Probabilistic Choice Induced by Strength of Preference

Abstract

Just as we formulate detailed theories of utility or preference, so too should we theorize carefully about strength of preference. Likewise, because behavior is inherently uncertain, we need a theoretical framework for understanding choice probabilities. This paper fleshes out the simple premise that more strongly preferred options are more likely to be chosen. The resulting distribution-free Fechnerian models (DFMs) eschew convenience assumptions underlying popular models like the logit and probit, revealing which aspects of a core decision theory do or do not remain invariant across different ways of constructing strengths of preference, as well as across different monotonic links between those strengths of preference and choice probabilities. We formulate DFMs in a unifying polyhedral geometric space that allows for direct comparisons of theories that can be as categorically different as, say, regret theory, expected utility theory, and lexicographic semiorders. The geometric representation also provides a nuanced perspective on theoretical parsimony beyond parameter counting. Through a series of examples, we demonstrate the derivation and mathematical characterization of DFMs for decision theories with and without utilities and the inferences one can draw from data. We show how DFMs provide a multi-layered quantitative approach to the identifiability of hypothetical constructs. We highlight specific cases where DFMs protect the researcher against mistaken conclusions caused by overspecified models.

Appendices

Appendix 1. DFMs for more general decision theories

In this section, we illustrate how DFMs work for theories without utility functions. We walk the reader through examples ranging from DFMs for well-known core theories to ideas for novel axiomatizations of strengths of preference. We start with a case where a decision theory generates strengths of preference directly.

Example 14

We briefly consider regret theory, as spelled out by Loomes et al. (1991) with choice options that follow the blue-print of Loomes and Sugden’s Table I (p. 430). The bottom three rows of our Table 5 indicate three choice alternatives,¹¹x, y, z. The right three columns denote three different states of the world, $SW_1, SW_2, SW_3$, and their probabilities. A decision maker who chooses x will receive $5 if the state of the world is $SW_1$, which has probability 0.2 of occurring, and $2 otherwise. A person who chooses y receives $4 if states of the world $SW_1$ or $SW_2$ happen, or $1 if $SW_3$ happens, which has a probability of 0.3 of occurring. Anyone who chooses z will receive $3 irrespective of the state of the world.

Let $i_m$ and $j_m$ denote the outcomes of options i and j in $SW_m$, and let $p_m$ denote the probability of the state $SW_m$. According to this version of regret theory, letting $\Psi$ be an odd function that is increasing in its first argument and that satisfies ${\Psi (a,c) > \Psi (a,b) + \Psi (b,c)}$, when $a> b > c$ are monetary amounts, the strength of preference for i over j is ${S(ij) = \sum _{m = 1}^{3} p_m \Psi (i_m,j_m)}$. Consider ${\Psi (i_m,j_m)=(i_m-j_m)^3}$. This yields ${S(xy)=-3.5}$, ${S(xz)=0.8}$, and ${S(yz)=-1.7}$. The resulting DFM forms a tetrahedron, which, using the analogous coordinate system P(xy), P(xz), P(yz) as before, is characterized by the constraints

$$\begin{aligned}&1\ge 1-P(xy)> 1 - P(yz)> P(xz) > \frac{1}{2}. \end{aligned}$$

(24)

These inequalities document how regret theory, on this domain, produces strengths of preference that are not compatible with the weak utility assumption. Namely, they imply $P(xz)>\nicefrac {1}{2}$, ${P(zy)>\nicefrac {1}{2}}$, and $P(yx)>\nicefrac {1}{2}$, which violates $\mathcal {WST}$.

The inequalities in Condition 24 are the facet-defining inequalities of the tetrahedron in the upper-left panel of Fig. 8. That tetrahedron is nested in the half-unit cube shown in the upper-right panel of Fig. 8 (the same as the right panel of Fig. 1, in a different angle of view, which is also the majority specification of an intransitive preference pattern). It also has no overlap with $\mathcal {WST}$, shown on the lower panels of Fig. 8, which we explain in Examples 15 and 16. The insight that this DFM is not contained in $\mathcal {WST}$, together with the earlier insight that DFMs need not contain $\mathcal {WST}$ (see, e.g., Fig. 1), implies the following formal result.

