aa r X i v : . [ ec on . T H ] F e b Identification in the Random Utility Model
Christopher Turansick ∗ February 11, 2021
Abstract
The random utility model is known to be unidentified. However,there are times when a data set is uniquely rationalizable by the ran-dom utility model. We ask the question for which data sets does therandom utility model have a unique representation. Our first resultcharacterizes which data sets admit a unique representation. Our sec-ond result provides a finite test which determines if a distribution ofpreferences is observationally equivalent to some other distribution ofpreferences. We then explore the implications of our results in thecontext of other random utility models.
When repeatedly faced with a decision, decision makers will frequently varytheir choice. This means that when a researcher observes choice data, theywill often observe a distribution of choices rather than a single choice. Acommon explanation for this behavior is the random utility model of Blockand Marschak (1959). In this model, a decision maker randomly draws apreference and then acts to maximize utility given the drawn preference.From a practical standpoint, identification of a choice model is importantas it allows for social planners and mechanism designers to perform proper ∗ Georgetown University; [email protected] Falmagne (1978) is the first able to fully characterize the behavioral content of therandom utility model. Barbera and Pattanaik (1986) and Fiorini (2004) discuss andprovide alternate proofs of the main result of Falmagne (1978). McFadden and Richter(1990) are able to provide an alternative characterization of the random utility model.
To begin, we review the standard random utility model (RUM). Let X be afinite set of alternatives. Let Π be the set of linear orders over X . We use π to denote an element of Π. We use the notation π ( A ) > π ( B ) to denote thatevery element of A is ranked higher than every element of B according to π .We place no further restrictions on how π ranks elements of A against otherelements of A . The same is true for elements of B . When A = { x } , we usethe notation π ( x ). Further, we call ∆(Π) the set of probability distributionsover Π. We say that an agent makes decisions according to RUM if theyare endowed with a ν ∈ ∆(Π) and, whenever they make a decision, theydraw a linear order according to this ν and then choose the maximal elementaccording to the drawn linear order.We consider stochastic choice data for each non-empty subset of X . Toformalize this, the data we consider is called a system of choice probabilities.A pair ( X, P ) is a system of choice probabilities if for all non-empty subsets A linear order is an asymmetric, transitive, and complete binary relation. of X , P A ( · ) defines a probability distribution over the elements of A . Asystem of choice probabilities captures the choice frequency of each elementof each non-empty subset A of X . We now define what it means for data tobe rationalizable by RUM. Definition 1 (Rationalizable) . We say that a system of choice probabilitiesis rationalizable if there exists some ν ∈ ∆(Π) such that for all non-empty A ⊆ X and all x ∈ A we have P A ( x ) = X π ∈ Π ν ( π ) { π ( x ) > π ( A \ { x } ) } . This definition says that the data is rationalizable if there exists someprobability distribution ν over linear orders of X such that choice probabilityof x in A is equal to the probability of drawing an order that ranks x above allother elements of A . Recall that Falmagne (1978) was the first to characterizedata that is rationalizable by RUM. The characterization relies on what arecalled the Block-Marschak polynomials, henceforth BM-polynomials, whichwere first introduced by Block and Marschak (1959). We state the definitionof the BM-polynomials here. Definition 2 (Block-Marschak Polynomials) . For a non-empty set A ⊆ X and an element x ∈ A , the BM-polynomial for x in A is given by q ( x, A ) = P A ( x ) − X A ( A ′ q ( x, A ′ )= X A ⊆ A ′ ( − | A ′ \ A | P A ′ ( x ) . For an interpretation of the BM-polynomials, we turn to a theorem pre-sented in Falmagne (1978). This theorem states that, for rationalizable data,the BM-polynomial q ( x, A ) is equal to the probability weight put on ordersthat rank x above all other elements of A and below all elements of X \ A .Alternatively stated, q ( x, A ) is equal to the probability weight put on or-ders for which the strict upper contour set of x is exactly X \ A . Thecharacterization of Falmagne (1978) states that all BM-polynomials mustbe non-negative. We are interested in characterizing when the rationalizingprobability distribution is unique.Our characterization combines the graphical representation of RUM pre-sented in Fiorini (2004) with the intuition of the counterexample to unique-ness presented by Fishburn (1998). As such, we review both