Advanced Microeconomics

Transcription

1 Advanced Microeconomics ECON Fall 2014

2 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market prices; - no strategic behavior. What we will do: - in many interesting situations, agents optimal behavior depends on the other agents behavior; - strategic behavior. Game theory provides a language to analyze such strategic situations; Countless number of examples! Auctions, Bargaining, Price competition, Civil Conflicts...

3 Introduction Road map Static Game: 1. With Complete Information (I); 2. With Incomplete Information (II). Dynamic Game: 1. With Complete Information (II-III); 2. With Incomplete Information (III).

4 Strategic Game with Pure Strategies N players with i I ; s S u i (s) payoff; S i pure strategy profile, s i S i ; i=1,..,n G I, {S i } i, {u i (s)} i strategic formof finite game with pure strategy.

5 Strategic Game with Mixed Strategies σ (S) (S i ) mixed strategyprofile, σ i (S i ); i=1,..,n u i (σ) = σ j (s j ) u i (s j ) expected utility; s S j=1,..,n G I, { (S i )} i, {u i (σ)} i strategic form of finite game with mixed strategy; Interpreting mixed strategies: - as object of choice; - as pure strategies of a perturbed game (see later in Bayesian Games); - as beliefs.

6 Equilibrium Concepts Nash Equilibrium it is assumed that each player holds the correct expectation about the other players behavior and act rationally (steady state equilibrium notion); Rationalizability players beliefs about each other s actions are not assumed to be correct, but are constrained by consideration of rationality; Every Nash equilibrium is rationalizable.

7 Rationalizability Definition In G, s i is rationalizableif there exists Z j S j for each j I such that: 1. s i Z i ; 2. every s j Z j is a best response to some belief µ j (Z j ). Common knowledge of rationality; An action is rationalizable if and only if it can be rationalized by an infinite sequence of actions and beliefs.

8 Example (1 - Rationalizability - See notes!)...

9 Strictly Dominance Definition s i is not strictly dominatedif it does not exist a strategy σ i : u i (σ i, s i ) > u i (s i, s i ), s i S i

10 Strictly Dominance A unique strictly dominant strategy equilibrium (D, D): It is Pareto dominated by (C, C ). Does it really occur??

11 Iterative Elimination of Strictly Dominated Strategies Definition Set S 0 = S, then for any m > 0 s i Si m any σ i such that: iff there does not exist u i (σ i, s i ) > u i (s i, s i ), s i S m 1 i Definition For any player i, a strategy is said to be rationalizable if and only if s i Si Si m. m 0

12 Example (2 - Beauty Contest - See notes!)...

13 Iterated Weak Dominance There can be more that one answer for iterated weak dominance; Not for iterated strong dominance.

14 Example (3 - Cournot vs Bertrand Competition - Proposed as exercise) Example n profit-maximizer-firms produce q i quantity of consumption good at a marginal cost equal to c > 0; demand function is P = max {1 Q, 0} with Q Find: 1. The rationalizable equilibria when n = 2; 2. The rationalizable equilibria when n > 2; q i ; i=1...n 3. Compare your results with the Bertrand competition outcome.

15 Nash Equilibrium Definition σ i (S i ) is a best responseto σ i (S i ) if: u i (σ i, σ i ) u i (s i, σ i ) for all s i S i Let B i (σ i ) (S i ) be the set of player i best response. Definition σ is a Nash equilibriumprofile if for each i I. σ i B i (σ i )

16 Nash Theorem Theorem (Nash (1950)) A Nash equilibrium exists in a finite game. Theorem (Kakutani Fixed Point Theorem) Let X be a compact, convex and non-empty subset of R n, a correspondence f : X X has a fixed point if: 1. f is non-empty for all x X ; 2. f is convex for all x X ; 3. f is upper hemi-continuous (closed graph).

17 Best Response Correspondence Example

18 The Kitty Genovese Problem/Bystander Effect n identical people; x > 1 benefits if someone calls the police; 1 cost of calling the police; What is the symmetric mixed strategy equilibriumwith p equal to the probability of calling the policy? In equilibrium each player must be indifferent between calling or not the police; If i calls the police, gets x 1 for sure; If i doesn t, gets: 0 with Pr (1 p) n 1 x with Pr 1 (1 p) n 1

19 The Kitty Genovese Problem/Bystander Effect Indifference when: x 1 = x (1 (1 p) n 1) Equilibrium symmetric mixed strategy is p = 1 (1/x) 1/(n 1)

20 Zero-Sum Game Definition A N-player game G is a zero-sum game(a strictly competitive game) if u i (s) = K for every s S. i=1,..,n

21 Zero-Sum Game Definition σ i (S i ) is maxminimizerfor player i if: min u i (σ i, σ i ) min u ( ) i σ i, σ i for each σi (S i ) σ i (S i ) σ i (S i ) A maxminimizer maximizes the payoff in the worst case scenario Theorem Let G be a zero-sum game. Then σ (S) is a Nash Equilibrium iff, for each i, σ is a maxminimizer.

22 Example (4 - All-Pay Auction - Proposed as exercise) Two players submit a bid for an object of worth k; b i [0, k] individual strategy space where b i is the bid; The winner is the player with the highest bid; If tie each player gets half the object, k/2; Each player pays her bid regardless of whether she wins; Find that: 1. No pure Nash equilibria exist; 2. The mixed strategy equilibrium is equal to the one represented here below.

23 Example (4 - All-Pay Auction - Proposed as exercise)

24 Extensive Form Games Representation of a Game Normal or strategic form; Extensive form. The Extensive form contains all the information about a game: who moves when; what each player knows when he moves; what moves are available to him; where each move leads. whereas a normal form is a summary representation.

25 Extensive Form Games Extensive Form Definition A treeis a set of nodes and directed edges connecting these nodes such that: 1. for each node, there is at most one incoming edge; 2. for any two nodes, there is a unique path that connect these two nodes. Definition An extensive form game consists of i) a set of players (including possibly Nature), ii) a tree, iii) an information set for each player, iv) an informational partition, and v) payoffs for each player at each end node (except Nature).

26 Extensive Form Games Extensive Form Definition An information setis a collection of points (nodes) such that: 1. the same player i is to move at each of these nodes; 2. the same moves are available at each of these nodes. Definition An information partitionis an allocation of each node of the tree (except the starting and end-nodes) to an information set. Definition A (behavioral) strategyof a player is a complete contingent-plan determining which action he will take at each information set he is to move.

27 Extensive Form Games Extensive Form vs Normal Form