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Probabilistic decision making
 

Probabilistic decision making

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    Probabilistic decision making Probabilistic decision making Document Transcript

    • Name: Shriraam Madanagopal<br />TERM PAPER<br />Engineering Economy –IE 5304<br />Topic: Probabilistic Decision Making<br />Probabilistic Decision Making:<br />Introduction:<br />In this diverse world, no two things are exactly the same. A decision maker is interested in both the advantages and the disadvantages for a given situation. Probability allows us to quantify the variability in the outcome of any experiment whose exact outcome cannot be predicted with certainty. Before we get introduced to the probability, we need to understand the concept of space of outcomes and the events on which it will be defined.<br />Sample Space:<br />A set of all possible outcomes of an experiment is called a sample space. This is because it usually consists of all the things that can happen when one takes a sample. Sample spaces are denoted by the letter ‘S’. To avoid misconception between the words experiment and outcome, we define these two words as follows.<br />An experiment may consist of the simple process of noting whether the switch is turned off or on; it may consist of determining the time it takes a car to accelerate to 40 miles per hour; or it may consist of a complicated process of finding the mass of an electron.<br />An outcome of an experiment may be a simple choice between two possibilities. It may be the result of a direct experiment or count, or it may be an answer obtained after extensive measurements and calculations.<br />When there is a study of outcomes of an experiment is seen, we usually identify various possibilities with numbers, points, or some other kinds of symbols.<br />Let us consider an example; if four contractors bid on a highway construction job and we let a, b, c and d denote that it is awarded to Mr.Smith, Mrs.Patric, Ms.Jasmine or Mr.Anderson. In this example the sample space is denoted S= {a,b,c,d}.<br />Also, if a government agency must decide where to locate two new computer research facilities and that for a certain purpose it is of the interest to indicate how many of them will be located in Texas and how many in California and thus we express the sample space as <br /> S= {(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2)}<br />The first coordinate represents the number of research facilities located in Texas and the second coordinate represents the number of research facilities located in California.<br />The Classical Probability Theory:<br />If there are n equally likely possibilities, of which one must occur and ‘S’ are regarded as favorable, or as a “success”, then the probability of success is given by S/n.<br />This can be explained with a suitable example.<br />What is the probability of drawing an ace from a well shuffled deck of 52 playing cards?<br />Solution:<br />There are S= 4 aces among the n=52 cards, so we apply the classical probability theory and get the result as S/n= 4/52 = 1/13 <br />The Frequency Interpretation of Probability:<br />The probability of an event (or an outcome) is proportion of times the event occurs in a long run of repeated experiments.<br />If we say that the probability of a flight from Texas to Illinois is 0.78, we mean that the flight has 78% chances to arrive on time.<br />In accordance with the frequency interpretation of probability, we estimate the probability of an event by observing what fraction of the time similar events have occurred in the past. Another point of view is to interpret probabilities as personal or subjective evaluations. Such subjective probabilities express the strength of one’s belief with regard to the uncertainties that are involved, and they apply especially when there is little or no direct evidence, so that there is no choice but to consider collateral individual evidence, educated guesses, and perhaps institution and other subjective factors. Subjective probabilities are best determined by referring to risk taking, or betting situations.<br />Conditional Probability:<br />Let us suppose assign a distribution function to a sample space and then learn that an Event E has occurred. The main doubt is how can we change the probabilities of the rest of the events? This question can be best answered by introducing the new probability for an event F the conditional probability of F given E and this can be expressed as P(FjE).<br />Conditional Probability can be best explained with a suitable example as follows:<br />Example : An experiment consists of rolling a die once. Let X be the outcome.<br />Let F be the event {X = 6}, and let E be the event {X > 4}. We assign the<br />distribution function m(ω) = 1/6 for ω = 1; 2; : : : ; 6. Thus, P(F) = 1/6. Now<br />suppose that the die is rolled and we are told that the event E has occurred. This<br />leaves only two possible outcomes: 5 and 6. In the absence of any other information,<br />we would still regard these outcomes to be equally likely, so the probability of F<br />becomes 1/2, making P(F|E) = 1/2. <br />It is important to note that any time we assign probabilities to real-life events; the resulting distribution is only useful if we take into account all relevant information. If we assigned a distribution function and then were given new information that determined a new sample space, consisting of the outcomes that are still possible, and caused us to assign a new distribution function to this space.