Multiattribute regret: theory and experimental study

Abstract

This paper generalizes the simple regret model by Bell in Operations Research 30(5), 961-981 and Loomes and Sugden in The Economic Journal 92(368), 805-824 to cope with the situation in which decision outcomes are multi-attributed. We propose a model that combines the simple regret model for ex ante preferences and the additive difference representation for ex post preferences. We first present a necessary and sufficient axiomatization of our model in Savage’s framework. The proposed model is composed of three types of functions. One is a value function for each attribute. The others are attribute-dependent and holistic regret-rejoicing functions that capture trade-offs among value-differences of chosen and forgone outcomes for each attribute and among all attributes, respectively. We then provide constructive procedures for those functions and methods for consistency checks of the procedures. We, finally, conduct an experiment to estimate those functions in the two attributes case by means of the constructive procedure and give an application of the model and our findings.

Appendices

Appendix A

This appendix provides proofs of all propositions except Proposition 3.

Proof of Proposition 1

Let $\left( \Phi ,p\right)$ be a Savagean SSA representation of $\left( \mathcal {A},\succ \right)$. To prove necessity of Axioms A1 and A2(K) for all integers $K>0$, we suppose that $\Phi$ has a continuous additive decomposition $\left( \varphi ;\Phi _{1},\ldots ,\Phi _{n}\right)$. Let $\omega =\left\{ E,E^{c}\right\}$ be a uniform 2-partition of S. Then for all $\varvec{x},\varvec{y}, \varvec{a},\varvec{b}\in \varvec{X}$,

$$\begin{aligned} \left\langle \varvec{x},\varvec{b}\right\rangle _{\omega } \succsim \left\langle \varvec{y},\varvec{a}\right\rangle _{\omega }\iff & {} p\left( E\right) \Phi \left( \varvec{x},\varvec{y}\right) +p\left( E^{c}\right) \Phi \left( \varvec{b},\varvec{a}\right) \ge 0 \\\iff & {} \Phi \left( \varvec{x},\varvec{y}\right) \ge \Phi \left( \varvec{a},\varvec{b}\right) \\\iff & {} \varphi \left( \sum _{i=1}^{n}\Phi _{i}\left( x_{i},y_{i}\right) \right) \ge \varphi \left( \sum _{i=1}^{n}\Phi _{i}\left( a_{i},b_{i}\right) \right) \\\iff & {} \sum _{i=1}^{n}\Phi _{i}\left( x_{i},y_{i}\right) \ge \sum _{i=1}^{n}\Phi _{i}\left( a_{i},b_{i}\right) . \end{aligned}$$

Since $\Phi _{i}$ for $i\in N_{n}$ are continuous,

$$\begin{aligned} \left\{ \left( \varvec{x},\varvec{y}\right) :\sum _{i=1}^{n}\Phi _{i}\left( x_{i},y_{i}\right) \ge \sum _{i=1}^{n}\Phi _{i}\left( a_{i},b_{i}\right) \right\}= & {} \left\{ \left( \varvec{x},\varvec{y} \right) :\left\langle \varvec{x},\varvec{a}\right\rangle _{\omega }\succsim \left\langle \varvec{y},\varvec{b}\right\rangle _{\omega }\right\} , \\ \left\{ \left( \varvec{x},\varvec{y}\right) :\sum _{i=1}^{n}\Phi _{i}\left( a_{i},b_{i}\right) \ge \sum _{i=1}^{n}\Phi _{i}\left( x_{i},y_{i}\right) \right\}= & {} \left\{ \left( \varvec{x},\varvec{y} \right) :\left\langle \varvec{a},\varvec{y}\right\rangle _{\omega }\succsim \left\langle \varvec{b},\varvec{x}\right\rangle _{\omega }\right\} \end{aligned}$$

are closed in $\varvec{X}^{2}$. Thus, Axiom A1 obtains.

To see necessity of Axiom A2(K), assume that $\left\{ \left( \varvec{x} ^{1},\varvec{y}^{1}\right) ,\ldots ,\left( \varvec{x}^{K}, \varvec{y}^{K}\right) ,\left( \varvec{w}^{1},\varvec{z} ^{1}\right) ,\ldots ,\left( \varvec{w}^{K},\varvec{z}^{K}\right) \right\}$ is balanced. Then we have, for $j\in N_{n}$,

$$\begin{aligned} \sum _{i=1}^{K}\Phi _{j}\left( x_{j}^{i},y_{j}^{i}\right) +\sum _{i=1}^{K}\Phi _{j}\left( w_{j}^{i},z_{j}^{i}\right) =0, \end{aligned}$$

so that

$$\begin{aligned}{} & {} \sum _{j=1}^{n}\left( \sum _{i=1}^{K}\Phi _{j}\left( x_{j}^{i},y_{j}^{i}\right) +\sum _{i=1}^{K}\Phi _{j}\left( w_{j}^{i},z_{j}^{i}\right) \right) \\{} & {} \quad =\sum _{i=1}^{K}\left( \sum _{j=1}^{n}\Phi _{j}\left( x_{j}^{i},y_{j}^{i}\right) +\sum _{j=1}^{n}\Phi _{j}\left( w_{j}^{i},z_{j}^{i}\right) \right) \\{} & {} \quad =0. \end{aligned}$$

