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The Bar Scene John and Joe Choose

The Bar Scene John and Joe Choose. The Bar Scene Nash Equilibrium and Mixed Strategies. The Bar Scene Nash Equilibrium and Mixed Strategies. No choice is dominated Nash equilibria? Two, but need coordination to get there A leader (1 st Mover)  Win/Win (except for Sally)

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The Bar Scene John and Joe Choose

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  1. The Bar SceneJohn and Joe Choose

  2. The Bar SceneNash Equilibrium and Mixed Strategies

  3. The Bar SceneNash Equilibrium and Mixed Strategies • No choice is dominated • Nash equilibria? • Two, but need coordination to get there • A leader (1st Mover)  Win/Win (except for Sally) • Tacit Coordination • Right goes to right/left goes to left • Tall goes to tall/short goes to short • Blonde goes to blonde/etc. Shared Values Simply Choice Shared Values Improve Outcomes

  4. The Bar SceneNash Equilibrium and Mixed Strategies • What to do? • Your expected payoff for any choice should be the same when his is the same Equal Bang per Buck? • Confront him with equal expected payoffs whatever he does …. and expect him to do the same to you • Let pS = probability you choose Sally, etc. • Then if he chooses Sally, EVS = 1 pS + 2 pJ + 2 pM • Or if he chooses Jane, EVJ = 2.1pS + 1.05pJ + 2.1pM • Since pJ = pM and pS + pJ + pM = 1 pS = .28 pJ = .36 pM = .36 EV = 1.72

  5. The Bar SceneNash Equilibrium and Mixed Strategies

  6. Golden Rules Do unto others as you would have others do unto you (Saint) Do unto others what others just did unto you (tit-for-tat) Do unto others before others do unto you (end-gaming) Do unto others what you believe they will do unto you (trust and reciprocity)

  7. Zero-Sum / Non – Zero Sum The City Game Playing for points (non – zero sum)  Split the map Playing to “win” (zero – sum) • Cover the map “Don’t have to outrun a bear” • Lots of Players / Repetitive Play • Cooperation ? • Turf battles ?

  8. Long-Run: Set Capacity Short-Run: Set Price • Problem 21.9 mc = 10 PA = 110 – XA – .5 XB PB = 110 – XB – .5 XA Set Capacity / Divide the market • XA = XB = 40 PA = PB = 50 ΠA = ΠB = 1600 Set the Price / Divide the market • PA = PB = 43.33 XA = XB = 44.44 ΠA = ΠB = 1481 Mix and Match (A sets XA; B sets PB) • XA = 46.15 PA = 44.62 ΠA = 1598 ~ 1600 • PB = 48.46 XB = 38.48 ΠB = 1480 ~ 1481

  9. Set Price or Output?: A Game • Nash equilibrium: Divide the market • Carving up markets (generally) more profitable than fixing prices (e.g., OPEC) • Are prices easier to monitor? (e-commerce, gas stations)

  10. First Mover Advantage • If A sets XA first, anticipating B’s best response … • Recall, X*B = 50 - .25 XA Then ΠA = [100 - XA - .5 (50 - .25 XA )] XA ΠA = 75 XA - .875 XA 2 XA = 42.86 PA = 47.50 ΠA = 1607 XB = 39.29 PB = 49.28 ΠB = 1543 • Pre-empt your competition with extra capacity • Total capacity greater with pre-emption

  11. Game Theory: Theory and Practice 10 10 1 • The Trust Game • Backward induction  Player 1: Take 10 • Actual Play • Player 1: 50:50 (Grab:Trust) • Player 2: 75:25 (Split:Greed) • EMV1(Trust) = .75 x 15 + .25 x 0 = 11.25 15 25 2 0,40

  12. Tie Your Hands • Can Player 2 alter his payoff when he’s greedy so Player 1 knows he won’t be greedy? • Social sanctions • Reputation • Intel: License competitors so your customers won’t fear your monopoly power … give up your monopoly power.

  13. Game Theory: Theory and Practice The Centipede Game: When to clear the table? • Suppose Player 2 has some prior belief, however small, that Player 1 is “kind” • Player 1: Cooperate to reinforce this belief • Player 2: Cooperate to get Player 1 to continue to cooperate, believing you believe he’s “kind” • As end-game approaches, things get tricky Do unto others before they do unto you • Small probability Player 1 is “kind” and Player 2 believes he’s kind  Lots of cooperation … up to endgame

  14. Aside on Tipping Small perturbation  Major consequences • Perfect Integration X O X O X O X O X O X O X O X O X O X O • Tipping X O X O X O X X X O X O X O X O X O X O • Complete Segregation • X X X X X X X X X X X O O O O O O O O O Document No: RM-6014-RC     Year: 1969     Pages: 89 Title:Models of Segregation. • Author(s):Thomas C. Schelling

  15. Game Theory: Theory and Practice Ultimatum Game Player 1: Divide $10 … anyway he wants Player 2: Take it (whatever Player 1 grants) orLeave it …and no one gets anything Role of roles • Random assignment of roles • Player 1 “meritorious” • Player 1 = “Seller” Player 2 = “Buyer”

  16. Threat Game / Chain Store Game • Potential Entrant: Don’t Enter … Game ends  (0,2) Enter … Incumbent moves • Incumbent: Acquiesce (1,1) Fight (-1,-1) Backward Induction • Incumbent shouldn’t fight so • Entrant should enter … but Rational Irrationality  Reputation

  17. Tie Your Hands • Can incumbent change his payoff for acquiescing so potential entrant knows he won’t acquiesce … and therefore won’t enter? • Can incumbent appear to change his payoff for acquiescing so potential entrant thinks he won’t acquiesce … and therefore won’t enter?

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