SPWeek6CaseStudies

Consider the following game:                                 Player 2                                  In     Out  Player 1      In     -2,-2    2, 0                      Out    0, 2     0, 0 (a) What is the Nash equilibrium of this game, or what are the Nash equilibriaof this game? (b) Does either firm have a dominate strategy (a strategy that is always abest response)? Which? (c) Suppose Player 1 could move before Player 2 and Player 2 could observe Player 1’s move. What do you think would happen?

Player 1 Cooperate (C) Defect (D) Cooperate (C) 3,3 8,0 Player 2 Defect (D) 0,8 1,1 In general, a combination of strategies is a Nash equilibrium if ... Every player is choosing a best response against the other players' strategies. Every player has a positive payoff. The players maximize the sum of their payoffs. The players choose identical strategies. If the game is repeated, which cooperative actions could benefit both players? O Both players choose C. Player 1 chooses C, Player 2 chooses D. O Player 1 chooses D, Player 2 chooses C. Both players choose D.

Solve for the Nash equilibrium (or equilibria) in each of the following games. (a) The following two-by-two game is a little harder to solve since firm 2’spreferred strategy depends of what firm 1 does. But firm 1 has a dominantstrategy so this game has one Nash equilibrium.                                         Firm 2                                 Launch       Don’tFirm 1      Launch     60, -10      100, 0                  Don’t       80, 30      120, 0 What is the Nash equilibrium of this simultaneous-move game? (b) What would the outcome of this game be if instead firm 1 moved first and then, after seeing what firm 1 chose, firm 2 chose it strategy? In this case firm 1 doesn’t necessarily need to choose a best response, but firm 2 must choose a best response since it moves second.

Player 1 Cooperate (C) Defect (D) If the game has a dominant strategy, what is it? There is none. If the game has a Nash equilibrium in pure strategies, what is it? There is none. Cooperate (C) 3,3 8,0 Cooperate (C) is a dominant strategy for both players. Defect (D) is a dominant strategy for both players. Cooperate (C) is a dominant strategy for 1, and Defect (D) is a dominant strategy for 2. C, C is the only Nash equilibrium. D, D is the only Nash equilibrium. C, C and D, D are both Nash equilibria. Player 2 Defect (D) 0,8 1,1

5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:ˆA writes down either number 1 or number 2, and B must guess which one.ˆIf the number that A has written down is i and B has guessed correctly, B receives i units from A.ˆIf B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.ˆSuppose B guesses 1 with probability p and 2 with probability 1 −p.ˆLet X1 denote B’s gain (or loss) in a game where A has written down 1.ˆLet X2 denote B’s gain (or loss) in a game where A has written down 2.(a)  Find the pmf of X1 and X2(b)  Find B’s expected gain for these two cases, E[X1] and E[X2].(c)  What value of p maximizes the minimum possible value of B’s expected gain?Now consider the expected loss of player AˆSuppose that A writes down 1 with probability q and 2 with probability 1 −q.ˆLet Y1 be A’s loss (or gain) if B chooses number 1.ˆLet Y2 be A’s loss (or gain) if B…

Aedri Quick Lesson in Game Theory A Nash Equilibrium is an outcome in which neither player is better off by changing their strategy. 2.7 Is a Dominant Strategy equilibrium also a Nash equilibrium? a) Yes b) No esc The Ice Cream Guys $3.99 $4.99 + Chucky's Chunky CCT: $20,000 $3.99 ICG: $20,000 CCT: $60,000 ICG: $10,000 tab Treats CCT: $10,000 $4.99 ICG: $60,000 CCT: $40,000 ICG: $40,000 The table above is the payoff matrix for the annual profit of the only two ice-cream-truck firms operating in Beach City. They are deciding the price of an ice cream cone. %3D caps lo 2.8 What is Chucky's dominant strategy? a) $3.99 b) $4.99 c) Not enough information hift 2.9 What is the dominant strategy equilibrium in this situation? a) Both charge $3.99 b) Both charge $4.99 c) CCT charges $3.99 and ICG charges $4.99 d) CCT charges $4.99 and ICG charges $3.99 2.10 Suppose these two firms engaged in collusion (which, of course, totally doesn't happen because it is against the law). Which outcome would…

rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.

es What are the necessary features of a prisoner's dilemma-type game? Check All That Apply Two prisoners must be locked in separate jail cells. There must be a unique Nash equilibrium. The Nash equilibrium outcome must be inefficient. The game must have exactly three players. The game must have a dominant strategy.

Consider this extensive form game. The top number is player 1’s payoff, themiddle number is player 2’s payoff, and the bottom number is player 3’spayoff.a. Describe the general form of a strategy for each player.b. Find the SPNE.

Player 1 b C d Player 2 X y 3,2 1,1 4,3 3,5 1,3 3,0 2,4 1,5 2,1 0,1 1,2 1,0 1,0 2,0 2,1 4,2 W N . Does this game have any strategies that are strictly dominated?

K In a game, there are two players: Player 1 and Player 2. They have two strategies to select from, A and B. Where the first payoff and the second payoff in every cell is for Player 1 and Player 2 respectively. Which of the following is an example of a game without a dominant strategy? (Check all that apply.) A. C. Player 1 Player 1 A B A B A 3,-3 1, 1 A 2,1 0,0 Player 2 Player 2 B 3, -3 5, -5 B 0,0 1,2 B. D. Player 1 Player 1 A B A B A 1,-1 -1, 1 Player 2 Player 2 A 3.5, -1.5 2.7, -1 B -1, 1 1,-1 B -5, 3 0,0 If the dominant strategy for Player 1 is selecting strategy A and the dominant strategy for Player 2 is strategy B, then A B is an example of a c From ame C Murat Çok Di id Kushn y My Hand