If $\sum _{j=1}^{n}\Phi _{j}\left( x_{j}^{i},y_{j}^{i}\right) +\sum _{j=1}^{n}\Phi _{j}\left( w_{j}^{i},z_{j}^{i}\right) \ge 0$ for $i\in N_{K-1}$ (i.e., $\left\langle \varvec{x}^{i},\varvec{w} ^{i}\right\rangle _{\omega }\succsim \left\langle \varvec{y}^{i}, \varvec{z}^{i}\right\rangle _{\omega }$ for $i\in N_{K-1}$), then

$$\begin{aligned} \sum _{j=1}^{n}\Phi _{j}\left( x_{j}^{K},y_{j}^{K}\right) +\sum _{j=1}^{n}\Phi _{j}\left( w_{j}^{K},z_{j}^{K}\right) \le 0, \end{aligned}$$

so $\left\langle \varvec{x}^{K},\varvec{w} ^{K}\right\rangle _{\omega }\succsim \left\langle \varvec{y}^{K}, \varvec{z}^{K}\right\rangle _{\omega }$. Hence, Axiom A2(K) obtains.

To prove sufficiency of Axioms A1 and A2(4), we suppose that they hold. Let $\omega =\left\{ E,E^{c}\right\}$ be a uniform 2 -partition of S. We note that, for all $\varvec{x},\varvec{y}, \varvec{z},\varvec{w}\in \varvec{X}$,

$$\begin{aligned} \left\langle \varvec{x},\varvec{w}\right\rangle _{\omega } \succsim \left\langle \varvec{y},\varvec{z}\right\rangle _{\omega }\iff & {} p\left( E\right) \Phi \left( \varvec{x},\varvec{y}\right) +\left( 1-p\left( E\right) \right) \Phi \left( \varvec{w},\varvec{z}\right) \ge 0 \\\iff & {} \Phi \left( \varvec{x},\varvec{y}\right) \ge \Phi \left( \varvec{z},\varvec{w}\right) . \end{aligned}$$

We thus define a binary relation $\ge ^{*}$ on $\varvec{X} ^{2}$ as follows: for all $\varvec{x},\varvec{y},\varvec{z}, \varvec{w}\in \varvec{X}$,

$$\begin{aligned} \left( \varvec{x},\varvec{y}\right) \ge ^{*}\left( \varvec{z },\varvec{w}\right) \iff \left\langle \varvec{x},\varvec{w} \right\rangle _{\omega }\succsim \left\langle \varvec{y},\varvec{z} \right\rangle _{\omega }\text {. } \end{aligned}$$

Then it clearly follows that, for all $\varvec{x},\varvec{y },\varvec{z},\varvec{w}\in \varvec{X}$,

$$\begin{aligned} \left( \varvec{x},\varvec{y}\right) \ge ^{*}\left( \varvec{z },\varvec{w}\right) \iff \Phi \left( \varvec{x},\varvec{y} \right) \ge \Phi \left( \varvec{z},\varvec{w}\right) , \end{aligned}$$

so that $\ge ^{*}$ is a weak order. Furthermore, $\ge ^{*}$ satisfies the following two conditions, understood as applying to all $\varvec{x},\varvec{y},\varvec{z},\varvec{w}$, $\varvec{a},\varvec{b},\varvec{c},\varvec{d}$, $\varvec{a}^{\prime }, \varvec{b}^{\prime },\varvec{c}^{\prime },\varvec{d}^{\prime }\in \varvec{X}$ and $i\in N_{n}$.

C1. $\left\{ \left( \varvec{x},\varvec{y}\right) \in \varvec{X} ^{2}:\left( \varvec{x},\varvec{y}\right) \ge ^{*}\left( \varvec{z},\varvec{w}\right) \right\}$ and $\left\{ \left( \varvec{x},\varvec{y}\right) \in \varvec{X}^{2}:\left( \varvec{z},\varvec{w}\right) \ge ^{*}\left( \varvec{x}, \varvec{y}\right) \right\}$ are closed in $\varvec{X}^{2}$.

C2. If

$$\begin{aligned}{} & {} \left( x_{i}\varvec{a}_{-i},x_{i}^{\prime }\varvec{a}_{-i}^{\prime }\right) \ge ^{*} \left( y_{i}\varvec{b}_{-i},y_{i}^{\prime } \varvec{b}_{-i}^{\prime }\right) , \\{} & {} \left( z_{i}\varvec{b}_{-i},z_{i}^{\prime }\varvec{b}_{-i}^{\prime }\right) \ge ^{*} \left( w_{i}\varvec{a}_{-i},w_{i}^{\prime } \varvec{a}_{-i}^{\prime }\right) , \\{} & {} \left( y_{i}\varvec{c}_{-i},y_{i}^{\prime }\varvec{c}_{-i}^{\prime }\right) \ge ^{*} \left( x_{i}\varvec{d}_{-i},x_{i}^{\prime } \varvec{d}_{-i}^{\prime }\right) , \end{aligned}$$

then $\left( z_{i}\varvec{c}_{-i},z_{i}^{\prime }\varvec{c} _{-i}^{\prime }\right) \ge ^{*}\left( w_{i}\varvec{d} _{-i},w_{i}^{\prime }\varvec{d}_{-i}^{\prime }\right)$.