here. We begin5ith the graphical construction presented in Fiorini (2004). Consider a graphwith nodes indexed by the elements of 2 X , the power set of X . We will usethe set indexing a node to refer to that node. There exists an edge betweentwo nodes A and B if one of the following is true.1. A ⊆ B and | B \ A | = 12. B ⊆ A and | A \ B | = 1In other words, the edge set of this graph is formed by applying the coveringrelation of ⊆ to X . Thus far, this is just the Hasse diagram of X . Now wewill assign weights to these edges. Assign q ( x, A ) to the edge connecting A and A \ { x } . Fiorini (2004) does not give a name to this graph, but we willrefer to it as the probability flow diagram . Figure 1 gives an example ofthe probability flow diagram for the set X = { a, b, c } .We now revisit the counterexample of Fishburn (1998) and explore theprobability flow diagram of the counterexample. Example 1 (Fishburn’s Counterexample) . Let X = { a, b, c, d } . Considerthe following probability distributions over linear orders on X . ν ( π ) = ( if π ∈ { a ≻ b ≻ c ≻ d, b ≻ a ≻ d ≻ c } otherwise ν ( π ) = ( if π ∈ { a ≻ b ≻ d ≻ c, b ≻ a ≻ c ≻ d } otherwiseThese two probability distributions induce the same system of choice proba-bilities. This is the counterexample that Fishburn (1998) uses to show that RUMdoes not necessarily have a unique representation. Below, Figure 2 shows thereduced probability flow diagram of this example. In the example above,since both probability distributions induce the same system of choice proba-bilities, they have the same probability flow diagram. The key feature of thisexample that we would like to draw attention to is focused at the node { c, d } . By reduced probability flow diagram we mean that we take the probability flow di-agram and remove each edge with zero weight and each node whose connected edges allhave zero weight. { a } { b } { c }{ a, b } { a, c } { b, c }{ a, b, c } q ( b , { a , b , c } ) q ( a , { a , b , c } ) q ( c , { a , b , c } ) q ( b , { a , b } ) q ( a , { a , b } ) q ( c , { a , c } ) q ( a , { a , c } ) q ( c , { b , c } ) q ( b , { b , c } ) q ( a , { a } ) q ( b , { b } ) q ( c , { c } ) Figure 1: The probability flow diagram for the set X = { a, b, c } .7ote that there are two edges with strictly positive weight that go into { c, d } and two edges with strictly positive weight that leave { c, d } . It turns outthat this two-in and two-out structure exactly characterizes non-uniquenessin RUM. We formalize this two-in and two-out structure in the next section. In this section, we present two characterizations for when stochastic choicedata arises from a unique distribution of preferences. The first characteriza-tion uses the BM-polynomials, which are constructed through basic opera-tions on the data, and the graphical intuition presented in the prior section.The second characterization is a finite test on a distribution of preferencesto see if it satisfies certain properties relating to upper contour sets. We nowintroduce the terminology needed to state our characterizations.
Definition 3 (Path) . We call a path ρ a finite sequence of sets { A i } | X | i =0 suchthat A i +1 ( A i for all i , A = X , and A | X | = ∅ . Fiorini (2004) notes that there is an obvious bijection between paths onthe probability flow diagram and the set of linear orders of X . The bijectionpairs the path { X, X \ { x } , X \ { x , x } , . . . , ∅ } with the order x ≻ x ≻ . . . . Further, when constructing a rationalizing probability distribution, theprobability weight associated with order π is derived directly from the edgeweights of the path corresponding to π . Similarly, we will be using thisbijection and the associated edge weights to study which orders can receive astrictly positive weight in a rationalizing probability distribution. This ideais captured graphically by the following definition. Definition 4 (Supported) . We call a path supported if for all i ∈{ , . . . , | X | − } , q ( A i \ A i +1 , A i ) > . As it turns out, there exists a rationalizing probability distribution whichputs strictly positive weight on a linear order π if and only if the path associ-ated with π is supported. Due to this fact, if a system of choice probabilitieshas multiple rationalizing distributions, it must be the case that the differingprobability weights are restricted to the set of orders for which the associatedpath is supported. As we mentioned prior, the characterization for unique- To see this, note that any order which has a path which is not supported must neces-sarily receive zero probability weight in a rationalizing probability distribution. { c } { d }{ c, d }{ b, c, d } { a, c, d }{ a, b, c, d } Figure 2: The reduced probability flow diagram for the Fishburn counterex-ample. 9ess relies on the idea of two-in and two-out. The definition of branchingformalizes this idea.
