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# Decision theory

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### Decision theory

1. 1. I . I . ! I j ! . ! . f : DECISION THEORY DEFINITION 1.1 : DECISION THEORY (DT) is a set of concepts, principles, tools and techniques that aid the decision maker in dealing with compl~x decision problems under uncertainty. .- COMPONENTS OF A DT PROBLEM: 1. THE DECISION MAKER 2. ALTERNATIVE COURSES OF ACTION This is the controllable aspect of the problem. 3. STATES OF NATURE OR EVENTS These are the scenarios or states of the environmen.tnot under the control of the decision maker. The events defmed should be mutually exclusive and collectively exhaustive. 4. CONSEQUENCES The consequences that must be assessed by the decision maker are measures of the net bertefit, payoff, cost or revenue received by the decision maker. There is a consequence (or vector of consequences) associated with each action-event pair. The consequences resutilmarized a decisionmatrix. a in . 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111II DECISION THEORY EDGAR L. DE CASTRO PAGE 1 ..
2. 2. .. " CLASSIFICATIONS OF DT PROBLEMS: 1. Single Stage Decision Problems A decision is m~de only once. ~" ~ 2. Multiple Stage/Sequential Decision Problems Decisions are made one after another. .. f: 3. Discrete DT Problems The alternative courses of actions and states of nature are finite. ./ 4. Continuous DT Problems The alternative courses of actions and states of nature are infinite. DT Problems can also be classified as those with or without experimentation. Experimentation is perfoaned to obtain additional information that will aid the decision maker. I. DISCRETE DECISION THEORY PROBLEMS DECISION TREES A discrete DT problem can be represented pictorially using a tree. diagram or decision tree. It chronologically depicts the sequence of actions and events as they unfold. A square node ( D) precedes the set of possible actions that can be taken by the decision maker. A round node (0 ) precedes the set of events or states of nature that could be encountered after a decision is made. The nodes are connectedby branches. (""- ) 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 DECISION THEORY EDGAR L. DE CASTRO PAGE 2 :.:.~.;. .' " '.
3. 3. '. '. . EXAMPLE: i. , I . .. DECISIONS UNDER RISK AND UNCERTAINTY Consider a DT problem with m alternative courses of actions and a maximum of n events or states of nature for each alternativecourse of action. Defme: Ai = alternative course of action i; i = 1, 2, . . .,m q)j=state of nature j; j = 1, 2, . . .,n The decision matrix of payoffs is given by : q)l q)2 ... q)11 Al v(A1,q)1) veAl ,q)2) ... v(Abn) A2 v(A2'1) v(A2'2) ... v(A2 '11) . . . . . . . . . . . . . . . Am v(Am '1) V(Am'2) ... v(An"l1 ) I" 11/1" III" III" 111111111111" 11/1111111111111111111111111111111" 111111111111111/1/11111111111111111111111/11/11111111111111111" 111/111111111"" II" 1111 "" lilli/II n" 111111 n n I" nnn 111111 nn" 1I11 "" III! 11111" 11111/1111111111111 DECISION THEORY EDGAR L. DE CASTRO PAGE 3 . .," ..... .......... .' ... . . . " '. ~ . .. .. '. . '.'. .','. ,',', ..' . .:.
4. 4. .~ ll1 ."j' " A. LAPLACE CRITERION This criterion is based on what is mown as the principle of insufficientreason. Her~the probabilitiesassociatedwith the'occurrence of the event is uriknown.We do not have,Su(.ticie~~,leason to ',. rjJj , " conclude that the probabilities are different. Hence we 'assume that all " events are equally likely, i.e. .... . 1 f.. . P( t/J = rjJ . ) = - ~.:-. } n ;-:.. ... Then, the optimal decision rule is to select action '51t corresponding to " .... ,,' B. MINIMAX (MAXIMIN) CRITERION This is the most conservative criterion since it is based on making the best out of the worst possible conditions. For each possible decision alternative, we select the worst condition and then select the alternative corresponding to the best of the worst conditions. The MINIMAX strategy is given by: min A; f max{V(Ai,rjJ L. r/J. J j } ] The MAXIMIN strategy is given by: max A; [min {v( Ai , rjJ ] j r/J j } 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 DECISION THEORY EDGAR L. DE CASTRO PAGE 4 . : .: .'. :'. ".: ,'; ~::..~.' '. . ..., :...:..: : ~.:r::... ,'.'.:.