C1 follows from Axiom A1. To see that C2 holds, we rewrite it in terms of $\succsim$ as follows: if

$$\begin{aligned}{} & {} \left\langle x_{i}\varvec{a}_{-i},y_{i}^{\prime }\varvec{b} _{-i}^{\prime }\right\rangle _{\omega } \succsim \left\langle x_{i}^{\prime }\varvec{a}_{-i}^{\prime },y_{i}\varvec{b} _{-i}\right\rangle _{\omega }, \\{} & {} \left\langle z_{i}\varvec{b}_{-i},w_{i}^{\prime }\varvec{a} _{-i}^{\prime }\right\rangle _{\omega } \succsim \left\langle z_{i}^{\prime }\varvec{b}_{-i}^{\prime },w_{i}\varvec{a} _{-i}\right\rangle _{\omega }, \\{} & {} \left\langle y_{i}\varvec{c}_{-i},x_{i}^{\prime }\varvec{d} _{-i}^{\prime }\right\rangle _{\omega } \succsim \left\langle y_{i}^{\prime }\varvec{c}_{-i}^{\prime },x_{i}\varvec{d} _{-i}\right\rangle _{\omega }, \end{aligned}$$

then $\left\langle z_{i}\varvec{c}_{-i},w_{i}^{\prime } \varvec{d}_{-i}^{\prime }\right\rangle _{\omega }\succsim \left\langle z_{i}^{\prime }\varvec{c}_{-i}^{\prime },w_{i}\varvec{d} _{-i}\right\rangle _{\omega }$. This claim follows from Axiom A2(4), since the set

$$\begin{aligned}{} & {} \left\{ \left( x_{i}\varvec{a}_{-i},x_{i}^{\prime }\varvec{a} _{-i}^{\prime }\right) ,\left( z_{i}\varvec{b}_{-i},z_{i}^{\prime } \varvec{b}_{-i}^{\prime }\right) ,\left( y_{i}\varvec{c} _{-i},y_{i}^{\prime }\varvec{c}_{-i}^{\prime }\right) ,\left( z_{i}^{\prime }\varvec{c}_{-i}^{\prime },z_{i}\varvec{c}_{-i}\right) ,\left( y_{i}^{\prime }\varvec{b}_{-i}^{\prime },y_{i}\varvec{b} _{-i}\right) ,\right. \\{} & {} \quad \left. \left( w_{i}^{\prime }\varvec{a}_{-i}^{\prime },w_{i} \varvec{a}_{-i}\right) ,\left( x_{i}^{\prime }\varvec{d} _{-i}^{\prime },x_{i}\varvec{d}_{-i}\right) ,\left( w_{i}\varvec{d} _{-i},w_{i}^{\prime }\varvec{d}_{-i}^{\prime }\right) \right\} \end{aligned}$$

is balanced.

It is well known (Wakker, 1989, Theorem III.6.6.) that a weak order $\ge ^{*}$ on $\varvec{X}^{2}$ that satisfies C1 and C2 is represented by a continuous additive representation, i.e., there exist n continuous functions $\Phi _{i}$ on $X_{i}\times X_{i}$ for $i\in N_{n}$ such that, for all $\varvec{x},\varvec{y},\varvec{z},\varvec{ w}\in \varvec{X}$,

$$\begin{aligned} \left( \varvec{x},\varvec{y}\right) \ge ^{*}\left( \varvec{z },\varvec{w}\right) \iff \sum _{i=1}^{n}\Phi _{i}\left( x_{i},y_{i}\right) \ge \sum _{i=1}^{n}\Phi _{i}\left( z_{i},w_{i}\right) . \end{aligned}$$

Furthermore, the $\Phi _{i}$ are unique up to similar positive transformations, i.e., for $i\in N_{n}$, $\Phi _{i}^{\prime }\left( x_{i},y_{i}\right) =\alpha \Phi _{i}\left( x_{i},y_{i}\right) +\beta _{i}$ for some $\alpha >0$ and $\beta _{i}$. Since $\Phi \left( \varvec{x}, \varvec{x}\right) =0$ for all $\varvec{x}\in \varvec{X}$, $\left( \varvec{x},\varvec{x}\right) \ge ^{*}\left( \varvec{y },\varvec{y}\right)$ and $\left( \varvec{y},\varvec{y}\right) \ge ^{*}\left( \varvec{x},\varvec{x}\right)$ for all $\varvec{x},\varvec{y}\in \varvec{X}$, so that $\sum _{i=1}^{n}\Phi _{i}\left( x_{i},x_{i}\right) =\sum _{i=1}^{n}\Phi _{i}\left( y_{i},y_{i}\right)$. We can thus assume by appropriate choices of $\beta _{1},\ldots ,\beta _{n}$, that $\Phi \left( x_{i},x_{i}\right) =0$ for all $x_{i}\in X_{i}$ and $i\in N_{n}$.

To assure that the $\Phi _{i}$ are skew-symmetric, we note by skew-symmetry of $\Phi$ that

$$\begin{aligned} \left( \varvec{x},\varvec{y}\right) \ge ^{*}\left( \varvec{z },\varvec{w}\right)\iff & {} \Phi \left( \varvec{x},\varvec{y} \right) \ge \Phi \left( \varvec{z},\varvec{w}\right) \\\iff & {} \Phi \left( \varvec{w},\varvec{z}\right) \ge \Phi \left( \varvec{y},\varvec{x}\right) \\\iff & {} \left( \varvec{w},\varvec{z}\right) \ge ^{*}\left( \varvec{y},\varvec{x}\right) \\\iff & {} \sum _{i=1}^{n}\Phi _{i}\left( w_{i},z_{i}\right) \ge \sum _{i=1}^{n}\Phi _{i}\left( y_{i},x_{i}\right) \\\iff & {} \sum _{i=1}^{n}\Phi _{i}^{\prime }\left( x_{i},y_{i}\right) \ge \sum _{i=1}^{n}\Phi _{i}^{\prime }\left( z_{i},w_{i}\right) , \end{aligned}$$

