Social Choice and Welfare

, Volume 48, Issue 2, pp 357–366 | Cite as

On the maximization of menu-dependent interval orders

Original Paper

Abstract

We study the behavior of a decision maker who prefers alternative x to alternative y in menu A if the utility of x exceeds that of y by at least a threshold associated with yandA. Hence the decision maker’s preferences are given by menu-dependent interval orders. In every menu, her choice set comprises of undominated alternatives according to this preference. We axiomatize this broad model when thresholds are monotone, i.e., at least as large in larger menus. We also obtain novel characterizations in two special cases that have appeared in the literature: the maximization of a fixed interval order where the thresholds depend on the alternative and not on the menu, and the maximization of monotone semiorders where the thresholds are independent of the alternatives but monotonic in menus.

1 Introduction

We consider the behavior of a decision maker who maximizes a menu-dependent preference defined by two functions. A utility function u assigns a utility u(x) to every alternative x. A threshold function \(\delta \) assigns a nonnegative number \(\delta (x,A)\) to every x in A, for every menu A. The decision maker prefers y to x in menu A if and only if the utility difference \(u(y)-u(x)\) exceeds the threshold \(\delta (x,A)\). Hence preferences are given by menu-dependent interval orders. The decision maker’s choice set in any menu comprises of all feasible alternatives which are undominated according to her preferences.

Our main result is a characterization of this model under the assumption that \(\delta \) satisfies the following monotonicity condition: \(\delta (x,A)\le \delta (x,B)\) if \(A\subseteq B\). We also offer novel characterizations of two specialized versions of our model which have been studied elsewhere: the maximization of interval orders (the case where \( \delta \) is independent of menus), and the maximization of monotone semiorders (the case where \(\delta \) is independent of alternatives).1

Our focus on monotonicity of thresholds aims to model a decision maker who is prone to making mistakes in larger menus. Literature has documented and studied a different potential consequence of large menus, namely choice deferral, whereby a decision maker may typically avoid choosing when presented with many alternatives (Iyengar and Lepper 2002; Gerasimou 2015). In our model larger menus lead to a different behavioral phenomenon in the form of more mistake-making. Nevertheless our model could still be useful in capturing an aspect of choice overload. We are interested in deterministic behavior as described by choice correspondences. One interpretation of choice correspondences is that they identify alternatives which form the support of random behavior. For instance, faced with menu A , the decision maker could be randomly choosing from c(A) as in Echenique (2015), never picking any alternative outside c(A). If this is the case, with more choosable alternatives on which to randomize, it may be more likely for the decision maker to switch choices when moving from smaller menus to larger ones. The greater likelihood of such switches is one potential consequence of choice overload reported in Chernev et al. (2015).

The role played by the thresholds in our model induces two kinds of deviations from pure utility maximization. The first kind arises from menu-dependence: x may be chosen in some menu at the expense of y, while the reverse may occur in another menu. A different kind of deviation occurs when, within a given menu A, the choice set contains x at the expense of y,  even though \(u(x)<u(y)\). This happens if \(\delta (x,A)\) is large enough and \(\delta (y,A)\) is low enough, so that some feasible alternative in A is preferred to y but no alternative in A is preferred to x. This latter kind of choice error, caused by the possibility that thresholds differ between alternatives, is an important difference between our work and the extant work on threshold-based models of satisficing, for instance Tyson (2008) and Frick (2015). In these models, if an alternative is in the choice set, then any other feasible alternative whose utility is higher is also in the same choice set. Consequently, any chosen alternative is revealed preferred to any unchosen alternative. In our model, this revelation does not hold. Our characterization exercise in Theorem 1 surmounts this difficulty using an existential weakening of the classical weak axiom of revealed preference.

2 Preliminaries

We adopt the standard deterministic choice framework. Given is a finite set X of alternatives, with \(2^{X}=\{A\subseteq X:A\not =\varnothing \}\) denoting the set of choice problems, or menus. A choice correspondence is a map \(c:2^{X}\rightarrow 2^{X}\) satisfying \( c(A)\subseteq A\) for every menu A. Let us record two fundamental axioms of choice theory for future reference.

