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- A game involves two players: player A and player B. Player A has three strategies a1, a2 and a3 while player B has three strategies b1, b2 and b3. Player B b1 b2 b3 a1 -40,30 70,20 -10,120 Player A a2 40,60 80,80 60,20 a3 -30,40 -50,110 150, -70 Assuming that this is a one-time game, answer the following questions: Is there any dominant strategy for each player? What is the secure strategy of each player. What is the Nash equilibrium of the game?2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.GAME Z Player A B1 A1 7, 13 A2 3,8 Player B B2 15, 10 9,16 A3 5,8 4,7 In Game Z (see table above), assuming players move simultaneously. Which of the following is true? Player A has a dominant strategy Player B has a dominant strategy Both players have dominant strategies O Neither player has a dominant strategy
- Two workers are on a production line. They each have two actions: exert effort, E, or shirk, S. Effort costs a worker e > 0 and shirking costs them nothing. If two workers do action E a lot of output is produced and the workers earn £3 each. If only one worker chooses action E less output is produced and they both earn £1. The workers earn nothing if they both shirk. (i) Describe this situation as a strategic form game (assuming the workers do not observe each other's effort choice when making their own decision). (ii) For what values of e does this game have strictly dominant strategies? (iii) Describe the Nash equilibria of this game for e = 0,1, 2, 3. (iv) The workers now are re-arranged into a production line. First worker 1 moves and then worker 2 moves. Worker 2 can now see worker l's effort level before they choose their effort. Draw this extensive form game. (v) Find the subgame perfect equilibria of the production-line game for c = 1/2 and c = 3/2.(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.ALBUS MINERVA N S Z MINERVA لی a C a b 3,3 ALBUS b 0,4 2,2 4 N S 15,2 1,3 13,1 For the above sequential game, both Albus and Minerva make moves in two possible situations. A number in purple indicates which node of the two for each of Albus and Minerva. Albus' strategy NS indicates a strategy of taking action N at node 1 and action 5 at node 4. Minerva's strategy ab indicates a strategy of taking action a at node 2 and action b at node 3. Adopt this format of writing strategies, find the subgame perfect Nash equilibrium. Fill in the blanks: The subgame perfect Nash equilibrium of this game is "Albus" Minvera"
- Consider the payoff matrix below, which describes a symmetric A B A 17,17 0,14 B 14,0 8,8 (a) Find all Nash equilibria of this game, including any mixed strategy equilibria. (b) Draw the expected payoff difference diagram for this game, including all the Nash equilibria you found in (a). is (c) What would you expect to occur most frequently when this game played as a lab experiment? Justify your answer using your answers from (a) and (b).GAME 5 Player B B1 B2 Player A A1 7,3 | 5, 10 A2 3, 8| 9, 6 In Game 5 above, O Neither player has a dominant strategy. O Player B has a dominant strategy. O Player A has a dominant strategy. O Both players have dominant strategies.The following table contains the possible actions and payoffs of players 1 and 2. Player 1 U M D L 2,10 2,5 8,7 Player 2 C 5,2 11,8 8,4 R 5,4 2,9 8,3 This is a simultaneous move game. In a pure strategy Nash equilibrium of this game, Player 1 receives a payoff of ✓and player 2 receives a payoff of If, instead of playing simultaneously, player 2 moves first, then in the Nash equilibrium player 1 receives a payoff of ✓and player 2 receives a payoff of
- (a) Consider a ROCK PAPER SCISSOR game. Two players indicate either Rock, Paper or Scissor simultaneously. The winner is determined by: Rock crushes Scissors, Paper covers Rock, and Scissor cut Paper. In the case of a tie, there is no payoff. In the case of a win, the winner collects 5 dollars. Write the payoff matrix for this game. (b) Find the optimal row and column strategies and the value of the matrix game. 3 2 4 -2 1 -4 5Consider the following simultaneous game: Player 1 U D Player 2 L 20,-10 -10, 20 R -10, 20 20,-10 Please indicate whether each of the following statements is true or false. Player 1 has a dominant strategy. This game has a Nash equilibrium. This game has a Nash equilibrium in pure strategies. Player 1's best response is D if player 2 plays R.a) Write out the extensive form of a game between a farmer (playing in the first round) and nature (playing a mixed strategy in the second round). Assume that the farmer can either pay a cash rent of $1500 for land (English system) or 1/2 of crop production to the landlord (sharecropping). Assume the farmer is planting corn and will produce 2 tons of corn. Assume that nature has a 50% chance of playing a strategy in which the price of corn is $3500/ton and a 50% chance of playing a strategy in which the price of corn is $500/ton. The farmer keeps any money left after paying cash rent and sells any corn left after paying the landlord in sharecropping. b)What is the most that a risk neutral farmer would be willing to pay for an accurate prediction of the price of corn in problem 1 before choosing whether to pay cash rent or sharecrop?