5. 5. C. SAVAGE MINIMAX REGRET CRITERION The MINIMAX rule is an extremely conservative type of decision rule. The savage MINIMAx regret criterion assUmes that a new .loss ,. I r ,. matrix is constructed id which v(Ai, (Jj) is replaced by r(A~,..(J)' which j 1. is defined by: .. max { v( Ak , (Jj )} - v( Ai , (Jj ), if v is profit Ak , r t. v(Ai,(J j) - min{v(Ak,(J )}, if v i~loss j II Ak .,/ . . Once the loss matrix is constructed using the above fonnula, we can now apply the MINIMAX criterion defined in b. D. HURWICZ CRITERION This criterion represents a range of attitudes from the most optimistic to the most pessimistic. Under the most optimistic conditions, one would choose the action yielding: max max {v( Ai , (Jj } Ai { I } t/> Under the most pessimistic conditions, the chosen action cOlTesponds to: . max min {v( Ai , (Jj } A. A.. t { '1') } 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 DECISIONTHEORY EDGARL. DE CASTRO PAGE5 " '.. ". ., : '..: ;.," .:j"...... . . . ..
6. 6. The Hurwicz criterionstrikes a balancebetween extremepessimism and extreme optimism by weighing the above conditions by respective I weights a and (1- a), where 0 < a <1. That is the action selected is t , that which yields:' . ., ma..'Xa m~ v( Ai , r/J ) + (1 - a )min v( Ai , r/J ) j j Ai { t/J j t/J j } .. [Note the above formulas represent the case where payoffs are expressed as profits] If a = 1, the decision rule is referred to as the MAXIMAX RULE, and if a = 0, the decision rule becomes the MAXIMIN RULE. For the case where the payoff represent costs, the decision rule is given by: min a min v( Ai , f/Jj ) + (1- a) max v( Ai , f/J ) j ~ { ~ ~ } E. BAYES' RULE Here, weasswne that the probabilities associated with each state of natureareknown.Let . . P{ljJ = ljJj} = Pj The action which minimizes (maximizes) the cost (profit) is selected. This is given by: The backward induction approach is used. With the aid of a decision tree, expected values are computed each time a round node is encountered and the above decision rule is utilized each time a square node is encountered ,i.e., a decision is made each time a square node is encountered. IIIIIIIIIIIII!IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 DECISIONTHEORY EDGARL. DE CASTRO PAGE6 . . . .: .: . .~ . .,". ':". .,',,1, .. . ','. i.L ,.
7. 7. .. I .' t, F. EXPECTED VALUE-VARIANCE CRITERION This is an extension of the expected value criterion. Here we simultaneously maximi~e profit and minimize the variance of the profit. . . If Z represents profit as a random variable with variance q:, then the criterion is given by: maximize E(z) - Kvar(z) where K is any specified constant. If Z represents cost: .' . ./ minimize E(z) + Kvar(z) G. DECISION MAKING WITH EXPERIMENTATION In some situations, it may be viable to secure additional information tp revise the original estimates of the probability of occurrence of the state of nature. DEFINITION 1.2 : Preposterior Analysis considers the question of deciding whether or not it would be worthwhile to get additional information or to perfonn further experimentation. DEFINITION 1.3 : Posterior Analysis deals with the optimal choice and evaluation of an action subsequent to all experimentation and testing using the experimental results. /1/1/1111111111111111111111/111111111111111111111111111111111111111111111/111111/11111111/1111111111111111111/111111111111111111/1/11111/1111111111111111111111111111111111111111111111111/1/111111111/1/111111111111111111111/1/111111/11111111111 DECISION THEORY EDGAR L. DE CASTRO PAGE 7 " ".':.:::: >. " :.-:'.:.:..
