Alternate Move Games
First, we will take a look at games played by two players who move one at a time. Such games are
called Alternate Move Games. This includes games of chance where we consider chance or nature
to be the second player.
Tic-Tac-toe This is a game where two players, X and O, take turns at placing their symbol, x and o
respectively, in a square of a 3 × 3 array as shown below.
x
x
→
x
→
→
o
→
o
x
x
o
x
→
o
x
o
x
x
→
o
x
x
o
o
o
x
x
→ x
x
o
o
o
x
The first player to get three marks in a row, horizontally, vertically or diagonally wins the game. If
neither player has three marks in a row by the time all 9 squares are full, the game is declared a draw.
Game Trees
A Move in a game is a single choice made by one of the players.
A Game Tree is a tree diagram showing all possible sequences of moves for a game.
Tic-tac-toe To make a game tree analysis of the game, we need notation for the different moves. We
number the squares on the tic-tac-toe grid as shown below.
1 2 3
4 5 6
7 8 9
We let X3 denote the move where player X puts the mark x in square 3 and we let O9 denote the move
where player O puts the mark o in square 9 etc..... .
We will consider a game in progress where the moves X1, O4, X9, O5, X6 have been made as shown
below.
x
o o x
x
Part of the tree diagram showing all possible moves for the rest of the game is shown below.
Exercise: Remainder of the game above (a) Draw a sequence of 3 × 3 array’s as shown above to
demonstrate what happens if the path followed by both players is
O2 → X8 → O3 → X7.
(b) Fill in the remaining part of this tree diagram starting at the node O8 on the path “Start → O8 →”.
1
O
X
O
X
Outcome
X wins
X3
O23
X7
O3
zz
zz
z
z
zz
X8
Draw
O8
33
33
33
33
O3
33
z
33
zz
z
33
zz
zz
X8
O7
O7
X2DD
DD
DD
DD
O8
O2
z
zz
z
zz
zz
X7
O3
O8
FFFF
FF
FF
O2
X8DD
DD
D
DD
D
O wins
X7
X wins
X3
X wins
O wins
X7
Draw
X8
X wins
X2
Draw
X7
X wins
O7
Start
%.
%% ..
%% ..
%% ..
%% ...
X2DD
O3
DD
%% ..
D
DD
%% ..
D
..
%%
..
O8
%%
..
%%
.. %%
. %%
O7 FF
X3
%%
FF
%%
FF
FF
%%
%%
X8DD
O2
%%
DD
%%
DD
DD
%%
%%
O3
%%
%%
%%
%%
%%
%%
%%
%%
%
O8?
O wins
O wins
X3
X wins
X wins
X3
X wins
X2
Draw
2
We follow the following rules when constructing game trees:
• As above, each node is labelled by the player who made the choice at that node.
• Each branch emanating from a node represents a possible choice made by the next player at that
node.
• Each final node is labelled by the outcome or pay-off to the players.
A complete game tree for Tic-tac-toe would be very large, starting with 9 choices for Player
X and 8 subsequent choices for Player O, 7 subsequent choices for Player X, 6 subsequent choices
for Player O, 5 subsequent choices for Player X, etc.. . The first five moves alone would generate
9 · 8 · 7 · 6 · 5 = 15, 120 different possible paths for those moves. One can compress such a tree using
symmetry or consider a partial game tree as we have done above.
A Partial Game Tree is a game tree showing only some of the possible sequences of moves for a game.
Example Football teams often replace certain players in special situations. One instance is near the
goal line, where the team with the ball may have its regular offense and a goal line offense. Similarly, the
team without the ball may have its regular defense and a goal line defense. The defense generally has
an opportunity to replace its players after the offense has. Therefore, we will assume that the offensive
coach first chooses regular or goal line offense and then the defensive coach selects regular or goal line
defense. Construct a game tree for the coaches’ choices.
The game of Nim involves two players who alternately remove objects, which we call pebbles, from one
or more piles according to some prescribed rules.
Example: Nim with 6 pebbles Consider a game of Nim where two players, Player A and Player
B, alternate removing either one or two pebbles from a single pile. The game starts with a pile of six
pebbles and the last player to take a pebble wins. A partial game tree is shown below where each node
is labelled by the number of pebbles left in the pile after the given player’s move. Complete this game
tree.
