2.1.1 Example Matching Pennies

1
Saddle Point
2
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 UL we call this common value the
  value of the game.

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1.3.1 Example
A
A
s
1
2
i
a
0
2
0
1
a
3
1
1
L
2
S
3
2
j
U

• L U

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1.3.2 Example
L
U

• Value of game is 2

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1.3.1 Theorem

• For 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

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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.

si
vij
Sj
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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 vij
  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
• vij min vij j1,2,...,n ( si)
• and from the definition of max we have that
• vij max viji1,2,...,m (Sj)

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• We therefore conclude that
• si vij Sj
• But since we already established that
• si Sj we conclude that vij 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 vij 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 (LU 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 LU)
• Assume that there is a saddle point, say vij.
  By definition then,
• Sj vij si .
• Since by definition L si and U Sj ,
• U Sj vij 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 2person zerosum 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.
• See lecture for solution.

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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.

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Twoperson Constantsum Games

• A twoperson constantsum game is a two player
  game in which, for any choice of both players
  strategies, the row players payoff and the
  column players payoff add up to a constant
  value, c.
• A twoperson zerosum game is a special case of
  this.
• A twoperson constantsum can be approached in
  the same way as a twoperson zerosum 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).
Soap Opera
Western
Comedy
(15,85)
(60,40)
Western
(35,65)
Network 1
(50,50)
(45,55)
Soap Opera
(58,42)
(14,86)
(70,30)
(38,62)
Comedy
Network 2
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• As with zerosum games we could just enter the
  payoffs to Player 1, with the understanding that
  Player 2 gets
• (100 Player 1s payoff)
• By subtracting 50 from all entries we can convert
  this to a zerosum game.
• In general by subtracting c/2, a twoperson
  constantsum game (where c is the constant sum)
  can be converted to a twoperson zerosum 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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We cannot guarantee
the existence of a solution satisfying both
principlesOne 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.
So, what do we do?
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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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Randomise your decisions, mate!
New Game
How should I randomise?
Payoff Table