<br />Let Ω =ω1+ω2….+ ωr be the original sample space with distribution function m(ωj) assigned.<br />Suppose we learn that the event E has occurred. We want to assign a new distribution function<br />m(ωj | E) to Ω to reflect this fact. Clearly a sample point ωj is not in E, we want m(ωj | E)=0.<br />Also it is reasonable to assume that the probabilities for ωk in E should have the same relative magnitudes that they have had before that we learnt that E has occurred.<br /> <br />m(ωk | E)= cm(ωk)<br />for all ωk in E, with c some positive constant. But we must also have<br /> ∑ m(ωk | E) = c ∑ m(ωk ) = 1<br /> E E<br /> Thus <br /> c=1÷∑ m(ωk ) = 1÷P(E)<br /> E<br />(Note that this requires us to assume that P(E) > 0.) Thus, we will define <br /> m(ωk | E)= m(ωk)÷P(E)<br />for ωk in E. We will call this new distribution the conditional distribution given E.<br />For a general event F, this gives<br /> P( F | E) = ∑ m(ωk | E) = ∑ m(ωk) ÷P( E) = P( F ∩ E) ÷ P(E)<br /> F ∩ E F ∩ E <br />If we call the P(E | F) , the conditional probability of F occurring given that E occurs and compute it using the formula<br /> <br /> P( F | E) = P( F ∩ E) ÷ P(E)<br />Bayes theorem: <br />Bayes theorem relates to the conditional and marginal probabilities of two random events. The posterior probabilities are calculated using Bayes’ theorem. It is accepted and used as a common interpretation in probability. Engineers and statisticians assign probabilities to random events according to their frequencies of occurrence or to subsets of populations as proportions of the whole, while Bayesians describe probabilities in terms of beliefs and degrees of uncertainty. <br />P(A) is the prior probability of A. It is " prior" in the sense that it does not take into account any information about B.<br />P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.<br />P(B|A) is the conditional probability of B given A.<br />P(B) is the prior or marginal probability of B, and acts as a normalizing constant.<br />Precisely Bayes' theorem in this form describes the way in which one's beliefs about observing 'A' are updated by having observed 'B'.<br />Role of emotions in decision making:<br />Emotions play a key role in a person who makes decisions. Once when a person is free from emotions and other related problems to the emotional side of the decision maker, the decision maker has a good chance of making a successful decision.<br />Adam Smith on Human attribute of sympathy: that we often derive sorrows from the sorrows of others, is a matter of fact which is too obvious to require any instance to prove it. The theory of modern sentiments Chapter1.<br />In contemporary models, this attitude is represented as intertemporial allocation of wealth. In the above mentioned models self control problem rooted in an emotional reasoning could negatively affect retirement savings and determine a sub-optimal allocation of wealth in the life cycle. If not going through self control problems, recent contributions have tried to model them and provide some advice on how to lessen sub-optimality due to compelling emotions when facing intertemporal choices. (among others, O'Donoghue-Rabin, 2000).<br />On the other hand we have researchers claiming emotions play a vital role in decision making.<br />In particular, when information structures embedded in the environment can be exploited<br />through simple search heuristics (Gigerenzer et al., 1999), emotions can fruitfully lead the<br />decision making process (ecological rationality).<br />The first and foremost thing that a decision maker should not indulge in is the aspect of a major emotion called Procrastination. Procrastination can lead to major discrepancy when one has to make a decision on time by calculating the risks involved in a decision. This can be illustrated with a suitable example of a person seeing a poisonous snake. The moment you see the snake you will have to act accordingly rather than becoming a victim by not taking a decision. The prime factor of consideration will be to save your life from snake rather being poisoned by the snake before you make a choice. Emotions on the other hand help us solve the frame problem, as they limit the range of possible consequences to be considered in a rational-decision process (e.g. de Sousa, 1987).<br />Ketelaar and Todd (2000) clearly explain how specific emotions might help to solve a problem of information to attend to in a given specific situation. Also gut feelings in some situations play a very vital role in making decisions with calculated amount of risks. These signals make people make approach-avoidance distinctions between options.