WARP: If \(B\subset A\) and \(c(A)\cap B\not =\varnothing \), then \(c(A)\cap B=c(B)\).2

\(\alpha \): If \(B\subset A\), then \(c(A)\cap B\subseteq c(B)\).3

Suppose that some alternative x chosen in menu A is feasible in a smaller menu B. WARP imposes two consistency conditions on behavior in B : (i) x must be chosen in B, i.e., \(c(A)\cap B\subseteq c(B)\), (ii) no alternative must be chosen in B unless it is also chosen in A, i.e., \( c(B)\subseteq c(A)\). It is well known (see for instance Moulin 1985) that WARP is equivalent to rationality, i.e., to the existence of some utility function \(u:X\rightarrow \mathfrak {R}\) such that \(c(A)=\arg \max _{a\in A}u(a)\) for every menu A. The axiom \(\alpha \) imposes only the first of the two consistency conditions which are embodied in WARP and it is at the heart of any model of behavior where the same objective is maximized across menus.

Anchors and dominant alternatives Fix a choice correspondence c,  a menu A and an alternative \(x\in A\). We will say that x is (i) an anchor in menu A if \(x\in c(S)\subseteq c(T)\) whenever \(x\in S\subset T\subseteq A\), and (ii) a dominant alternative in menu A if \(x\in c(B)\) whenever \(x\in B\) and \( c(B)\cap A\not =\varnothing \).4

It should be clear that if x is an anchor or a dominant alternative in A, then \(x\in c(A)\). If x is an anchor or a dominant alternative in menu A, then x is so in every smaller menu B where it belongs. The following example shows that anchors need not be dominant, and dominant alternatives need not be anchors.

Example 1

Consider the following choice correspondence on \(X=\{x,y,z\}\): \(c(\{x,y\})=c(\{x,y,z\})=\{x,y\}, c(\{y,z\})=\{z\}, c(\{x,z\})=\{x,z\}.\) Note that z is the anchor in \(\{y,z\}\) although it is not dominant in \(\{y,z\}\): it is feasible but not chosen in \(\{x,y,z\}\) where y is chosen. Also note that x is a dominant alternative in \(\{x,y,z\}\), a menu in which it is not an anchor as \(c(\{x,z\})\not \subseteq c(\{x,y,z\})\).

As the following proposition indicates, however, there is an intimate relationship between rationalizability and the statement that every chosen alternative should be an anchor and dominant. This result also gives two straightforward yet novel characterizations of standard rationalizability.

Proposition 1

The following statements are equivalent.
  1. 1.

    c satisfies WARP.

     
  2. 2.

    For every menu A and every \(x\in c(A), x\) is an anchor in A.

     
  3. 3.

    For every menu A and every \(x\in c(A), x\) is a dominant alternative in A.

     

Proof

To begin, suppose c satisfies WARP, fix a menu A and an alternative \( x\in c(A)\). If menus S and T are such that \(x\in S\subset T\subseteq A\), then \(c(A)\cap S\not =\varnothing \) and, by WARP, \(c(S)=c(A)\cap S\) and \( c(T)=c(A)\cap T\). Consequently \(x\in c(S)\subseteq c(T)\) and x is an anchor in A. Next, suppose that in every menu all chosen alternatives are anchors. Take a menu A, an alternative \(x\in c(A)\) and a menu B such that \(x\in c(B)\) and \(c(B)\cap A\not =\varnothing \). Let \(y\in c(B)\cap A, y\not =x\). Since y is an anchor in \(B, c(\{x,y\})\subseteq c(B)\). Furthermore, since x is an anchor in A\(x\in c(\{x,y\})\). Hence x is a dominant alternative in A. Finally, suppose that in every menu all chosen alternatives are dominant. Take two menus A and B such that \( B\subseteq A\) and \(c(A)\cap B\not =\varnothing \). By dominance of members of c(A) in \(A, c(A)\cap B\subseteq c(B)\). Analogously, by dominance of members of c(B) in \(B, c(B)\subseteq c(A)\). Hence c satisfies WARP. \(\square \)

Interval orders and threshold representations A binary relation P on X is an interval order if it has the following utility-threshold representation: xPy iff \(u(x)>u(y)+\delta (y)\) for some utility function \(u:X\rightarrow \)\(\mathfrak {R}\) and some threshold function \(\delta :X\rightarrow \mathfrak {R}_{+}\).5 Hence it is useful to think of an interval order P as assigning to each alternative x an interval \( [u(x),u(x)+\delta (x))\). An alternative is better than another if the interval of the former lies entirely to the right of the interval of the latter. The length of the interval is defined by the threshold and it is alternative-specific. An interval order P is a semiorder if the underlying threshold is constant across alternatives. The maximization of an interval order gives rise to a choice correspondence via \(c(A)=\{x\in A:\max _{a\in A}u(a)\le u(x)+\delta (x)\}\) for all A. Hence a feasible alternative is in the choice set if and only if no other feasible alternative exceeds it in utility by more than its threshold. Note that \( \varnothing \not =\arg \max _{a\in A}u(a)\subseteq c(A)\), and the choice correspondence is well defined.

