1. Games of pure conflict two-person constant sum games

2. Two person constant sum game Sometimes called zero-sum game. The sum of the players’ payoffs is the same, no matter what pair of actions they take.

3. Maximin strategy One way to play a game is to take a very cautious view. Your payoff from any action depends on other’s actions. In a two-player game, you might assume other player always does what is worst for you. Given that assumption, you would choose the strategy such that gives you the best payoff if the other player always does what is worst for you given your strategy.

4. Maximin in Constant Sum games In a two-person constant sum game, if the other player is doing his best response to your action, he is taking the action that is worst for you, given your action. A)Yes B)No C)Maybe

5. Why so? If the sum of your payoff and your opponent’s payoff is constant, then the larger is your opponent’s payoff, the smaller is yours. So if he is choosing the highest possible payoff for himself, given your action, he is taking the action that is worst for you, given your action.

6. In a two-player constant sum game, suppose that you know that the other player has a spy who will find out your strategy before he chooses his. Your best option is to use a maximin strategy. A)Yes B)No C)Maybe

7. Simple hide and seek 1,0 0, 1 0,1 1,0 Look Upstairs Look Downstairs Player 2 (Seeker) Player 1 (Hider) Hide upstairs Hide downstairs p Is this a constant sum game? A) Yes B) No

8. Mixed strategies and maximin Suppose you are hider, choosing a mixed strategy, and you believe that seeker will do what is worst for you, given your mixed strategy. This is not a silly assumption in a two-player zero sum game, because what is worst for you is best for your opponent. The maximin player will choose her best mixed strategy given that she believes opponent will respond with the strategy that is worst for her.

9. Clicker question Suppose that you are hider and you choose to randomize, hiding upstairs with probability.6. What strategy by seeker is worst for you? A)Look upstairs with probability.6 B)Look upstairs and downstairs with equal probability C)Look upstairs for sure D)Look upstairs with probability.4

10. Clicker question If you are hider and hide upstairs with probability.6 and seeker uses the strategy that is worst for you, what is your expected payoff? (Remember: Your payoff is 0 if he finds you, 1 if he doesn’t.) A).6 B).4 C).5 D).35

11. More generally If you are hider and you hide upstairs with probability p>1/2, what is the strategy for seeker that is worst for you? What is your expected payoff if he does that?

12. Where p is the probability you hide upstairs: If p>0, he will look upstairs always. He will not find you with probability 1-p. Your expected payoff is p x 0 + (1-p) x 1

13. What if you hide upstairs with p<1/2? What is worst thing that seeker can do to you? (He’ll look downstairs for sure.) What is your expected payoff?

14. If p<1/2, Seeker stategy that is worst for hider is Look downstairs always. Payoff to hider is p x 1+(1-p) x 0

15. Maximin for hide and seek

16. The pessimist’s view

17. Maximin strategy in hide and seek Randomize with p=1/2. There is a Nash equilibrium where both play p=1/2. If each player knew that the other had a spy who knew your strategy, both would choose to randomize with p=1/2.

18. Battle of Sexes 3,2 1,1 0,0 2,3 Bob Alice Movie A Movie B Movie A Movie B Is this a constant sum game? A)Yes B)No C)Maybe

19. Clicker Question In the Battle of the Sexes game, where Alice and Bob play only pure strategies, what is the strategy profile in which each plays a maximin strategy? A)Both go to A B)Both go to B C)Alice goes to A, Bob goes to B D)Alice goes to B, Bob goes to A E)There are two maximin strategy profiles. (Both go to A and both go to B)

20. Maximin strategies 3,2 1,1 0,0 2,3 Bob Alice Movie A Movie B Movie A Movie B What would Bob do if he thinks Alice will always do what is worst for Bob? His worst payoff if he goes to A is 0. His worst payoff if he goes to B is 1. His maximin strategy is therefore go to B. What is Alice’s maximin strategy?

21. If Alice and Bob both use maximin strategies in this game, is the outcome a Nash equilbrium? A)Yes B)No

22. In the Movie Choice game between Alice and Bob, suppose that Alice knows that Bob has a a spy who will find out Alice’s strategy before he chooses his. Alice’s best option is to use her maximin strategy. A)Yes B)No

23. What should Alice do if she knows that Bob will find out her strategy before she moves? A)Go to A B)Go to B C)Use a mixed strategy

24. Clicker question In a non-constant sum game, the strategy profile where each uses a maximin strategy must be a Nash equilibrium. A)Yes B)No

25. Working Some Problems

26. Penalty Kick.9,.1.5,.5.3,.7.8,.2 Jump Left Jump Right Goalkeeper Shooter Kick Left Kick right Is this a constant sum game? A) Yes B) No

27. The story When Goalie guesses wrong, Kicker either scores or misses the goal. When Goalie guesses right, Goalie sometimes but not always stops the ball. Kicker kicks harder and more accurately to the right than to the left. Both players know this. (Fairly realistic in professional soccer.)