3
Start
:6
eeeWWWWWWW
e
e
e
e
e
WWWWW
eeeee
WWWWW
eeeeee
WWWWW
eeeeee
WWW
A: E5
yE
yy EEE
EE
yy
y
E
yy
B: 4
yy
yy
y
y
yy
A: 3 A: E2
EE
EE
EE
E
B: 2 B: 1 B: 1
A: 1
A: 0
B: 0
B: E3
EE
EE
EE
E
A: E2 A: E1
E
E
EE
EE
EE
EE
EE
EE
B: 1
B Wins
A: 0
A: 0
A Wins A Wins A Wins
B: 0
B: 0
B Wins B Wins
A: 0
A Wins
B: 0
B Wins
Strategies
A strategy is a player’s planned choices of moves throughout the game. In the case of an alternative
move game it includes (planned) responses to all of the opponents possible moves.
Example In Tic-tac-toe, Player X might decide on the strategy “ on each move, choose the first available
spot on the list 1, 2, 3, 4, 5, 6, 7, 8, 9”, where the spots are labelled as below
Try playing a few games of tic-tac-toe with Player
X using this strategy against Player O who is attempting to win. You will find that this is not an
effective strategy.
1 2 3
4 5 6
7 8 9
An Optimal Strategy is a strategy that produces the best possible results against the most skillful
opponents. It may have a compact one line description or a move-by-move description of the form “If
. . . then . . . , and then if . . . then . . . etc...” or something in between. It may be that if the opponent
plays (skillfully) to win that a player is guaranteed to lose and an optimal strategy does not exist. (later
we will look at games where chance is involved and neither player is guaranteed to win.)
Example Let us consider the example of Tic-Tac-Toe shown above. We will use the partial game tree
developed above to figure out an optimal strategy for O assuming that X plays to win.
(a) What happens if O uses the strategy given in the previous example “on each move, choose the first
available spot on the list 1, 2, 3, 4, 5, 6, 7, 8, 9” and X plays skillfully?
Recall that the partial game tree started after the sequence of plays X1, O4, X9, O5, X6 had been made
as shown below.
x
o o x
x
Highlight the path(s) resulting from the above strategy on the graph below.
4
O
X
O
X
X3
O3
w
ww
w
w
ww
ww
O2
X7
O8
22
22
22
22
O3
22
w
22
ww
w
w
22
ww
ww
X8
O7
X2
O7
GG
GG
GG
GG
G
O8
O2
w
ww
w
w
ww
ww
O3
O8
X7
DDDD
DD
DD
X8
O2
GG
GG
GG
GG
G
Start
O7
$$,,
$$ ,,
$$ ,,
$$ ,,
X2
O3
GGGG
$$ ,,
G
,
GG
$$ ,
GG
$$ ,,
,
O8
$$
,,
$$
,, $$
, $$
O7 D
X3
DD
$$
DD
$$
DD
D
$$
O2
X8
$$
GG
GG
$$
GG
GG
$$
G
$$
O3
$$
$$
$$
O3
X2
$$
GGGG
$$
GG
GG
$$
G
$$
O7
$$ $$ O8 D
X3
X wins
DD
DD
DD
D
X7
O2
GG
GG
GG
GG
G
O3
(b) To find an optimal strategy for O, we should
start at the ends of the paths and work backwards.
We should eliminate paths leading directly to a win
by X. We trace these paths back to the last choice
made by O leading to the win and eradicate that
choice as an option for O. For example the path
highlighted in Green on the left shows that O2 is
not a good choice for O on the first move. Highlight the other paths/(choices for O) that should
be eliminated.
Out.
X wins
X8
Draw
O wins
X7
X3
X wins
(c) Draw the remaining paths on a tree, showing
the feasible options for O which lead to a win or a
draw.
X wins
O wins
X7
Draw
X8
X wins
X2
Draw
X7
X wins
O
X2
O wins
Start
X wins
X wins
X2
Draw
X7
Draw
X3
Xwins
5
Draw
O wins
O7
O7
Out.
X7
Draw
X2
Draw
O wins
(d) Use the same process to find a description of
X 0 s optimal strategy if it exists.
O wins
X2
X
We can now now see that O’s optimal strategy
is to play O3 on the first move and then play O7
unless x has played X7 on their first move, in which
case O should play O8 for a draw.
X wins
X3
O
DD
DDD
DD
D
O8
O3
O8
X7
DDDD
DD
DD
X8
DD
DD
DD
DD
O wins
X3
X
Chance as a player Many games involve chance which should also be used to evaluate strategies.
We will elaborate on this over the next few lectures. For the moment we can get a preview by considering
the example from football given above along with some extra information.
Example Football teams often replace certain players in special situations. One instance is near the
goal line, where the team with the ball may have its regular offense and a goal line offense. Similarly, the
team without the ball may have its regular defense and a goal line defense. The defense generally has
an opportunity to replace its players after the offense has. Therefore, we will assume that the offensive
coach first chooses regular or goal line offense and then the defensive coach selects regular or goal line
defense.
