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Solutions to Chapter 17 Exercises
SOLVED EXERCISES
S1. The businessman’s BATNA is probably quite low; holding onto the domain name may be worth
very little to him. Owning the domain name, which makes it easier for people to find the Alta Vista site, is
important to Compaq, so its BATNA is also, apparently, millions below the value of a negotiated
agreement. Compaq is probably quite impatient to make a deal. There are many firms in the “portal”
Part of any cooperative outcome is that all possible mutual gains should be exploited (and
exploited as quickly as possible). In this situation, a mutual gain is created by getting the
www.altavista.com domain name into Compaq’s hands as soon as possible. The name is certainly worth
S2. Ali makes offers at all even number totals and Baba makes offers at all odd number totals. The
sum of their BATNAs is $5.75, so the last round where any positive surplus remains is when the total
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
S3. (a) The government of Euphoria’s BATNA is 0; the government of Militia’s BATNA is 100.
(b) If the BATNAs were mutually known, the countries could reach agreement immediately
or could wait until October, regardless of which moves first. One of the reasons that actual negotiations
UNSOLVED EXERCISES
U1. The general formulas tell us that x = a + h(v – a – b) and y = b + k(v – a – b). Letting Donna’s
payoff equal x and Pierce’s equal y, then a = 132, b = 70, and v = 216. If Donna’s has 2.5 times as much
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
U2. (a) Player 1 makes offers in odd-numbered rounds, and Player 2 makes offers in
even-numbered rounds. When V = 4 and c = 1, there is nothing left to bargain over in round 5. Working
backward from round 5, the following splits (x, y) would be proposed and accepted:
The players will reach an agreement in the first round whereby Player 1 receives x = 2 and Player 2
receives y = 2.
(b) There is nothing left in round 6, thus—working backward—the following proposals will
be made and accepted:
In the first round the players agree to the split of x = 3 and y = 2.
(c) Following the same logic as part (a), when V = 10 the players agree in the first round to
(d) Following the pattern of part (b), when V = 11 the players agree in the first round to the
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(e) For any whole-number value of V, the players always agree to a split in the first round.
(f) In the final agreement, each player must receive at least her BATNA, or she will not
b = 0 b = 1 b = 2 b = 3
The final splits are x = xpartial + a and y = ypartial + b, where xpartial and ypartial are the agreed splits of V – a – b,
as given by part (e). For the possible pairs of BATNAs a and b when V = 4 and a + b < V, the final splits,
which the players agree upon in the first round, are
b = 0 b = 1 b = 2 b = 3
(g) When V = 5, the possible values of V – a – b given that a, b, and V are whole numbers
and a + b < V are
b = 0 b = 1 b = 2 b = 3 b = 4
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
The final splits are x = xpartial + a and y = ypartial + b, where xpartial and ypartial are the agreed splits of V – a – b,
as given by part (e). For the possible pairs of BATNAs a and b when V = 5 and a + b < V, the final splits,
which the players agree upon in the first round, are
b = 0 b = 1 b = 2 b = 3 b = 4
(h) When Player 1 and Player 2 have BATNAs of a and b, respectively, where a, b, and V are
whole numbers and a + b < V, they are really only bargaining over the surplus surplus of V – a – b. Each
The final split will include the players’ BATNAs, so that x = xpartial + a, y = ypartial + b. If V – a – b is
(i) As in part (h), when the players have BATNAs a and b, they will need to receive at least
that level of surplus to participate at all. They are then bargaining over V – a – b. The number of
alternating rounds of proposals from each of the two players depends on V – a – b and the rate at which
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
this quantity decays, c > 0. The number of rounds, N, in which the players bargain over a surplus of at
least c is given by
, where denotes the greatest integer less than or equal to .
V a b
N m m
c
- -
ê ú
=ê ú
ë û
ê ú
ë û
Note that N ≥ 0. After N rounds, there still may be a little remaining surplus, d, such that 0 ≤ d < c. The
precise value of d is given by
( )
mod .
V a b
d V a b c V a b c
c
- -
ê ú
= - - = - - - ×
ê ú
ë û
If N is even, Player 1 will receive d in round N + 1, and if N is odd, Player 2 will receive d in round N + 1.
Working backward from round N + 1, c is alternatively added to one side of the proposed split and then
2
2
N
x c d a
N
y c b
= × + +
= × +
If N is odd:
1
2
1
2
N
x c a
N
y c d b
+
= × +
-
= × + +
U3. (a) If B is twice as impatient as is A, then s will be twice as large as r, or s = 2r. In this case, x = s/
(b) When r = 0.01 and s = 0.02, the approximation formulas give x = 0.667 and y = 0.333. The
exact formulas show that when A makes the first offer, A gets x = (0.02 + 0.0002)/(0.01 + 0.02 + 0.0002)
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