where $\Phi _{i}^{\prime }\left( x_{i},y_{i}\right) =-\Phi _{i}\left( y_{i},x_{i}\right)$ for $i\in N_{n}$. Thus $\Phi _{i}^{\prime }\left( x_{i},y_{i}\right) =\alpha \Phi _{i}\left( x_{i},y_{i}\right)$ for some $\alpha >0$, so

$$\begin{aligned} \alpha \Phi _{i}\left( x_{i},y_{i}\right)= & {} -\Phi _{i}\left( y_{i},x_{i}\right) , \\ \alpha \Phi _{i}\left( y_{i},x_{i}\right)= & {} -\Phi _{i}\left( x_{i},y_{i}\right) . \end{aligned}$$

Therefore, $\alpha ^{2}\Phi _{i}\left( x_{i},y_{i}\right) =\Phi _{i}\left( x_{i},y_{i}\right)$, so $\alpha =1$. Hence, $\Phi _{i}\left( x_{i},y_{i}\right) =-\Phi _{i}\left( y_{i},x_{i}\right)$.

Hence, it easily follows that there exists a strictly increasing odd function $\varphi$ on $\mathbb {R}$ such that, for all $\varvec{x}, \varvec{y}\in \varvec{X}$,

$$\begin{aligned} \Phi \left( \varvec{x},\varvec{y}\right) =\varphi \left( \sum _{i=1}^{n}\Phi _{i}\left( x_{i},y_{i}\right) \right) \text {.} \end{aligned}$$

Also, the uniqueness of $\Phi _{i}$ for $i\in N_{n}$ follows from the uniqueness of additive conjoint measurement and the uniqueness of $\varphi$ follows from the uniqueness of $\Phi$ discussed in Step 1 in Sect. 2.2. This completes the sufficiency proof. $\square$

Proof of Proposition 2

Suppose that $\left( \mathcal {A},\succ \right)$ admits a Savagean SSA representation $\left( \Phi ,p\right)$ and $\Phi$ has a continuous nontransitive additive decomposition $\left( \varphi ;\Phi _{1},\ldots ,\Phi _{n}\right)$.

Necessity of Axiom A3 immediately follows from strict increasingness of $\varphi$, $v_{i}$, and $\tau _{i}$ for $i\in N_{n}$. For necessity of Axiom A4, assume that $\left\langle x_{i}\varvec{a}_{-i},y_{i}^{\prime } \varvec{b}_{-i}\right\rangle _{\omega }\succsim \left\langle y_{i} \varvec{a}_{-i},x_{i}^{\prime }\varvec{b}_{-i}\right\rangle _{\omega }$ and $\left\langle y_{i}\varvec{a}_{-i},z_{i}^{\prime }\varvec{b} _{-i}\right\rangle _{\omega }\succsim \left\langle z_{i}\varvec{a} _{-i},y_{i}^{\prime }\varvec{b}_{-i}\right\rangle _{\omega }$. Then

$$\begin{aligned} \varphi \left( \tau _{i}\left( v_{i}\left( x_{i}\right) -v_{i}\left( y_{i}\right) \right) \right) +\varphi \left( \tau _{i}\left( v_{i}\left( y_{i}^{\prime }\right) -v_{i}\left( x_{i}^{\prime }\right) \right) \right)= & {} 0, \\ \varphi \left( \tau _{i}\left( v_{i}\left( y_{i}\right) -v_{i}\left( z_{i}\right) \right) \right) +\varphi \left( \tau _{i}\left( v_{i}\left( z_{i}^{\prime }\right) -v_{i}\left( y_{i}^{\prime }\right) \right) \right)= & {} 0, \end{aligned}$$

which are rearranged to give

$$\begin{aligned} \tau _{i}\left( v_{i}\left( x_{i}\right) -v_{i}\left( y_{i}\right) \right) +\tau _{i}\left( v_{i}\left( y_{i}^{\prime }\right) -v_{i}\left( x_{i}^{\prime }\right) \right)= & {} 0, \\ \tau _{i}\left( v_{i}\left( y_{i}\right) -v_{i}\left( z_{i}\right) \right) +\tau _{i}\left( v_{i}\left( z_{i}^{\prime }\right) -v_{i}\left( y_{i}^{\prime }\right) \right)= & {} 0. \end{aligned}$$

Again, those two equations are rearranged to give

$$\begin{aligned} v_{i}\left( x_{i}\right) -v_{i}\left( y_{i}\right) +v_{i}\left( y_{i}^{\prime }\right) -v_{i}\left( x_{i}^{\prime }\right)= & {} 0, \\ v_{i}\left( y_{i}\right) -v_{i}\left( z_{i}\right) +v_{i}\left( z_{i}^{\prime }\right) -v_{i}\left( y_{i}^{\prime }\right)= & {} 0, \end{aligned}$$

which are additively combined to give

$$\begin{aligned} v_{i}\left( x_{i}\right) -v_{i}\left( z_{i}\right) +v_{i}\left( z_{i}^{\prime }\right) -v_{i}\left( x_{i}^{\prime }\right) =0. \end{aligned}$$

Hence, $\varphi \left( \tau _{i}\left( v_{i}\left( x_{i}\right) -v_{i}\left( z_{i}\right) \right) \right) +\varphi \left( \tau _{i}\left( v_{i}\left( z_{i}^{\prime }\right) -v_{i}\left( x_{i}^{\prime }\right) \right) \right) =0$, which means $\left\langle x_{i}\varvec{a} _{-i},z^{\prime }{}_{i}\varvec{b}_{-i}\right\rangle _{\omega }\succsim \left\langle z_{i}\varvec{a}_{-i},x_{i}^{\prime }\varvec{b} _{-i}\right\rangle _{\omega }$.