Proposition 7

Let $\mathcal{M}_\mathcal{S}$ be a DFM as defined in Definition 6. In general, $\mathcal {WST} \not \subseteq \mathcal{M}_\mathcal{S} \not \subseteq \mathcal {WST}$.

We proceed to DFMs for axiomatic theories that likewise need not evoke the concept of utility. We start with the axioms of transitivity and asymmetry. For the rest of this section, we assume that $\mathcal{D}$ contains all pairs of options of a given finite choice set.

Table 5 Stimuli for regret theory

Let $\mathcal {WST^*}$ denote $\mathcal {WST}$ (Condition 23) with the added constraint that $P(ij)\ne \nicefrac {1}{2}$ for all $ij\in \mathcal {D}$. Defining, as before,

$$\begin{aligned} i \!\succ \! j \! \quad \! \Leftrightarrow \quad S(ij)\!>\!0 \! \quad \!\Leftrightarrow \quad P(ij) \!>\! \frac{1}{2}, \qquad \forall ij \in \mathcal{D}, \end{aligned}$$

(25)

it is well known that $\mathcal {WST}^*$ is equivalent to transitivity of the preference relation $\succ$, namely,

$$\begin{aligned} i \succ j \quad \wedge \quad j \succ k \quad \Rightarrow \quad i \succ k, \qquad \forall ij, jk, ik \in \mathcal{D}. \end{aligned}$$

Writing $i \not \succ j$ to denote that $i \succ j$ does not hold, since ${S(ij)>0\Rightarrow S(ji)\not > 0}$, the preference relation $\succ$ defined in Condition 25 is asymmetric, i.e., ${i \succ j \Rightarrow j \not \succ i}$. This gives us the tools to better understand the geometry associated with $\mathcal {WST}$ and $\mathcal {WST^*}$, and axiomatic theories more broadly (for early related work, including geometric visualizations of $\mathcal {WST}$ and other types of stochastic transitivity, see Morrison, 1963).

Example 15

If $S(ij)~\ne ~0,~\forall ij \in \mathcal{D}$ then the preference relation $\succ$ defined in Condition 25 is complete, i.e., for all distinct choice options i, j it holds that either $i \succ j$ or $j \succ i$. A complete, asymmetric, transitive binary relation is a strict linear order (ranking, permutation) of the choice options. $\mathcal {WST}^*$ rules out $P(ij)=\nicefrac {1}{2}$, and therefore implies $S(ij)~\ne ~0$ through Condition 25. As a consequence, $\mathcal {WST}^*$ could be labeled the DFM for the collection of all strict linear orders through the lens of Condition 25. For three choice alternatives, this DFM forms a disjoint union of six open half-unit cubes (lower-left panel of Fig. 8). The figure uses the coordinate system P(xy), P(xz), P(yz). Characterizing the six component cubes of $\mathcal {WST^*}$ involves facet-defining inequalities of the form $P(ij)>\nicefrac {1}{2}$, or $P(ij)>0$, or $P(ij)<1$ (each with suitably selected ij).

While Example 15 constructed the DFM for complete, asymmetric, transitive preference relations, we now consider the DFM for a slightly different combination of axioms.

