Say that there are three travelers on a mythical journey and they happen to find a mysterious machine. This machine spea
Posted: Sat Mar 19, 2022 5:51 pm
Say that there are three travelers on a mythical journey and
they happen to find a mysterious machine. This machine speaks to
them, saying that it is a money-doubling machine. Any money put in
the machine will come out of the machine doubled. However, because
the machine is dedicated to fairness, the output of the machine
will be divided equally among all three of the travelers. The
travelers are so excited knowing that this machine could yield
untold riches.
a. Model this as a strategic form game with three players. Do
the travelers use the machine?
b. Now say that a centaur, named Arcturus, sees the travelers
and offers to help them. Arcturus says that each traveler can give
one gold coin to Arcturus or not (each traveler has at least one
gold coin). If all three travelers give Arcturus a gold coin, then
Arcturus will in turn put the money into the machine. If any
traveler does not give a gold coin to Arcturus, then Arcturus will
walk away with whatever money the other travelers have given to
Arcturus. The travelers scratch their heads, wondering why the
centaur is offering what seems to be a pretty bad deal. Should the
travelers take Arcturus's offer? (Arcturus is not a player in the
game; we take the centaur's actions as given.)
c. Arcturus vanishes before the travelers figure out what to do.
Then Deirdre the Wizard happens upon the travelers. Deirdre makes
them the following offer. Deirdre says that as a wizard, Deirdre
can hex the machine so that instead of dividing the machine's
output equally among the three travelers, the youngest of the
travelers will receive 80 percent of the machine's output while the
other two travelers receive only ten percent. Of course the older
travelers are not happy about this arrangement. Deirdre says that
the hex will be applied only if all three travelers agree. Do the
travelers accept Deirdre's offer? (Again, Deirdre is not a player
in the game.)
d. Deirdre vanishes before the travelers decide. Arcturus
returns, this time with a slightly different offer. Arcturus says
that each traveler can give one gold coin to Arcturus or not (each
traveler has at least one gold coin). If at least two travelers
give Arcturus a gold coin, then Arcturus will in turn put the money
into the machine. Otherwise, Arcturus will walk away with whatever
money Arcturus receives. This deal does not seem quite as bad as
Arcturus's earlier deal, but it still doesn't seem like a good
deal. Should the travelers take Arcturus's offer?
they happen to find a mysterious machine. This machine speaks to
them, saying that it is a money-doubling machine. Any money put in
the machine will come out of the machine doubled. However, because
the machine is dedicated to fairness, the output of the machine
will be divided equally among all three of the travelers. The
travelers are so excited knowing that this machine could yield
untold riches.
a. Model this as a strategic form game with three players. Do
the travelers use the machine?
b. Now say that a centaur, named Arcturus, sees the travelers
and offers to help them. Arcturus says that each traveler can give
one gold coin to Arcturus or not (each traveler has at least one
gold coin). If all three travelers give Arcturus a gold coin, then
Arcturus will in turn put the money into the machine. If any
traveler does not give a gold coin to Arcturus, then Arcturus will
walk away with whatever money the other travelers have given to
Arcturus. The travelers scratch their heads, wondering why the
centaur is offering what seems to be a pretty bad deal. Should the
travelers take Arcturus's offer? (Arcturus is not a player in the
game; we take the centaur's actions as given.)
c. Arcturus vanishes before the travelers figure out what to do.
Then Deirdre the Wizard happens upon the travelers. Deirdre makes
them the following offer. Deirdre says that as a wizard, Deirdre
can hex the machine so that instead of dividing the machine's
output equally among the three travelers, the youngest of the
travelers will receive 80 percent of the machine's output while the
other two travelers receive only ten percent. Of course the older
travelers are not happy about this arrangement. Deirdre says that
the hex will be applied only if all three travelers agree. Do the
travelers accept Deirdre's offer? (Again, Deirdre is not a player
in the game.)
d. Deirdre vanishes before the travelers decide. Arcturus
returns, this time with a slightly different offer. Arcturus says
that each traveler can give one gold coin to Arcturus or not (each
traveler has at least one gold coin). If at least two travelers
give Arcturus a gold coin, then Arcturus will in turn put the money
into the machine. Otherwise, Arcturus will walk away with whatever
money Arcturus receives. This deal does not seem quite as bad as
Arcturus's earlier deal, but it still doesn't seem like a good
deal. Should the travelers take Arcturus's offer?