A of N is thus . Period 2: B offers x2. A responds. 1 (1,0) x3=1 Y B N B x2 A A (0,0) N

0 Y (x2,1-x2) Price Analysis II 2018 Lecture 5 0 Period 3: A offers x3 = 1. B accepts. 183 Strategic Bargaining In Period 3 A gets a payoff of 1. In period 2, when replying to Bs offer of x2, the present-value to A of N is thus . What is the most B should offer to A?

Period 2: B offers x2. A responds. 1 B x2 A N 0 Y (x2,1-x2) Price Analysis II 2018 Lecture 5 184 Strategic Bargaining In Period 3 A gets a payoff of 1. In period 2, when replying to Bs offer of x2, the

present-value to A of N is thus . What is the most B should offer to A? x2 = . Period 2: B offers x2 = . A accepts. 1 B x2= A N 0 Y (,1- ) Price Analysis II 2018 Lecture 5

185 Strategic Bargaining Period 2: B offers x2 = . A accepts. 1 (x1,1-x1) 1 Y (1,0) x3=1 x1 A B Y B

N N 0 Period 1: A offers x1. B responds. B x2= A 0 N Y (,1- ) Price Analysis II 2018 Lecture 5 A

(0,0) 0 Period 3: A offers x3 = 1. B accepts. 186 Strategic Bargaining Period 2: B offers x2 = A accepts. 1 (x1,1-x1) 1 Y x1 A B

N 0 Period 1: A offers x1. B responds. B x2= A 0 In period 2 A will accept .. Thus B will get the payoff 1 - in period 2. What is the presentvalue to B in period 1 of N? N Y

(,1- ) Price Analysis II 2018 Lecture 5 187 Strategic Bargaining Period 2: B offers x2 = A accepts. 1 (x1,1-x1) 1 Y x1 A B N 0

Period 1: A offers x1. B responds. B x2= A 0 In period 2 A will accept .. Thus B will get the payoff 1 - in period 2. What is the presentvalue to B in period 1 of N ? (1 - ). N Y (,1- ) Price Analysis II 2018 Lecture 5

188 Strategic Bargaining Period 2: B offers x2 = A accepts. 1 (x1,1-x1) 1 Y x1 A B N 0 Period 1: A offers x1. B responds.

B x2= A 0 In period 2 A will accept .. Thus B will get the payoff 1 - in period 2. What is the presentvalue to B in period 1 of N ? (1 - ). N What is the most that A should offer to B in period 1? Y (,1- )

Price Analysis II 2018 Lecture 5 189 Strategic Bargaining Period 2: B offers x2 = A accepts. 1 (1-(1 - ), (1 - )) 1 Y x1 A B N 0

Period 1: A offers x1. B responds. B x2= In period 2 A will accept .. Thus B will get the payoff 1 - in period 2. What is the presentvalue to B in period 1 of N ? (1 - ). What is the most that A N should offer to B in period 1? 1 x1 = (1 - ); i.e. Y 0 x1 = 1 - (1 - ). (,1- ) B will accept. A

Price Analysis II 2018 Lecture 5 190 Strategic Bargaining Period 2: B offers x2 = . A accepts. 1 (1-(1 - ), (1 - )) x1=1- 1 (1-) A B Y (1,0) x3=1 Y

B N N 0 Period 1: A offers x1 = 1-(1 - ). B accepts. B x2= A 0 N Y (,1- ) Price Analysis II 2018 Lecture 5

A (0,0) 0 Period 3: A offers x3 = 1. B accepts. 191 Strategic Bargaining Notice that the game ends immediately, in period 1. Price Analysis II 2018 Lecture 5 Slide 192 Strategic Bargaining Alice gets 1 - (1 ) units of the pie. Bob gets (1 ) units. Which is the larger? x1 = 1 - (1 ) 1/2(1 - ) so Player A gets more than Player B if Player B is too impatient relative to Player A.

Price Analysis II 2018 Lecture 5 Slide 193 Strategic Bargaining Suppose the game is allowed to continue forever (infinitely many periods). Then using the same reasoning shows that the subgame perfect equilibrium results in Players 1 and 2 1 (1 ) respectively getting 1 and 1 pie units. Price Analysis II 2018 Lecture 5 194 Price Analysis II 2018 Lecture 5 195 Strategic Bargaining Player 1s share rises as and Player 2s share rises as and

Price Analysis II 2018 Lecture 5 196 Preface free course Online course Game Theory at the Stanford U. https://www.coursera.org/gametheory/auth/welcome FAQs: When does the class start? March 19th, 2012 (but the material will be pre-released by the weekend of March 10th). Will the text of the lectures be available? We hope to transcribe the lectures into text to make them more accessible for those not fluent in English. Stay tuned. Do I need to watch the lectures live? No. You can watch the lectures at your leisure. Can online students ask questions and/or contact the professor? Yes, but not directly. There is a

Q&A forum in which students rank questions and answers, so that the most important questions and the best answers bubble to the top. Teaching staff will monitor these forums, so that important questions not answered by other students can be addressed. Will other Stanford resources be available to online students? No. How much does it cost to take the course? Nothing: it's free! Will I get university credit for taking this course? No. Price Analysis II 2018 Lecture 5 Slide 197