Suppose that 3 players, 1, 2, and 3, are going to choose between three projects,

Question

Suppose that 3 players, 1, 2, and 3, are going to choose between three projects, denoted by X, Y , and Z. The players’ payoffs from each project are as follows: X Y Z

u1 2 1 0

u2 1 0 2

u3 0 2 1

(a) Suppose that the players choose by voting simultaneously for exactly one project (so each player i’s action set is Ai = {X, Y, Z}, corresponding to which project he or she voted for). If any project gets 2 or 3 votes, it is implemented. If all projects get 1 vote, then project X is implemented. Find all pure strategy subgame perfect Nash equilibria.

(b) Now suppose that the players vote sequentially: first player 1 votes and his or her vote is revealed. Then player 2 votes, and his or her vote is revealed. Finally, player 3 votes. Again, if any project gets 2 or 3 votes, then it is implemented and, if all 3 projects get 1 vote each, then project X is implemented. Find all subgame perfect Nash equilibria.

(c) Now suppose that the players first vote simultaneously between X and Y . The project that receives 2 or 3 votes is then revealed and pitted against Z, and all players vote simultaneously between the X/Y winner and Z. The alternative that gets 2 or 3 votes in that round is implemented. Find all subgame perfect Nash equilibria.

(d) Finally, suppose that the players first vote simultaneously between Y and Z. The project that receives 2 or 3 votes is then revealed and pitted against X, and all players vote simultaneously between the Y /Z winner and X. The alternative that gets 2 or 3 votes in that round is implemented. Find all subgame perfect Nash equilibria.

Explanation / Answer

a) Individual player i's payoff from each project is given as Ai={X,YZ} and the action for the 3 players are represented as u1,u2 and u3 respectively.

If simultanous voting is implemented for all the 3 projects for the 3 players,the observe that player 1 would choose project X as it would yield the maximum payoff for player 1 which is 2 out of all 3 projects.Player 2 on othe other hand,would prefer project Z because of th maximum payoff from all 3 projects which is also 2 and player 3 would vote for project Y for maximum payoff which is again 2 among all 3 projects.

Notice,in the above case that all the 3 projects (X,Y and Z) each gets one vote and hence,project X is implemented with payoffs for player 1,2 and 3 as {2,1,0} respectively since perfect information is not available

b) Now player 1 gets to vote first and if he/she votes for project X then his or her payoff would be 2 and in which case,player 2 could prefer project X with a payoff of 1 as if it chooses any other project other than X it would increase the probability of project X being chosen.He/she can also choose to vote for project Z with higher probability and a pay off of 2.Now,lastly,if player 2 chooses project X,player 3 would practically have no option as already project X received 2 votes which would result in its implementation which would be the subgame Nash Equilibrium in this case.But if player 2 votes for project Z then player 3 is better off voting for project Z with payoff of 1 and thus,project Z will be implemented with 2 votes from player 2 and 3.Therefore,in this instance,the subgame Nash equilibrium would be implementation of either project X or/and project Z.

Secondly,if player 1 votes for project Y with a payoff of 1 then player 2 would expectedly choose project Z with a relatively better payoff of 2 compared to 1 for project X.In this case,player 3 would definitely vote for project Z as well with a payoff of 1 to avoid the implementation of project X which would yield a payoff of 0 for him/her.Hence,the subgame Nash equilibrium in this case would be again the implementation of project Z with 2 vots from players 2 and 3 respectively.

Lastly,if player 1 chooses project Z with a payoff of 0(very rare or almost an impossibility) then player 2 would definitely vote for project Z and then player 3 can basically choose to vote for any of the projects as project Z already got the majority vote resulting in its implementation.Thus,the subgame Nash Equilibrium would result in the implementation of project Z again in this scenario.

c) Now,if the players vote simultaneously only between project X and Y,then both players 1 and 2 would vote for project X with an expectation of higher payoffs of 2 and 1 respectively compared to 1 and 0 for project Y.Only player Z would vote for project Y with the expected payoff of 2.But in this scenario,already project X would be the winner with 2 votes from players 1 and 2 making the outcome the subgame Nash Equilibrium for te 1st round of voting between project X and Y.

In the second round of voting between the winner project X and Z,observe that player 1 would stick to project X for a better payoff of 2 compared to 0 for voting project Z.Player 2 might posibly switch to project Z with a prospect of better payoff of 2 compared to 1 from voting for project X and player 3 in this case,would definitely vote for project Z with a payoff of 1 compared to 0 from choosing project X.This would again lead to a Nash Equilibrium for implementation of project Z with 2 majority votes.

d) Now,if the players simulataneously vote between projects Y and Z,then notice that players 2 and 3 would opt for project Z resulting in its majority preference imlying that project Z is the winner in this case which is the subgame Nash Equilibrium at the 1st stage of the voting.

In the second round of voting between the winning project Z and X, player 1 would prefer to switch to project X with a expected better payoff of 2 compared to 0 from the outcome of the 1st round of voting.However,notice that players 2and 3 would be reluctant to switch their votes to project X as both of their payoffs would decline to 1 and 0 from 2 and 1 respectively.Therefore,in this occassion,the subgame Nash Equilibrium stands in favour of project Z with its implementation.

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