Dealing with Imprecision in Consumer Theory: A New Approach to Fuzzy Utility Theory

Abstract

This chapter presents a new approach to dealing with imprecision in the Classical Consumer Utility Theory based on the concept of Marginal Rate of Substitution (MRS) and using the concept of fuzzy sets and fuzzy numbers. The methodology developed applies imprecision to MRS, whereas previous studies placed the imprecision factor on final utility values and functions. The chapter considers fuzzy elements applied to MRS and uses the necessary formulations to obtain the results in Utility Theory. In this fuzzy environment, the final consumer decision problem is framed as a fuzzy nonlinear programming problem, maintaining the classical structure in which consumers maximize their fuzzy utility subject to budget constraints, and showing that the consumer optimum choice is a fuzzy set. The chapter will also address the problem of aggregation of utility functions in order to offer a multi-criteria approach.

Appendix A. Additive Independence

1.1 Appendix A.1 Topological Approach by Debreu

If we assume a complete preference preordering $\succeq$ on $K=\prod _{i=1}^{k}C_i$ such that $\{C_i\in K\mid C_i\succeq C_h\}$ and $\{C_i\in K\mid C_i\preceq C_h\}$ are closed for all $C_h\in K$ (this will always occur if there is a finite number of criteria), it will be necessary to hold the following conditions for additive form on utility function:

• 1. The $n$ factors of $K$ are mutually utility independent.

• 2. More than two of them are essential.

If $K=\{C_1,...,C_k\}$ is the set of criteria, their factors $K_1,...K_n$ are separable spaces such that:

$$\begin{aligned} K=\prod _{f=1}^{n} K_f \end{aligned}$$

Though it would be possible for one factor to contain more than one criterion, in this case, the uni-criterion utility functions are built for each one, so that, each factor (component of total utility) corresponds with each criterion.

In general, a factor $K_f$ is essential if there exists a criterion $C_i$ included in another factor $K_{f'}$ such that not all criteria of $K_f$ are indifferent according to the preordering established by $c_i$.

As we said, in the case of consumer decision-making framework analyzed here, each factor corresponds to each criterion, therefore, as already shown, it is necessary to have as many summands as uni-criterion utility functions, that is, $n = k$, and $K_f=C_i$.

It is then possible to redefine the essential character of criteria: if $c_i$ is the set of possible values for criteria $C_i$, and $c_h$ represents the same for $C_h$, a criteria $C_i$ is essential if there exists:

$$\begin{aligned} C_{i}(x_1, x_2),C_{i}(x_1',x_2')\in c_i, C_{h}(x_1,x_2)\in c_h \text { such that} \end{aligned}$$

$$\begin{aligned} (C_{i}(x_1,x_2),C_{h}(x_1,x_2))\not \sim (C_{i}(x_1',x_2'), C_{h}(x_1,x_2)). \end{aligned}$$

The biggest limitation of this condition may be that it requires at least three essential factors, which means that it can only be applied in decision-making frameworks with three or more criteria. Nevertheless, it is easily satisfied in any consumer decision-making problem by all criteria involved.

For cases in which only two criteria are taken into account, it will be more accurate to use the Luce and Tukey axioms.

1.2 Appendix A.2. Algebraic Approach by Luce and Tukey

Adapting Luce and Tukey conditions to the consumer decision problem: If $C_1$ and $C_2$ are the two criteria involved in the decision-making process, with: $c_1=\{C_{1}(x_1,x_2), C_{1}(x_1',x_2'), C_{1}(x_1'',x_2''),...\}$ and $c_2{=}\{C_{2}(x_1,x_2),C_{2}(\!x_1',x_2'\!), C_2(x_1'',x_2''), ...\}$ the sets of values that can be reached by $C_1$ and $C_2$ respectively, $c_1 \times c_2$ is formed by pairs $(C_{1}(x_1,x_2),C_{2}(x_1,x_2))$, $(C_{1}(x_1,x_2),C_{2}(x_1',x_2'))$, $(C_{1}(x_1',x_2'),C_{2}(x_1,x_2))$, etc. Considering the binary relation $\succeq$, criteria will be additive if the following axioms are satisfied:

• I Ordering axiom: $\succeq$ is a weak order meeting the following axioms:

  • Reflexivity:

    $$\begin{aligned} (C_{1}(x_1,x_2),C_{2}(x_1,x_2))&\succeq (C_{1}(x_1,x_2),C_{2}(x_1,x_2)),\\ \forall C_{1}(x_1,x_2)&\in c_1, C_{2}(x_1,x_2)\in c_2. \end{aligned}$$

  • Transitivity:

    $$\begin{aligned} \text {If }(C_{1}(x_1,x_2),C_{2}(x_1,x_2))\succeq (C_{1}(x_1',x_2'),C_{2}(x_1',x_2')), \end{aligned}$$
    $$\begin{aligned} \text { and } (C_{1}(x_1',x_2'),C_{2}(x_1',x_2'))\succeq (C_{1}(x_1'',x_2''),C_{2}(x_1'',x_2'')), \end{aligned}$$
    $$\begin{aligned} \text {then: } (C_{1}(x_1,x_2),C_{2}(x_1,x_2))\succeq (C_{1}(x_1'',x_2''),C_{2}(x_1'',x_2'')). \end{aligned}$$

