## MCDM'07 - paper no. 16

#### Decision making problem with two incomparable criteria ? solution based on game theory

##### Maciej Wolny

###### Abstract:

In this paper the solution concept of decision making problem with two incomparable criteria is presented. The incomparability consists in having no premises to aggregate the assessments of the decision variants or to aggregate the relations of partial preferences, which are associated with considered criteria - it is possible to show the best variant with respect of the particular criterion, but there is no information which could enable the definition of relations between the considered criteria. Construction of the model of a multicriteria decision making problem is the starting point of consideration. The problem is presented in literature as a two-person zero-sum game; it is noticed, however, that the multicriteria decision making problem is not antagonistic. Our premise is the construction of the model on the ground of non-antagonistic game theory. The proposal is that the criteria play the game with themselves. It is not necessary but the player as a person can be identified with the decision maker who considers the problem from the point of view of one criterion. The set of strategy is defined by the set of decision variants for each player. The payoffs are defined by the assessments of the decision variants if the players use the same strategy (choose the same decision variants). Otherwise, they reach much worse result. Also they know that they must choose the same variant. The characteristic feature of the game is the problem of coordination between equilibria - each equilibrium is equivalent to one decision variant. Thus, the problem is: which equilibrium should be chosen? Harsanyi's and Selten's "A general theory of equilibrium selection in games" and their concept of risk domination is applied to answer this question. The consequences of the proposal's acceptance are that the solution of the primary bicriterial decision making problem is the decision variant which belongs to a non-dominated set of decision variants and for which the sum of the assessments of the decision variants is maximal (the assessments are subtracted for the minimized criteria and added for the maximized ones).

###### Keywords:

coordination game, incomparable criteria

###### Reference index:

Maciej Wolny, (2008), Decision making problem with two incomparable criteria ? solution based on game theory, *Multiple Criteria Decision Making* (3), pp.
251-259