Towards Foundations of Fuzzy Utility: Taking Fuzziness into Account Naturally Leads to Intuitionistic Fuzzy Degrees

The traditional utility-based decision making theory assumes that for every two alternatives, the user is either absolutely sure that the first alternative is better, or that the second alternative is better, or that the two alternatives are absolutely equivalent. In practice, when faced with alternatives of similar value, people are often not fully sure which of these alternatives is better. To describe different possible degrees of confidence, it is reasonable to use fuzzy logic techniques. In this paper, we show that, somewhat surprisingly, a reasonable fuzzy modification of the traditional utility elicitation procedure naturally leads to intuitionistic fuzzy degrees. 1 Formulation of the Problem Need to help people make decisions. In many practical situations, we need to make a decision, i.e., we need to select an alternative which is, for us, better than all other possible alternatives. If the set of alternatives is small, we can easily make such a decision: indeed, we can easily compare each alternative with every other one, and, based on these comparisons, decide which one is better. However, when the number of alternatives becomes large, we have trouble making decisions. Even in simple situations, when we are looking for cereal in a supermarket, there are usually so many selections that we just ignore most of them and go with a familiar one – instead of the optimal one. The situation is even more complicated if we are trying to make a decision not on behalf of ourselves, but rather on behalf of a company or a community. In this case, even comparing two alternatives is not easy: it requires taking into account interests of different people involved, so the decision making process becomes even more complicated. Traditional approach to decision making: the notion of utility. The traditional approach to decision making was originally motivated by the idea of money. We all know what money is, but when money was invented, it was a revolutionary idea that made economic exchange much easier. Indeed, before money was invented, people exchanged goods by barter: chicken for a shirt, jewelry for boots, etc. Thus, to make a proper decision, every person needed to be able to compare every two items with each other: how many chickens is this person willing to exchange for a shirt, how many boots for a golden earing, etc. For n goods, we have n · (n− 1) 2 ≈ n 2 2 possible pairs. So, each person had to have in mind a table of n/2 numbers. With money as a universally accepted means of exchange, all the person needs to do is to decide, for each of n items, how much he or she is willing to pay for 1 unit. So, to successfully make decisions, it is sufficient to know n numbers – the values of each of n items. Then, even when we want to barter, we can easily decide how many chickens are worth a shirt: it is sufficient to divide the price of a shirt by the price of a chicken. A similar idea can be used to compare different alternatives. All we need is to have a numerical scale, i.e., a 1-parametric family of “standard” alternatives whose quality increases with the increase in the value of the parameter. This can be the money amount. Alternatively, this can be the probability p of a lottery in which we something very good: the larger the probability, the more preferable the lottery. Then, instead of comparing every alternative with every other alternative, we simply compare every alternative with alternatives on the selected scale, and thus, for each alternative, we find the numerical value of the standard alternative which is equivalent to a given one. This numerical value is known as the utility u(a) of a given alternative a; see, e.g., [3, 4, 6, 8, 11]. In terms of utility, an alternative a is better than the alternative a′ if and only the utility u(a) of the alternative a is larger than the utility u(a′) of the alternative a′. Thus, once we have found the utility u(a) of each alternative, then it is easy to predict which alternative the person will select: he/she will select the alternative for which the utility u(a) is the largest possible. How to actually find the utility. From the algorithmic viewpoint, the fastest way to find the utility of a given alternative a based on binary comparisons is to use bisection. Usually, we have an a prior lower bound and an a priori upper bound for the desired utility u(a): u ≤ u(a) ≤ u. In other words, we know that the desired utility u(a) is somewhere in the interval [u, u]. In this procedure, we will narrow down this interval. Once an interval is given, we can compute its midpoint ũ = u+ u