Solution to Exercise 3.7 on Battleship from Part II of Fergusons Game Theory
3.7#15 Battleship
As noted in the problem description, Player I has two invariant strategies, [1, 2] and [2, 5] . To describe
Player IIs invariant strategies, call squares cfw_1,
Section 1.5
1.5#2 Player I holds a black Ace and a red 8. Player II holds a red 2 and a black 7. If the chosen cards
are of the same color Player I wins, if they dier Player II does. The amount won is the face value, in
dollars, of the winners card, with
Solutions to Homework 2, Math 432 Spring 2013 Exercises from Parts I and II of Fergusons
Game Theory
Section 5.5
5.5#3 Triplets.
Player must turn over exactly three coins. If position x corresponds to having a string of (x 1) tails
followed by a single he
Solutions to Exercises from Part I of Fergusons Game Theory
Section 1.5
1.5#2 Take-Away Game where you can remove 1-6 chips.
(a) The empty pile is a P-position, and clearly piles of sizes 1 through 6 are N positions, since one
can move from them to the em
(The value of V can also be obtained by plugging in p = 7/10 into either column 1s or column
4s linear function). Is strategy is (7/10, 3/10) and IIs strategy is (1/2, 0, 0, 1/2).
(b) Reduce by dominance to a 3 2 matrix game and solve:
0
8 5
8
4 6
12 4 3
(d) The equation (16) only gives an optimal strategy for Player II if the assumption that Player I
has an optimal strategy giving positive weight to each of the rows. holds (rst paragraph of
section 3, page II 27). Since this assumption is false in our ca
B
does not have a saddle point, so the safety level of Player II is
vII = Val(B ) =
16
0344
=
,
04+34
5
and its MM-strategy is q = (1/5, 4/5).
(0, 0) (2, 4 )
and hence there
(2 , 4 ) (3 , 3)
is one PSE at (2, 1) with payo (2, 4). This is the only SE in fa
Solutions to Homework 3 Exercises from Part II and Part III of Fergusons Game Theory
Section 4.7
4.7#1 Consider the game with matrix
0
A = 1
9
7
4
3
2
8
1
4
2 .
6
(a) A Bayes strategy against (1/5, 1/5, 1/5, 2/5) is the row which gives the best payo for P
(a) If n = 4 and Player I begins with S in the rst square, Player II responds with S in the fourth
square, resulting in a reasonable string, both of whose empty squares are unsafe.
(b) If n = 7, we claim Player I can force a win my placing an S in the 4th