ECON322 Game Theory Half II

Transcription

1 ECON322 Game Theory Half II Part 1: Reasoning Foundations Rationality Christian W. Bach University of Liverpool & EPICENTER

2 Agenda Introduction Rational Choice Strict Dominance Characterization of Rationality

3 ECON322 Half II: Program Half II Part 1: Reasoning Foundations Rationality Common Belief in Rationality Half II Part 2: Dynamic Games Modelling Dynamic Games Solution Concepts for Dynamic Games

4 ECON322 Half II: Organization Five lectures: 1 Friday, , 12h00-13h30, SCTH-MR 2 Friday, , 12h00-13h30, SCTH-MR 3 Friday, , 12h00-13h30, SCTH-MR 4 Friday, , 12h00-13h30, SCTH-MR 5 Monday, , 17h00-18h00, ULMS-SR1 Four tutorials: 1 Monday, , 12h00-13h00, ERB-HLT, & 17h00-18h00, ULMS-SR1 2 Monday, , 12h00-13h00, ERB-HLT, & 17h00-18h00, ULMS-SR1 3 Monday, , 12h00-13h00, ERB-HLT, & 17h00-18h00, ULMS-SR1 4 Friday, , 12h00-13h00, SCTH-MR

5 ECON322: Assessment Exam Two questions on Half I (by Y. Gu) Two questions on Half II (by C.W. Bach) Preparation for Half II Lecture Slides (download link: ) Exercises in the tutorials Reading of the book Borrowing at: Sydney Jones Library, Chatham Street Purchasing with special discount at: Blackwells University bookshop, 9 Brownlow Hill

6 ECON322 Half II: Required Reading Chapter 1: Introduction Chapter 2: Belief in the opponents rationality Chapter 3: Common belief in rationality Chapter 8: Belief in the opponents future rationality Chapter 9: Strong belief in the opponents rationality

7 Reasoning in Games In interactive situations ( games ) an agent must make a decision, while knowing that the outcome will not only depend on his choice, but also on the choice of other agents. Fundamental question: What choices are reasonable & why? The rather new discipline of epistemic game theory gives different possible answers to this question, i.e. develops different ways of reasoning. Heterogenity of agents & situations excludes the existence of a single universal concept for all agents & interactive situations.

8 Rationality as a Basic Concept Intuitively, an agent makes a choice, which he thinks will yield the best outcome for him. It is thus crucial what an agent believes his opponents to do. A choice is best for an agent, if it is optimal given his belief about his opponents choices. Accordingly, a choice is called rational for an agent, if it is optimal for some belief about his opponents choices. Hence, rationality is a very basic and weak concept. Rationality serves as the primitive, on which various reasoning concepts can be constructed.

9 Example I: Where to Locate My Pub? Story: Alice and Bob both want to open a new pub on Bold Street. Bold Street contains 600 houses, equally spaced. One person per house is assumed to visit the closest pub. There are seven possible locations for pubs: a, b, c, d, e, f, g. If Alice and Bob choose the same location, then each gets 300 clients. Question: Which location should Alice choose to maximize her benefits, i.e. the number of clients? It depends on the location Alice believes Bob to choose!

10 Example I: Where to Locate My Pub? Bold Street:

11 Example I: Where to Locate My Pub? Suppose Alice believes Bob to choose a. Then, b is optimal for Alice. Suppose Alice believes Bob to choose b. Then, c is optimal for Alice. Suppose Alice believes Bob to choose c. Then, d is optimal for Alice. Suppose Alice believes Bob to choose d. Then, d is optimal for Alice. Suppose Alice believes Bob to choose e. Then, d is optimal for Alice. Suppose Alice believes Bob to choose f. Then, e is optimal for Alice. Suppose Alice believes Bob to choose g. Then, f is optimal for Alice.

12 Example I: Where to Locate My Pub? The choices b, c, d, e, and f are therefore rational for Alice. What about the choices a and g? Whatever Alice believes Bob to choose, b is better than a, and f is better than g. The choices a and g are therefore irrational for Alice.