8. 8. ---.......----- DEFINITION 1.4 : Prior probabilities are the initial probabilities assumed without the benefit of experiment~tion. Posterior probabilities refer to the revised probability values obtamed after experimentation. . - Let: Pj =prior probability estimate of event (Jj P{Zkl(Jj} = conditional probability of experimental (jutcome Zk P{(Jj IZk} = posterior probability of event (Jj The experimental results are assumed to be given by Zk, k = 1, 2, ... 1. The conditional probability can be considered to be a measure of the reliability of the ~xperiment. The idea is to calculate the posterior probabilities by combining the prior probabilities and the conditional probabilities of experimental outcome Zk. The posterior probabilities are given by: m L P{Zk l(Ji}P{(Ji} i=1 Once the posterior probabilities are calculated, the original problem can be viewed as a multiple stage/sequential DT problem. The first stage involves the decision of whether to perform additional experimentation or not. Once this is decided, the outcomes of the experiment are considered together with the original set of decision alternatives and events. 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111/1111111111111111111111111111111111111111111111111111111111111111 DECISIONTHEORY EDGARL. DE CASTRO PAGE8 . '. .
9. 9. DEFINITION 1.5 : A perfect infonnation source would provide, with 100% reliability, which of the states of nature wouldoccur. . >'. Define: EPPI = expected profit from a perfect information source EVPI = expected value of the perfect infonnation source EP = Bayes' expected profit without experimentation Then: ., EVPI = EPPI - EP where: 11 E VPI = L Pj * max { v( Ai , fjJ ) } j . j=1 Ai EVPI is easily seen as a measure of the maximum amount a decision maker should be willing to pay for additional infonnation. Define: EVSI = expected value of sample information ENOS = expected net gain from sampling CAI = cost of getting additional information 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 DECISIONTHEORY EDGARL. DE CASTRO PAGE9 . '.' '. .. .... . .' .' ., . ..' ',' I"'. ," " .. ..
10. 10. , ., , Then: I , , " ,. - , ENGS = EVSI - CAI .. The information source would be viable if ENGS > O. II. CONTINUOUS DECISION THEORY. As previously mentioned, continuous decision theory problems refer to those where the number of alternatives and/or states of nature can be considered infmite. The optimization model in this case is given by: . max f(A) =J: v(A,t/J)htjJ(t/J)dt/J where: htjJ(t/J) = prior distribution function of the states of nature In the above model, it is assumed that no additional infonnation is available and the expectation is evaluated with respect to the prior distribution of the states of nature. If additional infonnation is available, we update the prior distribution of the states of nature by detennining its posterior distribution, which is nothing but the conditional distribution of the states of nature given the experimental outcome. Hence, the optimization converts to: IIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 DECISIONTHEORY EDGARL. DE CASTRO PAGE 10 . ..... ..:...:.... ... . ;...:..' :.. .... . . . . ..
11. 11. ------- . max f(A) = f: v(A,f/J)ht/>IZ=z (f/J)df/J where: '.. .jI I hfj)IZ=z (rjJ) = conditional distribution of the state of nature' given the experimental outcome :! I I ..J hZIfj)(z) = conditional distribution of the experimental outcome given the state of nature hz (z) = marginal distribution function of the experimental outcomes where: LEIBNIZ' RULE LEIBNIZ' Rule is applied to find the derivative'of a function which contains integrals. Consider a function in one variable A: d b ig b db da - fa g(A,rjJ)drjJ=fa ~rjJ+g(A,b)--g(A,a)- ciA 8A ciA ciA 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 DECISION THEORY EDGAR L. DE CASTRO PAGE 11 '. '.: '. . :