Trading Places: Two players, 1 and 2, each own a house. Each player i values his own house at v;. The value of player i's house to the other player, i.e., to player j ‡i, is v₁. Each player i knows the value v; of his own house to himself, but not the value of the other player's house. The values v; are drawn independently from the interval [0, 1] with uniform distribution. a. Suppose players announce simultaneously whether they want to ex- change their houses. If both players agree to an exchange, the exchange takes place. Otherwise no exchange takes place. Find a Bayesian Nash equilibrium of this game in pure strategies in which each player i ac- cepts an exchange if and only if the value v; does not exceed some threshold 0₁. b. How would your answer to (a) change if player j's valuation of player i's house were v? c. Try to explain why any Bayesian Nash equilibrium of the game de- scribed in (a) must involve threshold strategies of the type postulated

[Adverse Selection] Each of the two players receives an envelope, in which there is anamount of money that is equally distributed from $0, $1, $2, ..., $100. The amounts in twoenvelopes are independent. After receiving the envelope, each individual can check exactlyhow much money is put in his/her own envelope. Then each player has the option to exchangehis/her envelope for the other individual's prize. The decisions are made simultaneously. Ifboth individuals agree to exchange, then the envelopes are exchanged; otherwise, if at leastone player chooses not to exchange, each individual keeps his/her own envelope and receivesits attached sum of money.a. Model this game as a static Bayesian game (write the normal formrepresentation) and find the Bayesian Nash equilibrium.b. Consider a new game where the probability distribution of money in eachenvelope is changed. The amount is equal to $100 with probability 90%, and is equalto each number in $0, $1, $2, ... ,$99 with probability 0.1%.…

5 players must simultaneously choose a number from the interval [0, 9]. Let xi denote the number chosen by player i; x = (x1, . . . , x5) all the selected numbers; x−i = x\xi all the selected numbers except the one chosen by player i. If player i chooses the number xi ∈ [0, 9], while the others choose numbers x−i, then the utility of player i is given by ui(xi, x−i) = 4(9 − xi) + 2Σ5 j=1 xj. Find the Nash equilibrium of the game and the corresponding utilities. Would all players be better off if they signed a binding contract before the game, obligating them to choose the number 9?

Don't give poor answer

At a company, 20 employees are making contributions for a retirement gift. Each of the 20 employees is choosing how many dollars to contribute from the interval [0,20]. The manager of these 20 employees announces that she will contribute dd dollars for every dollar that an employee contributes. The payoff to employee ii who makes contribution of xixi​ dollars is bi(1+d)xi−xi, where bi>0.bi​(1+d)xi​−xi​, where bi​>0. Assume that d=4d=4, bi=0.25bi​=0.25 for i=1,2,…,10i=1,2,…,10, andbi=0.5bi​=0.5 for i=11,12,…,20i=11,12,…,20 What is the best contribution level of any employee ii for i=1,2,…,10i=1,2,…,10.   At a company, 20 employees are making contributions for a retirement gift. Each of the 20 employees is choosing how many dollars to contribute from the interval [0,20]. The manager of these 20 employees announces that she will contribute dd dollars for every dollar that an employee contributes. The payoff to employee ii who makes contribution of xixi​ dollars is bi(1+d)xi−xi,…

7

How would this exercise be solved step by step?