3 Maximizing monotone interval orders

We are interested in behavior that arises from maximizing a different interval order in every menu. To this end we will keep the underlying utility function of intact, but allow the threshold of an alternative to change monotonically when the menu changes.

Definition 1

A choice correspondence c admits a monotone interval order representation (MIOR) if there exist a utility function \( u:X\rightarrow \mathfrak {R}\) and a menu-dependent threshold function \(\delta :X\times 2^{X}\rightarrow \mathfrak {R}_{+}\) such that \(\delta (x,A)\ge \delta (x,B)\) whenever \(x\in B\subseteq A\), and for every menu \(A, c(A)=\{x\in A:\max _{a\in A}u(a)\le u(x)+\delta (x,A)\}.\)

MIOR models the maximization of an interval order \(P_{A}\) in every menu A, where \(P_{A}\) is defined by functions \(u(\cdot )\) and \(\delta (\cdot ,A)\). Because the threshold is allowed to change between menus, so is the comparison between alternatives. For \(x,y\in A\cap B\), it is possible for the decision maker to assess \(xP_{A}y\) but not \(xP_{B}y\). However a strict preference reversal of the form \(xP_{A}y\) and \(yP_{B}x\) is not allowed: given the nonnegativity of thresholds, if \(u(x)>u(y)+\delta (y,A)\), then \( u(y)\not >u(x)+\delta (x,B)\).

Note that if c admits a MIOR, then \(\arg \max _{a\in A}u(a)\subseteq c(A)\) for every menu A. However any other feasible alternative x whose utility does not differ from the maximal utility in A by more than the threshold \( \delta (x,A)\) is also included in c(A). Since the thresholds are lower in subsets, such x could drop out of the choice set in smaller menus where it belongs. Suppose for example that alternatives xy and menu A are such that \(u(x)+\delta (x,A)<\max _{a\in A}u(a)\le \max _{a\in A\cup \{y\}}u(a)\le u(x)+\delta (x,A\cup \{y\}).\) In this case the addition of y to menu A makes alternative x harder to beat. Even though \(x\not \in c(A)\) , as a result, \(x\in c(A\cup \{y\})\). As a result, this model fails \(\alpha \). Consider the following condition.

Anchor axiom: Every menu contains an anchor. In other words, for every menu A there exists some \(x\in A\) such that whenever \( x\in S\subset T\subseteq A, x\in c(S)\subseteq c(T)\).

We recall anchor alternatives must belong to the choice set, and that, by Proposition 1, WARP is equivalent to the statement that every chosen alternative is an anchor. Hence our Anchor axiom is an existential weakening of WARP as it only insists that some chosen alternative should be an anchor.

Theorem 1

A choice correspondence admits a monotone interval order representation if and only if it satisfies the Anchor axiom.

Proof

To begin, suppose that c admits a MIOR. Fix a set A and pick \(x\in \arg \max _{a\in A}u(a)\). We will show that x is an anchor in A. Now let menus S and T be such that \(x\in S\subset T\subseteq A\). First note that \( \max _{s\in S}u(s)=\max u_{t\in T}u(t)=u(x)\). Hence, since \(\delta \) is nonnegative valued, \(u(x)\le u(x)+\delta (x,S),\) implying that \(x\in c(S)\). Now for any \(y\in c(S), y\in T\), and \(u(x)\le u(y)+\delta (y,S)\le u(y)+\delta (y,T)\) by monotonicity of \(\delta \), and therefore \(y\in c(T)\). This establishes \(c(S)\subseteq c(T)\) and x is an anchor in A as desired.

Now suppose that c satisfies the Anchor axiom. We will construct a utility function u and a monotone threshold function \(\delta \) which together yield a MIOR for c. Define a binary relation \(P_{0}\) on X as follows: \( xP_{0}y\Leftrightarrow \exists A\) such that \(y\in A\) and \(c(A)\not \subseteq c(A\cup x).\) We claim that \(P_{0}\) is acyclic. To see this suppose towards a contradiction that \(x_{i}P_{0}x_{i+1}\) for \(i=1,...,n\), with \(x_{n+1}=x_{1}\). Hence for every i, there exists a menu \(A_{i}\) such that \(x_{i+1}\in A_{i}\) and \(c(A_{i})\not \subseteq c(A_{i}\cup x_{i})\). Let \(A=\cup _{i}A_{i}\). Note that A does not contain any anchor: for every \(a\in A\), there exists some i such that \(c(A_{i})\not \subseteq c(A_{i}\cup x_{i})\).