Let’s also assume that we have the following information about the offense’s success rate (percentage
of time offense scores a touchdown) in each of the four possible scenarios.
Regular Offense
Goal Line Offense
Regular Defense Goal Line Defense
50%
40%
60%
50%
(a) What strategy should the offensive coach choose here?
(b) What counterstrategy should the defensive coach choose?
6
A player has a A Winning Strategy if there is a strategy for that player, which guarantees a win for
the player no matter what the other player does. Note there can only be a winning strategy for one
player, but a winning strategy does not always exist.
Example Let us consider the game of Nim described above. Is there a winning strategy for Player A
or Player B.
Start :6
eeeYYYYYYYYY
eeeeee
YYYYYY
e
e
e
e
e
YYYYYY
eee
e
e
e
e
YYYY
e
eee
A: E4
A: E5
y EE
yy EEE
y
EE
EE
yy
yy
E
y
y
EE
EE
y
y
yy
yy
B: 4
>>
>>
>>
>>
A: 3 A: >2
>
>>
>>
>>
>
B: 2 B: 1 B: 1
A: 1
A: 0
B: 0
A: 0
A: 0
>>
>>
>>
>>
>>
>>
>
>
B: 1
B: 0
B: E2
EE
EE
EE
E
A: 2 A: 1 A: 1
A: >2 A: >1
>
>
B Wins
A Wins A Wins A Wins
B: 3
B: >3
B: 0
B: 1
B Wins B Wins
B: 0
A: 0
A Wins
B: 0
B: 0
B Wins B Wins B Wins
A: 0
A: 0
A Wins
A Wins
B: 0
B Wins
Show that if the players start with a pile of n stones in a game of nim with the same rules as above,
then the second player has a winning strategy if n is a multiple of 3 and the first player has a winning
strategy otherwise.
7
Football Draft [Straffin]
In Football, basketball and other professional sports, teams choose new players by a draft system which
involves sequential choices. From the pool of available players, the team with the worst win-loss record
in the previous season gets first choice. The second worst team gets second choice, and so on until all
teams have chosen. The procedure is then repeated for as many rounds as needed until all teams have
exhausted their choices. This system is designed with the goal of maintaining competitive balance in
the sport, thus maintaining the interest of the general public. The system however is open to strategic
play and in some cases leads to an example of a game similar to The Prisoner’s Dilemma (see later),
where all teams are worse off if every team pursues its self interest.
Example Let us consider the situation where we have two teams, the Penguins and the Polar Bears
and 4 players Albert(A), Brian(B), Colin(C), David(D). Suppose the teams’ preferences are given in
the following table:
Penguins Polar Bears
A
B
B
C
C
D
D
A
Here the Penguins most want Player A and least want Player D, while the Polar Bears most want Player
B and least want Player A. It is not unreasonable to assume that teams have different preferences, since
the Penguins may want a good quarterback, whereas the Polar Bears may have a good quarterback and
may need a strong linebacker.
Assuming that the Penguins have first choice, the game tree on the next page shows the possible choices
for both teams with the choices of the Penguins shown in red and those of the Polar Bears shown in
blue
If both teams follow a sincere drafting strategy the resulting choices will be: Penguins A, Polar Bears
B, Penguins C, Polar Bears D.
We might assign a payoff for each team to the game tree in the following way;
We assign a value to the players for each team by assigning 4 to their highest ranked player and 1 to
their lowest ranked player,
Penguins Polar Bears
A (4)
B (4)
B (3)
C (3)
C (2)
D (2)
D (1)
A (1)
(In reality, the weighting or value of the players to each team may be different.)
The payoff for each team is then the sum of the values they attach to the players they have drafted.
We have written the outcome or payoff for each path of the game tree as a pair of numbers with the
Penguins’ payoff listed first and the Polar Bears’ payoff listed second.
8
B
A.
C
.
..
..
..
..
..
..
..
