Suppose there are 4 agents i e (1, 2, 3, 4) and 4 objects je (a, b, c, d). Below is a matrix of probability shares. Show how to represent it as a lottery over deterministic object allocations. Agent Good abcd10.5 0.5 002 0.125 0 0.875 0 3 0.125 0.5 0.125 0.25 4 0.2500 0.75
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- Suppose there are 4 agents i e (1, 2, 3, 4} and 4 objects je {a, b, c, d). Below is a matrix of probability shares. Show how to represent it as a lottery over deterministic object allocations. Agent Good abcd10.5 0.5 0020.125 0 0.875 0 3 0.125 0.5 0.125 0.25 4 0.25 0 0 0.75The injured football player Bad news everyone! There is 1 second left in the game, and Tom Brady has injured himself. The matrices below depict the relative probabilities of winning givenan offensive and a defensive play call. (The row player is the New England Patriots and the column player is the opponent.) How much has the all star's home team probability of winning decreased due to the injury? Pass uny Patriots D Pass .4, .6 D Run .9,.1 .8,.2 .5,.5 Pass Run Opponent D Pass D Run .06, .94 .32, .68 .8,.2 .5,.5. Ayça and Barış are playing a game and following payoff matrix is for the payoffs of Ayça. Answer the questions according to the following payoff matrix. a) What is the probability that the value of the game is 10?
- 1. A dealer decides to sell a rare book by means of an English auction with a reservation price of 54. There are two bidders. The dealer believes that there are only three possible values, 90, 54, and 45, that each bidder’s willingness to pay might take. Each bidder has a probability of 1/3 of having each of these willingnesses to pay, and the probabilities for each of the two bidders are independent of the other’s valuation. Assuming that the two bidders bid rationally and do not collude, the dealer’s expected revenue is approximately ______. 2. A seller knows that there are two bidders for the object he is selling. He believes that with probability 1/2, one has a buyer value of 5 and the other has a buyer value of 10 and with probability 1/2, one has a buyer value of 8 and the other has a buyer value of 15. He knows that bidders will want to buy the object so long as they can get it for their buyer value or less. He sells it in an English auction with a reserve price which he must…You and a coworker are assigned a team project on which your likelihood or a promotion will be decidedon. It is now the night before the project is due and neither has yet to start it. You both want toreceive a promotion next year, but you both also want to go to your company’s holiday party that night.Each of you wants to maximize his or her own happiness (likelihood of a promotion and mingling withyour colleagues “on the company’s dime”). If you both work, you deliver an outstanding presentation.If you both go to the party, your presentation is mediocre. If one parties and the other works, yourpresentation is above average. Partying increases happiness by 25 units. Working on the project addszero units to happiness. Happiness is also affected by your chance of a promotion, which is depends on howgood your project is. An outstanding presentation gives 40 units of happiness to each of you; an aboveaverage presentation gives 30 units of happiness; a mediocre presentation gives 10 units…2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
- Imagine that a zealous prosecutor (P) has accused a defendant (D) of committing a crime. Suppose that the trial involves evidence production by bothparties and that by producing evidence, a litigant increases the probabilityof winning the trial. Specifically, suppose that the probability that the defendant wins is given by eD>(eD + eP), where eD is the expenditure on evidenceproduction by the defendant and eP is the expenditure on evidence production by the prosecutor. Assume that eD and eP are greater than or equal to0. The defendant must pay 8 if he is found guilty, whereas he pays 0 if heis found innocent. The prosecutor receives 8 if she wins and 0 if she losesthe case. (a) Represent this game in normal form.(b) Write the first-order condition and derive the best-response function foreach player.(c) Find the Nash equilibrium of this game. What is the probability that thedefendant wins in equilibrium.(d) Is this outcome efficient? Why?3. Find the saddle point, if it exists, for the following game. (b) Solve the following game by using the principle of dominance and find the probabilities of strategies for each player and the value of the game. Player B Player A II III IV V 3 4 4 II 2 4 III 4 4 IV 4 4 20 2420 8760Consider two firms who are engaged in a Research and Development (R&D) "con- test". Both firms simultaneously expend resources to try to win the contest (which may mean developing a superior product or developing a product before the com- petitor). If the two firms expend bị and b2, respectively, on R&D, the probability that firm 1 wins the contest is if b1 = b2 =0 BE otherwise P1(b1, b2) = where r is some exogenous constant, r E (0, 0). If firm 1 wins the contest, it will subsequently earn revenue of 1 (not including the cost of R&D, b1). If firm 1 loses the contest, it will earn zero revenue, and thus lose bj in total. Hence, firm l's expected profit is n'(b1,b2) = p1(b1,b2) - b1. Everything is symmetric for firm 2. i. How does p1(b1,0) depend on b1? Is it an equilibrium for both firms to spend nothing on R&D (b1 = b2 = 0)? Prove and explain your answer. For which values of r is n' (b,b2) concave in b1 when b2 > 0? ii. Consider the possibility of a symmetric pure-strategy…
- Consider “Providing a Public Good under Incomplete Information". (refer to pages 70-74, lecture notes.) If c and cz have the following distribution. 2. 0.5 1.2 C2 0.5 1.2 Prob 1/2 1/2 Prob 1/3 2/3 Find all Bayesian Nash Equilibria of this game. Select all cOTrect amswers. Don't copy Chegg give new answer ASAPSuppose the payoff matrix was modified such that: Confess Lie About Lie About Confess to Peter / MJ About Knowing Liking Peter Liking Peter Knowing Admit Being 2,2 3,1 8,12 3,14 Spidey Lie About 1,1 4,2 9,8 1,7 Spidey Lie About 4,5 8,6 14,18 4,14 Liking MJ Confess to 5,2 7,1 12,14 5,16 Liking MJTwo friends, Khalid and Mahmood, are going to a watch a world cup football match. They play a simple game in which they hold out one or two fingers to decide who will pay for the other's ticket. Khalid wins if the fingers held out add up to an even number; Mahmood wins if the fingers held out add up to an odd number. The price of the ticket is 25 OMR. Construct a payoff matrix for the game. Is there a unique Nash equilibrium in this game? Which strategy should a player use to maximize her chances of winning the game?