Lecture
Decision theory is a field of research involving the concepts and methods of mathematics , statistics , economics , management and psychology in order to study the patterns of people choosing ways to solve problems and problems, as well as ways to achieve the desired result.
A distinction is made between normative theory , which describes a rational decision-making process and a descriptive theory that describes decision-making practice.
The process of solving problems and tasks
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The rational process of solving problems and tasks includes the following stages, if necessary, performed simultaneously, in parallel, iteratively, with the return to the previous stages:
At the same time, the entire process and steps are carried out in a rational manner.
Ergodicity Problem
In order to make “strict” statistically reliable forecasts for the future, you need to get a sample of future data. Since this is impossible, many experts suggest that samples from past and current, for example, market indicators, are equivalent to a sample from the future. In other words, if you take this point of view, it turns out that the predicted indicators are only statistical shadows of past and current market signals. This approach reduces the analyst’s work to finding out how market participants receive and process market signals. Without the stability of the series it is impossible to draw sound conclusions. But this does not mean at all that the series should be stable in everything. For example, it can have stable variances and completely unsteady averages - in this case we will draw conclusions only about the variance, in the opposite case only about the average. Sustainability may be more exotic. The search for stability in the series is one of the tasks of statistics.
If decision makers (DM) believe that the process is not stationary (stable), and therefore ergodic , and even if they believe that the probability distribution functions of investment expectations can still be calculated, then these functions are “subject to sudden (that is, unpredictable) changes ”and the system is essentially unpredictable.
Decision making in the face of uncertainty
Uncertainty conditions are considered to be the situation when the results of the decisions made are unknown. Uncertainty is divided into stochastic (there is information about the probability distribution on the set of results), behavioral (there is information about the influence on the results of participants' behavior), natural (there is information only about possible results and there is no connection between decisions and results) and a priori (there is no information and about possible results). The task of substantiating decisions under conditions of uncertainty of all types, except a priori, reduces to narrowing the initial set of alternatives based on the information available to the decision maker. The quality of recommendations for making decisions in the context of stochastic uncertainty increases when taking into account such characteristics of the person as a decision maker, such as attitude to their wins and losses, risk appetite. Justification of decisions under conditions of a priori uncertainty is possible by constructing adaptive control algorithms [1] .
This area represents the core of decision theory.
The term “expected value” (now called mathematical expectation ) has been known since the 17th century . Blaise Pascal used this in the description of the famous bet contained in his work, Thoughts on Religion and Other Subjects , published in 1670 . The idea of the expected value is that in the face of many actions, when each of them can give several possible results with different probabilities, the rational procedure should identify all possible results, determine their values (positive or negative, income or expenses) and probabilities, then multiply the corresponding values and probabilities and add up to give the “expected value”. The action to be selected should give the highest expected value.
In 1738, Daniel Bernoulli published an influential article entitled Exposition of a New Theory on the Measurement of Risk, in which he uses the St. Petersburg paradox to show that the theory of expected value should be normatively incorrect. He also gives an example in which a Dutch merchant tries to decide whether to insure a cargo sent from Amsterdam to St. Petersburg in the winter, when it is known that there is a 5% chance that the ship and cargo will be lost. In his decision, he determines the utility function and calculates the expected utility , not the expected financial value.
In the 20th century, interest was reheated by the work of Abraham Wald ( 1939 ), indicating that the two central problems of orthodox statistical theory, namely, the testing of statistical hypotheses and the statistical theory of estimation , could both be regarded as specific special cases of a more general decision theory. This work introduced much of the “mental landscape” of modern decision theory, including loss functions, risk functions , admissible decision rules , a priori distributions , Bayesian decision rules , and minimax decision rules. The term "decision theory" began to be used directly in 1950 by E. L. Leman ru en .
The emergence of the theory of subjective probability from the works of Frank Ramsey , Bruno de Finetti , Leonard Savage and others, expands the possibilities of the theory of expected utility to situations where only subjective probabilities are available. At the same time, earlier in the economy it was generally assumed that people behave as rational agents, and thus the theory of expected utility also advanced the theory of real human behavioral decision-making at risk. The work of Maurice Allé and Daniel Ellsberg showed that this was not so obvious.
The perspective theory of Daniel Kahneman and Amos Tversky puts behavioral economics on a more solid footing of evidence . This theory indicated that in actual human decision-making (as opposed to normative) "loss is more sensitive than wins." In addition, people are more focused on the “changes” in the utility of their states than on the utility of the states themselves, and the assessment of the corresponding subjective probabilities is noticeably biased relative to each “reference point” inherent in each.
There are two types of descriptions of situations in which the exact outcome is unknown: risk and uncertainty. The situation is called a choice in risk conditions , when possible outcomes are known, and some of these outcomes are more favorable for the agent than others. For example, when making a bet there are two outcomes: the betting agent will either win or not, and the probability of winning can usually be calculated mathematically using formulas of varying complexity. Unlike risk-based choices, uncertainty choices imply an unknown set of outcomes. For example, if a bet is made with an agreement in a foreign language that is unfamiliar to the agent. According to the Avoidance of Uncertainty rule, an agent always prefers a choice in conditions of risk to a choice in conditions of uncertainty. As a rule, this can be achieved by turning uncertainty into risk by obtaining additional knowledge about the situation and using this knowledge by the agent. For example, in the example of betting in an unfamiliar language, uncertainty can be turned into a risk if the agent learns this language or uses a translation.