Definition 5 (Branching) . We call two paths ρ and ρ ′ branching if thereexists some i ≤ j with i, j ∈ { , . . . , | X | − } such that A ρi − = A ρ ′ i − , A ρj +1 = A ρ ′ j +1 , and for all m ∈ { i, . . . , j } , A ρm = A ρ ′ m . Unlike in the counterexample from Fishburn (1998), the definition ofbranching does not require the two-in and two-out to happen at the samenode. The definition of branching allows for two paths to go into the samenode, share a few common edges, and then split. We now have all theterminology we need to state our main theorem.
Theorem 1.
Suppose that a system of choice probabilities ( X, P ) is rational-izable. Then the rationalizing ν is unique if and only if the probability flowdiagram has no pairs of supported paths which are branching. We leave all proofs to the appendix. However, we discuss the intuitionof the proof here. To see the logic for necessity, first consider a node thatsatisfies two-in and two-out. Call the two-in edges a and b respectively. Callthe two-out edges c and d respectively. Then we can construct two disjointsets of paths that induce this two-in and two-out property. Consider thepair of paths { ( a, c ) , ( b, d ) } . By definition these two paths satisfy two-in andtwo-out at the considered node. Similarly, the pair of paths { ( a, d ) , ( b, c ) } satisfy two-in and two-out along the same edges as the first pair of paths.This shows that two supported branching paths imply non-uniqueness.To see the logic for sufficiency, we first note that if no pair of supportedpaths satisfy two-in and two-out, then every pair of supported paths thatsatisfy two-out must do so above any node at which they satisfy two-in.Similarly, any pair of supported paths that satisfy two-in must do so belowany node at which they satisfy two-out. These two facts together meanthat for every supported path there exists some edge for which any two-inhappens below that edge and every two-out happens above that edge. Theweight along this edge uniquely identifies the probability weight put on theorder associated with this path.Note that our Theorem 1 subsumes some known results. First, we knowfrom Block and Marschak (1959) that, when | X | ≤
3, any rationalization ofa system of choice probabilities is unique. We note that branching paths are10ot found unless | X | ≥ As an immediate corollary of our Theorem 1,we are able to show that, when | X | ≤
3, any rationalization of a system ofchoice probabilities is unique.McClellon (2015) shows that when | X | ≥
4, the issue of non-uniquenessis widespread. The result notes that if a system of choice probabilities isinduced by a probability distribution with full support over linear orders,then there is a different probability distribution that induces the same systemof choice probabilities. If a system of choice probabilities is induced by aprobability distribution with full support, then every path is supported. Sincethere are branching paths when | X | ≥
4, it follows immediately that anyrationalization is not unique in this case. This result can be extended to saythat if every BM-polynomial of a system of choice probabilities is strictlypositive, then the rationalizing distribution is not unique.We now move onto our second characterization. Intuitively, this char-acterization looks at the underlying structure of branching paths and thenrestates that structure in terms of properties of the upper contour sets of theassociated orders. These properties then can be checked with a finite test onthe rationalizing distribution. Before moving forward, we formalize what wemean by upper contour set.
Definition 6 (Upper Contour Set) . The weak upper contour set of someelement x ∈ X according to linear order π is the set of all elements y ∈ X such that π ( y ) ≥ π ( x ) . We write U π ( x ) = { y | π ( y ) ≥ π ( x ) } to denote the weak upper contour set of x according to π . With this definition, we are now able to state our second characterization.
