Consider the following two-player game. S; = [0, 1], for i = 1, 2. Payoffs are as follows S 100 if s1 + 82 if s1 = 82 u1 (81, 82) = u2 (81, 82) = 150 – [82 – 81 – . Part a: Describe B1. Explain. Part b: Describe B2. Explain.
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- Define a new game as a modified version of the game in Problem 1, which we will call the Matching Two Pennies game: each player choose heads (H) or tails (T) for two pennies. In this game, Player 1 wins if both coins show the same combination of heads and tails, while Player 2 wins if they do not. (a) Write the strategy sets S1 and S2 for this game. (b) Give the payoffs for each player in this Matching Two Pennies game, by defining T;(81, 82) for each player and for each strategy pair, either as a list or in matrix form.1. Contract Enforcement Between Buyer and Seller in a Supply Chain. Suppose that two firms, 1 and 2, must undergo a certain amount of investment so as to coordinate their efforts in order to enable the supplier to produce a good to be used as an input in the buyer's production process. Such investment may involve the costs incurred by both parties in the initial planning and design phase of the relationship and/or involve the firms in having to incur the associated costs of purchasing specialized equipment and developing certain production processes in order to satisfy the needs and specific requirements of both parties. In any case, suppose that the technology of the relationship and the structure of the underlying game are given by the following normal form representation of the game: Firm 1 Invest (I) Not Invest (N) Firm 2 Invest (I) Z₁, Z₂ (41, 35) X1, Y2 (92,-42) Not Invest (N) Y1, X2 (-32, 21) (0, 0) Further suppose that the players contract in a setting of court-imposed breach…Suppose d = 1/4. Before the game starts, you tell the players that you will offer each of them money m > 0 at each period both of them cooperate. The new per period payoff matrix is as follows: (d) m≥ 0.4 (e) m≥ 1/1/3 C D At least how much should you offer to make the strategy described in part 6 a SPNE? (a) m≥ 1/1/5 9 (b) m≥ 0.2 (c) m≥ 0.3 این C D 2+m, 2+m| -3, 3 3, -3 1, 1
- 1.1. What is a game in normal form? What is a strictly dominated strategy in a normal-form game? What is a pure-strategy Nash equilibrium in a normal-form game? 1.2. In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies? What are the pure-strategy Nash equilibria? L C R T 2.0 1,1 4,2 M 3,4 1,2 2,3 B 1,3 0,2 3,04. You have probably had the experience of trying to avoid encountering someone, whom we will call Rocky. In this instance, Rocky is trying to find you. It is Saturday night and you are choosing which of two possible parties to attend. You like Party 1 better and, if Rocky goes to the other party, you get a payoff 20 at Party 1. If Rocky attends Party 1, however, you are going to be uncomfortable and get a payoff of 5. Similarly, Party 2 gives you a payoff of 15, unless Rocky attends, in which case the payoff is 0. Rocky likes Party 2 better, but he is likes you. He values Party 2 at 10, Party 1 at 5, and your presence at either party that he attends is worth an additional payoff of 10. You and Rocky both know each others strategy space (which party to attend) and payoffs functions.3. Consider the game represented by the payoff matrix below: R S (12, 12) (8,-5) (25, 14) (50, 12) (26,70) (-2.6) (20, 100) (22,75) C (11,0) (3,25) (24, 26) (50,26) D (5,-10) (16,-8) (13,-8) (50,-20) A B a. Find the Nash equilibria of the game using any suitable method. b. Apply Iterated Elimination of Strictly Dominated Actions (IESDA) to this game. Explain the steps as necessary. Do the Nash equilibria found in part a. survive IESDA? c. Apply Iterated Elimination of (Weakly) Dominated Actions (IEDA) to this game by elim- inating all whakly dominated actions in each round. Explain the steps as necessary. Do the Nash equilibria found in part a, survive IEDA
- 3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤ 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X₂, be your…Consider the following game in extensive form, involving the chance parameter p. A P[T] = P (0,0) N U P[H] = 1-p h t B h d P[T] = P (-80, 80) (20,-20) (60,-60) t (0,0) n P[H] = 1-p (0,0) Suppose that the strategy profile (u, h) is played and player B receives an expected payoff of 12. What is the value of p? (Report your answer in decimal form, rounding if necessary to two places.)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)
- 6. Consider a partnership game between two people, in which each player simultaneously chooses an effort e; ≥ 0 to expend on a joint project, and receives in return a payoff π₁1 = == R(e1, €2) - e³ where the revenue R shared between the players is described by the function R(e₁, €2) = 100 (e₁ + ₂ + €502) €₂ 1 = R(e₁, 0₂) - π2 = (a) Find the best response functions BR₁(e₂) and BR₂(e₁), and the Nash Equilibrium (ei, e) for this game. (b) Find the payoff amounts ₁ and 7₂ at the NE. Note: recall that for all duopoly games, the payoff function for each player is of the form Tį = P(q1, 92)qi – Ci(qi) uch that the game payoff is equal to the profit (= Revenue - Cost).4. Two investors have each deposited 5 (all amounts are in $10,000s) with a bank. The bank has invested these deposits in a long-term project. If both investors make withdrawals at date 1 then each receives 3 and the game ends. If only one investor makes a withdrawal at date 1 then that investor receives 5, the other receives 1, and the game ends. Finally, if neither investor makes a withdrawal at date 1 then the project matures and the investors make withdrawal decisions at date 2. If both investors make withdrawals at date 2 then each receives 7 and the game ends. If only one investor makes a withdrawal at date 2 then that investor receives 9, the other receives 5, and the game ends. Finally, if neither investor makes a withdrawal at date 2 then the bank returns 7 to each investor and the game ends. Show the bi-matrix for dates 1 and 2.You and I play the following game. Hidden from you, I put a coin in my hand: with probability p it is a 10 pence coin and otherwise it is a 20 pence coin. You now guess which coin is in my hand: you guess it is 20 pence with probability s and otherwise you guess it is a 10 pence coin. You get to win the coin if you guess correctly and otherwise win nothing. What (in terms of p and s) is your expected gain in pence from playing this game once with me? Challenge: suppose we are going to play repeatedly and you want to maximise your gain and I wish to minimise my loss. What value of p should I choose and what value of s should you choose? (This question is somewhat ill-defined, but it does have an interesting possible answer.) (Note: anything labelled "challenge" will not be part of the hand-in.)