a.) How many Subgames does this Game have? b.) Convert this Game Tree into matrix form. c.) Find all Pure Strategy Nash Equilibria of this Game. d.) Find all Subgame Perfect Nash Equilibria of this Game. (Pure and Mixed)
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- Question 1 Consider the following game. Find all Nash equilibria, subgame perfect Nash equilibria, and weak perfect Bayesian equilibria in pure and mixed strategies. a P2 (12, 12) b (12,-8) (7.-3) P1 m d (-3,2) r P2 C (2,7) d (22,17)FOOP 6. (a) For the following extensive-form game: i. Identify the pure and mixed strategy Nash Equilibria. ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in part (a)? Explain (a couple of sentences should suffice). (3,1) A D B C (-2,-2) (2,5) D (0,7)5) Mixed strategy Nash equilibrium Consider a mixed strategy Nash equilibrium of the following coordination game: Player 2 Player 1 A B a 5.5 6.-2 b -2,6 1,1 a) In the above game, explain in words what condition player 1's probability p of playing strategy A must satisfy to induce player 2 to mix strategies between a and b in equilibrium. b) Solve for the mixed strategy Nash equilibrium where player 1 chooses A with probability P and player 2 chooses a with probability p, for 0 < p, P < 1.
- Question Completion Status: Consider Game 3: -10 -20 -20 3. 30 -30 -10 Write a normal form for this game. How many pure-strategy Nash Equilibria are there in this game? OA. 1 O8.2 OC3 O0.4 OE. 5 OF More than 5Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 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.a) Find the pure strategy Nash Equilibrium for the subgame starting at player l's second decision node. b) Find the Subgame Perfect Nash Equilibria for the entire game. 1 Withdraw Not 2 Withdraw Not Withdraw Not 10 2 6 2 10 Not Withdraw 2 Withdraw Not Withdraw Not 15 20 10 15 15 10 20 15 49
- а.) Your Crush Love Friendship You Love (100, 60) (30, 120) Friendship (60, 50) (40, 40) i. What is(are) the Nash equilibrium (equilibria), if any, under pure strategies in the above simultaneous game? i. What is the Nash equilibrium under mixed strategy? ii. What is the expected payoff for you and your crash respectively from the game under the mixed strategy in (ii.)?NE 2). Consider the following extensive form game between two players. (1,10) u D 1 B X (a) List all pure strategies of player 2. (b) Represent this game in normal form. (c) Find all pure-strategy Nash equilibria of this game. (d) Find all SPNE (in pure strategies) of this game. (6,3) (4,2) (5,1)(d) Consider a simultaneous-move game between two firms choosing to sell their product at either £6, £7 or £8. The actions and payoffs are given in the matrix below. Firm 2's Prices £6 £7 £8 Firm 1's prices £6 4, 5 3, 5 2, 1 £7 0,4 2, 1 3,0 £8 -1, 1 4, 3 0, 2 What are the Nash equilibria of this game? Game theory is often used by firms competing under an oligopoly as a means of determining their best strategy. Why is game theory a useful tool and which characteristics of an oligopoly make it particularly useful for firms competing in this market structure? One outcome of an oligopoly is that firms may have an incentive to collude. Explain some of the conditions that make collusion more likely to occur and how game theory can explain why collusive agreements often break down.
- AsapConsider the extensive form game shown below. At each terminal node, the first payoff belongs to player 1, the second payoff belongs to player 2, and the third payoff belongs to player 3. How many strategies does player 3 have in this game? Numerical answer (12 (2,2,1) 2 b₂ (22, (2,4,2) (4,2,3) b₁ 2 (3,3,3) C1 02/ 03 b2 b3 (2,2,1) (4,2,0) (2,0,2) 03 3 b3 (0,3,4) image 1P2: Hawk Dove P1: Hawk -25, -25 50, 0 Dove 0, 50 15, 15 Is the mixed-strategy NE of the above game an evolutionary stable strategy (ESS)? Show your work and explain your answer.