Theorem 2.
Suppose that a system of choice probabilities ( X, P ) is ratio-nalizable. Then the rationalizing ν is unique if and only if there are no pairsof orders π and π ′ satisfying the following.1. ν ( π ) > and ν ( π ′ ) > To see this, observe the following. A pair of branching paths share the node X , havediffering nodes somewhere below X , have a common node below their differing nodes,have another differing node below their common node, and then share the node ∅ . Thisrequires having five nodes which can only happen when | X | ≥ . There exists x, y, z ∈ X such that(a) π ( { x, y } ) > π ( z ) and π ′ ( { x, y } ) > π ′ ( z ) (b) x = y (c) U π ( z ) = U π ′ ( z ) (d) U π ( x ) = U π ′ ( y )Intuitively, the first condition of Theorem 2 captures the definition ofsupported path and the second condition captures the definition of a pairof branching paths. Our proof consists of showing that the existence ofa pair of supported branching paths is equivalent to the two conditions ofTheorem 2 holding. Necessity follows primarily from definitions. The logicfor sufficiency is as follows. We first suppose that the representation is notunique. Then, by Theorem 1, there must be a pair of supported branchingpaths. We then show that no matter how one allocates the weight from thesesupported branching paths, there will always be two orders which violate theconditions of Theorem 2.Before moving on, we note that both Theorem 1 and Theorem 2 rely onthe assumption that we have choice data on each nonempty subset of X .Theorem 1 relies on the BM-polynomials. Defining the BM-polynomials re-quires observing choice on each nonempty subset of X . Without well definedBM-polynomials, the edge weights for the probability flow diagram are notwell defined. Further, the proof of our Theorem 2 relies on Theorem 1. Assuch, Theorem 2 ceases to be valid in every case which Theorem 1 fails to bevalid. Although not studied here, there is a way to rectify this. Suppose thatthe choice domain is not complete and the system of choice probabilities isrationalizable. Then this means that there exist choice probabilities on theunobserved sets that lead to non-negative BM-polynomials. If we could verifythat the set of choice probabilities that lead to non-negative BM-polynomialsis singleton, then we would be able to apply our Theorem 1 and Theorem 2to the system of choice probabilities formed by combining the original choiceprobabilities with the choice probabilities on the unobserved sets. Further, ifthe set of choice probabilities that leads to non-negative BM-polynomials isnot singleton, we know immediately that there are multiple rationalizationsof the system of choice probabilities. This logic presents a clear path forstudying uniqueness on an incomplete choice domain.12 Other Random Utility Models
In this section, we study how our theorems relate to the uniqueness results oftwo other random utility models. In the case of the Luce model, we find thatuniqueness follows not from the preferences which are precluded from themodel but from the fact that a Luce representation does not explicitly mapto preference weights. In the case of the single crossing random utility model(SCRUM) of Apesteguia, Ballester, and Lu (2017), we find that uniquenessfollows from the restrictions that single-crossing puts on the support of ra-tionalizations.
To begin, we will briefly review the Luce model. We maintain the assumptionthat choice is observed on each nonempty subset of X . Definition 7 (Luce Representation) . A system of choice probabilities ( X, P ) has a Luce representation if there exists a weight function w : X → (0 , with P x ∈ X w ( x ) = 1 such that P A ( x ) = w ( x ) P y ∈ A w ( y ) for all x ∈ A ⊆ X . We know from Luce (1959) that if a system of choice probabilities has aLuce representation, then it is unique. Consider the following example.
