Modelling the expected probability of correct assignment under uncertainty
MMODELLING THE EXPECTED PROBABILITY OF CORRECTASSIGNMENT UNDER UNCERTAINTY
TOM DVIR , , RENANA PERES *, AND ZE´EV RUDNICK Abstract.
When making important decisions such as choosing health insur-ance or a school, people are often uncertain what levels of attributes will suittheir true preference. After choice, they might realize that their uncertaintyresulted in a mismatch: choosing a sub-optimal alternative, while anotheravailable alternative better matches their needs.We study here the overall impact, from a central planner’s perspective,of decisions under such uncertainty. We use the representation of Voronoitessellations to locate all individuals and alternatives in an attribute space.We provide an expression for the probability of correct match, and calculate,analytically and numerically, the average percentage of matches. We test de-pendence on the level of uncertainty and location.We find overall considerable mismatch even for low uncertainty - a possibleconcern for policy makers. We further explore a commonly used practice -allocating service representatives to assist individuals’ decisions. We showthat within a given budget and uncertainty level, the effective allocation is forindividuals who are close to the boundary between several Voronoi cells, butare not right on the boundary.
Introduction
Important decisions people make, such as choosing health insurance, or choosinga school, require complex considerations. In many cases these considerations arefurther complicated by the uncertainty, or error of individuals in understandingwhat levels of specific attributes match their true preferences. For instance, inchoosing a school, people might find it hard to specify what a ”good” school meansto them in terms of the level of specific attributes such as the number of Mathhours, intensiveness of the music program, vocational training, geographic location,or whether the athletics program should include Quidditch (Jenkin 2015) [1]. Aftertheir children start attending the school, they might realize that they find the Math a r X i v : . [ phy s i c s . s o c - ph ] A ug MODELLING ALLOCATION MATCH UNDER UNCERTAINTY program less demanding, the music program too intensive, or that a 20 minutewalk to school is more strenuous than they expected. Thus, they would have beenhappier with a school that has slightly different values on these attributes. Suchpatterns of post choice evaluation, regret, and disappointment have been empiricallydocumented in the past literature (e.g. Westbrook 1987 [2]; Inman, Dyer and Jia1997 [3]). Representing the relevant domain in the attribute space we say that whileindividuals might claim to know where their preferences are located in the space,there is often uncertainty as to their true desired location. Only after the choice,they might realize that their perceived location doesn’t match their true needs anddesires. This uncertainty might be a result of insufficient information about themeaning of different levels of attributes for them (e.g. what parental involvement,or an intensive music program require from them), or misconception as to whatthey really want.In a market with several alternatives, such uncertainty might result in a mis-match - that is, choosing an alternative that is sub-optimal, although there areother available alternatives which better match one’s real needs. For example,while parents might be certain that they want the school with the intensive Mathprogram, they might have actually been better off in a school with a less intensiveprogram. Therefore their true preference would be in a slightly different locationin the attribute space than what they initially thought they were. The choice lit-erature indicates that mismatches happen when the choice task is complicated, orwhen individuals do not have enough previous experience with the specific choicetask (Mosteller and Nogee 1951) [4]Considerable effort is invested in reducing uncertainty to avoid mismatch inimportant decisions. Financial planners are used to consult in choosing healthplans (McClanahan 2014) [5], and advisors assist in pension plan choice (PFau2016) [6]. Residents of major cities such as New York City employ expensive privateconsultants to assist in choosing a school (Harris and Fessenden 2017) [7]. From theperspective of the central planner that provides and supervises these services, toomany mismatches are undesirable. A large group of dissatisfied service recipientsmight cause a decrease in the overall social welfare, which, in turn might lead tosocial and economic consequences. Assuming that the central planner wants tomaximize the social welfare, as a goal by itself or in order to serve political andeconomic stability, it would better to minimize mismatches.Our goal in this paper is to study the overall impact, from the perspective of thecentral planner, of decisions under the uncertainty described above (which we termhereafter as ”uncertainty in preferences”). Similar to the school choice problem(Holmes Erickson 2017 [8]; Abdulkadiroglu et al. 2020 [9]), the decision scenarioswe model apply to high involvement, multiple attribute goods and services thatare monitored by a central planner. They can be credence/experience goods andservices, with a high importance for customer satisfaction and a high chance forpost-choice evaluation and regret. While some of their attributes (such as distanceor cost) might be very directional (a rational consumer will prefer zero distanceand zero cost), many other attributes (e.g. level of religiousness, intensity of themath program, hours of French per week etc.) are a matter of personal preferenceand can greatly vary between individuals. While the general formulation of theproblem can incorporate a large number of market conditions and variables, wewish to work with a restrained set of conditions that will enable us to focus on the
ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 3 effect of uncertainty. Therefore, we focus on the case of no supply constraints, nospecific market structure, and no interactions between individuals. Our modelingframework enables expansion to include these scenarios.We use the representation of Voronoi tessellations to describe an attribute spacewith different alternatives, each having its attraction basin. Individuals can bealso located in this space, according to their preference. Each individual has aperceived location, but since individuals might not correctly estimate the attributelevels that match their needs, this perceived location might be distant from theirtrue preference, up to a certain uncertainty factor. The uncertainty creates an errorin the perceived location of the individual, and hence can place the individual inthe attraction basin of another, sub-optimal alternative, causing a mismatch.We focus on the probability of correct match - that is, when the choice madeis indeed the best alternative for this individual. We provide an expression forthe probability for correct match, and show how it depends on the location in theattribute space and on the level of uncertainty. We give a formula for the averagepercentage of matches for low uncertainty level and use numerical simulation toextend the description for larger uncertainty.We then extend our model by including a policy to help individuals obtain thecorrect decision and avoid mismatches. In some cases the central planner might offer”front-desk” services, which provide help through face-to-face or phone meetings.Such services are effective but costly. We use our model to study how the authoritycan allocate service representatives to individuals within a given budget in a waythat will maximize the overall level of match.Our contribution is by studying decisions under uncertainty in prefer-ences from the perspective of the central planner . We draw inspiration from twostreams of literature: Decisions under uncertainty, and Matching theory. Decisionsunder uncertainty have been mostly modeled from the individual’s point of view,and focused on the information search of individuals (Branco, Sun and Villas-Boas2012) [10], on how they sample the choice alternatives (Chick and Frazier 2012) [11],how they use social influence to compensate for the missing information (Lopez-Pintado and Watts 2009)[12], and how they update their preferences based on eachadditional information bit they receive (Erdem and Keane 1996) [13]. These mod-els often consider factors such as expected utility from each alternative and riskaversion (Machina 1987) [14]. In choice modeling, random utility models were usedto describe uncertainty in choice, under the assumption that some attributes areunobserved and are represented as random variables (e.g. Ben-Akiva and Lerman1985 [15]), or, alternatively, that the decision-making individual considers each timeonly a subset of the attributes (Becker, DeGroot, and Marschak 1951 [16]). Workson post-choice evaluation (e.g. Inman, Dyer and Jia 1997) [3] emphasized factorssuch as satisfaction and regret. This body of literature focuses on uncertainty inone’s understanding of the true value of the suggested alternatives, or, as in therandom utility models, that the entire attribute space is not taken into accountduring the choice. Our focus is on an attribute space and a set of alternatives thatare entirely known to the individual, and the uncertainty in one’s understanding ofhis/her own needs and wants.The implications of choice from the central planner’s perspective have mostlybeen studied without relating to uncertainty. Studies in matching theory (Galeand Shapley 1961 [17]; Roth 1986 [18]) suggest algorithms for matching between
MODELLING ALLOCATION MATCH UNDER UNCERTAINTY individuals and outlets in various scenarios (schools, houses, hospital residency (seeS¨onmez and ¨Unver (2011) [19] for review), where slots are limited, requiring oneof the sides or both, to rank their mutual preferences. Recent works on matchinghave begun to incorporate uncertainty in various forms: Ehlers and Mass (2015)[20] describe a matching game where players are not sure about the preferences ofother players. Hazon et al. (2012) [21] study forecasting voting patterns, wherethe ranking of candidates for each voter is not fully known to an outside observer.Aziz et al. (2020) [22] study the case where the individuals themselves are notcertain in their rankings, but rather rank their preferences with a probability smallerthan 1. These models are usually characterized by: (1) assuming limited capacity(otherwise all individuals get what they want); and (2) not having a direct accessto the attributes, but rather to a rank ordering of alternatives. Their focus is tofind the best matching algorithm that will create stable equilibrium.Our modeling perspective draws from both streams - similar to the matchingmodels we deal with matching alternatives to individuals, from the perspective ofa central planner. Similar to the decision-under-uncertainty problems, our modeldeals directly with the attributes and does not use ranking of alternatives. How-ever, we do not focus on the individual level, but rather look at the entire set ofalternatives and individuals. We do not assume capacity constraints since, in thepresence of uncertainty, mismatches can occur even without capacity limitations.Our focus is not empirical estimation, or efficient matching algorithm but ratherto measure the probability for correct match and its dependence on various marketfactors. To the best of our knowledge, this work is the first to suggest a measurefor the overall probability of matches, and calculate analytically its average value.Our model enables studying specific policies for minimizing the mismatch, such asthe use of service representatives.
