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Strategic games (Normal form games) ECON5200 Advanced microeconomics Lectures in game theory Fall 2009, Part 1 17.09.2009 G.B. Asheim, ECON5200-1 1 Game theory studies multi-person decision problems, and analyzes agents that are rational (have well-defined preferences) reason strategically (take into account their knowledge and beliefs about what others do) Classification of games non-cooperative vs. cooperative games strategic vs. extensive games Games with perfect and imperfect information 17.09.2009 2 G.B. Asheim, ECON5200-1 Studies the outcome …of individual … of joint actions actions when there when there is is no external external enforcement. enforcement. (strict sense) no communication before the game non-cooperative game theory communication before the game (wide sense) 17.09.2009 G.B. Asheim, ECON5200-1 cooperative game theory 3 1 Incomplete information Imperfect information Complete information Perfect information “Almost perf.” info. Static Dynamic 17.09.2009 Lecture 1 Strategic games Lecture 1 Bayesian games Lecture 3 Extensive games (the general case) Lecture 2 Extensive games (multi-stage games) 4 G.B. Asheim, ECON5200-1 Solution concepts A systematic description of outcomes that may emerge in classes of games Game theory suggests reasonable solutions for classes of games and examines their properties Interpretation of solution concepts The evolutive (“steady state”) interpretation The deductive (eductive) interpretation Bounded rationality will not be treated in these lectures. 17.09.2009 5 G.B. Asheim, ECON5200-1 Rational behavior Decision-maker i chooses from a set Si of strategies. Decisions are made under uncertainty, where i is a finite set of uncertain states. The uncertainty may be Example strategic uncertainty relating to the strategies of others. exogenous uncertainty relating to the environment. A state-strategy pair (i, si) (i Si) leads to a consequence c C. A consequence function g: i Si C assigns a consequence to each state-strategy pair. 17.09.2009 G.B. Asheim, ECON5200-1 6 2 Rational behavior (cont) Let the decision-maker be endowed with a vNM utility function i : C . a subjective prob. distr. i over i. Example A (pure) strategy si is preferred to si if and only if i (ωi )i ( g (ωi , si)) i (ωi )i ( g (ωi , si)) ωi i ωi i Anscombe & Aumann’s decision-theoretic framework. It requires the availability of mixed strategies. A mixed strategy i is a obj. randomization over Si. The exp. utility of i: ( s ) (ω ) ( g (ω , s )) si S i 17.09.2009 i i i ω i i i i i 7 G.B. Asheim, ECON5200-1 A strategic game i Incomplete information Imperfect information Complete information Perfect information Definition 11.1 A strategic game consists of Lecture 1 Strategic games Static Dynamic Lecture 2 Extensive games (multi-stage games) Lecture 1 Bayesian games Lecture 3 Extensive games (the general case) a finite set N (of players) for each i N, a non-empty set Si (of strategies) for each i N, a payoff function ui on S jNSj: ( si , si ) S , ui ( si , si ) i ( g ( si , si )) A strategic game is finite if, for each i N, Si is finite. Interpretations: the game is played once? 17.09.2009 Consequence (or outcome) simultaneous actions? G.B. Asheim, ECON5200-1 8 Nash equilibrium for a strategic game Definition 14.1 A Nash equilibrium of a strategic game is a strategy profile s S with the property that, for each player i N, si S i , u ( si , si ) u ( si , si ) Alternative formulation: Define a set-valued function Bi (best-response fn): Bi ( si ) {si S i | si S i , u ( si , si ) u ( si , si)} The strategy profile s S is a Nash equilibrium if and only if , for each player i N, si Bi ( si ) 17.09.2009 G.B. Asheim, ECON5200-1 9 3 Can Nash equilibrium be used as a solution concept if the game is only played once? Yes, if each player can predict what each opponent will do. For each player, only one strategy survives iterative elimination of strictly dominated strategies. Through communication before the game starts, the players make a self-enforcing agreement (coordinate on an equilibrium). Given a common background, the players are able to co-ordinate on an equilibrium without communication before the game starts (Schelling, 1960, focal point). A unique Nash equilibrium is not sufficient. 