Example 16

When S(ij) is also allowed to be zero, the relation $\succ$ need not be complete. A binary relation $\succ$ is negatively transitive if and only if $i \not \succ j \wedge j \not \succ k \Rightarrow i \not \succ k$, for all i, j, k. If $\succ$ is defined as in Condition 25 then $\mathcal {WST}$ holds if and only if $\succ$ is negatively transitive. Asymmetric negatively transitive binary relations are strict weak orders. $\mathcal {WST}$ allows $P(ij)=\nicefrac {1}{2}$ for some pairs and therefore allows $S(ij)=0$ for some pairs. Consequently, $\mathcal {WST}$ could be labeled the DFM for the collection of all strict weak orders through the lens of Condition 25. In the case of three choice alternatives, there are 13 strict weak orders. Here, the DFM consists of the disjoint union of the six half-unit open cubes we just reviewed for the six strict linear orders, together with six open 2-dimensional half-unit squares and the point $(\nicefrac {1}{2}, \nicefrac {1}{2}, \nicefrac {1}{2})$. Figure 8 shows the 3-dimensional polytopes of this model in the lower-left panel, and the lower-dimensional polytopes in the lower-right panel, all in the coordinate system P(xy), P(xz), P(yz). Line segments of the form ${P(ij) = P(jk) = \nicefrac {1}{2}< P(ik) < 1}$ (written in a suitable coordinate system) are not included in this model because the corresponding preferences violate negative transitivity (i.e., $i \not \succ j$ and $j \not \succ k$ but $i \succ k$). These line segments represent “semiorders” with transitive preference and intransitive indifference (Luce , 1956), which we sketch in the next example. That example moves beyond all examples we have seen so far in that it explores how to convert axioms, here those for interval orders and semiorders, into novel axioms about strengths of preference.

Example 17

A binary relation $\succ$ is an interval order if and only¹² if

$$\begin{aligned}{}[ i \succ j \quad \wedge \quad k \succ \ell ] \quad&\Rightarrow \quad [ i \succ \ell \quad \vee \quad k \succ j ]. \end{aligned}$$

An interval order $\succ$ is a semiorder if and only if

$$\begin{aligned}{}[ i \succ j \quad \wedge \quad j \succ k ] \quad&\Rightarrow \quad [ i \succ \ell \quad \vee \quad \ell \succ k ]. \end{aligned}$$

These two axioms could inspire novel axioms about strengths of preference, such as

$$\begin{aligned}{} & {} [ S(ij)> 0 \quad \wedge \quad S(k\ell ) > 0 ] \quad \Rightarrow \\ {}{} & {} \,\,\,\,\,\,\quad \quad \qquad \qquad \max (S(i\ell ), S(kj)) \ge \min (S(ij), S(k\ell )), \end{aligned}$$

respectively

$$\begin{aligned}{} & {} [ S(ij)> 0 \quad \wedge \quad S(jk) > 0 ] \quad \Rightarrow \\ {}{} & {} \,\,\,\,\,\,\quad \quad \qquad \qquad \max (S(i\ell ), S(\ell k)) \ge \min (S(ij), S(jk)). \end{aligned}$$

We finish this section with a generalization of hierarchical decision strategies, such as “lexicographic semiorders” (see, e.g., Tversky, 1969).

Fig. 8

Regret DFM of Example 14 (upper-left panel), half-unit cube for intransitive preference $y\succ x$, $x \succ z$, $z \succ y$ (upper-right panel), and geometric representations of $\mathcal {WST^*}$ (lower-left panel) and $\mathcal {WST}$ (union of lower panels). Note. Dashed lines in the lower-right panel are for enhancing 3D visualization and have no special meaning

Example 18

One can readily imagine DFMs for various sorts of lexicographic decision processes, say in consumer choice. Consider a shopper comparing two computers or televisions who follows a hierarchy of attributes, say, price, then screen resolution, followed by screen size, etc. When considering an attribute, they stop the decision process when that attribute meets a stopping criterion, otherwise proceed to the next attribute. For instance, writing $S_n(ij)$ for the strength of preference for i over j according to attribute n, suppose that

$$\begin{aligned}{} & {} S_1(xy) = 2; \quad S_2(xy) = 0; \quad S_3(xy) = 1; \quad S_4(xy) = 50; \\ {}{} & {} S_5(xy) = -1; \quad S_6(xy) = -20; \end{aligned}$$

and

$$\begin{aligned}{} & {} S_1(wz) = 1; \quad S_2(wz) = 60; \quad S_3(wz) = 1; \\ {}{} & {} S_4(wz) = -30; \quad S_5(wz) = 15; \quad S_6(wz) = 10. \end{aligned}$$