C
D
(6, 6)
D
C
(5, 7)
B
D
(7, 5)
D
B
(5, 7)
B
C
(7, 5)
C
B
(6, 6)
C
D
(5, 3)
D
C
(4, 4)
A
D
(7, 5)
C
D
ii B >>
iiii
i
>
i
i
>
>>
iiii
>>
iiii
i
i
i
ii
i
i
i
i
A
D>
iii
>>
iiii
i
i
i
>>
iiii
>>
iiii
i
i
>
i
iii
Start
C
%% PPP
PPP
%%
PPP
PPP
%%
PPP
%%
PPP
B
PPP
%%
PPP
%%
PPP
PPP
%%
PPP
%%
PPP
PPP
%%
A
D
PPP
%%
PPP
PPP
%%
PPP
%%
PP
%%
C)
A
B>
%%
>>
))
>
%%
>>
))
>>
%%
))
%%
)
(4, 4)
A
B
C
D
))
%%
>>>
)
>>
%%
))
>>
%%
))
>
%%
))
(3, 5)
C
B
A
%%
))
%% ))
%% ))
A
(4, 4)
C
(7, 5)
A
(5, 3)
D
(5, 3)
B
(3, 5)
D
(6, 6)
A
(3, 5)
B
(6, 6)
A
(5, 3)
D
B>
A
>>
..
>
..
>>
>>
..
..
C
..
..
..
.
C
(5, 7)
A
(3, 5)
C>
A
B
(5, 7)
B
A
(4, 4)
D.
>>
>>
>>
>
9
A>
D
>>
>>
>>
>
B
To find a good or optimal strategy for the Penguins’ first pick in the football draft, we need
to think about the payoffs for both players. We should assume that both teams are seeking to maximize
their payoff.
We start by working backwards and eliminating those paths for which the second round choice for the
Penguins leads to a less desirable outcome (smaller payoff for the penguins). The new restricted game
tree is shown on the following page.
Next, the Penguins must think through how the Polar Bears will react to their first choice, assuming
that the Polar Bears are playing to maximize their payoff and know the preferences of the Penguins.
Use the restricted diagram below to identify which of the remaining paths the Polar bears will not take
on their first pick and eliminate those paths. Draw the resulting restricted diagram.
Lastly, using the diagram derived above, identify which player the Penguins should pick as their first
choice in order to maximize their payoff.
10
P
PB
P
PB
Payoff
B
}}
}}
}
}
}}
AA
C
AA
AA
AA
A
C
D
(6, 6)
B
D
(7, 5)
D
B
C
(7, 5)
A
C
D
(5, 3)
C
A
D
(7, 5)
D
A
C
(7, 5)
A
B
D
(5, 3)
B
AA
AA
AA
AA
A
D
(6, 6)
D
A
B
(6, 6)
}
}}
}
}}
}}
DA
B
AA
AA
AA
A
A
B
C
(4, 4)
A
C
(5, 7)
C
A
B
(5, 7)
}}
}}
}
}
}}
BA
Start
&&22
&& 22
&& 22
&& 222
&& 22
22
&&
22
&&
2
&&
C
&&
&&
&&
&&
&&
&&
&&
&&
&&
&&
&&
&
AA
AA
AA
A
}}
}}
}
}
}}
11
Tic Tac Toe
Although Tic Tac Toe does not have a winning strategy, if played by two players using perfect
strategy, they will always hold each other to a draw. Wikipedia has an explanation of how computers
play Tic Tac Toe, with perfect strategy by searching through all possible games for the best play. It
also has a guide to perfect strategy for mere mortals.
http://en.wikipedia.org/wiki/Tic-tac-toe
Hex
Hex is played on a board marked with a hexagonal grid similar to the one shown below. Any size
is possible, but the 11 × 11 grid is traditional. Each player has an allocated color, Red and Blue being
conventional. Players take turns placing a stone of their color on a single cell within the overall playing
board. The goal is to form a connected path of your stones linking the opposing sides of the board
marked by your colors, before your opponent connects his or her sides in a similar fashion. The first
player to complete his or her connection wins the game. The four corner hexagons each belong to two
sides.
Unlike Tic Tac Toe, Hex cannot end in a draw. There is a proof, due to John Nash, that a winning
strategy exists for the first player. However a specification of that winning strategy has not been shown
for boards larger than 9 × 9.
You will find games of hex online, in particular you will find a link in the “Games” section of our web
page. If you wish to try a game, a board has been provided.
Wikipedia has an informative page devoted to the game of Hex:
http://en.wikipedia.org/wiki/Hex (board game)
References for Game Theory in Sports
Mathematics beyond The Numbers: Gilbert, G, Hatcher, R.
Game Theory and Strategy, Phillip D. Straffin, The Mathematical Association of America, New Mathematical Library.
Mathletics, Wayne L. Winston, Princeton University Press.
Thinking Strategically, Avinash K. Dixit, Barry J. Nalebuff, W.W.Norton and Company.
Strategies and Games, Prajit K. Dutta, The MIT Press.
Sports Economics, R. D. Fort, Pearson.
Moneyball: The Art of Winning an Unfair Game, Michael Lewis.
12