Errors of the first and second kind
The separation of erroneous decisions into errors of the first and second kind is caused by the fact that the consequences of various kinds of erroneous decisions are fundamentally different in that the lost gain has less impact on the situation than the realized loss . For example, for an exchange trader, the consequences that stocks were not bought when they should have been bought differ from the consequences of the situation when stocks were bought, but they should not be bought. The first situation may mean lost profits , the second - direct losses right up to the ruin of the trader. Similarly, for a politician, a refusal to seize power in a revolutionary situation differs in consequences from a lost attempt to seize power. For a general, starting a military operation that will be lost is much worse than missing out on a situation where a successful operation could be carried out. At the same time, the classification of errors of the first and second kind is permissible only in situations where accurate accounting and analysis of risks is carried out. So, S. Gafurov noted for the situation of exchange brokers: “Many people believe that the strategic task of analytical services (unlike other divisions of investment companies) is not to increase profits, but to minimize possible losses. And this is a fundamental difference. From the point of view of game theory, the optimal decisions of analysts should differ from the optimal trading actions. It is assumed that the optimal strategies implemented in the recommendations of analysts are based on the principle of minimizing maximum losses ( minimax ), while for traders minimax is an unacceptable strategy (minimizing the maximum loss on the market is not to play), and in general, optimization of decisions of traders is formalized only from the point of view of the Bayesian approach. Hence the need for special functional units that provide a balance of strategies - fund managers. Companies expect unbiased forecasts and sound recommendations from stock analysts. Some properties of such forecasts are obvious: accuracy, reliability. Others, such as reproducibility, methodological correctness or robustness (independence of the forecast results from the coordinate system), often remain out of sight of both specialists making forecasts and those who evaluate these forecasts ” [2] .
Alternatives to Probability Theory
A very controversial issue is whether it is possible to replace the use of probability in solution theory with other alternatives. Proponents of fuzzy logic , the theory of possibilities , the theory of evidence of Dempster-Schafer and others support the point of view that probability is only one of many alternatives, and point to many examples where non-standard alternatives were used with obvious success. Defenders of probability theory point to:
The validity of a probabilistic measure was called into question only once - by J. M. Keynes in his treatise "Probability" (1910). But the author himself in the 30s called this work “the worst and most naive” of his works. And in the 30s he became an active supporter of Kolmogorov's axiomatics - R. von Mises and never challenged her. The finiteness of probability and countable additivity are strong limitations, but the attempt to remove them without destroying the buildings of the whole theory turned out to be futile. It was in 1974 that Bruno de Finetti, one of the most prominent critics of Kolmogorov's axiomatics, recognized.
Moreover, he showed the opposite in fact - the rejection of countable additivity makes integration and differentiation impossible, and therefore does not make it possible to use the apparatus of mathematical analysis in probability theory. Therefore, the task of abandoning countable additivity is not the task of reforming probability theory, it is the task of abandoning the use of mathematical analysis methods in the study of the real world.
Attempts to abandon the finiteness of probabilities led to the construction of probability theory with several probabilistic spaces, on each of which Kolmogorov's axioms were fulfilled, but in total the probability should no longer be finite. But so far, no substantial results are known that could be obtained within the framework of this axiomatics, but not within the framework of Kolmogorov's axiomatics. Therefore, this generalization of Kolmogorov's axioms so far has a purely scholastic character.
S. Gafurov believes that the fundamental difference between Keynes's probability theory (and, therefore, mathematical statistics) from Kolmogorov’s (Von Mises, etc.) is that Keynes considers statistics from the point of view of decision theory for non-stationary series. For Kolmogorov, Von Mises, Fischer, etc., statistics and probability are used for essentially stationary and ergodic (with correctly selected data) series - the physical world surrounding us.
It is known that the theory of fuzzy sets ( English fuzzy sets ) in a certain sense reduces to the theory of random sets, that is, to probability theory. The corresponding cycle of theorems is given in the books of A. I. Orlov , including those indicated in the list of references below.
The Paradox of Choice
In many cases, a paradox is observed when a larger choice can lead to a worse decision or, in general, to a refusal to make a decision. Sometimes this is theoretically explained by what is called “paralysis of analysis," real or perceived, and also, possibly, "rational ignorance . " Many researchers, including Sheena S. Aengar and Mark R. Lepper (Sheena S. Iyengar and Mark R. Lepper), published studies of this phenomenon. (Goode, 2001)
We also now have a central problem of choice - freedom of choice. [3] [ non-authoritative source? ] In the understanding of Barry Schwartz, the choice did not make us freer, but limited, did not make us happier, but constantly causes dissatisfaction.
Decision Making Modeling
A multifaceted model for studying various aspects of decision theory is business chess . Moreover, the use of existing chess computer programs is possible as expert systems .
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Decision theory
Terms: Decision theory