1. Using different equilibria to create incentives. Alex and Barry have a joint project. Each has rst to decide whether to invest $10 or zero (i.e., not to invest) into the project. They make these individual investment choices simultaneously. Once made, these investments are sunk. If noone invests, the project generates a total revenue of $0. If just one of them invests, then the project generates a (gross) total revenue of $15. If both of them invests, the project generates a (gross) total revenue of $30. Alex and Barry then use the following scheme to divide the total project revenue. Each player simulta neously writes down a 'share demand' on a piece of paper. The demands can be eithe %,%, or g. If the two 'share demands' add up exactly to one then each player is given his demand. Otherwise, all the money is thrown away. [For example, if at rst Alex and Barry each invest $10 then the project generates a gross total of $30. If Alex then writes down g and Barry writes down % then (since this adds up to one) Alex gets his demand of $24 (= % X $30) for a net prot of $14 (i.e., $24 minus his initial investment of $10), while Barry gets his demand of $6 (= 5 X $30) for a net prot of $4 (i.e., $6 minus his initial investment of $10). If Barry had demanded % while Alex was still demanding 4%, then the project money would have been thrown away and each would simply have lost his initial investment of $10.] Suppose for now that both Alex and Barry are told how much the project has generated before they make their 'share demands'. (a) Consider the subgame that follows Alex choosing to invest $10 and Barry choosing to invest $0. Find all the purestrategy NE of this subgame. (b) Is there a purestrategy subgame perfect equilibrium (SPE) of the whole game in which each player starts by investing $0? If so, explain the equilibrium strategies clearly. If not, explain why not clearly. (c) Is there a purestrategy SPE in which each player starts by investing $10? If so, explain the equilibrium strategies clearly. If not, explain why not clearly. Suppose now that, at the point at which each player has to make his 'share demand', he does not know what amount the other player has invested in the project and hence does not know how much the project has generated. (d) [6 points] Is there now a purestrategy SPE in which each player starts by investing $10. If so, explain the equilibrium strategies clearly. If not, explain why not