Since \(P_{0}\) is acyclic, it is contained in a linear order P by Szpilrajn (1930). Let m be the cardinality of X and order alternatives such that \( x_{m}Px_{m-1}P\cdot \cdot \cdot Px_{1}\). Define functions u and \(\delta \) as follows: \(u(x_{i})=2^{i}\) and
$$\begin{aligned} \delta (x_{i},A)=\left\{ \begin{array}{ll} U(A)-u(x_{i}) &{} \,\, \hbox { if } \,\, x_{i}\in c(A) \\ \frac{1}{2}(U(A)-u(x_{i})) &{} \,\, \hbox { if } \,\, x_{i}\in A\backslash c(A) \end{array} \right. \end{aligned}$$
where \(U(A):=\max _{a\in A}u(a)\) for every menu A. Set \(\delta (x_{i},A)=0\) if \(x_{i}\not \in A\). Note \(\delta (x_{i},A)\ge 0\). We will next establish the monotonicity of \(\delta \). Pick xB and A such that \(x\in B\subset A\). Note \(U(B)\le U(A)\). There are four cases to analyze.
  • Case 1: \(x\in c(B)\cap c(A)\). Then \(\delta (x,B)=U(B)-u(x)\le U(A)-u(x)=\delta (x,A).\)

  • Case 2: \(x\not \in c(B)\cup c(A)\). Then \(\delta (x,B)=\frac{1}{2} [U(B)-u(x)]\le \frac{1}{2}[U(A)-u(x)]=\delta (x,A).\)

  • Case 3: \(x\in c(A)\backslash c(B)\). Then \(\delta (x,B)=\frac{1}{2} [U(B)-u(x)]\le U(B)-u(x)\le U(A)-u(x)=\delta (x,A).\)

  • Case 4: \(x\in c(B)\backslash c(A)\). Let a be an anchor in A. Clearly, \( a\not \in B\) as \(c(B)\not \subseteq c(A)\). Since \(c(B\cup \{a\})\subseteq c(A)\) , \(c(B)\not \subseteq c(B\cup \{a\})\). This implies aPb for all \(b\in B\) and, by the construction of \(u, U(A)\ge 2U(B)\). Consequently \(\delta (x,B)=U(B)-u(x)\le \frac{1}{2}[U(A)-u(x)]=\delta (x,A)\) and the monotonicity of \(\delta (x,\cdot )\) is established.

Now pick an arbitrary set A. Let \(x_{i}\in A\) be the necessarily unique alternative such that \(u(x_{i})=U(A)\). We will show that \(x_{i}\) is an anchor in A. Suppose \(x_{i}\not \in c(A)\). Let y be an anchor in A. Then \(c(\{x_{i},y\})=\{y\}\) implying \(yPx_{i}\), a contradiction. Hence \( x_{i}\in c(A)\). Now take any S and T such that \(x_{i}\in S\subseteq T\). Since \(U(S)=U(T)=u(x_{i})\), we have \(x_{i}\in c(S)\cap c(T)\). Suppose \(s\in c(S)\backslash c(T)\) and let t be an anchor in T. Then \(t\not \in S\) and since \(c(S\cup t)\subseteq c(T), c(S)\not \subseteq c(S\cup t)\). This implies \(tPx_{i}\), a contradiction. Hence no such s exists and \( c(S)\subseteq c(T)\). This establishes that \(x_{i}\) is an anchor in A.

