Following the ideas of professor Raiffa, we can have the same attitude toward the subjective probabilities as with the objective probabilities, and we can use them freely in the theoretical constructions of the von Newman Utility theory. This is the subject of the chapter, evaluation of the subjective probability with the use of the stochastic programming. The probability is measured in an absolute scale in the context of the probability and measurement theory. Because of this, we can use the gambling approach to estimate the DM’s subjective probability as in the utility evaluations. Once again the authors solve the problem of best separation by using stochastic methods of the sets Au* and Bu*, (Au*nBu*)?Ø)). The difference with the previous chapter is that now they seek the existence of number (p), and not of function. This makes the problem easier to solve. However, the question remains the same, elimination of errors and uncertainty, and the way this is achieved in the stochastic programming.
Top1. Preferences Relations: Gambling Approach
In our everyday activity, we are constantly faced with the necessity of assessing uncertainty events and their effect on our life. For example, what will be the weather tomorrow, whether we should take a raincoat, whether to take a holiday and go on a trip, etc. On analyzing complex systems and processes, we face different forms of uncertainty while assessing various situations, parameters and others. On the one end of the spectrum, it is an uncertainty of a probability character just like the result after tossing a coin. The economic situation in 10 or 30 year's time is the uncertainty lying on the other end of the spectrum. Similar situations appear in complex biochemical processes as well, such as for example a subjective expert assessment of a biotechnological process of cultivation based on oblique factors or assessment of secondary metabolic product formation (See Chapter 9). So here raises the question whether we can rely on our subjective feelings and expectations in uncertainty while analyzing and decision-making in complex situations (so called uncertainty of second order) (Raiffa, 1986; Pfanzagl, 1971; Shmeidler, 1989; Larichev, 2010).
There are different opinions on that question. The most prevailing opinion is that if it is necessary to make a decision in a certain problem, than the subjective feelings, related to an existing uncertainty event have to be expressed by the language of the theory of probabilities and included in the decision-making process (Raiffa, 1986; Keeney, 1993; Pfanzagl, 1971). The idea that the subjective statistics and knowledge deserve trust has been around for centuries. It is known that a human being has great capacity to express precisely the nuances of the accumulated experience and to skillfully foresee the events in the everyday life. The psychologists, engineers, medical doctors, economists have everyday contact with such events and they freely use their subjective knowledge. The problem is that such empiric knowledge and predispositions are expressed mainly at verbal and difficult to measure level. Then the errors and contradictions with the aim of application of such knowledge and skill may increase in power. At such level, the human is lost in the multiattribute structure of the complex worldly and professional problems and has difficulties accounting for the nuances in his/hers empiric and long-term professional knowledge and the inter-relations in the structures of the sub-objectives of the main objective of the investigation. Without careful analysis, errors are often made and the influence of psychological factors as anger, fear, fatigue becomes a serious obstacle and source of errors. The mathematical measurement of the subjective knowledge in the respective scales and the description of the problem with models, reflecting mathematically precise these knowledge and the measurement scales, allow for the elimination of the mentioned above negative factors as a major influence in the decision process. Then a person, even if new in the professional field, makes decisions, if nothing else, in complete and non-contradictory agreement with his/hers personal knowledge and preferences.
The opinions of multitude scientists throughout the years is that a physiological mechanism in person’s brain accounts for the frequency probability expressed in his/hers life and transforms it into force, which has as a result the rationality of the behavior and in the formation of ideas. Such scientists are Piaget, Inhelder, Ramsey, Savage, Fishburn, Raiffa (Fishburn, 1970; Savage, 1954; Mengov, 2010). In Bulgaria research in the field is conducted by Mengov, Popchev, Tenekidjiev, and others (Mengov, 2010; Popchev, 1986; Tenekidjiev, 2004). New knowledge and mathematical techniques are sought, which in dialogue mode between the man and computer can extract the empirical skills of the researchers and realize them synchronously with the other theoretical skills.