Ann, Bob, Carl, Dora, Eddy and Frank are the contestants in a new TV show. They are placed on an island. The rules of the game are as follows. Ann goes first. She is given a bag that everyone knows contains six gold coins. Ann makes a proposal of how to allocate the six coins among the six contestants including herself. The contestants (including Ann) then vote "yes" or "no" on the proposal. If the proposal gets more than half the votes then the coins are allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than half the votes then Ann has to leave the island empty-handed and she is out of the game. In this case, the bag of six gold coins passes to Bob. He gets to make a proposal of how to allocate the coins among the remaining contestants (i.e., including Bob but excluding Ann) and the remaining contestants (i.e., including Bob but excluding Ann) then vote. As before, if the proposal gets more than half the votes then the coins are allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than half the votes then Bob has to leave the island empty-handed and is out of the game. In this case, the bag of six gold

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Ann, Bob, Carl, Dora, Eddy and Frank are the contestants in a new TV show. They are placed on an island. The
rules of the game are as follows. Ann goes first. She is given a bag that everyone knows contains six gold coins.
Ann makes a proposal of how to allocate the six coins among the six contestants including herself. The contestants
(including Ann) then vote "yes" or "no" on the proposal. If the proposal gets more than half the votes then the
coins are allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than
half the votes then Ann has to leave the island empty-handed and she is out of the game.
In this case, the bag of six gold coins passes to Bob. He gets to make a proposal of how to allocate the coins among
the remaining contestants (i.e., including Bob but excluding Ann) and the remaining contestants (i.e., including
Bob but excluding Ann) then vote. As before, if the proposal gets more than half the votes then the coins are
allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than half
the votes then Bob has to leave the island empty-handed and is out of the game. In this case, the bag of six gold
coins passes to Carl. And so on, with the same voting rules, with each failed proposal leading to expulsion of the
proposer,
and with the role of proposer being passed on alphabetically.
The following assumptions matter. The coins are indivisible, there is no other money on the island, and side
contracts to make payments off the island are not allowed. There are no abstentions; each surviving voter must
vote yes or no: whenever a voter is indifferent, they vote no. The players only care about the gold (and this
is common knowledge). For example, leaving empty handed because your proposal fails is the same as leaving
empty-handed because a successful proposal gives you no coins.
What offer should Ann make? (find the backward induction outcome of the game) Show all of your work.
Transcribed Image Text:Problem Ann, Bob, Carl, Dora, Eddy and Frank are the contestants in a new TV show. They are placed on an island. The rules of the game are as follows. Ann goes first. She is given a bag that everyone knows contains six gold coins. Ann makes a proposal of how to allocate the six coins among the six contestants including herself. The contestants (including Ann) then vote "yes" or "no" on the proposal. If the proposal gets more than half the votes then the coins are allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than half the votes then Ann has to leave the island empty-handed and she is out of the game. In this case, the bag of six gold coins passes to Bob. He gets to make a proposal of how to allocate the coins among the remaining contestants (i.e., including Bob but excluding Ann) and the remaining contestants (i.e., including Bob but excluding Ann) then vote. As before, if the proposal gets more than half the votes then the coins are allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than half the votes then Bob has to leave the island empty-handed and is out of the game. In this case, the bag of six gold coins passes to Carl. And so on, with the same voting rules, with each failed proposal leading to expulsion of the proposer, and with the role of proposer being passed on alphabetically. The following assumptions matter. The coins are indivisible, there is no other money on the island, and side contracts to make payments off the island are not allowed. There are no abstentions; each surviving voter must vote yes or no: whenever a voter is indifferent, they vote no. The players only care about the gold (and this is common knowledge). For example, leaving empty handed because your proposal fails is the same as leaving empty-handed because a successful proposal gives you no coins. What offer should Ann make? (find the backward induction outcome of the game) Show all of your work.
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