1. Suppose that there is a risky asset with liquidation value v which is a realization...

  

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1. Suppose that there is a risky asset with liquidation value v which is a realization of ~N(0,0%). Investors receive private signals about this liquidation value. In reality, investors receive signals according to s = +ĕ with N(0, 0). However, they think signals are distributed in the following way: ~N(0, 0). Assume that and è are independenty distributed. (a) Give an interpretation when c = 1 (b) Given an interpretation when 0 <c<1 =+c. ĕ with (c) Under what conditions would an investor buy 1 unit of the asset at a price of p (as opposed to doing nothing which gives a utilty of 0). You can assume that investor has linear utility for money. (d) Under what conditions would an investor (holding the asset) sell 1 unit of the asset (as opposed to doing nothing which gives a utilty of 0). Again, you can assume that investor has linear utility for money. Assume for simplicity that the investor selling the asset knows the true liquidation value. (e) Suppose that their is a continuum of investors i E [0, 1] that each hold a unit of the asset that are willing to sell under conditions in (d). And, their is a continuum of investors j = [0, 1] that are willing to buy a unit under conditions in (c). For a given price p, how many trades do you expect to happen as a function of CE [0, 1]. Give intuition for your answer. 2. Consider the standard ultimatum game (1 proposer and 1 responder). Can you give intuition for what you would expect to happen if people had reciprocity based preferences? 3. An experimenter hypothesizes that people are equally adverse to inequality both when it is an isn't in their favour. He decides to use the utlimatum game to explore this hypothesis. But, there is an observation problem with the ultimatum game in that it is very rare to see the proposer make offers bigger than half the pie so the experimentalist will not be able to observe what responders do when receive unequal proposals in their favour. So the experimenter decides to solve this problem by replacing the proposer with a computer who randomly selects a proposal (from all possible proposals) which will solve the observability problem. Using what you learned about social preferences explain what you would expect to observe about responders behaviour in such an experiment. 4. The ultimatum bargaining game experiment was designed to test theories of alternating-offers bargaining. The consistent failure of ultimatum bargaining to reach the subgame-perfect prediction has raised many fun- damental questions about bargaining models and individual preferences. One of the most puzzling behavior in the ultimatum game experiment is that of responders. Many responders reveal a preference for allocations that give both players less consumption. To understand this question, Andreoni, Castillo, and Petrie (AER 2003) consider a modified version of the ultimatum game, called the convex ultimatum game. There are two players, a proposer and a responder, who are to split 100 pounds between them. The proposer first specifies the proportion of the money that will go to the responder, 0 < a <1. The responder then determines how much money to divide, from 0 to 1 100 pounds, denoted by0 m < 100. Then the payoff functions for the proposer, TP, and the responder, TR, are Tp = (1-a)m TR = am The convex ultimatum game differs from the standard ultimatum game in which the proposer makes an offer, 0M 100, to the responder who then decides to either accept or reject the offer. (a) Draw the set of all possible divisions of 100 (including the case of the responder rejecting the offer so that both players receive nothing) for the standard ultimatum bargaining game in the 2-dimensional space of the proposer's payoff (x-axis) and responder's payoff (y-axis) The convex ultimatum game differs from the standard ultimatum game in which the proposer makes an offer, 0 <M<100, to the responder who then decides to either accept or reject the offer. (a) Draw the set of all possible divisions of 100 (including the case of the responder rejecting the offer so that both players receive nothing) for the standard ultimatum bargaining game in the 2-dimensional space of the proposer's payoff (x-axis) and responder's payoff (y-axis). (b) Consider a situation in which the proposer chooses 40% of the money to go the responder, i.e. a = 0.4, in the convex ultimatum game. Given this, draw the set of all possible divisions of 100 between the proposer and the responder. Considering the standard ultimatum game again, identify the payoff consequences of both rejecting and accepting the offer in the standard ultimatum game when the offer to the responder is 40. (c) Suppose that the responder is self-interested. That is, the responder's utility function is given by UR(TP,TR)=TR Draw the indifference curves over the space of (TP, TR) and determine the optimal decision by the responder in each case where the proposer offered £40 to the responder (in the standard ultimatum game) and where the proposer chooses 40% of the money to go the responder (in the convex ultimatum game). (d) Suppose that the responder has a non-monotonic preference such as UR(TRP) = TRẞp Draw the indifference curves when ẞ> 4/6 and determine the optimal decision by the responder in the case where the proposer chooses 40% of the money to go the responder in the convex ultimatum game. Conduct the similar analysis when < 4/6. Does this type of individual behave differently between the standard ultimatum game and the convex one? (e) Can you suggest any preference type for the responder that behaves differently between the convex ultimatum game and the standard one? 5. Consider the ultimatum game with 10 proposers and 1 responder. Each proposer offers a split of £10 to the responder. The responder can either reject all splits, in which case all players earn 0, or the responder can accept one of the proposed splits, in which case the chosen proposer and the responder earn according to the split and all other proposers earn 0. Discuss the predictions of inequality-aversion for this game. 1. Suppose that there is a risky asset with liquidation value v