Two competitive siblingsJuan and María-are deciding when to show up at their mom's house for Mother's Day. They are simultaneously choosing between times of 8:00 A.M., 9:00 A.M, 10:00 A.M. and 11:00 A.M. The payoffs to a sibling are shown in TABLE 4.2 and depend on what time he or she shows up and whether he or she shows up first, second, or at the same time. (Note that this is not a payoff matrix.) For example, if Juan shows up at 9:00 A.M. and Maria shows up at 10:00 A.M. then Juan's payoff is 8 (because he is first) and María's payoff is 4 (because she is second). Payoffs have the property that each would like to show up first but would prefer to show up second rather than show up at the same time. (They really do not like one another.) Furthermore, conditional on showing up first or at the same time or second, each prefers to show up later in the morning; note that payoffs are increasing as we move down a column that is associated with arriving later in the morning. TABLE 4.2 Juan and Maria on Mother's Day Time/Order of Arrival First Same Time Second 8:00 A.M. 7 -3 9:00 A.M. 8 -2 3 10:00 A.M. 1 4 11:00 A.M. Using the method deployed for Rock-Paper-Scissors, show that there is no Nash equilibrium. (Because you'll want to start getting used to solving games without a payoff matrix before you, l'd recommend trying to answer this question without constructing the payoff matrix. However, the answer in the back of the book does include the payoff matrix if you choose to work with it.)*
Two competitive siblingsJuan and María-are deciding when to show up at their mom's house for Mother's Day. They are simultaneously choosing between times of 8:00 A.M., 9:00 A.M, 10:00 A.M. and 11:00 A.M. The payoffs to a sibling are shown in TABLE 4.2 and depend on what time he or she shows up and whether he or she shows up first, second, or at the same time. (Note that this is not a payoff matrix.) For example, if Juan shows up at 9:00 A.M. and Maria shows up at 10:00 A.M. then Juan's payoff is 8 (because he is first) and María's payoff is 4 (because she is second). Payoffs have the property that each would like to show up first but would prefer to show up second rather than show up at the same time. (They really do not like one another.) Furthermore, conditional on showing up first or at the same time or second, each prefers to show up later in the morning; note that payoffs are increasing as we move down a column that is associated with arriving later in the morning. TABLE 4.2 Juan and Maria on Mother's Day Time/Order of Arrival First Same Time Second 8:00 A.M. 7 -3 9:00 A.M. 8 -2 3 10:00 A.M. 1 4 11:00 A.M. Using the method deployed for Rock-Paper-Scissors, show that there is no Nash equilibrium. (Because you'll want to start getting used to solving games without a payoff matrix before you, l'd recommend trying to answer this question without constructing the payoff matrix. However, the answer in the back of the book does include the payoff matrix if you choose to work with it.)*
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.9P
Related questions
Question
Let us see the example of Juan and María given but modify their preferences. It is still the case that they are competitive and are deciding whether to show up at their mom’s house at 8:00 A.M., 9:00 A.M., 10:00 A.M., or 11:00 A.M. But now they don’t mind waking up early. Assume that the payoff is 1 if he or she shows up before the other sibling, it is 0 if he or she shows up after the other sibling, and it is 1 if they show up at the same time. The time of the morning does not matter. Find all Nash equilibria.
Transcribed Image Text:Two competitive siblingsJuan and María-are deciding when to show up at their
mom's house for Mother's Day. They are simultaneously choosing between times of
8:00 A.M., 9:00 A.M, 10:00 A.M. and 11:00 A.M. The payoffs to a sibling are shown in
TABLE 4.2 and depend on what time he or she shows up and whether he or she
shows up first, second, or at the same time. (Note that this is not a payoff matrix.)
For example, if Juan shows up at 9:00 A.M. and Maria shows up at 10:00 A.M. then Juan's
payoff is 8 (because he is first) and María's payoff is 4 (because she is second).
Payoffs have the property that each would like to show up first but would prefer to
show up second rather than show up at the same time. (They really do not like one
another.) Furthermore, conditional on showing up first or at the same time or second,
each prefers to show up later in the morning; note that payoffs are increasing as we
move down a column that is associated with arriving later in the morning.
TABLE 4.2 Juan and Maria on Mother's Day
Time/Order of Arrival
First
Same Time
Second
8:00 A.M.
7
-3
9:00 A.M.
8
-2
3
10:00 A.M.
1
4
11:00 A.M.
Using the method deployed for Rock-Paper-Scissors, show that there is no Nash
equilibrium. (Because you'll want to start getting used to solving games without a
payoff matrix before you, l'd recommend trying to answer this question without
constructing the payoff matrix. However, the answer in the back of the book does
include the payoff matrix if you choose to work with it.)*
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