Midterm Review Questions

MIDTERM REVIEW QUESTIONS

The Midterm Exam will take place in class on Friday, Feb. 6. Please bring one or more blank
blue books and a pen to the exam. Please answer all questions completely, and, where applicable,
show your work. Answers without any indication of how you calculated them earn zero credit.
NOTE: In many of the problems below, I make the same STANDARD ASSUMPTION,
which is: THE PLAYERS ARE INDIVIDUALISTICALLY RATIONAL AGENTS WITH
COMPLETE (AND, IF RELEVANT, PERFECT) INFORMATION.

1. Define or explain the following terms, being sure to distinguish between the items in a list
of items. Feel free
to use examples in your explanations.
certainty/risk/uncertainty
utility/expected utility
Pareto inferior/Pareto superior/Pareto optimal
extensive form/normal form of a game
purely competitive/purely cooperative/mixed motive game
parametric/strategic rationality
complete/perfect information
pure/mixed strategy
information set
maximin/minimax pure strategy
saddle point
strict/weak dominance
equalizer mixed strategy
Nash equilibrium
equivalent strategy combinations
interchangeable strategy combinations
strong/weak sense of "collective action problem"
supergame/subgame
Two-Person Prisoner's Dilemma Supergame
C-TFT/D-TFT/C-/D-/
conditional cooperation/unconditional cooperation in PD Supergame
collectively stable strategy (stable against invasions by individuals and by clusters)
N-Person Prisoner's Dilemma (or Multi-person Prisoner's Dilemma)
N-Person Prisoner's Dilemma Supergame
Individualistic Rationality (IR)
Rational Self-Interest Theory (RST)
Methodological Individualism (MI) in the social sciences

2. In the sense in which we use the term, what is a game?

3. In Rock-Scissors-Paper, two players must simultaneously choose either Rock (R) or
Scissors (S) or Paper (P). Rock beats Scissors, Scissors beats Paper, and Paper beats Rock; any other
combinations are ties.
(a) Represent an outcome in which one player beats the other as having utility of 1 for the
winner and utility of -1 for the loser; represent a tie as utility of 0 to each player. Draw a
decision matrix with the three strategies (R, S, and P) for each player. Circle all pure strategy
Nash Equilibrium combinations. Explain how you found them. If there are none, explain
how you determined that there are none.
(b) There is a mixed strategy Nash equilibrium combination. It is the same for both players:
{<1/3, R>, <1/3, S>, <1/3, P>}. If one player plays this mixed strategy, what is the expected utility
of the other player's pure strategies against it. Show your work. (One calculation is sufficient.)
(c) Suppose one player did not play his/her Nash equilibrium mixed strategy. Suppose that
particular player liked S more than the other alternative choices, so s/he adopted the following mixed
strategy: {<1/4, R>, <1/2, S>, <1/4 P>}. Call this strategy the S-biased strategy. What would be the
best reply to the S-biased strategy? If one player played the S-biased strategy and one player played
the best response to it, what would be the expected utility for each player?

4. On the STANDARD ASSUMPTION, find all IR solutions to the following two-person,
zero-sum game:

	c₁	c₂	c₃	c₄
r₁	1,-1	2,-2	1,-1	2,-2
r₂	-4,4	-5,5	-3,3	3,-3
r₃	1,-1	3,-3	1,-1	1,-1
r₄	0,0	4,-4	-1,1	-3,3

5. (a) Explain why all possible outcomes of a two-person, zero-sum game are Pareto optimal.
(b) Explain why no zero-sum game can be a strong or weak CAP.

6. (a) The eponymous John Nash proved that all games have at least one Nash equilibrium
combination of strategies. What is the maximum number of pure strategy Nash equilibrium
combinations that there can be in a simple 2x2 game, if no two of the equilibrium combinations are
equivalent? Give an example.
(b) All two-person zero sum games have an IR solution. Why don't all games have an IR
solution? Use an example to explain your answer.

7. Find all pure and mixed Nash equilibrium combinations of strategies in the following two-
person games:
(a) Standard Convention Game
(b) Pareto Convention Game
(c) Chicken Game
(d) Battle of the Sexes/Hero.

8. Axelrod identifies four characteristics of C-Tit-for-Tat that he uses to explain its success in
his computer tournaments. Identify and explain each of the four characteristics, and explain how
each of them contributed to C-Tit-for-Tat's success in Axelrod's tournaments.

9. What is the Rendezvous Game? Identify two kinds of factors other than purely IR factors
can help to produce a "solution" to a Rendezvous Game. Give an example of each.

10. Assume that the Live-and-Let-Live System in the trenches in World War I was an
example of conditional cooperation (typically C-Tit-for-Two-Tats) on the part of the two armies.
(Ignore the issue of how individual soldiers were motivated to do their part in the conditionally
cooperative strategies of their respective armies. Just assume that they were rationally motivated to do
so.) What assumptions are necessary to show that the situation of the two armies was a 2-Person PD
Supergame and that a combination of conditionally cooperative strategies by the two armies was a
Nash equilibrium combination? Was each of the necessary assumptions satisfied in the trench
warfare in which the Live-and-Let-Live System arose? If any of the assumptions were not satisfied,
was it well enough approximated to make it seem plausible that the combination of conditionally
cooperative strategies a Nash equilibrium combination? Explain your answers. Use the information
provided by Axelrod and your best guesses about other relevant facts in your discussion of each
assumption.

