The 31 members of the board of the Student S. Inc. are about to take a secret ballot
whether to accept the merger proposal of Student G. Corp. Each member can vote to
accept the proposal, reject the proposal or to abstain. For the proposal to be accepted, 16
members must vote to accept it. All, the 31 members care about is being on the winning
side. That is, if the proposal is accepted, each member would prefer to accept it; and
if the proposal is rejected, each member would prefer to have rejected it. Find the two
Nash equilibria of this game. What would you predict the outcome will be and why?
To determine the Nash equilibria of this game, we need to analyze the preferences and strategies of the board members.
Let's represent the three choices as follows:
- A: Accept the proposal
- R: Reject the proposal
- Ab: Abstain from voting
Given that each member wants to be on the winning side, we can assume that they would prefer to vote in a way that maximizes their chances of being on the majority side.
Now, let's consider the possible scenarios:
1. No one abstains:
In this case, each member must choose between voting to accept or voting to reject the proposal. If 16 members vote to accept and 15 members vote to reject, the proposal will be accepted. However, if more than 16 members vote to reject, the proposal will be rejected. Since each member prefers to be on the winning side, it is rational for every member to vote to accept the proposal. This results in a Nash equilibrium where all members vote to accept.
2. Some members abstain:
If any member decides to abstain, the equilibrium would shift. By abstaining, a member avoids being on the losing side but also does not contribute to the majority. In this case, it is in the best interest of all remaining members to vote to accept the proposal, as one less vote against it would increase the chances of it being accepted. Therefore, the Nash equilibrium would be for all remaining members to vote to accept.
Based on the analysis, there are two Nash equilibria in this game:
- All members voting to accept the proposal
- All remaining members voting to accept if any member abstains
The predicted outcome would depend on the preferences and strategic decisions of the board members. However, considering that each member wants to be on the winning side, it is likely that they would vote to accept the proposal to maximize their chances of being part of the majority.
To find the Nash equilibria of this game, we need to analyze the strategies of the players and determine which combinations of strategies result in a stable outcome. In this case, we have three possible strategies for each player: accept the proposal, reject the proposal, or abstain.
Let's analyze the situation by considering the preferences of the individual players:
1. If the proposal is accepted, each member prefers to accept it. This means that all players voting for acceptance would be happy with the outcome.
2. If the proposal is rejected, each member prefers to have rejected it. Again, all players voting against the proposal would be satisfied.
3. If the proposal is abstained from, each member does not have a preference since they do not take a stand.
Based on these conditions, let's consider the possible outcomes:
1. If 16 or more members vote to accept the proposal, it will be accepted. In this case, all players who voted for acceptance will be on the winning side and satisfied. All other players who either rejected or abstained will not be satisfied.
2. If 16 or more members vote to reject the proposal, it will be rejected. In this case, all players who voted for rejection will be on the winning side and satisfied. All other players who either accepted or abstained will not be satisfied.
3. If 16 or more members abstain from voting, no outcome will be reached, and the proposal will not be accepted. In this case, all players will be unsatisfied as there is no winning side.
Now, let's determine the Nash equilibria:
1. If there are 16 or more members voting to accept the proposal, there is no incentive for any individual player to deviate from accepting, as it would not change the outcome. Similarly, no player would have an incentive to switch to rejection or abstention as it would result in an outcome they prefer less. This situation forms a Nash equilibrium.
2. If there are 16 or more members voting to reject the proposal, the same conditions apply. No player would want to switch to acceptance or abstention as it would result in a worse outcome for them. This is another Nash equilibrium.
Therefore, the two Nash equilibria of this game are when 16 or more members vote to accept the proposal or when 16 or more members vote to reject the proposal.
Predicting the outcome depends on the preferences of the individual members, which are not provided in the question. Without knowing individual preferences, we cannot determine with certainty which Nash equilibrium will be reached.