Unstable interactions in customers’ decision making: an experimental proof

Abstract

Understanding customers’ decision and behavior is the crux of marketing. Despite the broad applications of weighted sum approaches, like conjoint analysis, sophisticated methodological approaches are under-researched in this field. Conversely, multi-criteria decision making’s (MCDM) objective is to focus on and forecast these decisions. Based on the Choquet integral, this paper presents an effective and precise calculation method to understand real customers’ decision making and to overcome the weighted sum method’s limitations. We compare the weighted sum approach with stable and instable Choquet integral methods in three experimental studies. Our results indicate that the weighted sum approach is valuable in pure order-related applications. In quantitative comparisons, both the stable and instable Choquet integral approaches match the decision makers’ preferences more closely than the weighted sum approach. The paper demonstrates that well-developed traditional approaches have their merits and can still be applied in the right context.

Appendices

Appendix A: Formulation in the Choquet integral

Let ε be a positive value (ε ≠ 0). Equation (8) and (9) represent the inputs on the interaction of criteria i and j.

$$ \begin{array}{*{20}l} {w_{\mu } \left( {\text{ij}} \right) - w_{\mu } \left( {\text{i}} \right) - w_{\mu } \left( {\text{j}} \right) \le \varepsilon } \hfill \\ {\text{or}} \hfill \\ \begin{aligned} w_{\mu } \left( {\text{ij}} \right) - \, w_{\mu } \left( {\text{i}} \right) \, - w_{\mu } \left( {\text{j}} \right) \ge \varepsilon , \hfill \\ {\text{i}},{\text{j}} \in \left\{ {1, \ldots ,n} \right\} \hfill \\ \end{aligned} \hfill \\ \end{array} $$

(8)

$$ w_{\mu } (S) \le \, w_{\mu } \left( T \right),S \subseteq T,\forall S,T \in 2N $$

(9)

$$ w_{\mu } \left( T \right) \le \, 1\; for \, all\; T \subseteq \, N $$

(10)

Thus, a linear program with the objective function: max ε under the constraints (Eqs. 8–10) results in the fuzzy measure coefficients for each DM.

Appendix B: Formulation of the instable Choquet integral approach

For each alternative a_k, a linear program generates the best fuzzy measure that makes a_k the best with respect to the DM’s preference and the definition conditions.

With three criteria for a considered alternative a_k, the following equations describe the conditions (11), the veto effect level, the favor effect level (12), and the best position of a_k:

$$ \left\{ {\begin{array}{*{20}c} {w_{\mu } \left( {{\text{ij}}, {\text{k}}} \right) - w_{\mu } \left( {{\text{i}},{\text{k}}} \right) - w_{\mu } \left( {{\text{j}}, {\text{k}}} \right) \le \varepsilon , or w_{\mu } \left( {{\text{ij}}, {\text{k}}} \right) - w_{\mu } \left( {{\text{i}}, {\text{k}}} \right) - w_{\mu } \left( {{\text{j}}, {\text{k}}} \right) \ge \varepsilon ,} \\ {w_{\mu } \left( {S, k} \right) \le w_{\mu } \left( {, kT} \right),S \subseteq T, \forall S,T \in 2N} \\ {w_{\mu } \left( {T, k} \right) \le 1\;for all\; T \subseteq N} \\ \end{array} } \right. $$

(11)

$$ \left\{ {\begin{array}{*{20}c} {1 - \mathop \sum \limits_{t \ne i} \frac{1}{4}w_{\mu } \left( {t,k} \right) - \frac{1}{2}w_{\mu } \left( {ts, k} \right) \ge \frac{1}{2} + \varepsilon , i \in \left\{ {1,2,3} \right\}, t,s \in \left\{ {1,2,3} \right\} - \left\{ i \right\}} \\ {1 - \mathop \sum \limits_{t \ne i} \frac{1}{4}w_{\mu } \left( {t, k} \right) - \frac{1}{2}w_{\mu } \left( {ts, k} \right) \le \frac{1}{2} - \varepsilon , i \in \left\{ {1,2,3} \right\}, t,s \in \left\{ {1,2,3} \right\} - \left\{ i \right\}.} \\ \end{array} } \right. $$

(12)

$$ \left\{ {\begin{array}{*{20}c} {\mathop \sum \limits_{t \ne i} \frac{1}{4}w_{\mu } \left( {t_{i} ,k} \right) - \frac{1}{2}w_{\mu } \left( {i, k} \right) \ge \frac{1}{2} + \varepsilon , i \in \left\{ {1,2,3} \right\}, t \in \left\{ {1,2,3} \right\} - \left\{ i \right\}} \\ {\mathop \sum \limits_{t \ne i} \frac{1}{4}w_{\mu } \left( {t_{i} , k} \right) - \frac{1}{2}w_{\mu } \left( {i, k} \right) \le \frac{1}{2} - \varepsilon , i \in \left\{ {1,2,3} \right\}, t \in \left\{ {1,2,3} \right\} - \left\{ i \right\}.} \\ \end{array} } \right. $$

(13)

$$ C\mu (a_{k} ) \ge C\mu (a_{j} ) + \varepsilon ,\quad j \ne k $$

(14)

Let ε be a positive value (ε ≠ 0), a linear program with the objective function: max ε, under the above constraints, generates the fuzzy measure coefficients for each alternative a_k.

For the preference relation, the orness degree of a fuzzy measure is

$$ orness\left( {w_{\mu } } \right) = \frac{1}{n - 1}\mathop \sum \limits_{T} \frac{{\left( {n - t} \right)!t!}}{n!}w_{\mu } \left( {T,k} \right), t = \left| T \right|,\quad {\text{T}}\;{\text{a}}\;{\text{subset}}\;{\text{of}}\;{\text{N}}. $$

(15)

The preference relation across the set of alternatives is then a_k is at least as good as a_j if the orness degree of the fuzzy measure associated with a_k is less than the orness degree of the fuzzy measure associated with a_j.

Appendix C: Normalization of orness degrees in the instability case

Initially, we look for a range of fuzzy measures for every alternative that satisfies the associated constraints. If no solution is found, we drop the alternative and update the set of alternatives A to A1. By calculating the orness degree of each best fuzzy measure for each alternative in the set A1, we can rank the alternatives in A1. The alternative associated with the lowest orness degree is ranked first. Thereafter, we reconsider set A2 of dropped alternatives and we re-apply the same process, and so on. The alternatives in A2 should be ranked after the alternatives in set A1.

However, an alternative in A2 can have an orness degree lower than an orness degree of an alternative in A1. Thus, the orness degree values associated with the alternatives in different sets cannot be directly compared. We thus recalibrate the orness degree as described below.

Example:

• 1. The first calculation step results in two alternatives, A1 = {a₂, a₄}

  	Alternative		Rank
  	a2	a4
  Orness degree	.657	.600	a2 > a4

• 2. The second step results in A2 = {a₁} and the orness degree associated with the alternative a₁ is .243.

• 3. When A1 resulted in a₂ = .657 and a₄ = .6, the orness measure gives us an interval [0;∞]. Normalization per case needs to result in an interval [0;1]. If this is true, then the new reference point for A2 is the highest value of {a₂; a₄}. Thus, simply adding a₂ and a₁: .657 + .243 = .90 results in a new reference point and recalibrates A2. After that, a usual normalization results in the interval [0; 1].