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Game Theory I Notes

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Lecture 6: Decision Theory / Game Theory I 12 November 2010

Topics

• Tree Diagrams

• Decision Trees

• Decision Theory

• Game Theory

Reading

Key Points

• Tree diagrams

• Decision trees

• Decision criteria under risk

• Decision criteria under uncertainty

• Game theory Tree Diagrams

• Tree Diagram = A diagrammatic representation used for reviewing the sequential decisions open to decisionmakers

• Problems can be attached to different routes through the network

• Extensive form games

• Normal form games

• Perfect information

• 2-player zero-sum games

Example of a tree diagram Definitions

• Decision Nodes = Where the branch is selected by the decision-maker, represented by a square symbol

• Decision Theory = Playing against Nature

• Decision Tree = A tree diagram where some of the nodes are choice nodes and some are chance nodes

• Event Nodes = Where the branch is selected probabilistically, represented by a circle symbol

• Folding Back = The decision alternative to be selected is the one with optimal expected pay-off Decision Trees

• Decision Tree = A tree diagram where some of the nodes are choice nodes and some are chance nodes

• There are rewards associated with each outcome

• Decision Nodes = Where the branch is selected by the decision-maker, represented by a square symbol

• Event Nodes = Where the branch is selected probabilistically, represented by a circle symbol Example of a decision tree

• Game in Extensive Form = An n-person game where a tree represents the moves of the game

• Game in Normal Form = AN n-person game consisting of a set of strategies for each player and a set of corresponding pay-off functions

• Game Theory = Situations where the decision maker's outcome is affected by an opponent's actions

• Hurwicz = If are minimum and maximum pay-offs for strategy I, find where (with ) is your 'optimism-pessimism' index

• Laplace = Assume Nature's strategies are equiprobable and maximise expected pay-off

• Maximax = Choose the strategy whose maximum pay-off is highest

• Maximin = Choose the strategy whose minimum pay-off is highest

• Minimax-Regret = Apply minimax to the regret matrix

• Perfect Information = A game where all the information sets consist of one vertex only

• Regret Matrix = For row i and column j the regret placed in the cell is the amount by which you could have improved your pay-off had you known Nature's strategy

Decision Theory

• Decision Theory = Playing against Nature

• There are two distinct situations:

• Those in which we do not know the probabilities where Nature chooses her path

• Decisions under uncertainty

• Those in which we do know the probabilities where Nature chooses her path

• Decisions under risk

Decision Criteria Under Risk

• Decisions under risk are normally analysed using decision trees

• Situations may involve more than one decision so the tree will have more than one decision node

• Folding Back = The decision alternative to be selected is the one with optimal expected pay-off

• The conditional decisions at each node constitute a complete solution to the problem

• The value of chance nodes is the weighted sum of the values of the nodes it leads to (the weights being conditional probability) Example of a decision situation under risk

Course Notes Page 8

• Tree Diagram = A diagrammatic representation used for reviewing the sequential decisions open to decisionmakers

• Zero-Sum Two-Person Game = A game in which one player's gain is the other player's loss

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