<br />Anticipated Emotions:<br />Many researchers have made a measure to implement emotions to economics. One of the famous researcher Elter from 1996 to1998 was committed to the study of emotion and rationality and<br />tried to make an explicit link between emotion and economic theory<br />The most important commitment problems are solved by emotions which forms the back bone for economics. This concept was proposed by Frank in 1988. He has shown, for example, that players endowed with the emotion of guilt can sustain the cooperative outcome of a prisoner’s dilemma game.<br />Bell, 1982-1985; Loomes and Sugden, 1982-1986; Mellers, 1997-1999 did make a solid attempt to make people understand the importance of emotions for decision making and have included some measure of emotional dimension into their theories of decision making under risk and uncertainty. Anticipated emotions are not experienced at the moment of choice but are expected to occur when outcomes are experienced. These theories have focused on two counterfactual<br />emotions, namely disappointment and regret, which result from unfavorable comparisons<br />between alternative outcomes of the same option and between outcomes of alternative options<br />respectively. The core issue of these theories is the fact that individuals are motivated to avoid these negative emotions. The consequence of this is that subjects averse to regret or disappointment make decisions in a way that minimizes the likelihood of experiencing them.<br />Actual Emotions:<br />Actual emotions are the ones which are faced at the time of decision making. Feelings play a major role while making a decision. Feelings affect the subject learning process. Feelings hurl light on conditions involved in a situation that are congruent with their mood. Subjects in a negative affective state were found to acquire less positive information to which a situation is exposed. This concept is proposed by Bower and Cohen, 1982; Blaney, 1986. Secondly feelings affect what information is retrieved from memory. Tversky and kahneman in 1973-74 proposed that the ideas come to mind initially or most easily may influence judgment. People in a positive affective state were found to be more likely to think about positive outcomes and they were optimistic in their decisions- Isen et al., 1978-1982. The other way of feelings influences the choice of decision making strategy. The subjects in a positive affective, when compared with the subjects in the negative affective state tend to reduce the complexity of the given situation.<br />The above mentioned description gives an overall idea as to how emotions play a major role on the decision maker. Although emotions affect all domains of behavior, Leowenstein (2000) identifies three major categories of behavior which play a pivotal role in economics, namely bargain, temporal choice, making decision under risk and uncertainty.<br />Probabilistic Decision making under Uncertainty and risk factors:<br />The issue of fit between quantitative, probabilistic decision analysis processes and<br />game theoretical analysis arises frequently. This note attempts to outline how the two processes<br />fit together and can be used in a synergistic and integrated process. This note focuses on<br />decision techniques to deal with distinct and temporary issues, not the development of<br />policies or frameworks to guide the ongoing operations of a firm.<br />Game theory:<br />In many practical situations, it is required to take decisions in a situation where there are two or more parties opposite within the confliction interests and the action of one depends on the action of the another opponent’s choice. The outcome of the situation is controlled by the decisions of all the parties involved. Such a situation is termed as a competitive situation. Such problems occur frequently in economic, military, social, political, advertising and marketing by competing business forms. The mathematical expression for these above mentioned situations was proposed by Von Neumann.<br />A competitive situation is called a game if it has the following properties:<br />
      • If there are finite number of competitors called players
      • A list of finite or infinite number of possible courses of action is available to each player
      • A play is played when each player chooses one of his courses of action. The courses are assumed to be made simultaneously, so that no player knows his opponent’s choice until he has decided his course of action.
      • Every player i.e, combination of course of action is associated with an outcome, known as the pay off generally money, which determines a set of gains, one to each player. Here a loss is attributed to a negative gain.
      • A game involving n players is called n-person game.
      The value of the game theory is in understanding the likely outcomes of a business<br />issue when the outcome is dependent on actions taken by other parties with potentially<br />conflicting interests. In Game Theory we assume that every party acts rationally and<br />takes action based on their preferences.<br />
      • Game theory solver explains explain the basic listing of players, options available to them and their preferences.
