Transcript 2.1.1 Example Matching Pennies

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You should know by now…
The security level of a strategy for a player is
the minimum payoff regardless of what strategy
his opponent uses.
A player tries to choose among all strategies
available to him, the strategy that maximises the
security level.
That is, the option that gives the least worst
outcome.
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You should also know what a
saddle point is by now…
A solution (ai, Aj) to a zero-sum two-person
game is stable (or in equilibrium) if Player I
expecting Player II to Play Aj has nothing to
gain by deviating from ai
AND
Player II expecting Player I to Play ai has
nothing to gain by deviating from playing Aj.
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Principle II
The players tend to strategy pairs that
are in equilibrium, i.e. stable
An optimal solution is said to be
reached if neither player finds it
beneficial to change their strategy.
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1.3 Saddle Points
Let L denote the best (largest) security level of
Player I, and let U denote the best (smallest)
security level of Player II.
We shall refer to L as the lower value of the
game and to U as the upper value of the game.
If U=L we call this common value the value of
the game.
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1.3.1 Example
A1
A2
si
a1
0
2
0
a2
3
1
1
Sj
3
2
L
U
L = U
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A1
1.3.2 Example
A2
A3
si
a1
3
2
10
2
a2
4
1
5
1
a3
8
-1
-2
-2
Sj
8
2
10
L
U
•Value of game is 2
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1.3.1 Theorem
any zero-sum 2-person game we have L ≤ U.
Proof.
Consider the ith row and jth column of the payoff
matrix for some arbitrary choice of i and j.
By definition si is the smallest element of row i,
hence si ≤ vij.
 Similarly, by definition Sj is the largest element in
column j, hence Sj ≥ vij.
This implies that

si ≤ vij ≤ Sj , for all i and j
For
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By definition
L = max si
 = si for some i
and
U = min Sj
 = Sj for some j,
hence L= si
≤ vij ≤ Sj = U, so L ≤ U.
vij
Sj
si
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1.3.1 Lemma (Page 12)
For any zero-sum 2-person game, L = U implies the
existence of a pair (i*, j*) such that vi*j* = si* = Sj*.
i.e. There is an entry in the matrix that is both the
smallest in its row AND the largest in its column.
Proof: By definition, L= si for some i, call it i*, and
U = Sj for some j, call it j*.
Hence L= U implies the existence of a pair (i*, j*) such
that si* = Sj*.
From the definition of min it follows that
vi*j* ≥ min {vi*j: j=1,2,...,n} (= si*)
and from the definition of max we have that
vi*j* ≤ max {vij*:i=1,2,...,m} (=Sj*)
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We therefore conclude that
si* ≤ vi*j* ≤ Sj*
But since we already established that
si* = Sj* we conclude that vi*j* = si* = Sj*.
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1.3.1 Definition: Saddle Point
(Page 13)
An entry (i*, j*) of the payoff matrix is said
to be a saddle point iff vi*j* = si *= Sj*.
I.e. A saddle point is BOTH the smallest in
its row and the largest in its column .
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1.3.2 Theorem:
For any 2-person zero sum game, the lower
value of the game is equal to the upper
value of the game if and only if the payoff
matrix possesses a saddle point.
Proof.
Necessity (L=U implies the existence of a
saddle point) is provided by Lemma 1.3.1.
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Sufficiency (existence of a saddle point implies that
L=U):
Assume that there is a saddle point, say vi*j*. By
definition then,
Sj* = vi*j* = si* .
Since by definition L ≥ si* and U ≤ Sj* ,
U ≤ Sj* = vi*j* = si* ≤ L
But Theorem 1.3.1 claims that U ≥ L.
Hence it follows that U = L.
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Saddle Point
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Summary
If the players follow the two Principles (Best
Security Level and Equilibrium) and the payoff
matrix has a saddle point, then there is a pair of
pure strategies (one for each player) which is a
stable solution to the game. This solution is given by
the saddle point.
When we say a pure strategy we mean the player
uses one row (or one column) all the time.
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Example
Solve the 2–person zero–sum game whose payoff
matrix is below. ie. Find saddle points, if any. Find
the value of the game. State the strategies the
players should use, based on the philosophy given
earlier.
A
A
A
A
1
a1
a2
a3
a4
4
6
5

6
2
3
5
7
7
6
See lecture for solution.
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6
5
5
4
8 
9 
4 
5 

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Example
For the two-person zero-sum game whose payoff
matrix is given below, find the values of x for
which there is a saddle point. Solve the game for
these values of x.
See lecture for solution.
 x 8 3 


0
x
9




5 5 x 
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Two–person Constant–sum Games
A two–person constant–sum game is a
two player game in which, for any choice
of both players’ strategies, the row
player’s payoff and the column player’s
payoff add up to a constant value, c.
A two–person zero–sum game is a special
case of this.
A two–person constant–sum can be
approached in the same way as a two–
person zero–sum game.
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Example: In a certain time slot, two TV networks are
vying for 100 million viewers. They each have the same
three choices for that time slot. Surveys suggest the
following numbers of viewers would tune in to each
network (in millions).
Western
Western
(35,65)
Soap
Opera
Comedy
(15,85)
(60,40)
Soap Opera
(45,55)
(58,42)
(50,50)
Comedy
(38,62)
(14,86)
(70,30)
Network
2
Network
1
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As with zero–sum games we could just enter the
payoffs to Player 1, with the understanding that
Player 2 gets
(100 – Player 1’s payoff)
By subtracting 50 from all entries we can convert
this to a zero–sum game.
In general by subtracting c/2, a two–person
constant–sum game (where c is the constant sum)
can be converted to a two–person zero–sum game
and thus the same ideas can be used.
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Question:
What happens if we do not have a
saddle point?
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
So, what do we do?
We cannot guarantee the existence of a
solution satisfying both principles
One idea in this case is to think of playing
the game repeatedly and looking at
expected payoffs, rather than the actual
payoff on any one play of the game.
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The BIG Fix
Mixed Strategies
Each player will mix his/her decisions
using some probability distribution.
Thus on one play of the game Player 1 may
use strategy a2, on the next play a4, then a2,
then a2, then a1, ??? but ….
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New Game
Randomise your decisions,
mate!
Player 2
How should
I randomise?
Payoff Table
Player 1
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