Suppose the decision maker has some decision threshold $\delta$ and sequentially, i.e., lexicographically, considers attributes until $|S_m(ij)| > \delta$ for some m, then adopts the strength of preference according to that attribute. A decision maker with $\delta = 0$ stops the search process at the first attribute for both $\{x,y\}$ and $\{w,z\}$, with overall strengths of preference ${S(xy) = S_1(xy) = 2}$ and ${S(wz) = S_1(wz) = 1}$ and, hence $P(xy) > P(wz).$ A decision maker with $\delta = 10$ has strengths of preference ${S(xy)= S_4(xy) = 50}$ and ${S(wz) = S_2(wz) = 60}$ and, hence $P(xy) < P(wz).$ One can also design novel DFMs in which attribute-wise strengths of preference are somehow accumulated to weigh reasons in favor or against one option over another, similar to the “perceived relative argument” model of Loomes (2010).

Appendix 2. Proofs

Proof of Proposition 1

$$\begin{aligned} S(ij)> 0\Leftrightarrow & {} S(ji)<0 \\\Leftrightarrow & {} S(ji)< S(ij) \\\Leftrightarrow & {} P(ji)< P(ij) \\\Leftrightarrow & {} 1-P(ij) < P(ij) \\\Leftrightarrow & {} P(ij) > \frac{1}{2}. \end{aligned}$$

$\square$

Proof of Proposition 2

Let S be an odd, real-valued function on the domain $\mathcal{D}$ and let $\mathcal{D}'$ be any maximal subset of $\mathcal{D}$, such that S is one-to-one on $\mathcal{D}'$, i.e., for any distinct ij and $k\ell$ in $\mathcal{D}', S(ij) \ne S(k\ell )$. In particular, if S is a one-to-one function on $\mathcal{D}$ then $\mathcal{D}' = \mathcal{D}$. Let $\mathcal{D}'_{S^+}~=~\{ij~\in ~\mathcal{{D}'}~|~S(ij)>0\}$. If $ij \in \mathcal{D'}$ then $S(ij) \ne 0$ and therefore $ji \in \mathcal{D'}$, since $S(ji) = -S(ij)$. Therefore, for each $ij \in \mathcal{D'}$, exactly one of ij or ji is in $\mathcal{D'}_{S^+}$. As a consequence, $|\mathcal{D}'_{S^+}| = \frac{|\mathcal{D'}|}{2}$. We denote $|\mathcal{D}'_{S^+}|$ as n. Since S is one-to-one on $\mathcal{D}'$, there is a unique one-to-one function $f: X = \{1,2,\ldots ,n\} \rightarrowtail \mathcal{D}'_{S^+}$, which orders the elements of $\mathcal{D}'_{S^+}$ by decreasing strength of preference, i.e., such that $S(f(x)) > S(f({x+1}) )$ for $1 \le x \le n-1$. According to the Fechnerian property (6) for S, a binary choice vector P is in $\mathcal{M}_S$ if and only if

$$\begin{aligned} 1> & {} P(f(1)), \end{aligned}$$

(26)

$$\begin{aligned} P(f(x))> & {} P(f(x+1)) \text { for } 1 \le x \le n-1, \end{aligned}$$

(27)

$$\begin{aligned} P\left( f(n)\right)> & {} \frac{1}{2}, \end{aligned}$$

(28)

by construction. Each of these $n+1$ inequality constraints defines a half-space in $\mathbb {R}^{n}$. The intersection of these half-spaces is contained in $]0,1[^n$ and, therefore, forms a convex polytope of dimension n. Because they form $n+1$ nonredundant constraints describing a convex polytope, Inequalities 26–28 are facet-defining for that polytope. This construction of a polytope and of facets does not depend on the initial choice of $\mathcal{D}'$. As $\mathcal{D}'$ varies, the facet-defining inequalities (26–28) are stated in terms of different coordinates in $\mathbb {R}^{n}$, but in conjunction with the equivalence $S(ij) = S(k\ell ) \Leftrightarrow P(ij)= P(k\ell )$, they form the same convex polytope in $]0,1[^\mathcal{D}$. $\square$