      Game theory SolverStrategy <br /> Options <br />preferencesOutcomes PlayersTactics<br /> A typical game theory project would consider 20 – 25 different<br /> options. The output of the process is a set of outcomes<br /> one is called the “Natural Outcome”, resulting from all players <br /> following their natural self interest. Another is the “Best Attainable<br /> Outcome” (for the client), resulting from the client also taking subtle <br /> and possibly counter intuitive actions.<br />Probabilistic decision Analysis:<br />Probabilistic decision analysis processes have high value in cases where the value of different<br />outcomes is highly dependent on quantifiable external uncertainties that are not determined by<br />actions taken by identifiable players. Examples of such uncertainties could be the weather, the<br />foreign exchange rate, and the future price of oil. Uncertain future events are represented as<br />decision trees or probability distributions. <br />Decision Analysis can be well explained like the game theory block diagram.<br />Quantitative Decision AnalysisStructure<br />Expected Value<br />Options <br />Uncertainty <br />The input of a decision analysis process is the structure of an issue (influence diagram or<br />decision tree), with identified external uncertainties and decision variables.<br />Typically, a decision analysis would handle two to four actions (or decisions). The outcome of a<br />formal decision analysis is a recommended or preferred set of decisions for the client.<br />Similarities:<br />Although the analysis processes and tools are quite different, success of a project using either<br />methodology is quite dependent on participation by the right individuals from the client company<br />and skilful facilitation of the process. A strong knowledge elicitation methodology is essential to<br />either process and several facilitation tools can be applied equally well. Secondly, organizations that are receptive to objective and structured decision-making processes tend to like both processes, whereas purely intuitive decision cultures are not receptive.<br />Differences:<br />Game Theory has advantages in situations where the best course of action is dependent on<br />actions by other players. It is easier to apply where there are multiple value measures (where<br />a single decision criteria such as expected Net Present Value is not feasible). A typical<br />Game Theory project can be done in a little over a week, with 10hours of client team<br />involvement. Last, the Game Theory process is strong when there are too many outcomes to allow financial analysis (a typical case has 20 options, or 220 = 1 Million outcomes).<br />Decision Analysis is advantageous in cases where the decision uncertainty is caused<br />primarily by quantifiable uncertainties (e.g. the probability of success of a technology, the price of a commodity, an exchange rate), not dependent on the choices made by other<br />players. Decision Analysis requires the issue to be condensed into a small number of outcomes<br />and two or three decisions. Last, decision analysis provides a financial result (Expected<br />Value), which is often necessary to justify an investment.<br />Choosing the correct methodology:<br />Both the methodologies have their own application areas. In some instances, both could be used as a combined product. For issues pertaining to preferences where uncertainty and amount of calculated risks are involved game theory will be the preferred choice. This can be explained with a block diagram.<br />Decision Analysis block diagram<br />Game theory Solver<br /> <br /> Players <br />Strategy<br /> Options Tactics<br /> Preferences Outcomes <br />Quantitative Decision Analysis<br /> Uncertainties<br />A critical input for a game theory case is the preferences of the various players, including the<br />client company. The preferences are elicited from the client team. If the client has undertaken<br />quantitative analysis, taking into account uncertainties, than this quantitative analysis<br />influences the client’s preferences. However, it must be recognized that a client team will<br />always have a set of preferences, whether they’ve done formal analysis or not. Then the<br />game theory analysis will show what the possible outcomes will be.<br />Where feasible, we recommend that the Natural Outcome and the Best Attainable Outcome<br />be further analyzed using a Decision Analysis process. This may or may not result in a revised<br />set of preferences.<br />This can be explained with an example to make one understand probabilistic decision making. Let us consider a management which has a preference in order of priority. This issue has a major concern for labor relations.<br />
      • Cutting wages
      • Downsizing
      • Limiting early retirement
      If we evaluate the given situation using game theory, we find that there a possible outcome of a strike. Subsequent decision analysis shows that the cost of strike is very high which exceeds the benefits of the wage cut. The early retirement policy scheme is very expensive. After considering these evaluations the management makes a decision as follows.<br />
      • Limiting the early retirement
      • Downsizing
      • Not cutting wages.
