2Mixed strategy Nash Equilibrium A player using a mixed strategy chooses to ``randomizes’’ between ``pure strategies’’, assigning a specific probability to taking each possible pure strategy.Assume that if the other player is using a mixed strategy, your best response is to choose a strategy that maximizes your expected payoff.
3Matching Pennies: or Simple hide and seek_ Player 2 (Seeker)Heads Tailsq1-q1,-1-1, 1Heads(Hider)1-pTailsp-1, ,-1Player 1
4The game of matching pennies has two pure strategy Nash equilibriaOne pure strategy Nash equilibriumOne mixed strategy Nash equilibrium and no pure strategy Nash equilibriaTwo mixed strategy Nash equilibria and no pure strategy Nash equilibriaOne mixed strategy Nash equilibrium and two pure strategy Nash equilibria.
5Nash equilibrium in Matching Pennies Suppose Player 1 randomizes and plays Heads 2/3 of the time, what is Player 2’s best response?Heads for sureTails for sureRandomize with Probability of Heads 2/3.Randomize with Probability of Tails 2/3
6Mixed strategy as best response In a two-player, two-strategy game your best response is a mixed strategy with positive probabilities of playing both pure strategies only if your payoffs from the two pure strategies are equal.
7Mixed strategies for Hide-and-Seek Let pH be the probability that hider plays Heads and 1-pH the probability that hider plays Tails.When would seeker get the same payoff from playing Heads or Tails?Expected payoff to seeker from Heads ispH×1 +(1- pH) ×( -1)=2 pH -1.
8Best response Mapping 1 q=probability 2 chooses H 1/2 1/2 1 Player 2’s Reaction Function (in Red)1q=probability2 chooses HPlayer 1’sReactionFunction(in Green)1/21/21p= Probability 1 chooses H
9Nash equilibrium in Mixed Strategies Intersection of Reaction FunctionsEach is doing best response to other’s strategy
10A Fundamental TheoremSome games have no equilibrium in pure strategies: Examples: matching pennies; rock, paper scissorsEvery game in which there is a finite number of pure strategies has at least one mixed strategy equilibrium.
12Mixed strategy equilibrium In a mixed strategy equilibrium, all strategies that are assigned positive probability have equal expected value.You can use this fact to find mixed strategy Nash equilibria.
13Example: Advanced Hide and Seek When does Seeker have a mixed strategy best response. The payoffs to looking in the plains and looking in the forest must be the same.Where p is probability Hider is in the plains,Payoff to Plains is p3+(1-p)(-1)=4p-1.Payoff to Forest is -1p +(1-p)1=1-2p4p-1=1-2p if and only if 6p=2, p=1/3.
14Best response Mapping 1 q=probability 2 chooses Plains 1/3 1/3 1 Player 2’s Reaction Function (in Red)Player 1’sReactionFunction(in Green)1/31/31p= Probability 1 chooses Plains
15Expected Utility Theory of Choice Under Uncertainty Suppose that you face random outcomes. You assign a “utility” to each possible outcome in such a way that your choices among uncertain prospects are those that maximize “expected utility”.
16Expected utility Example: Utility of money Suppose you have a lottery that will with probability 1/4 win 10 million dollars and with probability ¾ will be worthless. You get just one chance to sell your ticket.Would you sell it for million dollars?YesNo
17Expected utility Example: Utility of money Suppose you have a lottery that will with probability 1/4 win 10 million dollars and with probability ¾ will be worthless. You get just one chance to sell your ticket.Would you sell it for 1 million dollars?YesNo
18Expected utility Example: Utility of money Suppose you have a lottery that will with probability 1/4 win 100 million dollars and with probability ¾ will be worthless. You get just one chance to sell your ticket.Would you sell it for 500 thousand dollars?YesNo
19Construct a utility scale Let u(10 million)=1 Let u(0)=0.Then ask question. How much money X for sure would be just as good as having a ¼ chance of winning 10 million and ¾ chance of 0?Then assign u(X)=(3/4)u(0)+(1/4)u(10,000,000)=(3/4)0+(1/4)1=1/4.
20Assigning utility to any income Lets choose a scale where u(0)=0 and u(10 million)=1.Take any number X. Find a probability p(X) so that you would just be willing to pay $X for a lottery ticket that pays 10 million with probability p(X) and 0 with probability 1-p(x).Assign utility p(X) to having $X.
21Field Goal or Touchdown? Field goal is worth 3 points.Touchdown is worth 7 points.Which is better? Sure field goal or probability ½of touchdown?
22Finding the coach’s von Neumann Morgenstern utilities Set utility of touchdown u(T)=1Set utility no score u(0)=0The 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 prob 1-p*) and having a sure field goal.Then u(F)=p*u(T)+(1-p*)u(0)=p*x1+(1-p*)x0=p*.
23Volunteers’ DilemmaN people observe a mugging. Someone needs to call the police. Only one call is needed. Cost of calling is c. Cost of knowing that the person is not helped is T. Should you call or not call? T>c>0. Many asymmetric pure strategy equilibria. Also one symmetric mixed strategy equilibrium.
24Mixed strategy equilibrium Suppose everybody uses a mixed strategy with probability p of calling.In equilibrium, everyone is indifferent about calling or not calling if expected cost from not calling equals cost from calling.Expected Cost of of not calling isT(1-p)N-1Expected cost of calling is c.Equilibrium has c= T(1-p)N-1 so 1-p=(c/T)1/N-1Then (1-p)N=(c/T)N/N-1 is the probability that nobody calls. This is an increasing function of N. So the morePeople who observe, the less likely that someone calls.
25Chicken Game 0, 0 0, 1 -10, -10 1, 0 Swerve Don’t Swerve Swerve Player 2q qSwerve Don’t SwerveP1-p0, 00, 1SwerveDon’t SwervePlayer 1-10, -101, 0Two Pure Strategy Nash equilibria
26Mixed StrategyWhen is Player 1 indifferent between the two strategies, Swerve and Don’t Swerve? Expected payoff from Swerve is 0. Expected payoff from Don’t Swerve is q-10(1-q). So Player 1 will use a mixed strategy best response only if 0=11q-10 or q=10/11. Similar reasoning inplies that in Nash equilibrium p=10/11. Crash occurs with probability 1/121.
27Battle of Sexes 3,2 1,1 0,0 2,3 Bob Movie A Movie B Movie A Alice BRA(A)=ABRA(B)=BBRB(A)=ABRB(B)=B
28Mixed Strategy Equilibrium Let p be probability Alice goes to movie A and q the probability that Bob goes to movie B.When is there a mixed best response for Alice?Expected payoff for Movie A for Alice is3(1-q)+ q1=3-2q.Expected payoff to Movie B for Alice is2q+(1-q)0=2qPayoffs are the same if 3-2q= 2q, so q=3/4.
29Similar for BobFrom the symmetry of the game, we see that a mixed strategy is a best response for Bob if p=3/4.In a symmetric mixed strategy, each goes to his or her favorite movie with probability ¾.Probability that they get together at Movie A is 3/4x1/4=3/16. Probability that they get together at Movie B is also 3/16. Probability that they miss each other is 5/8. Probability that each goes to favorite movie is 9/16. Probability that they each go to less preferred movie is 1/16.