13 Example II: Going to a Party Story: Alice and Bob are going together to a party tonight. Alice asks herself what colour she should wear. Alice prefers blue to green, green to red, and red to yellow. However, Alice dislikes most to wear the same colour as Bob. Let Alice s utilities be given as follows: blue: 4 green: 3 red: 2 yellow: 1 same colour as Bob: 0 Question: Which colours can Alice rationally choose for tonight s party?

14 Example II: Going to a Party Blue is optimal for Alice, if she believes Bob to pick any other colour than blue. Green is optimal for Alice, if she believes Bob to pick blue. Red is optimal for Alice, if she believes that with probability 0.6 Bob chooses blue and with probability 0.4 Bob chooses green. (Given this belief Alice gets utility 1.6 from blue and 1.8 from green) The colours blue, green, and red are therefore rational for Alice.

15 Example II: Going to a Party What about the colour yellow? To see that there is actually no belief such that yellow is optimal for Alice distinguish two exhaustive cases. Case 1: Suppose Alice s belief assigns probability of less than 0.5 to Bob choosing blue. Then, Alice expects utility of at least 2 from blue, hence yellow is not optimal. Case 2: Suppose Alice s belief assigns probability of at least 0.5 to Bob choosing blue. Then, Alice expects utility of at least 1.5 from green, hence yellow is not optimal. Therefore, yellow is irrational for Alice.

16 Agenda Introduction Rational Choice Strict Dominance Characterization of Rationality

17 Games Definition The tuple Γ N = ( ) I, (S i ) i I, (U i ) i I is called normal-form, where I is a finite set of players, S i is a set of strategies for player i I, U i : j I S j R is a utility function for player i I.

18 Belief About the Opponents Choices Definition Let Γ N be a normal-form, and i be a player. A belief for player i about the opponents strategies is a probability distribution b i : S i [0, 1] over the set of opponents strategy-combinations S i = j I\{i} S j. Note that the set of all probability distributions on some set X is often denoted by (X) := {p [0, 1] X : x X p(x) = 1}.

19 Expected Utility Definition Let Γ N be a normal-form, and i be a player with utility function U i. Suppose that player i entertains belief b i and chooses s i. The expected utility for player i is u i (s i, b i ) := b i (s i ) U i (s i, s i ) s i S i where (s i, s i ) = (s 1,..., s n ) j I S j.

20 Optimality Definition Let Γ N be a normal-form, and i be a player with utility function U i. Suppose that player i entertains belief b i. A strategy s i for player i is optimal, if u i (s i, b i ) u i (s i, b i ) holds for all strategies s i S i of player i.

21 Rationality Definition Let Γ N be a normal-form, and i be a player with utility function U i. A strategy s i for player i is rational, if there exists a belief b i for player i about the opponents strategies such that s i is optimal.

22 Illustration Bob L R U 10, 5 0, 3 Alice M 0, 2 10, 2 D 7, 3 7, 1 All three choices for Alice are rational. U is optimal for Alice, if she believes Bob to choose L. M is optimal for Alice, if she believes Bob to choose R. D is optimal for Alice, if she believes with probability 0.5 Bob to choose L and with probability 0.5 Bob to choose R. (Note that if a choice is not optimal for any probability-1 belief, then it can still be optimal for some belief with supp > 1)

23 Agenda Introduction Rational Choice Strict Dominance Characterization of Rationality

24 Randomizing Definition Let Γ N be a normal-form, and i be a player. A mixed strategy for player i is a probability distribution σ i : S i [0, 1] over the set S i of player i s strategies. Remark: It seems unnatural that people randomize when taking serious decisions. We assume players to make definite decisions. However, mixed strategies are used as technical tools for identifying the rational (pure) strategies in games.

25 Randomizing A player cannot gain anything from randomizing, but he can lose something. For an illustration recall Example 2 and consider the randomized choice σ Alice, where σ Alice (blue) = 0.5 and σ Alice (green) = 0.5. Suppose Alice believes with probability 0.25 Bob to choose blue and with probability 0.75 Bob to choose red. Then, Alice expects utility = 3 from blue, = 3 from green, and = 3 from σ Alice, i.e. Alice cannot gain from randomizing. Suppose Alice believes with probability 0.4 Bob to choose blue and with probability 0.6 Bob to choose red. Then, Alice expects utility = 2.4 from blue, = 3 from green, and = 2.7 from σ Alice, i.e. Alice is worse off randomizing. Randomizing is troublesome, since additional effort needs to be taken (e.g. tossing a coin, checking and remembering the outcome).

