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**Unformatted text preview: **Typical Final Questions — P830. NB: The point of this isri ’t to help you predict how you ’11 do on the ﬁnal -- it’s more to
bring up topics that we can discuss in the review sessions. Also, I ’ve put less of the ﬁrst
part of the course on here than will probably appear on the exam. It ‘s deﬁnitely not a
practice ﬁnal, in terms of content or length. 1. Current US policy on how to respond to a major attack from another state using
chemical or bacteriological weapons, is known as “calculated ambiguity” -- the US. says
it may or may not respond with nuclear weapons. (Using chemical or biological
weapons in response is not a realistic option -- the US does not stockpile biological
weapons any more and would not use chemical weapons.) (a) Draw a game tree showing a possible adversary state’s decision to use chemical or
bacteriological weapons or not, followed, if the adversary uses them, by a United States
decision to use nuclear weapons or something less. Put in payoffs showing the rank of
the outc0mes for both parties. (b) According to the method of rollback equilibrium, what would be the outcome in the
tree you have drawn? (0) Write a sentence justifying your assignment of payoff rankings to the United States. 3. (a) What should each person do in the game below? (Show or state your logic.) (Hint:
there are no mixed strategy Nash equilibria.) Column-chooser Row-chooser (b) Identify the efﬁcient outcomes in the game above. (0) Calculate Row chooser’s probability of choosing Up, in the game below Column-chooser
Leﬁ Ri ht Row-chooser 5. Label each of these statements true or false:
(a) In a zero sum game all outcomes are efﬁcient-[— (b) One need not worry aha“! the Winner’s Curse if everyone knows their true values for
the object being auctioned ' (0) Every matrix game has at least one mixed strategy Nash equilibrium}: (d) Every matrix game has at least one efficient outcomeT \
(e) For a strategy to be part of a Nash equilibrium, it must be a best response to anything the other player might do. F (1) Applying the method of rollback to a tree can sometimes produce an outcome that is
not a Nash equilibrium in the tree’s corresponding matrix game. F (g) Even when no two entries in a game matrix are equal, a single row can still contain
two Nash equilibria. {If (h) Game theory analyses assume that each person knows the other’s value for a
particular outcome F: 6. Below is a tree with two moves, Up and Down, for the ﬁrst player, and three, A, B, C,
and two moves, D and B, respectively, for the second. (SUPPLY YOUR OWN TREE
HERE.) Up and A gives 7, 1 Up and B 6, 5 Up and C 4, 8 Down and D 2, 6 Down and E 9, l (a) List each players’ strategies. (b) Write down a matrix that is equivalent to this tree. It isn’t necessary to solve the
game. (Hint: no player should have more than ten strategies and each might have fewer.) 7. Consider a game in which there is a prize worth $30. There are two contestants, A and
B. Each can buy a ticket worth $15 or $30 or not buy a ticket. They make these choices
simultaneously and independently. If no one has bought a ticket, no prize is awarded.
Otherwise the prize is awarded to the owner of the highest cost ticket bought, or if they
both bought the same price ticket, the prize is split equally between them. (a) Show this game in matrix form. (b) Is neither party buying a ticket a Nash equilibrium? (Just yes or no.)
(0) List all the pure strategy Nash equilibria. 8. In the usual chicken game, two thrill crazed teens decide whether to go straight or
swerve, and if one or both swerve there is no crash. Consider a modiﬁcation of the game
where you can stay straight or swerve east or swerve west. If they both stay straight OR
if they both swerve in the same direction, they crash. In order of each player’s preference
the kinds of outcomes are: S -- I go straight, he swerves 4 -- we both swerve in different directions 3 -- I swerve, he goes straight 2 -— we both stay straight I -- we both swerve in the same direction. Construct the matrix and identify the pure strategy equilibria.
9. Give a game whose Nash equilibrium is not efﬁcient, or say that this is impossible. 10. For any preference ranking of a group people over outcomes what is the minimum
possible number of Condorcet winners and what is the maximum? 11. In the US Senate, the vice president votes only when there is a tie. Assuming that no
senator abstains on any issue, does the vice president have equal, more or less power than
the other 100 senators? Explain why in a sentence, either intuitively or with reference to
the Shapley-Shubik measure of voting power. Suppose there were only 99 senators in the
Senate. 12. Here are the preferences of three voters over three proposals ABC N‘<><
KN‘<
‘<><N Two proposals will be put up against each other, and the winner will take on the third.
Choose the ﬁrst two proposals in a way such that y wins. 13. Here are ﬁve voters and their preferences over three proposals A,B&C D&E
X Y
y z
z x Who would win by the Borda count? Wh would win if 2 were absent? 2’: C Dny K W2 ”'1'
l y x j" 3%- State in a sentence why some people might see this as a criticism of the Borda count
method. 14. Here are three groups of voters, 9 voters in all, with their preferences over three
outcomes. Identify any Condorcet winner. Which proposal would win by plurality
voting? A,B,C&D E,F&G H&I x y z
y 3 y
z x x 15. Repeated games. For what values of p will the grim strategy used by both players be
an equilibrium in the following game: C D
C 4,4 2,9
D 9,2 3,3 16. A “nice” strategy in a PD tournament is a strategy that always cooperates when used
against itself. Give the sequence of moves produced by each strategy paired against
itself. Which of the following are nice: -\-/fit for tat (start C, do what the other did last time) -- Reverse tit for tat (start C, do the opposite of what the other did last time) —- the grim strategy -— cooperate at the start and keep cooperating unless the other has defected 10% or more
of the time. ...

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- Spring '03
- O' Neill