      From this situation, game theory was used to narrow down the cases that needed to be analyzed. Game theory provided a clear perspective on the likelihood of a strike and then a Decision analysis was used to get an optimal solution on the expected value alternative course of action as described in the block diagram.<br />Taking another example of making probabilistic decision making under uncertainty we can more understand this concept.<br />Subject images and feeling towards winning a lottery are likely to be the same whether the chance of winning is 1 on 1 million. Emotions in uncertain or risky situations seem to be sensitive to the possibility rather than the probability of strong positive or negative consequences, causing an overweight of very small probabilities-Loewenstein et al., 2001. Rottenstreich and Hsee (2001) found that the strong sensitivity to departure from impossibility and certainty and the insensitivity to changes in probability within a broad midrange of values is<br />even more dramatic for affect favorable for affect no favorable outcomes. <br />Introduction to Contextual variables lead to proximity of outcomes and see how choices affect them. Proximity variable can be defined over different dimensions like temporal, spatial and social factors. Proximity measures should not have any impact on a cognitive-driven decision process; however, they could play a role in affective driven one. The laboratory experimental method seems to be the most appropriate way to assess the impact of proximity on choices under uncertainty. In fact, relying on field data makes it disentangle, the impact that proximity has on the cognitive perception of risk and impact of proximity on choices under uncertainty. In fact relying on field data makes it disentangle, the impact it has on proximity of its cognitive perception of risk and impact that it has on the emotional system. When contextual aspect is considered many researchers question on do risk preferences change according to how outcomes are usually described? Do risk preferences change whether the resolution of uncertainty is immediate or delayed? Are the risk choices affected by the presence of other people at the moment of decision or at the moment of uncertainty resolution?<br />This paper is mainly concerned with empirical nature of the probabilistic decision making. The above mentioned examples provided the theoretical contributions to the economics and social psychology with the objective to develop a systematic approach and method to develop an empirical and theory validation model. This paper made me learn and go through various articles, journals and books from British council library pertaining to the decision making analysis and statistics journals and books helped on probabilistic decision making. However, this topic not only made me learn the existing theory but also the way these theories are involved in probabilistic decision making under risk and uncertainty of various dimensions.<br />A potentially useful and relatively unexplored source of data comes from websites where decisions under risk and uncertainty made me understand the most common and prevalent on line gambling which falls under probabilistic decision making.<br />Results:<br />Emotions are the fundamental element for decision making under risk and uncertainty. The way this paper was analyzed with emotions while making decision under risk and uncertainty made me feel a lot about how decision were made and how should decision be made in future concerning various parameters and emotional constraints.<br />References:<br />Operations Research –Prof .V.Sundaresan, Prof.K.S Ganapathy Subramanian, Prof.K.Ganesan<br />Miller and Freund’s Probability and Statistics for Engineers Edition 7<br />Operations Research – Taha edition 6<br />CSS online tutorial<br />Mellers, B.A., Schwartz, A., Ho, K., and Ritov, I. (1997). Decision affect theory: emotional reactions to the outcomes of risky options.<br />Mano, H. (1994). Risk-Taking, Framing Effects, and Affect. Organizational Behavior<br />and Human Decision Processes.<br />O'Donoghue, T. and Rabin, M. (2000). The Economics of Immediate Gratification.<br />Journal of Behavioral Decision Making.<br />Rubinstein, A. (2003). Instinctive and Cognitive Reasoning: A study of Response<br />Times. Tel-Aviv and NYU Working Paper.<br />Slovic, P., Finucane, M.L., Peters, E., and MacGregor, D.G. (2002). The affect heuristic.<br />In T. Gilovich, D. Griffin, and D. Kahneman (Eds.), Heuristics and Biases: The<br />psychology of intuitive judgment, (pp. 397-420), New York: Cambridge University<br />Press.<br />Loewenstein, G. and ODonoghue, T. (2004). Animal Spirits: A_ective and Deliberative<br />Processes in Economic Behavior. Social Science Research Network.<br />Loewenstein, G. (2000). Emotions in Economic Theory and Economic Behavior. The<br />American Economic Review, 90 (2), 426-432.<br />Ketelaar, T. and Todd, P.M. (2000). Framing our thoughts: ecological rationality<br />as evolutionary psychologys answer to the frame problem. In S.P. Davies and H. R.<br />Holcomb (Eds.), The evolution of minds: psychological and philosophical perspective,<br />Kluvier, Dordrecht.<br />Isen, A.M. and Geva, N. (1987). The influence of positive affect on acceptable level<br />of risk: the person with a large canoe has a large worry. Organizational Behavior and<br />Human Decision Processes, 39, 145-154.<br />Elster, J. (1998). Emotions and Economic Theory. Journal of Economic Literature,<br />36, 47-74.<br />Frank, R.H. (1988). Passions within reason: the strategic role of the emotions. New York:<br />W.W. Norton.<br />Bell, D.E. (1985). Disappointment in decision making under uncertainty. Operations<br />Research, 33, 1-27.<br />Bell, D.E. (1982). Regret in decision making under uncertainty. Operations Research,<br />30, 961-981.<br />