All that remains to show is that \(c(A)=\{x\in A:U(A)\le u(x)+\delta (x,A)\}\) . If \(a\in c(A)\), by construction of \(\delta , U(A)=u(a)+\varepsilon (a,A)\) and \(a\in \{x\in A:U(A)\le u(x)+\delta (x,A)\}\). Now take \(a\in A\) such that \(U(A)-u(a)\le \delta (a,A)\). If \(a=x_{i}\), then \(a\in c(A)\) as we showed above. Suppose \(a\not =x_{i}\) so that \(U(A)>u(a)\). If \(a\not \in c(A)\) then \(\frac{1}{2}[U(A)-u(a)]=\delta (a,A)\ge U(A)-u(a)\), which implies \( u(a)=U(A)\) a contradiction. Hence \(a\in c(A)\). This completes the proof. \(\square \)

Note on the literature

Theorem 5.5 in Aleskerov et al. (2007) gives a characterization of the more general model where the thresholds need not be monotone in menus. They use the following condition: every menu contains a fixed point. In other words, for every menu A there exists some \(x\in A\) such that whenever \( x\in B\subseteq A, x\in c(B)\).6 This axiom weakens \(\alpha \) exactly the same way our anchor alternative weakens WARP, by requiring not all but some chosen alternative to be chosen in smaller menus where it belongs. It is clear that every anchor alternative is also a fixed point, meaning our Anchor axiom implies the Fixed Point axiom. In addition, Aleskerov et al. (2007) also define and give an analysis of MIOR. In particular, they note that if c admits a MIOR, then it necessarily satisfies the classical \(\gamma \) axiom, which we will define shortly. However they do not give a characterization result analogous to our Theorem 1.

4 Special cases and connections with the literature

In this section we will study two specialized versions of MIOR: (i) the maximization of an interval order in every menu, and (ii) the scenario where the thresholds are independent of alternatives, yet still depend on the menus monotonically.

4.1 Maximization of an interval order

Consider the special case of our model where menu dependence disappears. We recall the definition here for convenience.

Definition 2

A choice function c admits an interval order representation (IOR) if it admits a monotone interval order representation and if the associated threshold is constant across menus: \(\delta (x,A)=\delta (x,B)\) for all xA and B.

If c admits an IOR, we will simply write \(\delta (x)\) to denote the menu-independent threshold of x. In our next result we provide characterizations of IOR using the following axioms.

Path Independence: \(c(A\cup B)=c(c(A)\cup c(B))\) for all A and B.

\(\gamma \): \(c(A)\cap c(B)\subseteq c(A\cup B)\) for all A and B.

Weak Arrow axiom: For every A, there exists some \(x\in c(A)\) such that if \(x\in B\subset A\) then \(c(B)=c(A)\cap B\).

Dominant Alternative axiom: Every menu contains a dominant alternative. In other words, for every menu A there exists some \( x\in A\) such that if \(x\in B\) and \(c(B)\cap A\not =\varnothing \), then \(x\in c(B)\).

Path Independence and \(\gamma \) are well-known conditions.7 The Dominant Alternative axiom is new and, in view of Proposition 1, it offers a weakening of WARP, as does our Anchor axiom from the previous section. The Weak Arrow axiom appears in Payró and Ülkü (2015), and it offers yet a different existential weakening of WARP. The following result features these four axioms, together with \(\alpha \) and the Anchor axiom which we have recorded in the previous section.

Theorem 2

The following are equivalent:
  1. 1.

    c admits an interval order representation.

     
  2. 2.

    c satisfies \(\alpha \) and the Anchor axiom.

     
  3. 3.

    c satisfies \(\alpha \) and the Weak-Arrow axiom.

     
  4. 4.

    c satisfies Path Independence, \(\gamma \) and the Dominant Alternative axiom.

     

Proof

Step 1: Suppose that c admits an IOR and let u and \(\delta \) be the associated utility and threshold functions. That c satisfies \(\alpha \) follows from the fact that c maximizes the same objective across menus. Fix a menu A,  an alternative \(x\in \arg \max _{a\in A}u(a)\) and menus S and T such that \(x\in S\subset T\subseteq A\). Note \(u(x)=\max _{a\in S}u(a)=\max _{a\in T}u(a)\). It follows directly from these observations that \( x\in c(S)\), and that if \(s\in c(S)\) then \(u(s)+\delta (s)\ge u(x)\) and \( s\in c(T)\) as well. Hence x is an anchor in A.

Step 2: Next suppose that c satisfies \(\alpha \) and the Anchor axiom. We will show that c satisfies the Weak Arrow axiom. Pick A and let x be an anchor in A. If \(x\in B\subset A\), then \(c(A)\cap B\subseteq c(B)\) by \( \alpha ,\) and \(c(B)\subseteq c(A)\) since x is an anchor. Hence \( c(B)=c(A)\cap B\) as desired.

