Economics S4415
Columbia University
Summer 2001
Professor Dutta
Solutions to Problem Set 1
Chapter 1
1.4 Consider the purchase of a house. By carefully examining each of
the four components of a game situation - group, interaction, rationality,
and strategy - discuss whether this quali…es as a game.
Answer: Group - seller and potential buyers (bidders); interaction
- only one of the bidders is going to eventually buy the house and will
do so at his winning bid and this bid will be what the seller gets from
selling the house; strategy - for bidders a determination of how much
they are willing to go up to and for the seller a determination of how
much he is willing to come down to; rationality - a strategy for the bidder
that is consistent with the maximization of some objective (such as the
expected gain from buying the house) and similarly a strategy for the
seller that is consistent with the maximization of some objective (such
as the expected gain from selling the house).
3.1 Show in detail that player 2 has a winning strategy in Nim if the
two piles of matches are balanced. (Your answer should follow the formalism I set up in the text; in particular, every con…guration of matches
should be written as (m; n) and removing matches should be represented
as a reduction in either m or n.)
Answer: Initially, the con…guration of matches is (m; m) for some
number m. Suppose that after player 1’s move the con…guration is (k; m)
where k is a number between 0 and m ¡ 1. If k = 0, then player 2 can
remove the m matches from the second pile and win the game. If k > 0,
then player 2 can move the game to (k; k). Now it is player 1’s turn
to move; suppose he moves the game to (l; k). If l = 0, player 2 can
remove the k matches from the second pile and win the game. Else.
he can move the game to (l; l). In this fashion, after no more than m
moves of player 1, the game must arrive at a con…guration such as (0; p)
where p > 0. Then player 2 removes the last p matches from the second
pile. (In this answer I simpli…ed the exposition by assuming that player
1 always removes from the …rst pile and I can do this without loss of
generality since after every move of player 2 there is exactly the same
number of matches in each pile. Hence, if player 1 removes from the
second pile then player 2 can remove from the …rst so as to keep the
piles balanced.)
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3.10 Finally, show that if the two piles are unbalanced, player 1 has
a winning strategy.
Answer: Player 1 can balance the piles in his …rst move. Thereafter, by question 3.9, he has a winning strategy. This strategy involves
matching the other player - in order to kep the two piles balanced unless
player 2 has a) moved the game to (m; 0), or b) (m; 1), where m ¸ 2.
In the former case player removes all but one match from the only pile
remaining. In the latter case he removes all m matches from the larger
pile.
3.12 Suppose voter 3’s preferences were the following:
N ÂBÂA
(instead of N Â A Â B as in the text). What would be the outcome
of truthful voting in this case? What about strategic voting?
Answer: Truthful voting second round - A wins against N but B
loses to N. Truthful voting …rst round - A wins against B.
Strategic voting second round - A wins against N but B loses to
N. Strategic voting …rst round - A wins against the e¤ective alternative
choice of N.
Chapter 3
1.1 Consider the game of Battle of the Sexes. How would you modify
the payo¤s to (f,o) and (o,f) to re‡ect the following: the husband is
unhappiest when he is at the opera by himself, he is a little happier if he
is at the football game by himself (he is happier still if he is with his wife
at the opera and is the happiest if they are both at the football game)?
(Likewise, the wife is unhappiest when she is at the football game by
herself, she is a little happier if she is alone at the opera, happier still if
she is with her husband at the football game and is the happiest if they
are both at the opera.)
Answer: Modi…ed Battle of the Sexes
HnW f
o
f
3; 1 0; ¡1
o ¡1; 0 1; 3
Consider the following model of price competition. Two …rms set
prices in a market whose demand curve is given by:
Q= 6¡p
where p is the lower of the two prices. If …rm 1 is the lower priced
…rm, then it is …rm 1 that meets all of the demand and conversely if …rm
2 is the lower priced out…t. For example, if …rms 1 and 2 post prices
2
equal to 2 and 4 dollars respectively, then …rm 1, as the lower priced …rm,
meets all of the market demand - and hence sells 3 units. If the two …rms
post the same price p; then they each get half the market, i.e., they each
get 6¡p
. Suppose that prices can only be quoted in dollar units, i.e.,
2
prices can be 0, 1, 2, 3, 4, 5 or 6 dollars. Suppose, furthermore, that
costs of production are zero for both …rms.