To prove sufficiency of Axioms A3 and A4, we suppose that Axioms A3 and A4 hold. Monotonicity of $\Phi _{i}$ follows from Axiom A3 and Proposition 1.

Let $\omega =\left\{ E,E^{c}\right\}$ be a uniform 2-partition. Then it follows from Proposition 1 that

$$\begin{aligned}{} & {} \left\langle x_{i}\varvec{a}_{-i},w_{i}\varvec{b}_{-i}\right\rangle _{\omega } \succsim \left\langle y_{i}\varvec{a}_{-i},z_{i}\varvec{ b}_{-i}\right\rangle _{\omega } \\{} & {} \quad \iff p\left( E\right) \varphi \left( \Phi _{i}\left( x_{i},y_{i}\right) \right) +p\left( E^{c}\right) \varphi \left( \Phi _{i}\left( w_{i},z_{i}\right) \right) \ge 0 \\{} & {} \quad \iff \varphi \left( \Phi _{i}\left( x_{i},y_{i}\right) \right) \ge \varphi \left( \Phi _{i}\left( z_{i},w_{i}\right) \right) \\{} & {} \quad \iff \Phi _{i}\left( x_{i},y_{i}\right) \ge \Phi _{i}\left( z_{i},w_{i}\right) . \end{aligned}$$

We thus define a binary relation $\ge _{i}$ on $X_{i}\times X_{i}$ as follows: for all $x_{i},y_{i},z_{i},w_{i}\in X_{i}$,

$$\begin{aligned} \left( x_{i},y_{i}\right) \ge _{i}\left( z_{i},w_{i}\right) \iff \left\langle x_{i}\varvec{a}_{-i},w_{i}\varvec{b}_{-i}\right\rangle _{\omega }\succsim \left\langle y_{i}\varvec{a}_{-i},z_{i}\varvec{b} _{-i}\right\rangle _{\omega }\text { for some }\varvec{a},\varvec{b} \in \varvec{X}. \end{aligned}$$

We show that $\ge _{i}$ satisfies the following four conditions, understood as applying to all $x_{i},y_{i},z_{i}$, $x_{i}^{\prime },y_{i}^{\prime },z_{i}^{\prime }\in X_{i}$:

• D1. $\ge _{i}$ is a weak order, i.e., it is complete and transitive.

• D2. If $\left( x_{i},y_{i}\right) \ge _{i}\left( x_{i}^{\prime },y_{i}^{\prime }\right)$, then $\left( y_{i}^{\prime },x_{i}^{\prime }\right) \ge _{i}\left( y_{i},x_{i}\right)$.

• D3. If $\left( x_{i},y_{i}\right) \ge _{i}\left( x_{i}^{\prime },y_{i}^{\prime }\right)$ and $\left( y_{i},z_{i}\right) \ge _{i}\left( y_{i}^{\prime },z_{i}^{\prime }\right)$, then $\left( x_{i},z_{i}\right) \ge _{i}\left( x_{i}^{\prime },z_{i}^{\prime }\right)$.

• D4. $\left\{ \left( x_{i}^{\prime },y_{i}^{\prime }\right) :\left( x_{i}^{\prime },y_{i}^{\prime }\right) \ge _{i}\left( x_{i},y_{i}\right) \right\}$ and $\left\{ \left( x_{i}^{\prime },y_{i}^{\prime }\right) :\left( x_{i},y_{i}\right) \ge _{i}\left( x_{i}^{\prime },y_{i}^{\prime }\right) \right\}$ are closed in $X_{i}\times X_{i}$.

Completeness of $\ge _{i}$ follows from completeness of $\succsim$. Transitivity of $\ge _{i}$ and conditions D2-D3 are rewritten in terms of $\succsim$ as follows:

D1*. If $\left\langle x_{i}\varvec{a}_{-i},w_{i}\varvec{b} _{-i}\right\rangle _{\omega }\succsim \left\langle y_{i}\varvec{a} _{-i},z_{i}\varvec{b}_{-i}\right\rangle _{\omega }$ and $\left\langle z_{i}\varvec{a}_{-i},w_{i}^{\prime }\varvec{b}_{-i}\right\rangle _{\omega }\succsim \left\langle w_{i}\varvec{a}_{-i},z_{i}^{\prime } \varvec{b}_{-i}\right\rangle _{\omega }$, then $\left\langle x_{i} \varvec{a}_{-i},w_{i}^{\prime }\varvec{b}_{-i}\right\rangle _{\omega }\succsim \left\langle y_{i}\varvec{a}_{-i},z_{i}^{\prime }\varvec{b} _{-i}\right\rangle _{\omega }$.

D2*. If $\left\langle x_{i}\varvec{a}_{-i},y_{i}^{\prime }\varvec{b} _{-i}\right\rangle _{\omega }\succsim \left\langle y_{i}\varvec{a} _{-i},x_{i}^{\prime }\varvec{b}_{-i}\right\rangle _{\omega }$, then $\left\langle y_{i}^{\prime }\varvec{a}_{-i},x_{i}\varvec{b} _{-i}\right\rangle _{\omega }\succsim \left\langle x_{i}^{\prime } \varvec{a}_{-i},y_{i}\varvec{b}_{-i}\right\rangle _{\omega }$

D3*. The same as Axiom A4.

