8. The Borda Count is a common way of making a choice among more than two alter- natives. As defined in the text, each member of society assigns a rank to the social alternatives (1,2,...), with 1 corresponding to first choice, and so on. The ranks each alternative gets are summed over all the individuals. The alternative with the low est sum wins; the second choice has the second-lowest sum, and so forth. Set up an example to show that the Borda Count violates the independence of irrelevant alter natives. Hint: Use a three-person society with three alternatives and rank them using
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- Heterogeneity in an endowment economy Suppose we have two typesof households: A and B. The utility maximization problem for a consumerof type i is max InCi,t+β ln Ci,t+1 Ct,Ct+1subject to Ci,t +Ci,t+1/ 1 + rt = Yi,t + Yi,t+1 / 1 + rt Note that the A and B households have the same discount rate and the same utility function. The only thing that is possibly different is their endowments. 1, Write down the Euler equation for households A and B.The Condorcet paradox illustrates Arrow’simpossibility theorem by showing that pairwisemajority votinga. is inconsistent with the principle of unanimity.b. leads to social preferences that are not transitive.c. violates the independence of irrelevantalternatives.d. makes one person in effect a dictator.Define the Various Mathematical Implications of Homoskedasticity?
- Suppose two bidders compete for a single indivisible item (e.g., a used car, a piece of art, etc.). We assume that bidder 1 values the item at $v1, and bidder 2 values the item at $v2. We assume that v1 > v2. In this problem we study a second price auction, which proceeds as follows. Each player i = 1, 2 simultaneously chooses a bid bi ≥ 0. The higher of the two bidders wins, and pays the second highest bid (in this case, the other player’s bid). In case of a tie, suppose the item goes to bidder 1. If a bidder does not win, their payoff is zero; if the bidder wins, their payoff is their value minus the second highest bid. a) Now suppose that player 1 bids b1 = v2 and player 2 bids b2 = v1, i.e., they both bid the value of the other player. (Note that in this case, player 2 is bidding above their value!) Show that this is a pure NE of the second price auction. (Note that in this pure NE the player with the lower value wins, while in the weak dominant strategy equilibrium where both…Three players (Allen, Mark, Alice) must divide a cake among them. The cake is divided into three slices.The table below shows the value of each slice in the eyes of each of the players. S1 S2 S3 Allen $7.00 $6.00 $5.00 Mark $4.00 $4.00 $4.00 Alice $5.00 $4.00 $6.00 Which of the slices does Allen deem fair? Group of answer choices S1 and S2 S1 and S3 S2 and S3 S1, S2, and S3 S1 only3.17. Consider the social choice function that selects as a winner thecandidate (or candidates) with the most second-place votes.(a) Show that this method violates monotonicity.(b) Show that this method violates the Pareto criterion.(c) Show that this method is anonymous and neutral.
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