Chapters 2 and 3: Common Belief in Rationality Andrés Perea Maastricht University Period 5, 2012/2013 Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 1 / 78
What is game theory about? In game theory, we study situations where you must make a choice, but where the …nal outcome also depends on the choices of others. Examples are everywhere: Negotiating about the price of a car, choosing a marketing strategy for your …rm, bidding in an auction, discussing with your partner about what TV program to watch this evening. Key question: What choice would you make, and why? This depends crucially on how you reason about the opponent! Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 2 / 78
Example: Where to locate my pub? g a c e b d f v v v v v v v 100 100 100 100 100 100 Story In a village called Longstreet, you and your opponent must both choose a location for your pub. Possible locations: a , b , c , d , e , f , g . Between every two locations, there are 100 thirsty men living. All at equal distance from each other. Every man will visit the pub that is nearest to his house. You wish to attract as many customers as possible. What location would you choose, and why? Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 3 / 78
g a c e b d f v v v v v v v 100 100 100 100 100 100 Which location is optimal for you, depends on your belief about the opponent’s location: If you believe your opponent chooses a , then location b is optimal for you. If you believe your opponent chooses b , then location c is optimal for you. If you believe your opponent chooses c , d or e , then location d is optimal for you. If you believe your opponent chooses f , then location e is optimal for you. If you believe your opponent chooses g , then location f is optimal for you. Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 4 / 78
Beliefs diagram Opponent’s choices Your choices � 1 a �������� a � 1 �������� b b c c - d d PPPPPPPP e e P q PPPPPPPP f f P q g g Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 5 / 78
g a c e b d f v v v v v v v 100 100 100 100 100 100 We call b , c , d , e and f rational choices for you, since they are all optimal for some belief about the opponent’s choice. Location a can never be optimal for you: Whatever location your opponent chooses, choosing b is always better for you! We say that your choice a is strictly dominated by your choice b . Similarly, your choice g is strictly dominated by your choice f . Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 6 / 78
g a c e b d f v v v v v v v 100 100 100 100 100 100 Consider your choice b . Choosing b is optimal if you believe that your opponent chooses a . But choosing a is irrational for your opponent. So, this belief is unreasonable! If you take your opponent seriously, then you must believe that your opponent chooses rationally too. So, you must believe that your opponent does not choose a or g . But then, choosing b can no longer be optimal for you, as choosing c is always better! Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 7 / 78
g a c e b d f v v v v v v v 100 100 100 100 100 100 So, if you believe that the opponent chooses rationally, then you believe that he will not choose a or g , and then b is no longer optimal for you. Similarly, if you believe that the opponent chooses rationally, then choosing f is no longer optimal for you, as e would always be better. Hence, the choices b and f are rational for you, but unreasonable. Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 8 / 78
g a c e b d f v v v v v v v 100 100 100 100 100 100 On the other hand, c , d and e can still be optimal if you believe that the opponent chooses rationally. Choosing c is optimal for you if you believe that the opponent rationally chooses b . Choosing d is optimal for you if you believe that the opponent rationally chooses d . Choosing e is optimal for you if you believe that the opponent rationally chooses f . Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 9 / 78
Beliefs diagram Opponent You You a a a �������� 1 �������� 1 b �������� 1 b �������� 1 b c c c - - d d d PPPPPPPP PPPPPPPP e e e q q PPPPPPPP PPPPPPPP f f f q q g g g Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 10 / 78
g a c e b d f v v v v v v v 100 100 100 100 100 100 But are all of the choices c , d and e reasonable ? No! If you believe that the opponent chooses rationally, it seems reasonable to believe that the opponent believes that you choose rationally! If the opponent believes that you choose rationally, then he could rationally choose c , d and e , but not b and f . So, to believe that the opponent chooses b or f is unreasonable. But if you believe that the opponent will only choose c , d or e , you must choose d ! Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 11 / 78
Opponent You You a a a 1 � 1 � ����� ����� 1 � 1 � b b b ����� ����� c c c - - d d d PPPPP PPPPP e e e q P P q PPPPP PPPPP f f f P q P q g g g So, location d is the only candidate for a reasonable choice. But is d really reasonable? Yes! Consider, namely, the belief hierarchy that starts at your choice d . Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 12 / 78
In this belief hierarchy: You believe that the opponent chooses d . You believe that the opponent believes that you choose d . You believe that the opponent believes that you believe that the opponent chooses d . And so on. It consists of a …rst-order belief, a second-order belief, a third-order belief, and so on. If you hold this belief hierarchy, then you believe that the opponent chooses rationally, you believe that the opponent believes that you choose rationally, you believe that the opponent believes that you believe that the opponent chooses rationally, and so on. We say that you express common belief in rationality. Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 13 / 78
g a c e b d f v v v v v v v 100 100 100 100 100 100 Summarizing Your choices a and g are irrational. Your choices b and f are rational, but can no longer be optimal if you believe that the opponent chooses rationally. Your choices c and e can be optimal if you believe that the opponent chooses rationally, but can no longer be optimal if you believe, in addition, that the opponent believes that you choose rationally. You can rationally choose d if you express common belief in rationality. In particular, under common belief in rationality there is only one choice you can rationally make, namely d ! Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 14 / 78
Example: Going to a party blue green red yellow same color as Barbara 4 3 2 1 0 Story This evening, you are going to a party together with your friend Barbara. You must both decide which color to wear: blue, green, red or yellow. Your preferences for wearing these colors are as in the table. These numbers are called utilities. You hate wearing the same color as Barbara: If you both would wear the same color, your utility would be 0. What color should you choose, and why? Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 15 / 78
blue green red yellow same color as Barbara 4 3 2 1 0 Again, what color is optimal for you depends on your belief about Barbara’s choice: If you believe that Barbara wears blue, then green is optimal for you. If you believe that Barbara wears green, then blue is optimal for you. If you believe that Barbara wears red, then blue is optimal for you. If you believe that Barbara wears yellow, then blue is optimal for you. Does this mean that red and yellow are irrational for you? Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 16 / 78
blue green red yellow same color as Barbara 4 3 2 1 0 Suppose you believe that, with probability 0.6, Barbara chooses blue , and that, with probability 0.4, she chooses green . If you would choose blue, your expected utility would be ( 0 . 6 ) � 0 + ( 0 . 4 ) � 4 = 1 . 6 . If you would choose green, your expected utility would be ( 0 . 6 ) � 3 + ( 0 . 4 ) � 0 = 1 . 8 . If you would choose red, your utility would be 2. If you would choose yellow, your utility would be 1. So, choosing red is optimal for you if you hold this probabilistic belief about Barbara’s choice. In particular, red is a rational choice for you. Andrés Perea (Maastricht University) Chapters 2 and 3 Period 5, 2012/2013 17 / 78
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