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Math 166, Fall 2015, Robert
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Ch G- Game Theory
Strictly Determined Games
A game is any situation in which players compete against one another. It can
be used to model competition between individual people or organizations, and
it can even model competition between non-sentient things like the weather patterns. The underlying assumption in game theory is that all parties involved
are intelligent enough to make the best possible decision.
Example 1 A company needs to hire a new employee and has narrowed their
canididates down to two equally qualified people, Todd and Maria. The company decides to call them in for one more interview. The canididates know the
following
• If neither canididate is dressed well for the interview, there is a 50% chance
that the company will hire Todd
• If Todd dresses well but Maria does not, there is a 95% chance that the
company will hire Todd
• If Maria dresses well and Todd does not, there is a 5% chance that the
company will hire Todd
• If both candidates dress well, there is a 50% chance that the company will
hire Todd
What is the best strategy for both candidates?
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Math 166, Fall 2015, Robert
Williams
A two-person constant-sum game is a game in which
1. there are two players
2. the row player must choose one of the row strategies and the column player
must choose one of the column strategies
3. if the row player chooses strategy i and the column player chooses strategy
j, the row player recieves ai,j and the column player gets the constant
minus ai,j
Unless otherwise stated, games are assumed to be zero-sum games. That is,
whatever one player gains the other loses. The matrix containing the results for
these different strategies is called the payoff matrix
If the payoff matrix has an entry, say ai,j that is the minimum entry in its
row and the maximum entry in its column, then ai,j is called a saddle point
or an equilibrium point. A game with a saddle point is said to be strictly
determined. Note that a game may have more than one saddle point.
The amount that the row player wins (or loses) in a zero-sum game is called the
payoff value. A game with a payoff value of zero is called fair.
Example 2 Recall the payoff matrix for our previous example was


0.5 0.95


0.05 0.5
Is this game strictly determined?
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Math 166, Fall 2015, Robert
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Example

3


1. 2

1

0
3 Find the saddle points for the following payoff matrices:

0 −2 5


1
4 3

−2 0 7
−1


2.  1

−1

2


1
3. 

3

4
0
1
1
1



−1

0
3
−5
5
−7
−10
1
9
1



−5


−2

1
Which of these games are strictly determined?
Mixed Strategy Games
When a game is not strictly determined, players have a meaningful choice in
strategy. In this case, we can write the strategy used by each player as the
vector of probabilities that the player will choose each possible strategy. For
example, if the row player has two strategies and chooses the first one with
probability 0.25 and the second one with probability 0.75, we would write that
h
i
the row player uses strategy 0.25 0.75 .
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Math 166, Fall 2015, Robert
Williams
If a game has a payoff matrix A, the row player uses strategy p, and the column
player uses strategy q, then the expected value of the

a1,1 a1,2


h
i  a2,1 a2,2
E(p, q) = pAq = p1 p2 . . . pm 
..
 ..
 .
.

am,1
am,2
game is
...
...
..
.
...
 
q1
 
 
a2,n   q2 
 
..   .. 
 
. 
 . 
a1,n
am,n
qn
Example 4 In the previous example, one of the payoff matrices given was for
the game Rock-Paper-Scissors:
Rock

Rock
Paper
Scissors
0
−1
1
0
−1
1
0


 1

Scissors
−1
Paper





1. If the row player always picks rock and the column player uses the strategy
h
i
q T = 0.25 0.1 0.65 , what is the expected outcome of the game?
h
iT
2. If the column player picks uniformly at random (i.e. strategy 13 13 13 ),
h
i
and the row player uses strategy p = x y z , what is the expected outcome of the game?
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Math 166, Fall 2015, Robert
Williams
Notice that the expected outcome changes with the strategies chosen by each
player. There exists an optimal strategy for each player, p̄ and q̄, that gives
that player the best possible outcome against a skilled opponent. If both players
play optimally, the expected outcome of the game, E(p̄, q̄) is called the value of
the game. In the previous example, we saw that when the column player chose
between his strategies uniformly at random, the expected outcome of the game
did not depend on the choice made by the row player. We can quickly check
that any other choice of strategy will allow the row player to choose a strategy
that gives him a positive expected value, and so this strategy is optimal.
One strategy dominates another strategy if the first strategy gives the player a
better outcome than the second regardless of the oponent’s play. A dominated
strategy can be safely removed from the first player’s choices, as a rational player
would never choose it.
Example
5 Remove

5 −2 8
0


 2 −6 2 −1


 3 −1 −1 1

−2 5
6
2
all dominated strategies from the following payoff matrix:
3


3


1

4
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Math 166, Fall 2015, Robert
Williams
Example 6 A bag contains three marbles: a large marble, a medium marble,
and a small marble. The first player looks into the bag and picks one of the
three marbles without showing it to the second player. The second player, only
seeing the two marbles left, picks a marble and then guesses whether his marble
is smaller than or larger than the first player’s marble. Find the optimal strategy
for the second player in this game. What is the value of the game?
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