17.09.2009 N {1, … , n} 10 G.B. Asheim, ECON5200-1 Example: An auction Player i’s valuation vi, where v1 … vn 0. The players submits bids simultaneously. The object is given to the players submitting the highest bid (if there are several players with the highest bid, then the winner is the one with the lowest index. First price auction: The winner pays his bid. Second price auction: The winner pays the highest bid among the non-winners. Equilibria in a first Equilibria in a second price auction? price auction? 17.09.2009 11 G.B. Asheim, ECON5200-1 Existence of Nash equilibrium Lemma 20.1 (Kukutani’s fixed point theorem) Let X be a compact convex subset of n, and let f : X X be a set-valued function for which for all x X the set f(x) is non-empty and convex. the graph of f is closed Then there exists x X such that x f(x). Proposition 20.3 A strategic game has a Nash equilibrium if for all i N, Proof of Proposition 20.3 the strategy set Si is non-empty and compact. the payoff function ui is continuous the payoff function ui is quasi-concave on Si. 17.09.2009 G.B. Asheim, ECON5200-1 12 4 A Bayesian game Incomplete information Imperfect information Complete information Perfect information Definition 25.1 A Bayesian game consists of Lecture 1 Strategic games Static Lecture 2 Extensive games (multi-stage games) Dynamic a finite set N (of players) Lecture 2 Bayesian games Lecture 3 Extensive games (the general case) for each i N, a set Ai (of actions) for each i N, a set Ti (of types) for each i N, a prob. distr. pi on T i N Ti that satisfies, for all ti Ti , t T pi (t i , ti ) 0 i j i j for each i N, a vNM utility function ui on the set A T, where A jNAj, and where ui(a, t) is player i’s payoff if (a, t) is realized. 17.09.2009 13 G.B. Asheim, ECON5200-1 Nash equilibrium for a Bayesian game An ex post perspective. A Nash equilibrium of a Bayesian game is a Nash equilibrium for the strategic game defined as follows: The set of players is the set of all pairs (i, ti) for i N and ti Ti. For each player (i, ti), the set of strategies is Ai. For each player (i, ti), the payoff function is defined by u(i ,ti ) (a ) pi (t i , ti ) u (a , , a(n ,tn ) ), t i , ti ) i (1,t1 ) t i j i T j t T pi (t i ji j 17.09.2009 14 G.B. Asheim, ECON5200-1 Example 27.2 (t1B , t2B ); pi (t1B , t2B ) 1 / 4 (t1B , t2S ); pi (t1B , t2S ) 1 / 4 B S B S B 2, 2 0, 0 B 2, 1 0, 0 S 0, 0 1, 1 S 0, 0 1, 2 (t1S , t2B ); pi (t1S , t2B ) 1 / 4 (t1S , t2S ); pi (t1S , t2S ) 1 / 4 B S B S B 1, 2 0, 0 B 1, 1 0, 0 S 0, 0 2, 1 S 0, 0 2, 2 Is a(i ,t ) B and a(i ,t ) S for both i a Nash equilibrium for this Bayesian game? B i 17.09.2009 S i G.B. Asheim, ECON5200-1 15 5 Example: A second price auction N {1, … , n} Ai Ti V pi (v1 , , vn ) nj 1 (v j ) for some probability distribution over V. The payoff u(i ,v ) (a ) equals the expectation of the random variable whose value given (v1 , , vn ) is vi max jN \{i} a j (v j ) if player i wins. 0 otherwise. Equilibria? i 17.09.2009 16 G.B. Asheim, ECON5200-1 The mixed extension Definition 32.1 A mixed extension of a strategic game consists of Incomplete information Imperfect information Complete information Perfect information Lecture 1 Strategic games Static Dynamic Lecture 2 Extensive games (multi-stage games) Lecture 1 Bayesian games Lecture 3 Extensive games (the general case) a finite set N (of players) for each i N, a set (Si) (of mixed strategies), where (Si) is the set of prob. distributions on Si. for each i N, a payoff function Ui on jN(Sj) assigning to each jN(Sj) the expected value: U i ( ) sS jN j ( s j ) ui ( s ) 17.09.2009 G.B. Asheim, ECON5200-1 17 Mixed strategy Nash equilibrium Definition 32.3 A mixed strategy Nash equilibrium of a strategic game is a Nash Proof of Proequilibrium of its mixed extension. position 33.1 Proposition 33.1 Every finite strategy game Proof of has a mixed strategy Nash equilibrium. Lemma 33.2 Lemma 33.2 Let G be a finite strategic game. Then iN(Si) is a mixed strategy Nash equilibrium if and only if, for each player i, every pure strategy assigned positive probability by i is Can be used as primitive a best response to i . definition (cf. Def. 44.1). 