26 Utility with Randomizing Definition Let Γ N be a normal-form, and i be a player with utility function U i. Suppose that player i chooses σ i, and that his opponents choose according to s i. The randomizing-utility for player i is V i (σ i, s i ) = s i S i σ i (s i ) U i (s i, s i ) where (s i, s i ) = (s 1,..., s n ) j I S j.

27 Expected Utility with Randomizing Definition Let Γ N be a normal-form, and i be a player with utility function U i. Suppose that player i entertains belief b i and chooses σ i. The expected randomizing-utility for player i is v i (σ i, b i ) = b i (s i ) V i (σ i, s i ) s i S i = ( ) b i (s i ) σ i (s i ) U i (s i, s i ) s i S i s i S i where (s i, s i ) = (s 1,..., s n ) j I S j.

28 Strict Dominance: The Pure Case Definition Let Γ N be a normal-form, and i be a player. A strategy s i for player i is strictly pure-dominated, if there exists some strategy s i S i of player i such that U i (s i, s i ) < U i (s i, s i ) holds for every opponents strategy combination s i S i.

29 Strict Dominance: The Randomized Case Definition Let Γ N be a normal-form, and i be a player. A strategy s i for player i is strictly randomized-dominated, if there exists some mixed strategy σ i (S i ) of player i such that U i (s i, s i ) < V i (σ i, s i ) holds for every opponents strategy combination s i S i.

30 Strict Dominance Definition Let Γ N be a normal-form, and i be a player. A strategy s i for player i is strictly dominated, if s i is either strictly pure-dominated or strictly randomized-dominated.

31 Example I: Where to Locate My Pub? Story: Alice and Bob both want to open a new pub on Bold Street. Bold Street contains 600 houses, equally spaced. One person per house is assumed to visit the closest pub. There are seven possible locations for pubs: a, b, c, d, e, f, g. If Alice and Bob choose the same location, then each gets 300 clients. Question: Which location choices obtain with strict dominance?

32 Example I: Where to Locate My Pub? Neither b, nor c, nor d, e, nor f are strictly dominated for Alice: U Alice (b, a) U Alice (c Alice, a) for all c Alice {a, b, c, d, e, f, g}, U Alice (c, b) U Alice (c Alice, b) for all c Alice {a, b, c, d, e, f, g}, U Alice (d, d) U Alice (c Alice, d) for all c Alice {a, b, c, d, e, f, g}, U Alice (e, f ) U Alice (c Alice, f ) for all c Alice {a, b, c, d, e, f, g}, U Alice (f, g) U Alice (c Alice, g) for all c Alice {a, b, c, d, e, f, g}. a is strictly dominated by b for Alice, as U Alice (a, c Bob ) < U Alice (b, c Bob ) for all c Bob {a, b, c, d, e, f, g}. g is strictly dominated by f for Alice, as U Alice (g, c Bob ) < U Alice (f, c Bob ) for all c Bob {a, b, c, d, e, f, g}. Hence, SD Alice = {b, c, d, e, f }, and analogously, it can be shown that, SD Bob = {b, c, d, e, f }.

33 Example II: Going to a Party Story: Alice and Bob are going together to a party tonight. Alice asks herself what colour she should wear. Alice prefers blue to green, green to red, and red to yellow. However, Alice dislikes most to wear the same colour as Bob. Let Alice s utilities be given as follows: blue: 4 green: 3 red: 2 yellow: 1 same colour as Bob: 0 Question: Which colour choices obtain with strict dominance for Alice?

34 Example II: Going to a Party Neither blue, nor green, nor red are strictly dominated for Alice: U Alice (blue, green) U Alice (c Alice, green) for all c Alice {blue, green, red, yellow}, U Alice (green, blue) U Alice (c Alice, blue) for all c Alice {blue, green, red, yellow}, U Alice (red, blue) > U Alice (blue, blue), U Alice (red, green) > U Alice (green, green), and U Alice (red, yellow) > U Alice (yellow, yellow), hence there is no choice for Alice better than red against all of Bob s choices. yellow is strictly dominated by 0.5 blue green for Alice, as U Alice (yellow, c Bob ) < V Alice (0.5 blue green, c Bob ) for all c Bob {blue, green, red, yellow}. Hence, SD Alice = {blue, green, red}.