Example 2 (Luce) . Let X = { a, b, c, d } . Consider the system of choiceprobabilities induced by putting equal weight on each linear order over X . Itis easy to see that this system of choice probabilities has a Luce representationwhere w ( a ) = w ( b ) = w ( c ) = w ( d ) . In the example above, the system of choice probabilities is induced by aprobability distribution over linear orders. Therefore, the system of choiceprobabilities is rationalizable. Further, each linear order over X is givenstrictly positive weight. This means that each BM-polynomial for the inducedsystem of choice probabilities is strictly positive. As mentioned prior, when | X | ≥ .2 The Single Crossing Random Utility Model We now turn our attention to SCRUM. SCRUM puts an additional restrictionon the underlying structure of X in that X is endowed with some naturaland exogenous linear order ≻ . We use ≻ instead of π here to denote thedifference between the linear order with which X is endowed and the linearorders π that compose a rationalization of the system of choice probabilities.For a rationalization of a system of choice probabilities, we consider the setof linear orders with strictly positive weight. Definition 8 (Support) . Let S ν = { π | ν ( π ) > } be the set of linear orderswith strictly positive weight under rationalization ν . Then we call S ν thesupport of ν . We say that the system of choice probabilities is rationalizable by SCRUMif there exists some RUM rationalization of the system, ν , such that S ν canbe ordered so that it satisfies the single-crossing property with respect to ≻ .Recall that the single-crossing property is as follows. Definition 9 (Single-Crossing Property) . We say that a rationalization ν satisfies the single-crossing property if S ν can be ordered in such a way thatfor all x ≻ y , π i ( x ) > π i ( y ) implies π j ( x ) > π j ( y ) for all j ≥ i . Apesteguia, Ballester, and Lu (2017) show that a SCRUM rationalizationis unique. We return to the counterexample of Fishburn (1998) to study theimplications of this uniqueness result.
Example 3 (Fishburn and SCRUM) . Let X = { a, b, c, d } . Consider thesystem of choice probabilities induced by the following two distributions overlinear orders. ν ( π ) = ( if π ∈ { a ≻ b ≻ c ≻ d, b ≻ a ≻ d ≻ c } otherwise ν ( π ) = ( if π ∈ { a ≻ b ≻ d ≻ c, b ≻ a ≻ c ≻ d } otherwiseSuppose the natural order on X is a ≻ b ≻ c ≻ d . Then the system of choiceprobabilities is SCRUM rationalized by ν . Now suppose that the naturalorder on X is a ≻ b ≻ d ≻ c . Then the system of choice probabilities isSCRUM rationalized by ν . X is a ≻ b ≻ c ≻ d , ν satisfiesthe single-crossing property and ν does not. Similarly, when the naturalorder on X is a ≻ b ≻ d ≻ c , ν satisfies the single-crossing property and ν does not. This observation can be extended to show that if ν satisfies thesingle-crossing property then ν necessarily does not. The same can be saidreversing the roles of ν and ν . Recall that the probability flow diagram ofthe Fishburn counterexample has the two-in and two-out structure. In thestandard RUM environment, there are no restrictions put on the assignmentbetween inflows and outflows of branching paths. However, the addition ofthe single-crossing property in SCRUM necessarily pins down the assignmentof inflows to outflows. This is exactly how SCRUM recovers uniqueness. In this paper we have given conditions that characterize when a stochas-tic choice data set is induced by a unique distribution of preferences. Ourfirst characterization gives conditions on the underlying graphical structureof the choice data. This characterization shows that non-uniqueness ariseswhen multiple inflows can be assigned to multiple outflows. Our second char-acterization provides a finite test that can be performed on a distribution ofpreferences.When we compare our uniqueness results to the uniqueness results of twoother random utility models, we find that these other models restore unique-ness in two ways. The first model restores uniqueness by not defining anexplicit map between the model representation and a probability distributionover linear orders. The second model restores uniqueness by putting restric-tions on the flow assignment problem described in our Theorem 1. Whilenot explored here, a third method of restoring uniqueness is by restrictingthe set of preferences which are randomized over. This method is used inthe random expected utility model of Gul and Pesendorfer (2006). Theirmodel restricts randomization to the set of expected utility functions insteadof allowing for randomization over the set of weak orders over lotteries.We see two potential extensions of our paper. Recall that our results relyon the assumption that we observe stochastic choice data on each nonemptysubset of our choice environment. While discussing our results, we lay out15 clear path for extending them to the case where we observe data on asmaller collection of sets. The second extension we propose is the extensionof our uniqueness results to an infinite choice environment. This extensionhas possible implications for models which allow for random preferences overlotteries.When looking at stochastic choice data, identification of the underlyingdistribution of preferences is important as it allows decision theorists to givebehavioral meaning to the various aspects of their model. Further, it is bestto not put meaningful restrictions on choice behavior when we are able torecover the distribution of preferences without such restrictions.