11. Critically evaluate Taylor's claim on p. 78 [Reader, p. 24] that all other equilibria of the
PD Supergame are either equivalent to or Pareto-dominated by at least one of those that he considers.

12. For each of the following two-person games, draw the Schelling diagram of an n-person
analogue, and explain why it is analogous to the two-person game:
(a) Prisoner's Dilemma
(b) Standard Convention Game
(c) Pareto Convention Game
(d) Chicken Game

13. In a section of his book that we did not read, Taylor says that there are Chickens nesting
in the N-Person Prisoners' Dilemma supergame. To see what he means, consider an N-person PD
supergame, and consider the two strategies D and B_m (where m < N1). Draw a Schelling diagram
for the two strategies, with L = D and R = B_m. (It is not necessary to calculate numerical utilities for
the different combinations of strategies, the shape of the diagram is what is important, particularly at
the point at which there are m other agents choosing B_m. Is the matrix an N-person Chicken Game?
Explain your answer.

14. "Expressway traffic is flowing toward the city during morning rush hour. Suddenly,
brake lights flash, cars slow down, traffic crawls. An accident has occurred in one of the outbound
lanes. Why is it the citybound traffic jams up? Drivers have reduced their speed to get a glimpse of
the wreckage on the other side of the divider. Eventually, large numbers of commuters spend an
extra ten minutes driving, for a ten-second look. It costs each driver ten minutes to get his look. But
he pays ten seconds for his own look and nine minutes, fifty seconds for the curiosity of the drivers
ahead of him. It is a bad bargain." [from the back-cover of Schelling's Micromotives and
Macrobehavior].
(a) David is one commuter who has just, at this moment, reached the scene of the accident.
He must decide in the next three seconds whether or not to slow down and look. Is he, at that precise
moment, in an MPD [Multi-Person Prisoners' Dilemma] with the other commuters? Explain your
answer in a way that shows you are familiar with Schelling's definition of an MPD. [Hint: David has
already been delayed 9 minutes and 50 seconds by the commuters in front of him who stopped to
look.]
(b) Suppose there available at no charge an audio tape that commuters can use while they
sleep to eliminate the urge to slow down and look at a traffic accident. In such circumstances, the
urge to slow down and look is replaced by a compulsion to keep their eyes on the road straight ahead
and not to slow down. The tape works after only one night of use, and has no adverse side effects. If
all commuters use the tapes, then there will be no more delays due to drivers slowing down to look at
an accident scene. All commuters, including John, have received a copy of the tape in the mail. Each
is trying to decide whether or not to use the tape. Assume that all the commuters would prefer to
look at the traffic than not, if the cost were only a 10 second delay in their commute. Thus, if there
were no other commuters, each commuter would not use the tape. But all commuters would be
willing to forego looking at an accident, if by doing so they could save 10 minutes of commute time.
Thus, if using the tape would save them 10 minutes of commute time in such circumstances, they
would all use the tape. Explain how the decision situation of John and the other commuters might
well be an MPD. Draw a Schelling diagram which illustrates how the situation could satisfy the four
conditions of an MPD.

15. What is the "Folk Theorem" of Non-Cooperative Game theory? Illustrate it with an
example.

16. This question concerns the example of the Potato Farmer (PF) and the Broccoli Farmer
(BF) discussed in class. Take the STANDARD ASSUMPTION as given. Also suppose that the
situation is one in which the crops are perishable, so they must be consumed when they come in, and
that each year the potato crop comes in well before the broccoli crop (which, in turn, comes in well
before the next year's potato crop). Thus, PF must decide whether to share some of this year's
potatoes with BF before BF must decide whether to share some of this year's broccoli with PF (and
BF will have to decide whether to share this year's broccoli with PF well before PF must decide
whether to share some of next year's potato crop with BF).
(a) One-Shot Interaction. Assume only a one-time interaction (each player has an
opportunity to share once), with utilities given by the matrix below. (Perhaps one farmer plans to
move away next year.) Explain why the game is a Two-Person Prisoner's Dilemma.

PLAYER 2 (Broccoli Farmer)

		Share	Don't Share
PLAYER 1 (Potato Farmer)	Share	8,8	3,10
	Don't Share	10,3	5,5

(b) Fixed-Length Supergame. Suppose that BF plans to move away in four years, but that 
they both expect to have four years of potential sharing. Use backward induction to explain why IR 
agents would not share, even though they could potentially obtain the benefits of four years of 
trading.

(c) Supergame of Indefinite Length. Suppose that PF and BF have the potential for sharing 
for the indefinite future (e.g., neither plans to move away and both are young and in good health). 
Suppose that the long-term interaction satisfies the conditions for a PD Supergame with discount 
factor = .99 for each agent. 
(i) Prepare a supergame matrix with the following strategies for each agent: D-; C-TFT; and 
D-TFT.
(ii) Is the combination (B,B) a Nash Equilibrium in the matrix in (i) above? Explain.
(iii) Is there an IR solution to the matrix in (i) above? Explain.
(iv) Is the combination (B,B) a Nash equilibrium in the full supergame (not limited to the 
matrix in (i) above, which shows only three of the infinitely many possible supergame 
strategies)? Explain.
(v) Is there an IR solution to this supergame? Explain. 
(vi) What additional factors might produce a solution to this supergame? Explain.