3.2 Show that the strategy of posting a price of $5 (weakly) dominates
the strategy of posting a price of $6. Does it strongly dominate as well?
Answer: If the rival’s price is $4 or less, each of the two strategies
yields a pro…t of 0. If the rival’s price is $5, then the latter has a zero
pro…t but the former has a strictly positive pro…t of 25 . Finally, if the
rival’s price is $6, then the former strategy yields a pro…t of 5 while the
latter yields a pro…t of 0. Hence the former strategy (weakly) dominates
the latter.
3.4 Is there a dominant strategy for player 1? Explain.
Answer: No, there is no dominant strategy. Consider any price
between p = 2 and p = 6, and suppose that …rm 2 happens to match
that price. In that case, …rm 1 would have been better o¤ pricing at
p ¡ 1 (work the pro…t …gures out for yourself). On the other hand p = 1
is not a dominant strategy because if the rival …rm prices at 6, then …rm
1 would have been better o¤ pricing at 3 instead. Clearly a price of 0 is
not a dominant strategy either.
3.9 Show that the strategy of volunteering for 1 hour (weakly) dominates the strategy of volunteering for 2 hours. Does it strongly dominate
as well?
Answer: The strategy of volunteering for 1 hour (strongly) dominates the strategy of volunteering for 2 hours if and only if
q
1+y¡1>
q
2+y¡2
That inequality, after some rearranging, is equivalent to
q
1+y+1>
or, squaring both sides
q
q
2+y
2+y+2 1+y >2+y
and that last inequality always holds.
Chapter 4
1.5 Consider voter 1. There are two strategies that involve truthvoting in the second stage, i.e., AAN and BAN . Does AAN dominate
BAN - or vice-versa?
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Answer: Yes - switching a …rst-round vote from B to A might get A
passed in a circumstance where it might not have passed otherwise (i.e.,
when only one of the other two voters votes for A). Then A would go on
to win the second round of voting and that outcome would be strictly
preferred by this voter to the alternative under which B gets elected
in the …rst round but then loses the second round election. If the vote
switch makes no di¤erence to the …rst round election’s outcome, then it
makes no di¤erence to the overall outcome - and hence payo¤.
1.9 Based on your answer to the previous two questions can you give
a reason why - in any price competition model - a duopoly …rm would
never want to price above the monopoly price? (Hint: when can a
duopoly …rm, that prices above the monopoly price, make positive profits? What would happen to those pro…ts if the …rm charged a monopoly
price instead?).
Answer: The monopoly price, say m, dominates any price, say m+k
above that level. If the rival prices at or above m + k then, by de…nition,
the …rm has strictly higher pro…ts from pricing at m (why?). If the rival
prices between m and m + k; again the monopoly price yields a higher
pro…t (since the pro…t is 0 for price m + k). Finally, if the rival prices
below m, then both prices yield zero pro…ts.
2.3 a) What is the dominant strategy for Africa? b) What is the
IEDS solution?
Answer: a) If B has not been vetoed, then elect B by vetoing the
other remaining candidate. If B has already been vetoed, then veto
A (and thereby elect H). b) The US vetoes B (and Africa plays its
dominant strategy); H gets elected.
4.3 Give an example of a game that has an outcome to IEDS although
no player has a dominant strategy. Do this for both i) strong as well as
ii) weak domination.
Answer: The following game has an IEDS solution of U p; Right
- regardless of whether we use strong or weak dominance to eliminate
strategies. There is however no dominant strategy.
1n2 Lef t Center Right
U p 0; 0 1; 1
1; 2
Down 1; 0 0; 1
0; 0
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