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A Omitted Proofs
A.1 Preliminary Definitions
Definition 10 (In-branching) . We call two paths ρ and ρ ′ in-branching ifthere exists some i ∈ { , . . . , | X | − } such that A ρi = A ρ ′ i and A ρi − = A ρ ′ i − Definition 11 (Out-branching) . We call two paths ρ and ρ ′ out-branchingif there exists some i ∈ { , . . . , | X | − } such that A ρi = A ρ ′ i and A ρi +1 = A ρ ′ i +1 Definition 12 (Branching Section) . We call a collection of sets, { A i , . . . , A j } , a branching section of paths ρ and ρ ′ if A ρi − = A ρ ′ i − , A ρj +1 = A ρ ′ j +1 , and for all m ∈ { i, . . . , j } , A ρm = A ρ ′ m . .2 Proof of Theorem 1 We begin by showing necessity. We proceed by contraposition. Suppose thereare two supported paths, ρ and ρ ′ , that are branching. This means that thesetwo paths share some set of common nodes { A n , . . . , A m } such that A ρn − = A ρ ′ n − and A ρm +1 = A ρ ′ m +1 . Consider the following two paths, respectively ρ ′′ and ρ ′′′ , ( A ρ , . . . , A ρm , A ρ ′ m +1 , . . . , A ρ ′ | X | ) and ( A ρ ′ , . . . , A ρ ′ m , A ρm +1 , . . . , A ρ | X | ).Note that the node set and the edge set of ρ ∪ ρ ′ are the same as the nodeset and edge set of ρ ′′ ∪ ρ ′′′ . Let r be the minimum flow along the edge setof ρ ∪ ρ ′ . Without loss, let r be the flow of an edge that belongs to the edgeset of ρ and ρ ′′ . We will now construct two different rationalizations. Weconstruct ν as follows.1. Let ν ( π ρ ) = r .2. For all q ( · , · ) on the edge set of ρ , let q ( · , · ) = q ( · , · ) − r . For all q ( · , · )not on the edge set of ρ , let q ( · , · ) = q ( · , · ).3. Initialize at i = 0.4. Let s be the smallest strictly positive q i ( · , · ). Choose some edge whichhas flow equal to s . Since inflow equals outflow (see explanation belowthe algorithms), this edge is a part of some path from X to ∅ with alledges along the path having strictly positive flow. Fix this path andcall it γ .5. Let π i denote the linear order that is bijectively associated with γ . Set ν ( π i ) = s .6. For all edges along path γ , let q i +1 ( · , · ) = q i ( · , · ) − s . For all edges notalong path γ , let q i +1 ( · , · ) = q i ( · , · ).7. If there is strictly positive flow anywhere along the graph, return tostep 4. If not, terminate the algorithm.We construct ν as follows.1. Let ν ( π ρ ′′ ) = r .2. For all q ( · , · ) on the edge set of ρ ′′ , let q ( · , · ) = q ( · , · ) − r . For all q ( · , · )not on the edge set of ρ ′′ , let q ( · , · ) = q ( · , · ).18. Initialize at i = 0.4. Let s be the smallest strictly positive q i ( · , · ). Choose some edge whichhas flow equal to s . Since inflow equals outflow, this edge is a part ofsome path from X to ∅ with all edges along the path having strictlypositive flow. Fix this path and call it γ .5. Let π i denote the linear order that is bijectively associated with γ . Set ν ( π i ) = s .6. For all edges along path γ , let q i +1 ( · , · ) = q i ( · , · ) − s . For all edges notalong path γ , let q i +1 ( · , · ) = q i ( · , · ).7. If there is strictly positive flow anywhere along the graph, return tostep 4. If not, terminate the algorithm.Note that we know from Fiorini (2004) that we have inflow equals outflowon this graph at the start of each of these algorithms. Since each iterationof the algorithm subtracts out a fixed amount from each edge of a givenpath, we have inflow equals outflow at each stage of this algorithm. Thismeans that this algorithm terminates with zero flow along the graph. To seethis, suppose not. Then there is positive flow somewhere along the graphat termination. Since we have inflow equals outflow, we can follow thispositive flow all the way to the nodes X and ∅ . This then shows that thereis some path with strictly positive flow, thus contradicting termination ofour algorithm. Further, this algorithm assigns q ( x, A ) to orders that rank x exactly at the top of A . Thus, we know from Falmagne (1978) that ν and ν rationalize the system of choice probabilities. Now note that sincethere is an edge that is shared between ρ and ρ ′′ which has flow equal to r , ν puts zero weight on π ρ ′′ while ν puts weight equal to r on π ρ ′′ . Thusthese two rationalizations are different, meaning that there is not a uniquerationalization. By contraposition, we have proven necessity.Now we prove sufficiency. Suppose no two supported paths are branching. Claim 1.
