Advanced Microeconomics ECON5200 - Fall 2014
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...
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).
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.
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.
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.
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.
Example (1 - Rationalizability - See notes!)...
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
Strictly Dominance A unique strictly dominant strategy equilibrium (D, D): It is Pareto dominated by (C, C ). Does it really occur??
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
Example (2 - Beauty Contest - See notes!)...
Iterated Weak Dominance There can be more that one answer for iterated weak dominance; Not for iterated strong dominance.
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.
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 )
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).
Best Response Correspondence Example
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
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) http://en.wikipedia.org/wiki/murder_of_kitty_genovese
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
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.
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.
Example (4 - All-Pay Auction - Proposed as exercise)
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.
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).
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.
Extensive Form Games Extensive Form vs Normal Form