Economics 211: Dynamic Games by David K
Economics 211: Dynamic Games by David K. Levine Last modified: January 6, 1999 © This document is copyrighted by the author. You may freely reproduce and distribute it electronically or in print, provided it is distributed in its entirety, including this copyright notice. Basic Concepts of Game Theory and Equilibrium Course Slides Source: \Docs\Annual\99\class\grad\basnote7.doc A Finite Game an N player game iN= 1K PS() are probability measure on S finite strategy spaces σ ∈≡Σ iiPS() i are mixed strategies ∈≡×N sSi=1 Si are the strategy profiles σ∈ΣΣ ≡×N i=1 i ∈≡× other useful notation s−−iijijS ≠S σ ∈≡×ΣΣ −−iijij ≠ usi () payoff or utility N uuss()σσ≡ ∑ ()∏ ( ) is expected iijjsS∈ j=1 utility 2 Dominant Strategies σ σ i weakly (strongly) dominates 'i if σσ≥> usiii(,−− )()( u i' i,) s i with at least one strict Nash Equilibrium players can anticipate on another’s strategies σ is a Nash equilibrium profile if for each iN∈1,K uu()σ = maxσ (σ' ,)σ− iiii'i Theorem: a Nash equilibrium exists in a finite game this is more or less why Kakutani’s fixed point theorem was invented σ σ Bi () is the set of best responses of i to −i , and is UHC convex valued This theorem fails in pure strategies: consider matching pennies 3 Some Classic Simultaneous Move Games Coordination Game RL U1,10,0 D0,01,1 three equilibria (U,R) (D,L) (.5U,.5L) too many equilibria?? Coordination Game RL U 2,2 -10,0 D 0,-10 1,1 risk dominance: indifference between U,D −−=− 21011ppp222()() == 13pp22 11,/ 11 13 if U,R opponent must play equilibrium w/ 11/13 if D,L opponent must
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