Step 3: Now suppose that c satisfies \(\alpha \) and the Weak Arrow axiom. Take menus A and B and let x be the alternative identified by the axiom in \(A\cup B\). Note that \(x\in c(A\cup B)\) and x is an alternative identified by Weak Arrow in every subset of A where it belongs. By \(\alpha , c(A\cup B)\subseteq c(A)\cup c(B)\). Hence \(c(c(A)\cup c(B))=c(A\cup B)\cap (c(A)\cup c(B))=c(A\cup B)\) and Path Independence is established. Now suppose \(y\in c(A)\cap c(B)\). If \(x\in A\), then \(c(A)\subseteq c(A\cup B)\) and \(y\in c(A\cup B)\). Similarly if \(x\in B\). Hence c satisfies \(\gamma \). Now we will show that c satisfies the Dominant Alternative axiom. Define a binary relation \(\succ \) on X as follows: \(x\succ y\) iff for some menu A , \(x\in c(A)\) and \(y\in A\backslash c(A)\). It suffices to show that \(\succ \) is acyclic, so that every menu A contains some \(\succ \)-maximal alternative. Any such alternative is a dominant anchor in A by definition of \(\succ \). Suppose, towards a contradiction, that \(\succ \) is cyclic, i.e., there exist alternatives \(x_{1},..,x_{n},x_{n+1}\) such that \( x_{i}\succ x_{i+1}\) for all i and \(x_{n+1}=x_{1}\). Hence there exist menus \(A_{i}\), \(i=1,...,n\), such that \(x_{i}\in c(A_{i})\) and \( x_{i+1}\in A_{i}\backslash c(A_{i})\). Set \(A_{0}=A_{n}\). Let \(a_{i}\) be an alternative identified by the Weak Arrow axiom in \(c(A_{i})\), for all i. Since \(x_{i}\not \in c(A_{i-1}), x_{i}\not \in c(\{a_{i-1},x_{i}\})\). Furthermore, by \(\alpha , x_{i}\in c(\{a_{i},x_{i}\})\). Now consider the menu \(S=\{x_{1},...,x_{n},a_{1},...,a_{n}\}\). By \(\alpha , x_{i}\not \in c(S)\) since \(x_{i}\not \in c(\{a_{i-1},x_{i}\})\) and therefore \(c(S)\subseteq \{a_{1},...,a_{n}\}\). Hence the alternative in S identified by the Weak Arrow axiom is some \(a_{i}\), and \(c(\{a_{i},x_{i}\})=c(S)\cap \{a_{i},x_{i}\} \). This implies \(x_{i}\in c(S)\), a contradiction.

Step 4: To finish the proof, suppose c satisfies Path Independence, \( \gamma \) and the Dominant Alternative axiom. Define xPy iff \(x\not =y\) and \( \{x\}=c(\{x,y\})\). Clearly P is asymmetric. Using Theorem 3 in Plott (1973) we conclude that P is also transitive, hence a strict partial order. Furthermore its maximization rationalizes c: \(c(A)=\{x\in A:yPx\) for no \(y\in A\}\). Pick wxyz such that wPxyPz. We need to show that wPz or yPx. By definition of \(P, \{w\}=c(\{w,x\})\) and \( \{y\}=c(\{y,z\})\). Now \(\alpha \) gives \(c(\{w,x,y,z\})\subseteq \{w,y\}\). If x is a dominant alternative in \(\{x,z\}\), then \(y=c(\{x,y,z\})\), for \( x\not \in c(\{x,y,z\})\) by \(\alpha \). This implies yPz. Similarly, if z is a dominant alternative in \(\{x,z\}\), then wPx. Hence, P is an interval order. Using Fishburn’s characterization of interval orders (Fishburn 1970), there exist a utility \(u:X\rightarrow \mathfrak {R}\) and a threshold \(\delta :X\rightarrow \mathfrak {R}_{+}\) such that \(c(A)=\{x\in A:u(x)+\delta (x)<u(y)\) for no \(y\in A\}\) and c admits an IOR. \(\square \)

Note on the literature

Our Theorem 2 follows two distinct characterizations of the maximization of interval orders. The characterization result in Schwartz (1976, Theorem 3) relies on the following two axioms: (1) \(c(A)\cap c(B)=c(A\cup B)\cap A\cap B \) for every A and B, and (2) if \(B\subset A, c(A)\not \subseteq B\) and \(c(B)\not \subseteq c(A)\), then \(c(A\backslash B)\subseteq c(A)\). Aleskerov et al. (2007, Corollary 3.5) show that Path Independence, \(\gamma \) and the following condition together yield yet another characterization: if \( c(A)\cap (B\backslash c(B))\not =\emptyset \), then \(c(B)\cap (A\backslash c(A))=\emptyset \).8

4.2 Monotone semiorders

Frick (2015) studies the special case of our MIOR in which the thresholds are not alternative-specific.

