Roommates Voting: Three roommates need to vote on whether they will adopt a new rule and clean their apartment once a week or stick to the current once-a-month rule. Each votes "yes" for the new rule or "no" for the current rule. Players 1 and 2 prefer the new rule while player 3 prefers the old rule.
a. Imagine that the players require a unanimous vote to adopt the new rule. Player 1 votes first, then player 2, and then player 3, the latter two observing the previous votes. Draw this as an extensive-form game and find the Nash equilibria.
b. Imagine now that the players require a majority vote to adopt the new rule (at least two "yes" votes). Again player 1 votes first, then player 2, and then player 3, the latter two observing the previous votes. Draw this as an extensive-form game and find the Nash equilibria.
c. Now imagine that the game is as in part (b), but the players put their votes into a hat-so that the votes of earlier movers are not observed by the later movers-and the votes are counted after all have voted. Draw this as an extensive-form game and find the Nash equilibria. In what way is this result different from the result in (b)?