Proof of Proposition 3

Let $S_1,S_2\in \mathcal{S}$ be strength-of-preference functions on $\mathcal{D}$, and suppose ${P\in \mathcal{M}_{{S}_1} \cap \mathcal{M}_{{S}_2}}$. Then, by definition

$$\begin{aligned} S_1(ij)> S_1(k\ell ) \Leftrightarrow P(ij)> P(k\ell ) \Leftrightarrow S_2(ij) > S_2(k\ell ). \end{aligned}$$

Therefore, by the construction in the proof of Proposition 4, $\mathcal{M}_{S_1}$ and $\mathcal{M}_{S_2}$ form identical polytopes nested in  $]0,1[^\mathcal{D}$. Since there are only finitely many rankings (with possible ties) of $\frac{|\mathcal {D}|}{2}$ many choice probabilities from largest to smallest, there are also only finitely many such convex polytopes possible. $\square$

Proof of Proposition 4

Now, suppose that $\mathcal{S}$ is maximal. Since we have already shown that the convex polytopes under consideration are disjoint, we only need to show that they form a complete partition of  $]0,1[^\mathcal{{D}}$. Let $P\in ~]0,1[^\mathcal{D}$ be a binary choice vector on $\mathcal{D}$. To show that there exists a Fechnerian model family $\mathcal{M}_S$ containing P, define a strength-of-preference function S on $\mathcal{D}$ by $S(ij)=P(ij)-1/2,$ for all $ij\in \mathcal{D}$. Then P satisfies Condition 6, hence P is a Fechnerian model for S. $\square$

Appendix 3. Facet-defining inequalities for panels 3 and 4 of Fig. 2

In the third panel from the left, the left tetrahedron is defined by

$$\begin{aligned} 0< & {} P(xy)< P(xz)< P(yz)< \frac{1}{2}< P(zy)< P(zx) \\ {}< & {} P(yx) < 1, \end{aligned}$$

i.e., $S(xy)< S(xz)< S(yz)< 0< S(zy)< S(zx) < S(yx)$. In that panel, the right tetrahedron is defined by

$$\begin{aligned} 0< & {} P(yz)< P(xz)< P(xy)< \frac{1}{2}< P(yx)< P(zx) \\ {}< & {} P(zy) < 1, \end{aligned}$$

i.e., $S(yz)< S(xz)< S(xy)< 0< S(yx)< S(zx) < S(zy)$. These are on opposite sides of the line segment defined by $0< P(yz) = P(xz) = P(xy) < \frac{1}{2}$, which is also the convex hull of $\Big (0,0,0\Big )$ and $\Big (\frac{1}{2},\frac{1}{2},\frac{1}{2}\Big )$.

In the right-most panel of Fig. 2, the left tetrahedron is defined by

$$\begin{aligned} 0< & {} P(xy)< P(yz)< P(xz)< \frac{1}{2}< P(zx)< P(zy) \\ {}< & {} P(yx) < 1, \end{aligned}$$

i.e., $S(xy)< S(yz)< S(xz)< 0< S(zx)< S(zy) < S(yx)$. In that panel, the left tetrahedron is defined by

$$\begin{aligned} 0< & {} P(yz)< P(xy)< P(xz)< \frac{1}{2}< P(zx)< P(yx) \\ {}< & {} P(zy) < 1, \end{aligned}$$

i.e., $S(yz)< S(xy)< S(xz)< 0< S(zx)< S(yx) < S(zy).$ These are on opposite sides of the triangle defined by $0< P(yz) = P(xy)< P(xz) < \frac{1}{2}$.