Let ρ and ρ ′ be supported out-branching paths. Let i ∈ { , . . . , | X |− } be such that A ρi = A ρ ′ i and A ρi +1 = A ρ ′ i +1 . Then for all j ≤ i , no supportedpath may be in-branching at A j for either ρ or ρ ′ .Proof. Suppose not. Then, without loss of generality, there exists some sup-ported path ρ ′′ such that ρ ′′ and ρ are in-branching and there exists j ≤ i A ρj = A ρ ′′ j and A ρj − = A ρ ′′ j − . Construct supported path ρ ′′′ asfollows. ρ ′′′ = ( A ρ ′′ , . . . , A ρ ′′ j , A ρj +1 , . . . , A ρi , A ρ ′ i +1 , . . . , A ρ ′ | X | )By construction, ρ and ρ ′′′ are supported paths which are branching. This isa contradiction. Thus our claim is proven. Claim 2.
Let ρ and ρ ′ be supported in-branching paths. Let i ∈ { , . . . , | X | − } be such that A ρi = A ρ ′ i and A ρi − = A ρ ′ i − . Then for all j ≥ i , no supportedpath may be out-branching at A j for either ρ or ρ ′ .Proof. Claim 2 is an alternative statement of Claim 1.Together, Claim 1 and Claim 2 state that for every supported path ρ thereexists some i ∈ { , . . . , | X | − } , such that A i is in ρ , with all supported out-branching paths doing so at or above A i and with all supported in-branchingpaths doing so strictly below A i . This means that the edge associated with q ( A i \ A i +1 , A i ) belongs to no supported path other than ρ . We know fromFalmagne (1978) that any rationalizing ν must put probability weight on theset of orders ranking A i \ A i +1 exactly at the top of A i equal to q ( A i \ A i +1 , A i ).Since ρ is the unique supported path that contains q ( A i \ A i +1 , A i ), it must bethe case that the order π associated with ρ must have ν ( π ) = q ( A i \ A i +1 , A i ).This can be said about all such orders. Thus no pair of supported paths beingbranching implies that the rationalizing representation must be unique. Thuswe have proven our theorem. A.3 Proof of Theorem 2
We begin by proving necessity of the conditions on ν . We proceed by con-traposition. Let ν put strictly positive probability weight on two orders π and π ′ satisfying condition 2 of the theorem. This means that there exist x, y ∈ X such that x = y and U π ( x ) = U π ′ ( y ). Let A x ∈ ρ be such thatmax( π, A x ) = x . Define A y similarly. Let i be such that A i − = A x . By U π ( x ) = U π ′ ( y ) and construction of the probability flow diagram, A i = A i .By U π ( z ) = U π ′ ( z ), it must be that A z = A z . This implies there is some nodeafter A i at which ρ and ρ ′ out-branch. Call this node A j . Thus A i − = A i − , A j +1 = A j +1 , and for all m ∈ { i, . . . , j } , A m = A m . This means that ρ and ρ ′ are a pair of supported branching paths. Thus ν is not unique by Theorem1, and by contraposition the conditions on ν are necessary.20e now show the sufficiency of the conditions on ν . We proceed bycontraposition. Suppose that ν is not unique. Then, by Theorem 1, there is apair of supported branching paths on the probability flow diagram of ( X, P ).Recall the definition of branching path. With this definition in mind, we callthe length of a branching section j − i . Let l be the minimum length of allbranching sections of all pairs of supported branching paths. Note that l iswell defined because X is