(Game theory, equilibrium matrix) A corporate board for a large company is made
ID: 3276543 • Letter: #
Question
(Game theory, equilibrium matrix)
A corporate board for a large company is made up of five members: Jane, Tom, Sally, Emily, and Jason. Each member has a strict hierarchy: Jane is senior to Tom, Tom is senior to Sally, Sally is senior to Emily, Emily is senior to Jason. The board must decide how to split a 5,100,000 dollar bonus among the five members of the board. The rules of board are as follows.
The most senior member proposes a distribution among the members. After observing the proposal, the five board members vote on whether to accept the proposal or reject it. If the proposal is rejected, then the most senior member is fired and no longer participates in the process. In the case of a tie, the proposer’s vote decides whether to accept or reject. If a proposal is rejected, then the next most senior member now proposes a distribution .If the proposal is accepted, then the distribution is received by the members of the board.
Each board member cares about three main factors. (1) They do not want to be fired (2) If they are not fired, then they want to get the most dollars they possibly can. (3) Each board member would prefer to fire a board member, if all other results would otherwise be equal.
Finally the board members do not trust each other and will not follow any promises except for the proposal.(Assume that any offer must be in whole dollars 0, 1, 2, .....)
Find the Nash equilibrium for this game.
Explanation / Answer
Suppose If members are fired one by one till only the last two members Emily and Jason are left. Now Emily shall propose nothing to Jason and the vote shall go 1:1 and Emily's proposal would be accepted, as in case of tie proposer's vote is considered. Hence, Jason otherwise does not expect to gain any money. However, Sally is also aware of this and hence when Sally, Emily and Jason are left, he would propose 1 dollar to Jason and nothing to Emily, remaining for himself. Jason knows that Emily rejects the proposal and Sally votes for it and hence his vote is deciding. Now if he rejects the 1 dollar, he knows that now Emily shall propose the distribution and he would get nothing. So, Jason votes for Sally's proposal. Now Tom also knows that if his proposal is rejected then Sally shall propose and Emily gets nothing, Jason gets 1 dollar. So, to get Emily's vote he proposes 1 dollar to Emily and rest for himself. Emily would get nothing if it rejects Tom's proposal because if Tom is ousted then Sally proposes nothing to Emily and the proposal is accepted. Jane hence knows that if she is ousted then Tom divides the wealth as 5099999:0:1:0 and the proposal would be accepted. Hence, Jane would want the vote of Sally and Jason and would propose 1 dollar to each of them and the remaining to herself. Hence optimally the distribution would be 5099998:0:1:0:1.
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