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5. (6 marks total) Corporate Division Suppose that there are five executives on the board of directors of a major financial corporation: the CEO and four vice-presidents. After a particularly profitable year, the five executives learn that there is a surplus of 100 million dollars to be shared among the board members, and they call a meeting to decide how to split up the money. After some deliberation, they agree on the following setup: . A proposal about how to split up the money is presented, and then voted upon. A proposal is just a way to split up the 100 million dollars. For simplicity (and because these executives are all very rich) we will assume that the proposal units are in millions of dollars. For instance, one possible proposal for the CEO would be to keep 60 million for herself, and give 10 million to each of the four vice-presidents, another possibile proposal is 100 million for the CEO and zero for everyone else . After each proposal is presented, there is a vote. In order for a proposal to pass, it needs to be agreed on by 50% or more of the board members that are voting (the proposer is allowed to vote for their own proposal) » If the proposal passes, then the money will be split up accordingly; if the proposal is rejected then the board member who made that proposal will lose their right to vote in the future. If the proposal is rejected, the next board member in line will then make a new proposal and the process repeats ·The order for proposals is as follows: CEO → VP1 → VP2 → VP3 → VP4 Assuming that board members are perfectly rational, how will the money be divided? Explain

math   Statistics-And-Probability   
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The problem can be solved if we think it from the back end. Suppose if CEO, VP1, and VP2 lose their voting rights than only VP3 and VP4 will be left to vote and the chance for putting up a proposal will be of VP3 He will propose to keep all the money with himself and give VP4 nothing. Since each proposal requires 50% or more votes to be accepted even if VP4 votes in against of this proposal VP3 will vote in its support and hence proposal will get passed. So, in any case, VP4 does not want the vote to between the last two members i.e. between VP3 and VP4 as he will get nothing Now consider that there are three Voters left i.e. VP2, VP3, and VP4. Now the chance of making Proposal lies with VP2. For acceptance of his proposal, he should get 50% or more votes. VP2 knows that if his proposal is rejected than VP3 will be given a chance to make proposal and VP4 will not get a single penny. So he decided to give VP4 1 million dollars and keep the rest with him. In this case, he secures the vote of VP4 and thus can pass his proposal with a 66.6% majority. In this case, VP3 doesnt receive a single penny Now consider that there are 4 voters left that is VP1 VP2 VP3 and VP4. According to the seniority, VP1 has the right to make a proposal. He knows that apart from his vote he needs just one more vote for his proposal to pass. He also knows that VP3 does not want the decision to pass to VP2 because in that case, he will not get a single penny as mentioned in the above case. So what VP1 does is he gives 1 million dollars to VP3 to secure his vote and keeps the rest 99 million dollars with him. By doing so he gets a 50% majority and his proposal passes. In this case, VP2 and VP4 do not get a Single Penny Now we are back to our initial case where the CEO, as well as all the other VPs, have their voting rights intact. After considering all the above cases the CEO knows that if his proposal gets rejected he loses his voting right and the authority of making proposal passes on to VP1 and in that case, VP2 and VP4 will not get a single penny. So they do not want that. What he does is he gives 1 million dollars each to VP2 and VP4 who otherwise wouldnt have received anything in case the proposal of CEO got rejected and the authority moved to VP1. By doing so he secures the vote of VP2 as well as VP4 and together with his vote obtains a 60% majority. Hence his proposal gets selected and he keeps 98 million dollarsSo final distribution would be CEO: 98 million dollars VP1: Zero dollars VP2 1 million dollars VP3: Zero dollars VP4 1 million dollars

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