# Decision Theory is a body of knowledge and related analytical techniques Decision is an action to be taken by the Decision Maker Decision maker is a person,

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Decision Theory is a body of knowledge and related analytical techniques Decision is an action to be taken by the Decision Maker Decision maker is a person, a group of people or a committee who makes the final choice among the alternatives Alternative is one of the courses of action attaining the objectives States of Nature (mutually exclusive and exhaustive) are future events NOT under the control of the DM Consequence is an interaction between a decision and a state of nature (a result of a course of action or of a decision taken) Payoff is a benefit that accrues from a consequence Decision Theory

risk means uncertainty for which the probability distribution is known Under certainty – DM completely certain as to which state of nature will occur Under uncertainty – DM has no knowledge of the probabilities of the occurrences of the states of nature Under risk – DM has knowledge about probabilities of the occurrences of the states of nature Decision Making Expected-Value Criterion = Sum of weighted pay-offs for that action weight = probability assigned to a state of nature by DM

Under certainty – DM completely certain as to which state of nature will occur Products Sates of nature GOOD MODERATE POOR A10 1.5 0.4 B20 10 - 6 C22 7 - 10 Pay-off Matrix Pay-off in crores of INR

Under uncertainty – DM has no knowledge of the probabilities of the occurrences of the states of nature Products Sates of nature GOOD MODERATE POOR A10 1.5 0.4 B20 10 - 6 C22 7 - 10 Pay-off Matrix OPTIMIST (Aggressive) – Maximax Criterion Max pay-off of each alternative 10 20 22

Products Sates of nature GOOD MODERATE POOR A10 1.5 0.4 B20 10 - 6 C22 7 - 10 Pay-off Matrix PESSIMIST (Conservative) – Maximin Criterion Min pay-off of each alternative 0.4 - 6 - 10

Coefficient of Optimism (Hurwicz's Index, index of optimism) a)Choose an index of optimism a, 0 ≤ a ≤ 1 b)1 means optimistic and 0 means pessimistic b) Multiply largest payoff (row-wise) by a and the smallest by (1 – a) d) Pick action with largest sum. NEITHER AN OPTIMIST NOR A PESSIMIST (Middle of the road) Sates of nature GOOD MODERATE POOR A 10 1.5 0.4 B 20 10 - 6 C 22 7 - 10 Pay-off Matrix Let a = 0.7 Weighted pay-off 10 * 0.7 + 0.4 * 0.3 = 7.12 20 * 0.7 + (- 6) * 0.3 = - 0.4 22 * 0.7 + (- 10) * 0.3 = 12.4 max min

REGRET CRITERION (Savage's Opportunity Loss) Sates of nature GOOD MODERATE POOR A 22 – 10 = 12 10 – 1.5 = 8.5 0.4 – 0.4 = 0 B 22 – 20 = 2 10 – 10 = 0 0.4 – (- 6) = 6.4 C 22 – 22 = 0 10 – 7 = 3 0.4 – (- 10) = 10.4 REGRET TABLE Maximum Regret of each alternative 12 6.4 10.4 Minimise over all regrets Regret = payoff of what would have been the best decision in the circumstances the payoff for the actual decision in the circumstances minus

Under risk – DM has knowledge about probabilities of the occurrences of the states of nature Sates of nature GOOD MODERATE POOR Probabilities 0.3 0.5 0.2 A10 1.5 0.4 B20 10 - 6 C22 7 - 10 Pay-off Matrix EV of each alternative 3.83 8.6 8.1 EV of A = 10 * (0.3) + 1.5 * (0.5) + 0.4 * ( 0.2) = 3.83

EV of Perfect Information = EV under certainty – maximum EV under risk Sates of nature GOOD MODERATE POOR Probabilities 0.3 0.5 0.2 A10 1.5 0.4 B20 10 - 6 C22 7 - 10 EV under certainty 0.4 * 0.2 = 0.08 10 * 0.5 = 5 22 * 0.3 = 6.6 Total = 11.68 EV of Perfect Information = 11.68 – 8.6 = 3.08

New scheme Sates of nature – Market share 1% 3% 7% 19% Introduce - 5 - 3 22 40 Not introduce 0 0 0 0 Pay-off Matrix Pay-off in crores of INR Probabilities 0.2 0.1 0.4 ???

Decision Tree Decision node – from which only one path of action may be taken State of Nature node – only one of the alternatives may happen in future

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