Math 167
Homework 8
December 9, 2008
Game Theory
Thomas Ferguson
Section II.3.7
Problem 15.
Battleship.
The game of Battleship, sometimes called Salvo, is played on two square
boards, usually
10
by
10
. Each player hides a fleet of ships on his own board and tries to sink the opponent’s
ships before the opponent sinks his. (For one set of rules, see http://www.kielack.de/games/destroya.htm,
and while you are there, have a game.)
For simplicity, consider a
3
by
3
board and suppose that Player I hides a destroyer (length 2 squares)
horizontally or vertically on this board. Then Player II shoots by calling out squares on the board, one at
a time. After each shot, Player I says whether the shot was a hit or a miss. Player II continues until
both squares of the destroyer have been hit. The payoff to Player I is the number of shots that Player II
has made. Let us label the squares from
1
to
9
as follows
1
2
3
4
5
6
7
8
9
The problem is invariant under rotations and reflections of the board. In fact, of the
12
possible positions
for the destroyer, there are only two distinct invariant choices available to Player I: the strategy,
[1
,
2]
*
,
that chooses one of
[1
,
2]
,
[2
,
3]
,
[3
,
6]
,
[6
,
9]
,
[8
,
9]
,
[7
,
8]
,
[4
,
7]
and
[1
,
4]
, at random with probability
1
/
8
each,
and the strategy,
[2
,
5]
*
, that chooses one of
[2
,
5]
,
[5
,
6]
,
[5
,
8]
and
[4
,
5]
, at random with probability
1
/
4
each. This means that invariance reduces the game to a
2
by
n
game where
n
is the number of invariant
strategies of Player II. Domination may reduce it somewhat further. Solve the game. Consider the mis`
ere
version of the take-away game of Section 1.1, where the last player to move loses. The object is to force
your opponent to take the last chip. Analyze this game. What are the target positions (P-positions)?
Solution
Ferguson II.5.9
Problem 1.
The Silver Dollar.
Player II chooses one of two rooms in which to hide a silver
dollar. Then, Player I, not knowing which room contains the dollar, selects one of the rooms to search.
However, the search is not always successful. In fact, if the dollar is in room #1 and I searches there,
then (by a chance move) he has only probability
1
/
2
of finding it, and if the dollar is in room #2 and I
1