D4*. The two sets,

$$\begin{aligned}{} & {} \left\{ \left( x_{i}^{\prime },y_{i}^{\prime }\right) :\left\langle x_{i}^{\prime }\varvec{a}_{-i},y_{i}\varvec{b}_{-i}\right\rangle _{\omega }\succsim \left\langle y_{i}^{\prime }\varvec{a}_{-i},x_{i} \varvec{b}_{-i}\right\rangle _{\omega }\right\} , \\{} & {} \left\{ \left( x_{i}^{\prime },y_{i}^{\prime }\right) :\left\langle x_{i} \varvec{a}_{-i},y^{\prime }w_{i}\varvec{b}_{-i}\right\rangle _{\omega }\succsim \left\langle y_{i}\varvec{a}_{-i},x_{i}^{\prime } \varvec{b}_{-i}\right\rangle _{\omega }\right\} , \end{aligned}$$

are closed in $X_{i}\times X_{i}$.

For D1*, $\left\{ \left( x_{i}\varvec{a}_{-i},y_{i}\varvec{a} _{-i}\right) ,\left( z_{i}\varvec{a}_{-i},w_{i}\varvec{a} _{-i}\right) ,\left( y_{i}\varvec{a}_{-i},x_{i}\varvec{a} _{-i}\right) ,\left( w_{i}\varvec{b}_{-i},z_{i}\varvec{b} _{-i}\right) ,\left( w_{i}^{\prime }\varvec{b}_{-i},z_{i}^{\prime } \varvec{b}_{-i}\right) ,\right.$

$\left. \left( z_{i}^{\prime }\varvec{b}_{-i},w_{i}^{\prime } \varvec{b}_{-i}\right) \right\}$ is balanced. By Axiom A2(3), D1* holds true. For D2*, $\left\{ \left( x_{i}\varvec{a}_{-i},y_{i} \varvec{a}_{-i}\right) ,\left( x_{i}^{\prime }\varvec{a} _{-i},y_{i}^{\prime }\varvec{a}_{-i}\right) ,\right.$

$\left. \left( y_{i}^{\prime }\varvec{b}_{-i},x_{i}^{\prime } \varvec{b}_{-i}\right) ,\left( y_{i}\varvec{b}_{-i},x_{i}\varvec{ b}_{-i}\right) \right\}$ is balanced. By Axiom A2(2), D2* holds true. D3* follows from Axiom A1.

It follows from Krantz et. al. (1971, Chapter 4) that Conditions D1-D4 hold if and only if there exists a strictly increasing continuous function $v_{i}$ on $X_{i}$ such that, for all $x_{i},y_{i},z_{i},w_{i}\in X_{i}$,

$$\begin{aligned} \left( x_{i},y_{i}\right) \ge _{i}\left( z_{i},w_{i}\right) \iff v_{i}\left( x_{i}\right) -v_{i}\left( y_{i}\right) \ge v_{i}\left( z_{i}\right) -v_{i}\left( w_{i}\right) . \end{aligned}$$

By definition of $\ge _{i}$,

$$\begin{aligned} \left( x_{i},y_{i}\right) \ge _{i}\left( z_{i},w_{i}\right) \iff \Phi _{i}\left( x_{i},y_{i}\right) \ge \Phi _{i}\left( z_{i},w_{i}\right) . \end{aligned}$$

Hence, $\phi _{i}\left( x_{i},y_{i}\right) =\tau _{i}\left( v_{i}\left( x_{i}\right) -v_{i}\left( y_{i}\right) \right)$ for some strictly increasing odd function $\tau _{i}$.

The uniqueness of $u_{i}$ for $i\in N_{n}$ follows from the uniqueness of additive difference measurement and the uniqueness of $\tau _{i}$ for $i\in N_{n}$ follows from the uniqueness of $\phi$ for $i\in N_{n}$ in Proposition 1. This completes the proof. $\square$

Proof of Proposition 4

By the hypotheses of Axiom B2, we have

$$\begin{aligned} \Phi \left( \varvec{y}_{I}\varvec{a}_{J},\varvec{x}_{I} \varvec{b}_{J}\right) +\Phi \left( \varvec{z}_{I}\varvec{b}_{J}, \varvec{y}_{I}\varvec{c}_{J}\right)= & {} 0, \\ \Phi \left( \varvec{y}_{I}\varvec{a}_{J},\varvec{y}_{I} \varvec{b}_{J}\right) +\Phi \left( \varvec{z}_{I}\varvec{b}_{J}, \varvec{x}_{I}\varvec{c}_{J}\right)= & {} 0. \end{aligned}$$

