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January 20 Lecture Econ 171

January 20 Lecture Econ 171. The game of matching pennies has . two pure strategy Nash equilibria One pure strategy Nash equilibrium One mixed strategy Nash equilibrium and no pure strategy Nash equilibria Two mixed strategy Nash equilibria and no pure strategy Nash equilibria

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January 20 Lecture Econ 171

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  1. January 20 LectureEcon 171

  2. The game of matching pennies has • two pure strategy Nash equilibria • One pure strategy Nash equilibrium • One mixed strategy Nash equilibrium and no pure strategy Nash equilibria • Two mixed strategy Nash equilibria and no pure strategy Nash equilibria • One mixed strategy Nash equilibrium and two pure strategy Nash equilibria.

  3. If a game has at least one pure strategy Nash equilibrium A) It must also have at least one mixed strategy Nash equilibrium that is not a pure strategy Nash equilibrium B) It will not have a mixed strategy Nash equilibrium C) It might or might not have a mixed strategy Nash equilibrium that is not a pure strategy equilibrium D) It must have a dominant strategy for each player.

  4. Field Goal or Touchdown? • Field goal is worth 3 points. • Touchdown is worth 7 points. Which is better? Sure field goal or probability ½ of touchdown?

  5. Finding the coach’s von Neumann Morgenstern utilities • Set utility of touchdown u(T)=1 • Set utility no score u(0)=0 The utility of a gamble in which you get a touchdown with probability p and no score with probability 1-p is pu(T)+(1-p)u(0). What utility u(F) to assign to a sure field goal? Let p* be the probability such that the coach is indifferent between scoring a touchdown with probability p* (with no score with prob1-p*) and having a sure field goal. Then u(F)=p*u(T)+(1-p*)u(0)=p*x1+(1-p*)x0=p*.

  6. Advanced Hide and Seek Hider’s Choice Plains Forest q 1-q p 3,-3 -1,1 Plains 1-p Forest Seeker’s Choice

  7. Guard and shoot game Guard protects q 1-q Left Right P 1-p Left Right B, -B D, -D Shooter uses C, -C A, -A Probabilities of scoring: A>C , B>D, B>C

  8. Nash Demand Game There are N dollars to be divided among 2 players. Each player gets to make a demand. The demand must be a positive integer. If sum of demands is not greater than N, each gets what she demanded. If sum of demands exceeds N, both get nothing.

  9. Nash demand game, N=3

  10. A Game of Chicken Two teenagers have their fathers’ Buicks and decide to play Chicken. There are two strategies, Swerve and Don’t Swerve. If neither swerves, disastrous consequence for both. Like the hawk-dove game. Alternative story. Two countries have mobilized their armies with threats and counterthreats. Will one of them yield? If not, there is a disastrous war.

  11. A Chicken payoff matrix

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