which is a realization of ~N(0,0%). Investors receive private signals about this liquidation value. In reality, investors receive signals according to s = +ĕ with N(0, 0). However, they think signals are distributed in the following way: ~N(0, 0). Assume that and è are independenty distributed. (a) Give an interpretation when c = 1 (b) Given an interpretation when 0 <c<1 =+c. ĕ with (c) Under what conditions would an investor buy 1 unit of the asset at a price of p (as opposed to doing nothing which gives a utilty of 0). You can assume that investor has linear utility for money. (d) Under what conditions would an investor (holding the asset) sell 1 unit of the asset (as opposed to doing nothing which gives a utilty of 0). Again, you can assume that investor has linear utility for money. Assume for simplicity that the investor selling the asset knows the true liquidation value. (e) Suppose that their is a continuum of investors i E [0, 1] that each hold a unit of the asset that are willing to sell under conditions in (d). And, their is a continuum of investors j = [0, 1] that are willing to buy a unit under conditions in (c). For a given price p, how many trades do you expect to happen as a function of CE [0, 1]. Give intuition for your answer. 2. Consider the standard ultimatum game (1 proposer and 1 responder). Can you give intuition for what you would expect to happen if people had reciprocity based preferences? 3. An experimenter hypothesizes that people are equally adverse to inequality both when it is an isn't in their favour. He decides to use the utlimatum game to explore this hypothesis. But, there is an observation problem with the ultimatum game in that it is very rare to see the proposer make offers bigger than half the pie so the experimentalist will not be able to observe what responders do when receive unequal proposals in their favour. So the experimenter decides to solve this problem by replacing the proposer with a computer who randomly selects a proposal (from all possible proposals) which will solve the observability problem. Using what you learned about social preferences explain what you would expect to observe about responders behaviour in such an experiment. 4. The ultimatum bargaining game experiment was designed to test theories of alternating-offers bargaining. The consistent failure of ultimatum bargaining to reach the subgame-perfect prediction has raised many fun- damental questions about bargaining models and individual preferences. One of the most puzzling behavior in the ultimatum game experiment is that of responders. Many responders reveal a preference for allocations that give both players less consumption. To understand this question, Andreoni, Castillo, and Petrie (AER 2003) consider a modified version of the ultimatum game, called the convex ultimatum game. There are two players, a proposer and a responder, who are to split 100 pounds between them. The proposer first specifies the proportion of the money that will go to the responder, 0 < a <1. The responder then determines how much money to divide, from 0 to 1 100 pounds, denoted by0 m < 100. Then the payoff functions for the proposer, TP, and the responder, TR, are Tp = (1-a)m TR = am The convex ultimatum game differs from the standard ultimatum game in which the proposer makes an offer, 0M 100, to the responder who then decides to either accept or reject the offer. (a) Draw the set of all possible divisions of 100 (including the case of the responder rejecting the offer so that both players receive nothing) for the standard ultimatum bargaining game in the 2-dimensional space of the proposer's payoff (x-axis) and responder's payoff (y-axis) The convex ultimatum game differs from the standard ultimatum game in which the proposer makes an offer, 0 <M<100, to the responder who then decides to either accept or reject the offer. (a) Draw the set of all possible divisions of 100 (including the case of the responder rejecting the offer so that both players receive nothing) for the standard ultimatum bargaining game in the 2-dimensional space of the proposer's payoff (x-axis) and responder's payoff (y-axis). (b) Consider a situation in which the proposer chooses 40% of the money to go the responder, i.e. a = 0.4, in the convex ultimatum game. Given this, draw the set of all possible divisions of 100 between the proposer and the responder. Considering the standard ultimatum game again, identify the payoff consequences of both rejecting and accepting the offer in the standard ultimatum game when the offer to the responder is 40. (c) Suppose that the responder is self-interested. That is, the responder's utility function is given by UR(TP,TR)=TR Draw the indifference curves over the space of (TP, TR) and determine the optimal decision by the responder in each case where the proposer offered £40 to the responder (in the standard ultimatum game) and where the proposer chooses 40% of the money to go the responder (in the convex ultimatum game). (d) Suppose that the responder has a non-monotonic preference such as UR(TRP) = TRẞp Draw the indifference curves when ẞ> 4/6 and determine the optimal decision by the responder in the case where the proposer chooses 40% of the money to go the responder in the convex ultimatum game. Conduct the similar analysis when < 4/6. Does this type of individual behave differently between the standard ultimatum game and the convex one? (e) Can you suggest any preference type for the responder that behaves differently between the convex ultimatum game and the standard one? 5. Consider the ultimatum game with 10 proposers and 1 responder. Each proposer offers a split of £10 to the responder. The responder can either reject all splits, in which case all players earn 0, or the responder can accept one of the proposed splits, in which case the chosen proposer and the responder earn according to the split and all other proposers earn 0. Discuss the predictions of inequality-aversion for this game.

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Answering the Finance and Economic Game Theory Questions 1 Investment Decisions Based on Perceived Asset Value and Private Signals a Interpretation when c1 When c1 the noise in the investors signal ab... View the full answer