By Proposition 1 and strict increasingness of $\varphi$, those two are rearranged to give

$$\begin{aligned} \Phi _{I}\left( \varvec{y}_{I},\varvec{x}_{I}\right) +\Phi _{J}\left( \varvec{a}_{J},\varvec{b}_{J}\right)= & {} \Phi _{I}\left( \varvec{y}_{I},\varvec{z}_{I}\right) +\Phi _{J}\left( \varvec{c} _{J},\varvec{b}_{J}\right) , \\ \Phi _{I}\left( \varvec{y}_{I},\varvec{y}_{I}\right) +\Phi _{J}\left( \varvec{a}_{J},\varvec{b}_{J}\right)= & {} \Phi _{I}\left( \varvec{x}_{I},\varvec{z}_{I}\right) +\Phi _{J}\left( \varvec{c} _{J},\varvec{b}_{J}\right) , \end{aligned}$$

which are additively combined to get

$$\begin{aligned} \Phi _{I}\left( \varvec{x}_{I},\varvec{y}_{I}\right) =\Phi _{I}\left( \varvec{x}_{I},\varvec{z}_{I}\right) +\Phi _{I}\left( \varvec{z}_{I},\varvec{y}_{I}\right) . \end{aligned}$$

By the conclusion of Axiom B2, we have

$$\begin{aligned} \left\langle \varvec{x}_{I}\varvec{d}_{J},\varvec{y}_{I} \varvec{d}_{J},\varvec{z}_{I}\varvec{d}_{J}\right\rangle _{\omega }\sim & {} \left\langle \varvec{y}_{I}\varvec{d}_{J}, \varvec{z}_{I}\varvec{d}_{J},\varvec{x}_{I}\varvec{d} _{J}\right\rangle _{\omega } \\\iff & {} \Phi \left( \varvec{x}_{I}\varvec{d}_{J},\varvec{y}_{I} \varvec{d}_{J}\right) +\Phi \left( \varvec{y}_{I}\varvec{d}_{J}, \varvec{z}_{I}\varvec{d}_{J}\right) +\Phi \left( \varvec{z}_{I} \varvec{d}_{J},\varvec{x}_{I}\varvec{d}_{J}\right) =0 \\\iff & {} \varphi \left( \Phi _{I}\left( \varvec{x}_{I},\varvec{y} _{I}\right) \right) +\varphi \left( \Phi _{I}\left( \varvec{y}_{I}, \varvec{z}_{I}\right) \right) +\varphi \left( \Phi _{I}\left( \varvec{z}_{I},\varvec{x}_{I}\right) \right) =0 \\\iff & {} \varphi \left( \Phi _{I}\left( \varvec{x}_{I},\varvec{z} _{I}\right) +\Phi _{I}\left( \varvec{z}_{I},\varvec{y}_{I}\right) \right) \\{} & {} =\varphi \left( \Phi _{I}\left( \varvec{x}_{I},\varvec{z} _{I}\right) \right) +\varphi \left( \Phi _{I}\left( \varvec{z}_{I}, \varvec{y}_{I}\right) \right) , \end{aligned}$$

where we substituted the last expression of $\Phi _{I}\left( \varvec{x}_{I},\varvec{y}_{I}\right)$ in the preceding paragraph for the second equation. Since $\Phi _{I}\left( \varvec{x}_{I}, \varvec{z}_{I}\right)$ and $\Phi _{I}\left( \varvec{z}_{I}, \varvec{y}_{I}\right)$ can be arbitrary, the last equation implies that $\varphi$ is a linear function. $\square$

Proof of Proposition 5

This follows from the argument in the preceding paragraph. $\square$

Proof of Proposition 6

Similar to the proof of Proposition 5. $\square$

Proof of Proposition 7

Suppose that $\left\langle x_{i}\varvec{b}_{-i},y_{i} \varvec{c}_{-i}\right\rangle _{\omega }\sim \left\langle y_{i} \varvec{a}_{-i},z_{i}\varvec{b}_{-i}\right\rangle _{\omega }$ for some $\varvec{a},\varvec{b},\varvec{c}\in \varvec{X}$. Then

$$\begin{aligned} \Phi _{i}\left( x_{i},y_{i}\right) +\Phi _{i}\left( y_{i},z_{i}\right) =\Phi _{-i}\left( \varvec{a}_{-i},\varvec{b}_{-i}\right) +\Phi _{-i}\left( \varvec{b}_{-i},\varvec{c}_{-i}\right) \end{aligned}$$

By Axiom B2(i), $\left\langle x_{i}\varvec{b}_{-i},y_{i} \varvec{c}_{-i}\right\rangle _{\omega }\sim \left\langle z_{i} \varvec{a}_{-i},y_{i}\varvec{b}_{-i}\right\rangle _{\omega }$, so that

$$\begin{aligned} \Phi _{i}\left( x_{i},z_{i}\right) =\Phi _{-i}\left( \varvec{a}_{-i}, \varvec{b}_{-i}\right) +\Phi _{-i}\left( \varvec{b}_{-i},\varvec{ c}_{-i}\right) . \end{aligned}$$

Hence $\Phi _{i}\left( x_{i},y_{i}\right) +\Phi _{i}\left( y_{i},z_{i}\right) +\Phi _{i}\left( z_{i},x_{i}\right) =0$. $\square$

Appendix B

Method for Eliciting the indifference values. In the measurement of $v_i$ with the loss frame, $x_i^{-k-1}$ was elicited through choices between $A=<x_i^{-k-1} a_{-i},x_i^{-k+1} a_{-i} >_\omega$ and $B=x_i^{-k} a_{-i}$. We stopped the measurement of $v_i$ with the loss frame when $-k-1 \ge 4$ and $x_1^{-k-1} \ge 800$m ($x_2^{-k-1} \ge 4.8$kg), which is $80\%$ of the reference point of the gain frame (1000 m in attribute 1 and 6 kg in attribute 2), or $-k-1$ reached 51. Figure 9 gives an example of the procedure for the elicitation of $x_1^{-1}$ through comparisons between $A=<x_1^{-1} 3\text{ kg },0\text{ m } 3\text{ kg }>_\omega$ and $B=$50 m 3 kg. From the response in Fig. 9(a), the indifference value should be somewhere between 50 m and 70 m. Then in Fig. 9 (b), a second step was presented using a narrower range (e.g., 50 m to 70 m in increments of 4 m). From the response in Fig. 9(b), we recorded as indifference value the midpoint between 66 and 70, that is, 68. Figure 10 gives another example of the procedure for the elicitation of $x_1^{-1}$ in first step. From the response in Fig. 10 (a), the indifference value must be larger than 150 m. Then as in Fig. 10 (b), a second screen of first step was presented using a wide range (e.g., 150 m to 250 m in increments of 20 m).