17.09.2009 G.B. Asheim, ECON5200-1 18 6 Interpretations of mixed strategies Mixed strategies as an object of choice Mixed strategy Nash equilibrium as a steady state. Mixed strategies as pure strategies of an extended game. Mixed strategies as pure strategies of a perturbed game. Mixed strategies as beliefs. 17.09.2009 Mixed strategy Nash equilibrium as a steady state Mixed strategies as pure strat. in a perturbed game G.B. Asheim, ECON5200-1 19 Rationalizability Definition A set of strategy profiles Z iN Z i has the best response property if, for all i N and si Zi, there exists a i ( j i Z j ) such that si Bi ( i ). Definition 55.1 A strategy si Si is rationalizable if and only if there exists a set of strategy profiles with the best resp. prop., Z iN Z i , such that si Z i . 17.09.2009 G.B. Asheim, ECON5200-1 20 Further results on rationalizability Lemma Every strategy used with positive probability by some player in a mixed strategy Nash equilibrium, is rationalizable. Proposition Let G be a finite strategic game. Then there exists, for any i N, a rationalizable strategy for i. Observation Even in a strategic game with a unique Nash equilibrium, common knowledge of the players' rationality does not imply that the players will play this Nash equilibrium. 17.09.2009 G.B. Asheim, ECON5200-1 21 7 Never-best response and strict domination Definition 59.1 A strategy si Si of player i in the finite strategic game G is a never-best response if there is no i (jiSj) such that si Bi(i). Definition 59.2 A strategy si Si of player i in the finite strategic game G is strictly dominated if there is a mixed strategy i (Si) such that, si S i , sS i (si)ui ( si , si) ui ( si , si ) i i Lemma 60.1 A strategy si Si of player i in the finite strategic game G is a never-best response if and only if it is strictly dominated . 17.09.2009 G.B. Asheim, ECON5200-1 22 Iterated elimination of strictly dominated strategies Definition 60.2 The set of X S of strategy profiles in the finite strategic game G survives iterated elimination of strictly dominated strategies if X iNXi and there is a collection of sets (( X tj ) jN )Tt0 that satisfies the following conditions for each j N. X 0j S j and X Tj X j X tj1 X tj for each t 0, ... , T 1. For each t 0, ... , T 1, every strategy of player j in X tj \ X tj1 is strictly dominated in the game Gt, where, for each i N, i’s strategy set is restricted to X it . No strategy in X Tj is strictly dominated in the game GT. 17.09.2009 G.B. Asheim, ECON5200-1 23 Iterated elimination of strictly dominated strategies (cont.) Lemma (i) If Z iNZi has the best response property, then Z X. (ii) X iNXi has the best response property. Proposition 61.2 If X iNXi survives iterated elimination of strictly dominated strategies in the finite strategic game G, then, for each j N, Xj is the set of player j’s rationalizable strategies. 17.09.2009 G.B. Asheim, ECON5200-1 24 8 Weak domination R 1, 0 B 1, 0 0, 1 Definition 62.1 A strategy si Si of player i in the finite strategic game G is weakly dominated if there is a mixed strategy i (Si) such that, s i S i , s i S i , L 1, 1 T sS i (si)ui ( si , si) ui ( si , si ) sS i ( si)ui ( s i , si) ui ( si , si ) i i i i Iterated elim. of weakly dom. str. is dubious because the result depends on the the order of elimination, hard to give the procedure an epistemic foundation. 17.09.2009 G.B. Asheim, ECON5200-1 25 9