finite. Choose a pair of supported paths ρ and ρ ′ such that ρ and ρ ′ have a branching section of length l . Let { A i , . . . , A j } be that branching section. Because l is the minimal length of supportedbranching sections, there is no k ∈ { i + 1 , j − } such that { A i , . . . , A k } or { A k , . . . , A j } are supported branching sections. We know from Fiorini (2004)and Falmagne (1978) that the probability flow diagram satisfies inflow equalsoutflow. Since there is no supported out-branching path in { A i , . . . , A j − } ,it must be the case that inflow into A i equals outflow from A j .Let M x,A be the set of linear orders on X that rank x exactly at the topof A . M x,A = { π | π ( X \ A ) > π ( x ) > π ( A \ x ) } We know from Falmagne (1978) that q ( x, A ) = ν ( M x,a ) for all rationalizing ν . Since ρ and ρ ′ are supported, the total outflow from A j is strictly greaterthan the inflow into A i from the edge belonging to ρ . This means that ν cannot assign weight onto orders in M A ρi − \ A i ,A ρi − equal to the total outflowfrom A j . Claim 3.
There are two orders, π and π ′ , satisfying the following.1. π M A ρi − \ A i ,A ρi − π ′ ∈ M A ρi − \ A i ,A ρi − max( π, A j ) = max( π ′ , A j ) ν ( π ) > and ν ( π ′ ) > Proof.
Since { A i , . . . , A j } is a branching section of two supported paths,there are at least two orders whose paths path through { A i , . . . , A j } whichhave positive weight under ν . Further, there must be at least two orderswhose paths in-branch at A i and have positive weight under ν . Similarly,there must be at least two orders whose paths out-branch at A j and havepositive weight under ν . There are two cases.21. There is some supported edge leaving A j such that no order in M A ρi − \ A i ,A ρi − with positive weight under ν has a path containing thatedge.Since inflow at A i equals outflow form A j , it must be the case thatsome order not in M A ρi − \ A i ,A ρi − has positive weight and has pathcontaining the prior mentioned edge. Call this order π . Then any π ′ ∈ M A ρi − \ A i ,A ρi − with ν ( π ′ ) > π satisfy the above conditions.2. For every supported edge leaving A j , there is some order in M A ρi − \ A i ,A ρi − with positive weight under ν whose path contains thatedge.In this case, choose some order π M A ρi − \ A i ,A ρi − such that ν ( π ) > π passes through { A i , . . . , A j } . Theexistence of such an order is guaranteed by inflow equals outflow. Byinflow at A i equals outflow at A j , the path corresponding to π passesthrough A j . Choose some π ′ ∈ M A ρi − \ A i ,A ρi − such that ν ( π ′ ) > π ′ does not have the same edge leaving A j asthe path corresponding to π . The existence of such a π ′ is guaranteedby the supposition. Then π and π ′ satisfy the conditions of the claim.By the definition of ρ π and ρ π ′ , both of these paths are branching andsupported. Now let x = min( π, X \ A i ) and y = min( π ′ , X \ A i ). By π M A ρi − \ A i ,A ρi − and π ′ ∈ M A ρi − \ A i ,A ρi − , x = y . By A i ∈ ρ π and A i ∈ ρ π ′ , U π ( x ) = U π ′ ( y ). Let z = max( π, A j ). By max( π, A j ) = max( π ′ , A j ), U π ( z ) = U π ′ ( z ). By j ≥ i , π ( { x, y } ) > π ( z ) and π ′ ( { x, y } ) > π ′ ( z ). Thus theconditions on νν