### Game Theory 2 Notes

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Lecture 7: Decision Theory / Game Theory II
19 November 2010

• Decision Making Under Uncertainty: When Nature's Probabilities are • Taha (13.2 and 13.4) • Decision making under uncertainty
Estimated • Expected value criterion
• Game Theory for Non-Strictly Determined Two-Person Zero-Sum Games • Expected value adjusted for risk criterion
Decision Making Under Uncertainty: When Nature's Probabilities are Estimated • Strictly determined games
• We may be sufficiently confident to assign specific subjective probabilities • Dominance
• i.e. using method of relative frequencies of occurrence • Non-strictly determined games
• We would then have a probabilistic decision matrix • Mixed strategies
• This is not the same as using the Laplace criterion
• Finding optimal mixed strategies
• We could apply the "folding back" method but this is strictly for objective probabilities
• Non-strictly determined games when m=n=2
• Non-strictly determined games when m>2, n=2
• Two further methods for decision making under uncertainty:
• Non-strictly determined games when m=2, n>2
• Expected Value Criterion
• Expected Value Adjusted for Risk Criterion • Rugby post diagrams

Expected Value Criterion
• We evaluate the expected pay-off for each of the player's strategies Definitions
○ Expected Pay-off: • A mixed strategy = A combination of pure strategies in a
certain proportion
• Perhaps the best criterion for making long run decisions
• A Strictly Determined Game = A game where there is a
• Under achievement and over achievement counter balance each other
• Conservative Strategy = A strategy where from all the
○ Example of the expected value criterion
available strategies it is the one where the smallest
• A decision maker may wish to minimise the risk of getting a low value in addition to maximising the expected • Dominant = If every value in a row of the pay-off matrix
value is greater to or equal to the corresponding values of
• i.e. they might have two objectives: another row
• To maximise the expected value • Saddle Point = A cell of the pay-off matrix which is the
minimum in one direction and the maximum in the other
• To minimise the risk of getting too low a value
direction
• One way to achieve this is to minimise the standard deviation • Strictly Dominant = If every value in a row of the pay-off
matrix is greater than the corresponding values of
• We can combine this and the maximising expectation:
 Maximise another row, the former strategy strictly dominates the
latter one
• Where k is an arbitrary (but suitable) non-negative constant
• The higher the value of k, the more the decision maker wants to avoid risk
Formulae
• Expected Pay-off:
Example values of k • Maximise

○ Example of the expected value adjusted for risk criterion
Game Theory for Non-Strictly Determined Two-Person Zero-Sum Games
• Saddle Point = A cell of the pay-off matrix which is the minimum in one direction and the maximum in the other
direction
• A Strictly Determined Game = A game where there is a saddle-point
• The value of the game is the value in the saddle-point cell

• Not all two-person zero-sum games are strictly determined
• A zero-sum game is fair only if its value is zero

Dominance
• Strictly Dominant = If every value in a row of the pay-off matrix is greater than the corresponding values of
another row, the former strategy strictly dominates the latter one
• Dominant = If every value in a row of the pay-off matrix is greater to or equal to the corresponding values of
another row
○ Example demonstrating dominance

Non-Strictly Determined Games and Mixed Strategies
• A mixed strategy = A combination of pure strategies in a certain proportion
• A mixed strategy can push up the values of the minimum that A can guarantee to win
• Conservative Strategy = A strategy where from all the available strategies it is the one where the smallest payoff

○ An example of a game with a mixed strategy solution

• Any pair of pure strategies will not be optimal and will not provide a stable solution if the two players are playing
rationally
An example demonstrating that any pair of pure strategies will not be optimal and will not provide a stable
solution if the two players are playing rationally

Finding Optimal Mixed Strategies
• We generate a pay-off table for the m x n game
• This in an LP problem:
• m+1 unknowns

• A's Problem:
• The solution that the value that B will pay-out will be v:

Course Notes Page 11

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