Decision Analysis. An insightful study of the decision-making process under uncertainty and risk

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1 Decision Analysis An insightful study of the decision-making process under uncertainty and risk

2 Decision Theory A philosophy, articulated by a set of logical axioms and a methodology, for analyzing the complexities inherent in decision problems in order to identify the best alternative from among several alternatives. I cannot decide what to do!

3 Decisions, based upon. Instinct Intuition Subjective Judgement Experience Quantitative Analysis Yes, sell all my oil and buy nuclear power industries. Decision-making in action!

4 Problem complexity may be caused by Large number of alternatives (combinatorial) more than one decision-maker (theory of games) multiple attributes (criteria) (AHP) risk and uncertainty sequential decisions - (decision trees) long time horizons high stakes

5 The Analysis of Decision Analysis i.e. the road ahead Decision making under uncertainty Decision making under risk without experimentation with experimentation (sampling) (Bayesian) Decision Trees (time permitting)

6 The Decision Table PROBABILITIES STATES OF NATURE P 1 P 2 P n S 1 S 2 S N A 1 O 1,1 O 1,2 O 1,N ALTERNATIVES A 2 0 2,1 O 2,2 O 2,N... A M O M,1 O M,2 O M,N OUTCOMES, PAYOFFS OR UTILITIES

7 Decision Table Example Unit cost = $10 Selling price = $15 Unit profit = $ 5 probability = Alternatives State of Nature Demand is 0 Demand is 1 Demand is 2 Stock Stock Stock Stock Demand is 3 Profit Matrix

8 Decision Making under Uncertainty Mom, what if you don t know the probability distribution for the states of nature? What do you do then? P 1, P 2,, P n are unknown

9 Decision-making without probabilities S 1 S 2 S 3 S 4 states of nature Pay attention! This is a great decision matrix. alternatives A A A A A maximize payoff

10 Dominance If A i is preferred over A j for all States of Nature, then A i dominates A j and A j can be eliminated as an alternative of choice. Must investigate all possible pairwise comparisons. For M alternatives, there are MC 2 = M! / [(M-2)! 2!] comparisons. If M = 6; then 6!/[4!2!] = 15 comparisons

11 Decision-making without probabilities S 1 S 2 S 3 S 4 A A A A 3 is always preferred over A 4. Therefore A 4 is dominated by A 3. A A Dominance

12 .Optimism Maximax principle: Select the alternative that maximizes the maximum possible outcome i j { } Max Max O "It is demonstrable," said he, "that things cannot be otherwise than as they are; for as all things have been created for some end, they must necessarily be created for the best end. - Voltaire s Candide A S ij I am wild and adventurous. The best of all possible outcomes will occur.

13 Decision-making without probabilities S 1 S 2 S 3 S 4 A A A Max A A Dominance 2. Optimism: maximize maximum payoff

14 Pessimism Maximin principle: Select the alternative that maximizes the minimum possible outcome I am timid and unadventurous. The worst of all possible outcomes will occur. A i S j { } Max Min O ij

15 Decision-making without probabilities S 1 S 2 S 3 S 4 A A A A A Max Min Dominance 2. Optimism: maximize maximum payoff 3. Pessimism: maximize minimum payoff

16 The Hurwicz Principle Outlook is somewhere between extreme pessimism and extreme optimism The degree of optimism (α) of the decision maker can be measured on a scale from 0 to 1. For each alternative i, the Hurwicz criterion is given by α { ( )} max O ( ) i, j + (1 α) min Oi, j S S j { } Select the alternative that maximizes this criterion j Note: The glass is neither half empty nor half full because it is bigger than it need be.

17 Professor Hurwicz Regents Professor Emeritus Department of Economics University of Minnesota Leo Hurwicz received his LL.M. from Warsaw University - Poland in He teaches in the areas of theory, welfare economics, public economics, mechanisms and institutions, and mathematical economics. Professor Hurwicz's current research includes comparison and analysis of systems and techniques of economic organization, welfare economics, game-theoretic implementation of social choice goals, and modeling economic institutions.

18 Decision-making without probabilities S 1 S 2 S 3 S 4 Max Min Hurwicz (α =.8) A A A A A Dominance 2. Optimism: maximize maximum payoff 3. Pessimism: maximize minimum payoff 4. Hurwicz (coefficient of optimism) (α) maxi-max + (1- α) maxi-min

19 The degree of optimism Hurwicz Alpha A1 A2 A3 A4 A5

20 Minimax Regret The Savage principle: Compute a regret matrix by finding for a given state of nature, the difference between each profit and the maximum profit. Then for each alternative, find the maximum regret Select that alternative that minimizes the maximum regret I get it. The regret is the difference in what we get versus what we could have gotten if we had chosen the best alternative for that state of nature.