28. Finding N.E. goalie strategy Suppose Goalie goes Left with probability p Payoff for Shooter from shooting Left is.3p+.8(1-p)=.8-.5p Payoff for Shooter from shooting Right is.9p +.5(1-p)=.5+.4p Shooter will randomize if.8-.5p=.5+.4p Solve this equation to find p=1/3

29. Finding N.E. Shooter strategy If Shooter goes Left with probability q: Payoff to Goalie from jumping Left is.7q+.1(1-q)=.1+.6q Payoff to Goalie from jumping Right is.2q+.5(1-q)=.5-.3q Goalie will randomize if.1+.6q=.5-.3q which implies that q=4/9.

30. Does this game have a Nash equilibrium in which Kicker mixes left and right but does not kick to center?

31. If there is a Nash equilibrium where kicker never kicks middle but mixes between left and right, Goalie will never play middle but will mix left and right (Why?) If Goalie never plays middle but mixes left and right, Kicker will kick middle. (Why?) So there can’t be a Nash equilibrium where Kicker never kicks Middle. (See why?)

32. Problem 4: For what values of x is there a mixed strategy Nash equilibrium in which the victim might resist or not resist and the Mugger assigns zero probability to showing a gun?

33. Problem 7.7, Find mixed strategy Nash equilibria

34. c dominates a and y dominates z A mixed strategy N.E. strategy does not give positive probability To any strictly dominated strategy Look at reduced game without these strategies

35. Problem 7.7 Find mixed strategy Nash equilibia For player 1, Bottom strictly dominates Top. Throw out Top Then for Player 2, Middle weakly dominates Right. Therefore if Player 1 plays bottom with positive probability, player 2 gives zero Probability to Right. There is no N.E. in which Player 1 plays Bottom with zero probability, (Why?) (If he did, what would Player 2 play? Then what would 1 play?)

36. A Nash equilibrium is any strategy pair in which the defense defends against the outside run with probability.5 and the offense runs up the middle with probability.75. No matter what the defense does, The offense gets the same payoff from wide left or wide right, So any probabilities pwl and pwr such that pwl+pwr=.25 will be N.E. probabilities for the offense. Problem 8, Chapter 7

37. Problem 10. Each of three players is deciding between the pure strategies go and stop. The payoff to go is 120, where m is the number of players that choose go, and the payoff to stop is 55 m (which is received regardless of what the other players do). Find all Nash equilibria in mixed strategies. Let’s find the “easy ones”. Are there any symmetric pure strategy equilibria? How about asymmetric pure strategy equilibria? How about symmetric mixed strategy equilibrium? Solve 40p^2+60*2p(1-p)+120(1-p) 2 =55 40p 2 -120p+65=0

38. What about equilibria where one guy is in for sure and other two enter with identical mixed strategies? For mixed strategy guys who both Enter with probability p, expected payoff from entering is (120/3)p+(120/2)(1-p). They are indifferent about entering or not if 40p+60(1-p)=55. This happens when p=1/4. This will be an equilibrium if when the other two guys enter with Probability ¼, the remaining guy is better off entering than not. Payoff to guy who enters for sure is: 40*(1/16)+60*(3/8)+120*(9/16)=92.5>55.

39. RockPaperScissors Rock0,0-1,12,-2 Paper1,-10,0-1,1 Scissors-2,21,-10,0 Advanced Rock-Paper-Scissors Are there pure strategy Nash equilibria? Is there a symmetric mixed strategy Nash equilibrium? What is it?

40. RockPaperScissors Rock0,0-1,12,-2 Paper1,-10,0-1,1 Scissors-2,21,-10,0 Finding Mixed Strategy Nash Equilibrium Let probabilities that column chooser chooses rock, paper, and scissors be r, p, and s=1-p-r Row chooser must be indifferent between rock and paper This tells us that -p+2(1-p-r)=r-(1-p-r) Row chooser must also be indifferent between rock and scissors. This tells us that –p+2(1-p-r)=-2r+p We have 2 linear equations in 2 unknowns. Let’s solve. They simplify to 4r+4p=3 and 4p=2. So we have p=1/2 and r=1/4. Then s=1-p-r=1/4.

41. Expected 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”.

42. Expected 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 2.5 million dollars? A)Yes B)No

43. Expected 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? A)Yes B)No

44. Expected 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? A)Yes B)No

45. Construct 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.

46. Assigning 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.

47. 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?

48. 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 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*.

49. Clicker question What is the strategy profile in which goalie and shooter both use maxmin strategies? A) Shooter shoots Left, Goalie jumps Right B) Shooter shoots Left, Goalie jumps Left C) Shooter shoots Right, Goalie jumps Right D) Shooter shoots Right, Goalie jumps Left E) None of the above