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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 beCourse 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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