The space of attributes and Voronoi tessellations
Our goal is to calculate the overall impact, from the perspective of the centralplanner, of decisions under uncertainty in preferences. To do so, we want to de-fine a measure for the probability of a correct match for every possible individualpreference, and then calculate its average value over a population. As explainedabove, most matching algorithms (S¨onmez and ¨Unver (2011) [19] assume limitedcapacity, and the criterion for the optimal overall match is a stable equilibriumthat is, there is no pair of individuals who would be better-off by switching thealternatives they were assigned with. Therefore, these algorithms do not provide acontinuous metric for the probability of a correct match. Individual level decisionmodels that incorporated uncertainty (Erdem and Keane 1996 [13]; Ben-Akiva andLerman 1985 [15]) were used more for empirically estimating one’s utility and rarelyprovide an overall view of all the individuals and alternatives. The representationwe seek is one that considers the entire attribute space and range of alternatives,allows representation of the alternatives as well as the individuals, provides a con-tinuous measure of the match probability as a function of uncertainty, and can beeasily expanded to incorporate cases of interventions of the central planner, changesin the alternatives, and population changes.To do so, we define a space A of attributes. Each dimension in this space isa numerical representation of a single attribute in the relevant context (e.g. levelof religiousness, level of parental involvement, geographic location of the school). ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 5
The space is a K dimensional box with boundaries, representing the range of eachattribute.In this space we place J alternatives, (such as the various schools) giving toeach alternative a point P j in this K -dimensional space.The location of an alter-native represents its performance on each of the attributes. Each alternative hasits attraction basin, and these partition the space of attributes A into a Voronoitessellation (Obake and Suzuki 1997 [23]; De Leeuw 2005 [24]).The construction divides the space of attributes into Voronoi cells, which are thebasins of attraction: D j = { x ∈ A : dist( x, P j ) ≤ dist( x, P k ) , ∀ k = 1 , . . . , J } . These are convex polyhedra, with disjoint interiors, whose union is all of A , seeFigure 1a.Individuals (say, the students, or their parents) are represented as points inthe attribute space A . The location of an individual i in this space, denoted by x , represents the true desire, or the ”ideal” product of the individual (that is, ahypothetical alternative which should maximize individual i ’s utility). It reflectsboth the desired level of attributes, as well as the importance of the attribute tothe individual. We want to match individuals to the alternative which most closelymatches their preferences. A closest match would be an alternative P j so that thedistance between the individual’s location x ∈ A is not greater than the distance toany other product, i.e., that resides within the same Voronoi cell. In utility terms,one can say that the utility derived from each actual alternative j can be representedas a function of the proximity of individual i to the location of alternative j .We assume that the location of the alternatives in space is known to the indi-viduals and is also known to the central planner. This is a reasonable assumptionsince consumers these days have wide access, through social media, customer re-views, and other online resources to the specifications of the alternatives in theirchoice set (Bronnenberg, Kim and Mela 2016[25]). Modeling uncertainty.