Fig. 9

Example of the elicitation of $x_1^{-1}$

Fig. 10

Another example of the elicitation of $x_1^{-1}$ in first step

In the measurement of $\tau _i$ with the loss frame, the procedure was largely the same as the elicitation method for $v_i$. $z_i^{-1}$ was elicited through choices between $A=<z_i^{-(k+1)} a_{-i}^0,z_i^0 a_{-i}^0 >_\omega$ and $B=<z_i^0 a_{-i}^0,z_1^{-k} a_{-i}^{-1} >_\omega , k=0,...,-3$, for attribute 1 (distance) and $k=-1,...,-3$, for attribute 2 (weight). Figure 11 gives an example of the procedure for the elicitation of $z_1^{-1}$ through comparisons between $A=<z_1^{-1} 0\text{ kg },50\text{ m } 0 \text{ kg }>_\omega$ and $B=<50\text{ m } 0\text{ kg },50\text{ m } 0.5\text{ kg } >_\omega$. From the response in Fig. 11(a), the indifference value should be somewhere between 50 m and 70 m. Then in Fig. 11(b), a second step was presented using a narrower range (e.g., 50 m to 70 m in increments of 4 m). From the response in Fig. 11(b), We recorded as indifference value the midpoint between 66 and 70, that is, 68. Figure 12 gives another example of the procedure for the elicitation of $z_1^{-1}$ in first step. From the response in Fig. 12 (a), the indifference value must be larger than 150 m. Then as in Fig. 12 (b), a second screen of first step was presented using a wide range (e.g., 150 m to 250 m in increments of 20 m).

Fig. 11

Example of the elicitation of $z_1^{-1}$ in first step

Fig. 12

Another example of the elicitation of $z_1^{-1}$ in first step

In the measurement of $\psi _i$ with the loss frame, the procedure was the largely similar. We elicited the value of $A_{-k}$ for which indifference held between $A=<x_i^{-k} a_{-i},x_i^1 a_{-i} >_{\sigma k}$ and $B=x_i^0 a_{-i}, k=2,...,n$, where $\sigma _k= \{A_{-k}, A_{-k}^c \}$ and $\{x_i^{-k},x_i^1,x_i^0\}$ were the outcomes of the standard sequence elicited in the measurement of $v_i$ with the loss frame. The probability of $A_{-k}$ was assumed to be $A_{-k}/100$. Figure 13 gives an example of the procedure for the elicitation of $A_{-2}$ through comparisons between $A=<x_1^{-2}$ 3 kg,0 m 3 kg$>_{\sigma _k }$ and $B=$50 m 3 kg, and $A_{-3}$ through comparisons between $A=<x_1^{-3}$ 3 kg,0 m 3 kg$>_{\sigma _k }$ and $B=$50 m 3 kg. In this example, the elicited value for $x_1^{-2}$ and $x_1^{-3}$ were 100 m and 150 m, respectively. From the response in Fig. 13(a), the indifference value should be somewhere between 0 and 10 white balls. Then in Fig. 13(b), a second step was presented using a narrower range (e.g., 0 to 10 white balls in increments of 2). From the response in Fig. 13(b), We recorded as indifference value the midpoint between 6 and 8, that is, 7. As shown in Fig. 13(c), a first step for the elicitation of $A_{-3}$ was passed with a message for participants (e.g., “There is only one response held consistency. You don’t need to choose here. This screen will be changed in a few seconds.”), because $A_{-3}$ should be less than $A_{-2}$. Then in Fig. 13(d), the second step was presented using a narrower range (e.g., 0 to 8 white balls in increments of 2). From the response in Fig. 13(d), We recorded as indifference value the midpoint between 6 and 8, that is, 7.

Fig. 13

Example of the elicitation of $A_{-2}$ and $A_{-3}$ in attribute 1 when $x_1^{-2}=$100 m and $x_1^{-3}=$150 m

Figure 14 gives an example of the procedure for the investigation of axiom B1(i) through comparisons between $A=<$333.3 m 3 kg,166.7 m 3 kg,1000 m 3 kg, 0 m 3 kg$>_\omega$ and $B=<$0 m 3 kg, 333.3 m 3 kg, 166.7 m 3 kg, 1000 m 3 kg$>_\omega$. As showed in Fig. 14(a), each row represents $\omega =\{$white ball, yellow ball, green ball, blue ball$\}$. The order of attributes in each row was randomized. Each row was presented for 5 s. Figure 14(b) gives an example of presented a first row. After the last row was presented, as in Fig. 14(c), participants were asked to choose between A and B by clicking the left or right box.

Fig. 14

Example of the investigation of axiom B1(i) for distance

Appendix C

See Fig. 15.

Fig. 15

Examples of the scene of the experiment