Definition 3

A choice correspondence c admits a monotone semiorder representation (MSR) if it admits a monotone interval-order representation and if \(\delta (x,A)=\delta (y,A)\) for all A and all \(x,y\in A\).

Frick (2015) shows that MSR is characterized by the following axiom.

Occasional Optimality: For every A, there exists some dominant alternative \(x\in A\), which additionally satisfies the following condition: if \(x\in B\) and \(y\in A\) then \(c(B)\subseteq c(B\cup \{y\})\).

Now consider the following condition which strengthens both Anchor and Dominant Alternative axioms.

Dominant Anchor axiom: Every menu contains a dominant anchor, i.e., an anchor which is also a dominant alternative.

The next result shows that these two axioms are equivalent.

Lemma 1

A choice correspondence c satisfies Occasional Optimality if and only if it satisfies the Dominant Anchor axiom.

Proof

Suppose that c satisfies our axiom and fails Occasional Optimality: there is no alternative as described by Occasional Optimality in some menu A. Say \(\{x_{1},...,x_{n}\}\) is the set of dominant anchors in A. By definition, for every \(x_{i}\), there exist an alternative \(y_{i}\in A\) and a menu \(B_{i}\) such that \(x_{i}\in B_{i}\) and \(c(B_{i})\not \subseteq c(B_{i}\cup \{y_{i}\})\). This implies that \(y_{i}\) is the unique dominant anchor in \(B_{i}\cup \{y_{i}\}\) for every i. Now let \(S=A\cup B_{1}\cup \cdot \cdot \cdot \cup B_{n}\). We will argue that S does not contain a dominant anchor. Say z is a dominant anchor in S. If \(z\in B_{i}\) for some i, we must have \(c(B_{i})\subseteq c(B_{i}\cup \{y_{i}\})\), a contradiction. Hence \(z\not \in B_{i}\) for any i. This implies \(z\in A\) and, in particular, that z is a dominant anchor in A as well. Hence \( z=x_{i}\in B_{i}\) for some i, a contradiction. In the other direction, suppose c satisfies Occasional Optimality and pick some x which is as described by Occasional Optimality in A. We will show that x is a dominant anchor in A as well. That x is a dominant alternative follows directly from the first requirement of Occasional Optimality. Say \(x\in S\subset T\subseteq A\). Then \(x\in c(S)\) by the first requirement of Occasional Optimality. Let \(T\backslash S=\{y_{1},...,y_{n}\}\). Set \(S_{0}=S\), and \(S_{i}=S_{i-1}\cup \{y_{i}\}\) for \(i=1,...,n\). By the second requirement of Occasional Optimality, \(c(S)=c(S_{0})\subseteq c(S_{1})\subseteq \cdot \cdot \cdot \subseteq c(S_{n})=c(T)\) and x is an anchor in A. \(\square \)

Hence the Dominant Anchor axiom provides a new characterization of MSR.

Corollary 1

A choice correspondence admits a monotone semiorder representation if and only if it satisfies the Dominant Anchor axiom.

Proof

The proof follows from Lemma 1 above and from Theorem 2.4 in Frick (2015). \(\square \)

5 Concluding remarks

We would like to conclude with a quick remark on the nature of our axioms. Our axiomatic analysis uses existential statements in the Anchor, the Dominant Alternative and the Dominant Anchor axioms.9 These axioms impose a testable structure on behavior which could in principle be useful in applications, especially with a finite domain of alternatives. Proposition 1 indicates that these axioms are suitable relaxations of rationality. In fact, in the case of single-valued choice, these axioms collapse to standard rationality.

We would also like to point out that we do not offer these axioms as normatively desirable properties of choice behavior. Instead, we think of them as descriptive devices related to an interesting deviation from full rationality, in which the decision maker becomes more mistake-prone in larger menus. They are also appealing because, as we show in Theorem 2, they prove useful in characterizing the maximization of interval orders.

Footnotes

  1. 1.

    Schwartz (1976, Theorem 3) and Aleskerov et al. (2007, Corollary 3.5) characterize the maximization of interval orders. Frick (2015, Theorem 2.4) characterizes the maximization of monotone semiorders.

  2. 2.

