2. Two players have to divide a perfectly divisible cake of size 2. Half ofthe cake is made of chocolate, while the other half is made of marzipan. Ifa piece of cake contains a units of chocolate cake and y units of marzipan,player 1 gets utilityU(x, y) = x + y,from this piece, while player 2 gets utilityV(x, y) = 2x+y,from this piece.That is, player 1 cares only about the total size of the piece, while player2 is values chocolate twice as much as marzipan.The players use the divide-and-choose protocol one player divides thecake into two, and the other chooses one of the two pieces. More precisely,the divider splits the cake into two pieces (x, y) and (1-x,1- y) (where0≤x≤ 1 and 0 ≤ y ≤ 1. The chooser may choose either the piece (x, y) orthe piece (1-2, 1-y).a) Use backwards induction to prove (or argue for) the the following claim:"In any subgame perfect equilibrium, the chooser is indifferent between thetwo pieces."Solve by backwards induction, for the subgame perfect equilibrium when:b) player 2 divides.c) player 1 divides.d) Does a player prefer to be a divider or a chooser? Explain why, briefly.

Answered: 2. Two players have to divide a…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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2. Two players have to divide a perfectly divisible cake of size 2. Half of
the cake is made of chocolate, while the other half is made of marzipan. If
a piece of cake contains a units of chocolate cake and y units of marzipan,
player 1 gets utility
U(x, y) = x + y,
from this piece, while player 2 gets utility
V(x, y) = 2x+y,
from this piece.
That is, player 1 cares only about the total size of the piece, while player
2 is values chocolate twice as much as marzipan.
The players use the divide-and-choose protocol one player divides the
cake into two, and the other chooses one of the two pieces. More precisely,
the divider splits the cake into two pieces (x, y) and (1-x,1- y) (where
0≤x≤ 1 and 0 ≤ y ≤ 1. The chooser may choose either the piece (x, y) or
the piece (1-2, 1-y).
a) Use backwards induction to prove (or argue for) the the following claim:
"In any subgame perfect equilibrium, the chooser is indifferent between the
two pieces."
Solve by backwards induction, for the subgame perfect equilibrium when:
b) player 2 divides.
c) player 1 divides.
d) Does a player prefer to be a divider or a chooser? Explain why, briefly.

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A utility function is a mathematical representation of an individual's preferences over different possible outcomes or states of the world. It is used in economics and decision theory to model how people make choices under uncertainty. The utility function assigns a numerical value to each possible outcome, representing the level of satisfaction or "utility" that the individual would derive from that outcome. By comparing the utilities of different outcomes, the individual can then make a rational decision about which one to choose.

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