Linear partial information

Linear partial information (LPI) is a linear modeling method for practical decisions based on previously fuzzy information. The theory was developed in 1970 by Edward Kofler in Zurich .

We always create concepts, properties, ideas or models of our reality with incomplete information . This also applies to our colloquial language and logical considerations. Considered time it changes uncertainty (fuzziness Engl.) Our reality continuously. But, although we live in the area of incomplete information, rational decisions have to be made in our decision- making situations that take account of this uncertainty and avoid wrong decisions on the basis of only seemingly constant knowledge. This leads to so-called soft modeling .

Complete information is not available for many practical decisions. However, it is often possible to determine prognoses , cautious strategies, fuzzy equilibrium points and stability conditions. For example, in investment models in portfolio decisions, in economic planning , but also in strategic conflict situations. The following applies: the more complex the decision-making situation is, the softer, i.e. with greater uncertainty, the corresponding model must be designed; only with increasing certainty may the fuzziness of the model be gradually reduced.

In decision-making situations , the uncertainty of the distribution of the possible scenarios as well as the final results (outputs) are taken into account.

Every activity is based on decisions that have to be made in a world of blurring and uncertainty of data, terms and laws . The “fuzziness” of the world is a rule , not an exception. The optimality of our decisions, which we want to achieve using classic methods under these conditions, must be questioned. All of this forces us to create so-called soft (fuzzy) models. The more complex a system under consideration, the greater the degree of uncertainty in the data and the more softly it has to be modeled, said Lotfi Zadeh . The soft model has three important features :

The probability of occurrence of the model is generally greater in comparison with the sharp model.
It's more stable over time.
It allows an adaptive method with regard to new information with a corresponding adjustment to new requirements.

The soft modeling from the point of view of the great thinkers

Bertrand Russell claims, “... traditional logic speaks of precise terms . Unfortunately, these are not applicable in our environment, at most in an imaginary heavenly reality. In our conditions, there is nothing left to do but to vaguely determine the model data according to the circumstances. The probability of occurrence of the models and the possibility of adapting to further information should be taken into account ”.

Albert Einstein proclaims, "Sharp claims about our reality are false, or vice versa, correct ideas about our reality do not lead to sharp claims." This actually represents the Russell formulation in another form.

In another assertion, the fact of the reality of the decision-making elements is expressed analogously to the concepts of the state of rest and motion in mechanics .

LPI fuzzy equilibrium and stability strategies

Although complete information is not available in many practical decisions, it is still possible to identify fuzzy equilibrium points and stability conditions. For example, in fuzzy investment models, portfolio decisions, in economic planning models but also in fuzzy strategic conflict situations and cooperative negotiations . In everyday decisions, a “ modus vivendi ” is often sought with mutual tolerance - which must also lead to fuzzy equilibrium strategies and stability conditions. The terms of fuzzy equilibrium points (single-stage stability) and multi-stage stability are interpreted on the basis of the optimization principles for fuzzy data. In the decision-making aspect, the MaxEmin principle is used for the average value and the prognostic decision-making principle (PEP) for one-off decisions, taking into account the decision-maker's individual risk tolerance . In fuzzy multi-level decisions, the search for stabilizing strategies is linked to learning and control aspects in the adaptive process.

Balance points and stability with sharp data

Every optimal strategy has the property of an equilibrium point. Deviating from this strategy generally leads to disappointment . This applies to single-scenario as well as multi-scenario decisions with a sharp distribution ( risk situations , normal distribution ). For example, the determination of a maximum expected value in investment models, portfolio decisions, even in multi-stage decisions. The equilibrium points also have the "deviation property" in strategic games. Assuming that the decision results belong to the stability range of the decision maker, the equilibrium strategies are regarded as stability strategies. From our considerations it follows that the credibility of the sharp models and thus also of the decision-making aspects associated with them are questionable.

Fuzzy equilibrium and stability conditions with fuzzy data

The fuzziness, "LPI-based" with great credibility , is regarded as a disturbance quantity. The optimal strategies are determined by means of the MaxEmin principle for the average value and the prognostic decision-making principle (PEP) in the case of one-off decisions in the area of the LPI disturbance volume, taking into account the individual risk tolerance of the decision maker.

Selected bibliography

Edward Kofler - Equilibrium Points, Stability and Regulation in Fuzzy Optimization Systems under Linear Partial Stochastic Information (LPI), Proceedings of the International Congress of Cybernetics and Systems , AFCET, Paris 1984, pp. 233-240)
Edward Kofler - Decisions with linear partial information (Decision Making under Linear Partial Information). Proceedings of the European Congress EUFIT, Aachen, 1994, p. 891-896.
Edward Kofler - Linear Partial Information with Applications. Proceedings of ISFL 1997 (International Symposium on Fuzzy Logic ), Zurich, 1997, p. 235-239.
Edward Kofler - decisions when the distribution of conditions is partially known, magazine for OR, Vol. 18/3, 1974
Edward Kofler - Extensive games with incomplete information, in Information in der Wirtschaft, Gesellschaft für Wirtschafts- und Sozialwissenschaften, Volume 126, Berlin 1982