35 Agenda Introduction Rational Choice Strict Dominance Characterization of Rationality

36 Characterization of Rationality Pearce s Lemma: The rational strategies are exactly those strategies that are not strictly dominated.

37 Application Four ways to rationality: 1 Identify all rational strategies: find a belief on the opponents strategies such that the respective strategy is optimal. 2 Identify all irrational strategies: show that the respective strategy is not optimal for any belief on the opponents strategies. 3 Identify all strategies that are not strictly dominated: find an opponents strategy-combination such that there is no strategy that is better than the respective strategy. 4 Identify all strategies that are strictly dominated: show that the respective strategy fares worse than some other strategy for all opponents strategy-combinations. Note: For rational strategies it is often easier to find a supporting belief. For irrational strategies it is often easier to show strict dominance.

38 A Basic Lemma Basic-Lemma Let I be some index set, 0 α i 1 for all i I such that i I α i = 1, x R, and y i R for all i I. If x < i I α iy i, then there exists i I such that x < y i. Proof: Using contraposition, suppose that x y i for all i I. Then, α i x α i y i holds for all i I. It directly follows that 1 x = i I α ix i I α iy i.

39 Two Useful Facts Remark 1 Let Γ N be a normal-form, i I be a player, s i, s i S i be strategies of player i, and σ i (S i ) be a mixed strategy of player i. 1 If s i is strictly dominated by s i, then u i(s i, b i ) < u i (s i, b i) for all beliefs b i (S i ). 2 If s i is strictly dominated by σ i, then u i (s i, b i ) < v i (σ i, b i ) for all beliefs b i (S i ). Proof of (1): By definition U i (s i, s i ) < U i (s i, s i ) holds for all s i S i. Let b i (S i ) be some belief for player i. Then, and b i (s i ) U i (s i, s i ) b i (s i ) U i (s i, s i ) for all s i S i, b i (s i ) U i(s i, s i ) < b i(s i ) U i(s i, s i ) for all s i supp(b i). Hence, u i (s i, b i ) = s i S i b i (s i ) U i (s i, s i ) < s i S i b i (s i ) U i (s i, s i ) = u i (s i, b i ). Proof of (2): Analogously to the pure case.

40 Pearce s Lemma Theorem (Pearce s Lemma) Let Γ be a normal-form, i be a player, and s i be some strategy of player i. s i is rational, if and only if, s i is not strictly dominated.

41 Proof of the Direction (Epistemic Foundation) Case 1: Case 2: Let s SD i be a strategy of player i that is strictly dominated. Suppose that s SD i is strictly dominated by another strategy s i. Remark 1 then implies that u i (s SD i, b i ) < u i (s i, b i) holds for all beliefs b i (S i ). Hence, there exists no belief b i (S i ) such that s SD i optimal, and s SD i therefore is irrational. can be Suppose that s SD i is strictly dominated by a mixed strategy σ i. Remark 2 then implies that u i (s SD i, b i ) < v i (σ i, b i ) holds for all beliefs b i (S i ).

42 Proof of the Direction (Epistemic Foundation) Observe that by associativity, commutativity, and distributivity it holds that v i (σ i, b i ) = ( ) s i S i b i (s i ) s σ i S i i(s i ) U i (s i, s i ) = ( ) s i S i σ i (s i ) s b i S i i(s i ) U i (s i, s i ) = s i S i σ i (s i ) u i (s i, b i ) Hence, u i (s SD i, b i ) < s i S i σ i (s i ) u i (s i, b i ) holds for all beliefs b i (S i ). Let b i (S i) be some belief. As 0 σ i (s i ) 1 for all s i S i, there exists, by Basic-Lemma, some strategy s i S i such that u i (s SD i, b i ) < u i(s i, b i ). Therefore, s SD i cannot be optimal given belief b i. As the belief b i has been chosen arbitrarily, ssd i is irrational.