  • $\succeq$ is closed:

    $$\begin{aligned} (C_{1}(x_1,x_2),C_{2}(x_1,x_2))\succeq (C_{1}(x_1',x_2'),C_{2}(x_1',x_2')) \end{aligned}$$
    $$\begin{aligned} \text { or } (C_{1}(x_1,x_2),C_{2}(x_1,x_2))\preceq (C_{1}(x_1',x_2'),C_{2}(x_1',x_2')), \text { or both}. \end{aligned}$$

  • Definition: $* \quad (C_{1}(x_1,x_2),C_{2}(x_1,x_2))\sim (C_{1}(x_1',x_2'),C_{2}(x_1',x_2'))\text { only if }:$

    $$\begin{aligned} (C_{1}(x_1,x_2),C_{2}(x_1,x_2))\succeq (C_{1}(x_1',x_2'),C_{2}(x_1',x_2')) \text { and } \end{aligned}$$
    $$\begin{aligned} (C_{1}(x_1,x_2),C_{2}(x_1,x_2))\preceq (C_{1}(x_1',x_2'),C_{2}(x_1',x_2')). \end{aligned}$$

    $*\quad (C_{1}(x_1,x_2),C_{2}(x_1,x_2))\succ (C_{1}(x_1',x_2'),C_{2}(x_1',x_2'))$ only if:

    $$\begin{aligned} \lnot [(C_{1}(x_1',x_2'),C_{2}(x_1',x_2'))\succeq (C_{1}(x_1,x_2),C_{2}(x_1,x_2))]. \end{aligned}$$

• II Solution to equations:

  • For each $C_{1}(x_1,x_2)\in c_1$ and $C_{2}(x_1,x_2), C_{2}(x_1',x_2')\in c_2$, the equation $(C_{1}(x_1^{*},x_2^{*}),C_{2}(x_1,x_2))=(C_{1}(x_1,x_2),C_{2}(x_1',x_2'))$ has a solution $C_{1}(x_1^{*},x_2^{*})\in c_1$ and,

  • For each $C_{1}(x_1,x_2),C_{1}(x_1',x_2')\in c_1$ and $C_{2}(x_1,x_2)\in c_2$, the equation $(C_{1}(x_1,x_2),C_{2}(x_1,x_2))=(C_{1}(x_1',x_2'),C_{2}(x_1^{*},x_2^{*}))$ has a solution $C_{2}(x_1^{*},x_2^{*})\in c_2$.

• III Cancelation:

  $$\begin{aligned} \text {For all }C_{1}(x_1,x_2),&C_{1}(x_1',x_2'), C_{1}(x_1'',x_2'') \in c_1\text { and }C_{2}(x_1,x_2), C_{2}(x_1',x_2'),\\&C_{2}(x_1'',x_2'') \in c_2, \end{aligned}$$
  $$\begin{aligned} \text {if }(C_{1}(x_1,x_2),C_{2}(x_1'',x_2''))\succeq (C_{1}(x_1'',x_2''),C_{2}(x_1',x_2')) \end{aligned}$$
  $$\begin{aligned} \text {and } (C_{1}(x_1'',x_2''),C_{2}(x_1,x_2))\succeq (C_{1}(x_1',x_2'),C_{2}(x_1'',x_2'')), \end{aligned}$$
  $$\begin{aligned} \text {then: }(C_{1}(x_1,x_2),C_{2}(x_1,x_2))\succeq (C_{1}(x_1',x_2'),C_{2}(x_1',x_2')) .\end{aligned}$$

• IV Archimedean axiom: if $\{C_{1}(x_1,x_2)_i, C_{2}(x_1,x_2)_i\}, i=0, 1, 2,...$ is a non-trivial and increasing dual standard sequence, for each $C_{1}(x_1',x_2')\in c_1, C_{2}(x_1',x_2')\in c_2$, then, there exist two integers (positive or negative) $m$ and $n$ such that:

  $$\begin{aligned}(C_{1}(x_1,x_2)_n,C_{2}(x_1,x_2)_n)&\succeq (C_{1}(x_1',x_2'),C_{2}(x_1',x_2'))\\&\succeq (C_{1}(x_1,x_2)_m,C_{2}(x_1,x_2)_m).\end{aligned}$$

  An infinite sequence $\{C_{1}(x_1,x_2)_t, C_{2}(x_1,x_2)_t\}, t=0, 1, 2,...$, with $C_{1}(x_1,x_2)_t\in c_1, C_{2}(x_1,x_2)_t\in c_2$ is a dual standard sequence (dss) when $(C_{1}(x_1,x_2)_m, C_{2}(x_1,x_2)_n)= (C_{1}(x_1,x_2)_p, C_{2}(x_1,x_2)_q)$ for $m+n=p+q$ for any $m, n, p, q$ integer, positive, negative or null. A dss will be trivial if $C_{1}(x_1,x_2)_t = C_{1}(x_1,x_2)_0$, or $C_{2}(x_1,x_2)_t = C_{2}(x_1,x_2)_0$.