We add uncertainty to this representation: individuals,being sure they know what they want, locate themselves in a perceived place, whichis distant from their true location in the attribute space up to an uncertainty factor ρ . A common distinction is made in literature between uncertainty which assumesthe probability of each alternative is known, and ambiguity where the individualalso needs to assess the probability distribution from which the alternatives aredrawn (Kahn and Sarin 1988) [26]. In this paper we do not deal with ambigu-ity. We assume that all alternatives are available and their properties are known.We describe uncertainty in the desired level of attributes, that is, uncertainty inpreferences, and not in product location, product performance, or influence of un-controlled factors.Our setup has appeared in Computer Science, in the ”nearest-neighbor searchproblem”, which returns the nearest neighbor of a query point x in a set of points P in R d . Both the data (the set of points P ) and the query (the point x ) may beuncertain. For instance (see Beskales et al. 2008 [27]), in location-based services,a user may request the locations of the nearest gas stations. To protect the user’sprivacy, an area that encloses the user’s actual location may be used as the queryobject, while gas stations (the data objects P ) have deterministic locations. In MODELLING ALLOCATION MATCH UNDER UNCERTAINTY contrast to our goals, this literature focused on algorithmic and complexity aspectsof the problem, see for instance the recent paper by Agrawal (2006) [28] and thereferences there.Due to the uncertainty ρ , an individual with a true location x , has a perceivedlocation at a point around x . The perceived location is a point randomly drawnfrom a uniformly distributed ball of radius ρ around the true location x . Note thatneither the individual nor the central planner know the true location x . All theyknow is the perceived location. Even if individuals are aware of the uncertainty ρ , they can not reconstruct the drawing process. The uniformity assumption isrequired for the convenience of the formal analysis, and makes sense for a finitespace and for the general case, where we assume zero information on the preferences.Thus, rather than a point in the space of attributes, we actually have a ball B ( x, ρ ) of all points at distance at most ρ which define a possible perceived locationof the individual whose true location is x . By taking the shape of a ball, we assumethat the uncertainty is equal in all dimensions. This is a reasonable assumption fora general space, with no specific information on the dimension. However, even ifthe uncertainty is not equal in all dimensions, the uncertainty ball can be regardedas the circumscribed ball where ρ is the uncertainty in the dimension with themaximal uncertainty.Note, that while the uncertainty ball is uniform across attributes, and has asingle radius for the entire population, the random draw of the perceived locationgenerates heterogeneity across individuals: the perceived location is drawn for eachand every individual separately, and therefore the actual error, namely, the dis-tance between the perceived location and the true location varies across individualsand across attributes. The uncertainty ρ can therefore be regarded as the maximumpossible error in the perception.The point x usually lies in a unique Voronoi cell D j which gives the correctmatch, while the ball B ( x, ρ ) may intersect with some other cells.The probability P ρ ( x ) that the individual whose true location is the point x is assigned to thecorrect Voronoi cell to which it belongs (that is, the cases where the choice of theindividual is indeed optimal) is the relative area (or volume) of the ball which liesin that cell, see Figure 1b:(1) P ρ ( x ) = vol( D j ∩ B ( x, ρ ))vol B ( x, ρ ) . As illustrated in Figure 1, P ρ ( x ) strongly depends on the distribution of prod-ucts in the attribute space, on the distance from the cell boundaries, and on therelationships between ρ and the location within the Voronoi cell. The probability for correct match.
From the perspective of the central plannerwhich provides and supervises the services, a key measure of interest would be theeffect of the uncertainty in individuals’ preferences on the overall mismatch for theentire population. A key measure we calculate is the average probability of correctmatch (cid:104) P ρ (cid:105) :(2) (cid:104) P ρ (cid:105) := 1vol( A ) (cid:90) A P ρ ( x ) dx that is the average of P ρ ( x ) over the entire attribute space - all the locations x ,and all the Voronoi cells D j . We seek to describe its variation as we change the ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 7 uncertainty factor ρ . For a uniformly distributed population in the attribute space (cid:104) P ρ (cid:105) is given by:(3) (cid:104) P ρ (cid:105) = 1vol( A ) (cid:88) j (cid:90) D j vol ( B ( x, ρ ) ∩ D j )vol B ( x, ρ ) dx where B ( x, ρ ) is the ball around x of radius ρ ,and (cid:82) D j dx means integration withina Voronoi cell j (see Supplementary Information Part 1 Proposition 1 for details).To provide an intuition as to how to compute this integral, recall that for agiven individual in location x , when x is distanced more than ρ to the boundary,a match will always be obtained. However, when x is closer to the boundary than ρ , the probability for a mismatch grows. As illustrated in Figure 2a, for eachcell, there is only a finite danger zone, around its boundaries, where a mismatchcan occur. The cumulative area of the danger zones of all cells depends on twofactors: 1) the size of ρ , 2) the total length of the boundaries between cells (in ageneral K dimensional space the danger zone will be the relevant volume, and thelength will be in dimension K −