21 Leonard Savage Born: 20 Nov 1917 in Detroit, Michigan, USA Died: 1 Nov 1971 in New Haven, Connecticut Savage wrote on the foundations of statistics which led him into deep philosophical questions both about statistics and knowledge in general. The other main direction of his work was to study gambling as a source to stimulate problems in probability and decision theory. Savage's book The Foundations of Statistics (1954) is perhaps his greatest achievement. The book considers subjective probability and utility. It starts with six axioms, which are both motivated and discussed, and from these are deduced the existence of a subjective probability and a utility function. Another important work by Savage is How to gamble if you must : Inequalities for stochastic processes in 1965, written jointly with L Dubins. Other articles written by Savage relate to statistical inference, in particular the Bayesian approach. He introduced Bayesian hypothesis tests and Bayesian estimation.

22 Regret Matrix S 1 S 2 S 3 S 4 A A A A max regret Regret: A A A A

23 Decision-making without probabilities S 1 S 2 S 3 S 4 A Max 19 Min 8 Hurwicz (α =.8) 16.8 Regret 5 A A A A Dominance 2. Optimism: maximize maximum payoff 3. Pessimism: maximize minimum payoff 4. Hurwicz (coefficient of optimism) (α) maxi-max + (1- α) maxi-min 5. Savage: minimize maximum regret

24 Laplace Criterion (Principle of insufficient reason) It is sometimes suggested (initially by Laplace) that in the absence of any evidence to the contrary, one might as well assume that all futures are equally likely. Principle of Insufficient Reason: Assume all possible futures are of equal probability then select the alternative that maximizes expectation.

25 Laplace The first edition of Laplace's Théorie Analytique des Probabilités was published in The first book studies generating functions and also approximations to various expressions occurring in probability theory. The second book contains Laplace's definition of probability, Bayes's rule (so named by Poincaré many years later), and remarks on moral and mathematical expectation. The book continues with methods of finding probabilities of compound events when the probabilities of their simple components are known, then a discussion of the method of least squares, Buffon's needle problem, and inverse probability. Applications to mortality, life expectancy and the length of marriages are given and finally Laplace looks at moral expectation and probability in legal matters. Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France

26 Decision-making without probabilities p 1 = p 2 = p 3 = p 4 =.25 S 1 S 2 S 3 S 4 A Max 19 Min 8 Hurwicz (a =.8) 16.8 Regret 5 Laplace 13.5 A A A A Dominance 2. Optimism: maximize maximum payoff 3. Pessimism: maximize minimum payoff 4. Hurwicz (coefficient of optimism) (a) maxi-max + (1-a) maxi-min 5. Savage: minimize maximum regret 6. Laplace: Assume equi-likely outcomes

27 Which Alternative? S 1 S 2 S 3 S 4 A Max 19 Min 8 Hurwicz (a =.8) 16.8 Regret 5 Laplace 13.5 A A A

28 The end of the first example and the beginning of the next That was a really great example. I can t wait until the next one.

29 monthly heating cost avg winter temp < >35 0 optimism pessimism a =.5 Hurwicz Savage Laplace Electric Baseboard Heat Pump Central Gas Solar Panels

30 < >35 0 Regret Matrix regret Max

31 What are the pitfalls? I see some problems here.

32 Maxi-max or Not all is well! I want A 1! No A 2??? S 1 S 2 A 1 -$1000 $1000 A Max

33 Maxi-min or more Not all is well! I want A 2! No A 1??? S 1 S 2 A 1 0 $1000 A 2 $1 $1 Min 0 1

34 More Maxi-min -lack of column linearity 1 2 min A 1 $ 5 $8 5 A 2 $10 $2 2 Say you receive an additional $4 if S 2 occurs: 1 2 min A 1 $ 5 $12 5 A 2 $10 $ 6 6

35 Hurwicz! What can be said about Hurwicz? S 1 S 2 S 3 S N Max Min A A Gosh! I am indifferent between A 1 and A 2 regardless of the value of my coefficient of optimism.

36 What about regret? Will I regret that too? S 1 S 2 S 3 A A max A A

37 What about regret? Will I regret that too? S 1 S 2 S 3 A A max A A S 1 S 2 S 3 A A A max A A A

38 There is always good old Laplace. Ignorance is little justification to assume all outcomes are equi-likely! Try using subjective probabilities rather than a toss of the dice. I didn t know. Equi-likely assumes least knowledge - most influenced by additional information.

39 The Spreadsheet and it came to the engineering manager in a dream that the decision problem under uncertainty can be analyzed using a spreadsheet.

40 This way to Decision Making under Risk I cannot wait to find out more about decision making?

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