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Game Theory 2 Notes

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

Topics

• Decision Making Under Uncertainty: When Nature's Probabilities are Estimated

• Game Theory for Non-Strictly Determined Two-Person Zero-Sum Games

Reading

• Taha (13.2 and 13.4)

Decision Making Under Uncertainty: When Nature's Probabilities are Estimated

• We may be sufficiently confident to assign specific subjective probabilities

• i.e. using method of relative frequencies of occurrence

• We would then have a probabilistic decision matrix

• This is not the same as using the Laplace criterion

• We could apply the "folding back" method but this is strictly for objective probabilities

• Two further methods for decision making under uncertainty:

• Expected Value Criterion

• Expected Value Adjusted for Risk Criterion Expected Value Criterion

• We evaluate the expected pay-off for each of the player's strategies
○ Expected Pay-off:

• Perhaps the best criterion for making long run decisions

• Under achievement and over achievement counter balance each other
○ Example of the expected value criterion Expected Value Adjusted for Risk Criterion

• A decision maker may wish to minimise the risk of getting a low value in addition to maximising the expected value

• i.e. they might have two objectives:

• To maximise the expected value

• To minimise the risk of getting too low a value

• One way to achieve this is to minimise the standard deviation

• We can combine this and the maximising expectation:
 Maximise

• 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 Example values of k

○ 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 A can receive, A receives the largest
○ 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

Course Notes Page 11

v:

Key Points

• Decision making under uncertainty

• Expected value criterion

• Expected value adjusted for risk criterion

• Saddle points

• Strictly determined games

• Dominance

• Non-strictly determined games

• Mixed strategies

• Finding optimal mixed strategies

• Non-strictly determined games when m=n=2

• Non-strictly determined games when m>2, n=2

• Non-strictly determined games when m=2, n>2

• Rugby post diagrams

Definitions

• A mixed strategy = A combination of pure strategies in a certain proportion

• A Strictly Determined Game = A game where there is a saddle-point

• Conservative Strategy = A strategy where from all the available strategies it is the one where the smallest payoff A can receive, A receives the largest

• Dominant = If every value in a row of the pay-off matrix is greater to or equal to the corresponding values of another row

• Saddle Point = A cell of the pay-off matrix which is the minimum in one direction and the maximum in the other direction

• 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 Formulae

• Expected Pay-off:

• Maximise

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