    This version of the weak axiom of revealed preference appears in Arrow (1959).

  3. 3.

    This axiom appears in Chernoff (1954). It is referred to as \(\alpha \) in Sen (1971) and as Heredity in Aleskerov et al. (2007).

  4. 4.

    To the best of our knowledge, the concept of anchors is new. Dominant alternatives are related to occasionally optimal alternatives, a concept which appears in Frick (2015). In particular, every occasionally optimal alternative is dominant, but not vice versa. In Sect. 4.2, we will show that occasionally optimal alternatives are precisely those anchor alternatives which are also dominant.

  5. 5.

    Equivalently P is an interval order if it satisfies asymmetry (if xPy then not yPx) and the following intervality condition: if wPx and yPz, then wPz or yPx. See Fishburn (1970).

  6. 6.

    This condition also appears as the Bliss Point axiom in Masatlıoğlu and Nakajima (2013).

  7. 7.

    See, for instance, Plott (1973) who shows that Path Independence and \(\gamma \) together characterize the maximization of strict partial orders.

  8. 8.

    See also Fishburn (1975). Also recall that Path Independence and \(\gamma \) are equivalent to the maximization of a strict partial order, as shown by Plott (1973). Since every interval order is a strict partial order, interval order representation requires a stronger axiomatic system.

  9. 9.

    Besides the work we have already cited, other prominent examples of existential axioms include the Reducibility axiom in Manzini and Mariotti (2012) and the WARP-LA axiom in Masatlıoğlu et al. (2012).

References

  1. Aleskerov F, Bouyssou D, Monjardet B (2007) Utility maximization, choice and preference, 2nd edn. Springer, BerlinGoogle Scholar
  2. Arrow K (1959) Rational choice functions and orderings. Economica 26:121–127CrossRefGoogle Scholar
  3. Chernev A, Böckenholt U, Goodman J (2015) Choice overload: a conceptual review and meta analysis. J Consum Psychol 25(2):333–358CrossRefGoogle Scholar
  4. Chernoff H (1954) Rational selection of decision functions. Econometrica 22:422–443CrossRefGoogle Scholar
  5. Echenique F, Saito K(2015) General Luce Model, CalTech SS Working Paper 1407Google Scholar
  6. Fishburn P (1970) Utility theory for decision making. Wiley, New YorkGoogle Scholar
  7. Fishburn P (1975) Semiorders and choice functions. Econometrica 43:475–477Google Scholar
  8. Frick M (2015) Monotone threshold representations. Theor Econ (forthcoming)Google Scholar
  9. Gerasimou G (2015) Indecisiveness, undesirability and overload revealed through rational choice deferral. MPRA Working Paper 67290Google Scholar
  10. Iyengar S, Lepper MR (2002) When choice is demotivating: can one desire too much of a good thing? J Personal Soc Psychol 79(6):995–1006CrossRefGoogle Scholar
  11. Manzini P, Mariotti M (2012) Choice by lexicographic semiorders. Theor Econ 7:1–23CrossRefGoogle Scholar
  12. Masatlıoğlu Y, Nakajima D (2013) Choice by iterative search. Theor Econ 8:701–728CrossRefGoogle Scholar
  13. Masatlıoğlu Y, Nakajima D, Özbay E (2012) Revealed attention. Am Econ Rev 102(5):2183–2205CrossRefGoogle Scholar
  14. Moulin H (1985) Choice functions over a finite set: a summary. Soc Choice Welf 2:147–160CrossRefGoogle Scholar
  15. Payró F, Ülkü L (2015) Similarity-based mistakes in choice. J Math Econ 61:152–156CrossRefGoogle Scholar
  16. Plott C (1973) Path independence, rationality and social choice. Econometrica 41:1075–1091CrossRefGoogle Scholar
  17. Schwartz T (1976) Choice functions, revealed preference. J Econ Theory 13:414–427CrossRefGoogle Scholar
  18. Sen A (1971) Choice functions and revealed preference. Rev Econ Stud 38:307–317CrossRefGoogle Scholar
  19. Szpilrajn E (1930) Sur l’extension de l’ordre partiel. Fundam Math 16:386–389Google Scholar
  20. Tyson C (2008) Cognitive constraints, and the satisficing criterion. J Econ Theory 138:51–70CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und Geometrie at Vienna University of TechnologyViennaAustria
  2. 2.Department of EconomicsCentro de Investigación Económica (CIE), ITAMCiudad de MéxicoMexico

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