MATH 1340, HOMEWORK #8 SOLUTIONS
DUE THURSDAY, APRIL 13
1. [TP 4.13] Consider a two-player game in which the players simultaneously show a penny, either
heads up or tails up. If both players show heads, then both players lose their pennies to a lucky
third party, and if both players show tails, each player keeps his or her own penny. If both players
show different sides, then the player who shows heads gets both coins.
(a) Write down the two-by-two matrix for this game.
Solution. Let H denote heads and T denote tails.
H
T
H (2, 2) (4, 1)
T (1, 4) (3, 3)
(b) Is Chicken or Prisoner’s Dilemma or neither a model for this game?
Solution. Prisoner’s Dilemma is a model for this game. This is under the assumption that each
player would prefer to let a third player get both pennies instead of the other player getting both
pennies. If this preference was reversed then the game of chicken would be a model for this game. (c) Do the players have a dominant strategy?
Solution. Yes, showing heads is a dominant strategy for both players since 4 > 3 and 2 > 1.
(d) Is there a Nash equilibrium?
Solution. Both players showing heads is a Nash equilibrium as it is a result of dominant strategies.
2. [TP 4.16] Kathryn and Nadia each plan to throw a New Year’s Eve party; each one has a back-up
date as well, and the two back-up dates do not conflict. Ideally, Kathryn hopes that she can throw
the New Year’s Eve party, and that Nadia will choose a different date. But if that doesn’t happen,
she really wants to be able to attend Nadia’s party, even though she’ll be very jealous if Nadia’s
party is on New Year’s and she has to choose a different date.
(a) If Nadia feels the same way as Kathryn, write down a 2 × 2 ordinal game that models the
situation. What, if anything, does the model predict will happen?
Solution. Each person choosing a different date is denoted by C (indicating cooperation) and choosing to throw a New Year’s Eve party is denoted by N (indicating non-cooperation).
C
N
C
N
(3, 3) (2, 4)
(4, 2) (1, 1)
This is like the game of Chicken and there is no expected outcome although (3, 3) suggests itself
which is unstable as it is not a Nash equilibrium. Thus, both of them will choose to throw a party
on a different date.
1
2
DUE THURSDAY, APRIL 13
(b) Suppose that Nadia’s first priority is that her party is on New Year’s Eve, and would absolutely hate it if Kathryn gets to throw the New Year’s Eve party and she is forced to choose a
different date. Write down a new 2 × 2 ordinal game that models the situation. What, if anything,
does the model predict will happen?
Solution. Let Nadia be the Row player and let Kathryn be the Column player. Then the game is
modelled as follows:
C
N
C
N
(3, 3) (1, 4)
(4, 2) (2, 1)
In this case (4, 2) is a Nash equilibrium obtained by a dominant strategy for Nadia. Hence, Nadia
would throw a New Year’s Eve party and Kathryn would choose a different date to throw a party.
(c) For both scenarios above, is the game Prisoner’s Dilemma, Chicken, or neither?
Solution. In the first scenario, the game is Chicken. In the second scenario, it is neither.
3. [TP 4.17] Consider the following hypothetical situation. NASA plans to launch a manned vehicle
into space, but the engineers feel that it is unsafe. NASA has the options to launch or not, and the
engineers have the option to go public with their reservations or not. Assume that NASA’s first
priority is that the engineers remain silent (because NASA honestly feels that they are wrong),
and, as a second priority, NASA would rather launch than not launch. Additionally, assume that
the engineers have a first priority of preventing the launch, and a second priority of going public
with their reservations. Model this as a 2 × 2 game, and, in a few sentences, explain what outcome
is predicted by the existence of dominant strategies.
Solution. For NASA, C would mean that it does not launch (indicating cooperation with the
engineers) and N would mean that it launches the manned vehicle into space (indicating noncooperation with the engineers). For the engineers, C would stand for remaining silent (indicating
cooperation with NASA) and N would stand for going public with their reservations (indicating
non-cooperation with NASA). Let NASA be Row and the engineers be Column. Then the model
is as follows:
C
N
C
N
(3, 3) (1, 4)
(4, 1) (2, 2)
This is exactly like Prisoner’s Dilemma. N is dominant for both Column and Row and the expected outcome is NASA going ahead with the launch and the engineers going public with their
reservations. This is a Nash equilibrium and hence stable.
4. [TP 4.19] Do there exist 2 × 2 ordinal games with a Nash equilibrium that is not the result of
dominant strategies by Row and Column? Give an example or prove that one does not exist.
Solution. Yes, there exist 2 × 2 ordinal games with a Nash equilibrium that is not the result of
dominant strategies by Row and Column. The game of Chicken is one such example and the
proposition in section 4.5 shows that there are two Nash equilibria but no dominant strategies for
either player.
5. [TP 4.21] Find all Nash equilibria for the following 3 × 3 game, and for each outcome that is not
a Nash equilibrium, explain why it is not.
MATH 1340, HOMEWORK #8 SOLUTIONS
C
N
V
3
C
N
V
(1, 4) (2, 5) (3, 3)
(4, 8) (5, 9) (6, 2)
(7, 6) (8, 7) (9, 1)
Solution. (8, 7) is the only Nash equilibrium because it is the result of dominant strategies. V is
a dominant strategy for Row player and N is a dominant strategy for Column player. Any other
entry is not a Nash equilibrium because atleast one of the players can unilaterally make a change
to their dominant strategy and yield a better outcome.
6. [TP 4.25] In 1960 William Newcomb, a physicist, posed the following problem: Suppose there
are two boxes labeled A and B. You have a choice between taking box B alone or taking both A
and B. God has definitely placed $1,000 in box A. In box B, He placed either $1,000,000 or nothing,
depending upon whether He knew you’d take box B alone (in which case He placed $1,000,000 in
box B) or take both (in which case He placed nothing in box B). The question is: Do you take box
B alone or do you take both? You can answer this if you want to, but that’s not the point of this
exercise. In fact, hundreds of philosophical papers have been written on this problem. Most people
think the answer is obvious, although they tend to split quite evenly on which answer is obvious
and which answer is clearly wrong.
(a) Give an argument that suggests you should take both boxes.
Solution. Taking both boxes will ensure that we will end up with atleast $1,000. This way we don’t
risk a chance of ending up with no money.
(b) Give an argument that suggests you should take box B alone.
Solution. The expected value seems higher if we consider that the probability of Box B containing
nothing is the same as the probability of the box containing $1,000,000.
(c) Indicate which argument you find most compelling and why.
Solution. If we are not aware of what God is planning to do, then it seems better to take both boxes
in the hopes of getting more than $1,000 and ensuring that we don’t end up with nothing.
(d) Consider the following 2 × 2 ordinal game:
C
N
C (3, 4) (1, 3)
N (4, 1) (2, 2)
Prove that Row has a dominant strategy of N. Now suppose that we change the rules of the game
so that Row chooses first, and then Column-knowing what Row did-chooses second. Explain why,
even though Row has a dominant strategy of N in the game with the usual rules, Row should choose
C in this version of the game where Row moves first.
Solution. Row has a dominant strategy of N because 4 > 3 and 2 > 1.
Let us assume Row chooses first. If Row chooses C, then Column will choose C as 4 > 3. If Row
chooses N, then Column will choose N, as 2 > 1. The two possible outcomes are: both players
choosing C or both players choosing N. Row would prefer if both players choose C as 3 > 2. Hence
Row would choose C, even though N is a dominant strategy for Row.