1. In the one-dimensional case, where we onlyhave one attribute, and the boundary consists of isolated points, the ”length” ofthe boundary will be the number of points). For example, in Figure 2a, describinga two dimensional space, this factor will be the total length of all the internalboundary segments between cells. When ρ = 0, clearly P ≡ ρ (cid:29) ρ is small enough to disregard overlap of danger zones from differentcells, the volume of the total danger zone is approximately (to leading order) givenby ρ times the total area of the internal boundaries ( ∂ int D ). For the special caseof a single attribute (one dimensional space), the attribute space is an interval ofthe size length( A ), and the Voronoi cells are segments within this interval. Theboundaries are single points, so the total area of the boundaries is given directlyby the number of alternatives J . We give an analytic formula for (cid:104) P ρ (cid:105) for thecase of a small ρ (namely, ρ smaller or equal to half of the smallest segment (SeeSupplementary Information Part 1 Proposition 2):(4) (cid:104) P ρ (cid:105) = 1 − J − ρ length A . For higher dimensions K ≥
2, we compute the match probability for the firstvariation of (cid:104) P ρ (cid:105) , that is for the slope at ρ = 0, which is the v in the expansion (cid:104) P ρ (cid:105) ∼ − vρ . The notation f ( ρ ) ∼ g ( ρ ) as ρ → ρ → f ( ρ ) /g ( ρ ) = 1.In dimension K ≥
2, the mean probability for correct assignment (cid:104) P ρ (cid:105) for ρ small is(5) (cid:104) P ρ (cid:105) ∼ − c K vol A (cid:88) j vol K − ( ∂ int D j ) · ρ, ρ (cid:38) MODELLING ALLOCATION MATCH UNDER UNCERTAINTY where(6) c K = 12 Γ (cid:0) K + 1 (cid:1) √ π Γ (cid:0) K +32 (cid:1) = π m ( m +1) ( m +1 m ) , K = 2 m even m +2 (cid:0) m +1 m (cid:1) , K = 2 m + 1 odd . Here, Γ is the Gamma function, thus c = , c = π , c = , etc. SeeSupplementary Information Part 1 Proposition 3 for the proof. When ρ is large,we can no longer disregard the overlap of the different danger zones, and we relyon numerical calculation of equation (3).Note that in the special case of ( K = 1), Eq (5) reduces to Eq (4) as (cid:104) P ρ (cid:105) =1 − ( (cid:80) j ∂ int D j ) · ρ , once we note that c K = . The boundary of an interiorinterval consists of 2 points, so that ∂ int D j = 2 for the J − ∂ int D j = 1 for the two intervals at the boundary of the space - j = 1, and j = J .Eq (5) reveals the dominance of the cell boundaries on the match probability.It predicts that as ρ increases, (cid:104) P ρ (cid:105) decreases linearly, with a slope that dependsstrongly on the length of the boundaries between the different Voronoi cells.To extend the above analysis for the all values of ρ we numerically calculate (cid:104) P ρ (cid:105) for the two dimensional case. We represent the market as a two dimensionalgrid, with 6 alternatives located as shown in Figure 1 (the results are robust acrosslocation choices). We then execute three steps: first we assign for each grid point thebest matched alternative. Second, we evaluate P ρ ( x ) by measuring the percentageof points having the same alternative in a sphere of radius ρ . Finally, we average P ρ ( x ) over the entire grid to obtain (cid:104) P ρ (cid:105) .Figure 2a shows P ρ ( x ) for the setting described in Figure 1, for ρ = 0.075. Whilemost of the attribute space enjoys a perfect probability for a match, near the bound-aries the probability decreases. Panel b describes (cid:104) P ρ (cid:105) as a function of ρ for thesame market configuration, comparing the small ρ approximation to numerical cal-culations. The slope of (cid:104) P ρ (cid:105) vs. ρ that is obtained from the approximation matchesprecisely the result of the numerical simulation. Both analytical and numericalcalculations show that the probability for a correct match rapidly decreases with ρ .While for the approximation, the decrease is linear, the numerical simulations showthat for large values of ρ , the decrease is attenuated, saturating at (cid:80) Jj =1 (vol D j ) .To illustrate the implications of the mismatch think of the opening example ofchoosing a school. In this setting, With ρ = 0 .
15, 20% of the population will bedissatisfied, on average, with their choice, while there is another available schoolwhich matches their needs.Note, that the matching in the above analysis is binary, that is, a mismatchhappens when not assigning an individual with the true alternative, regardless ofhow the assigned alternative is close to the individual in the attribute space (thisis an assumption in some of the literature on post-purchase evaluation e.g. Inman,Dyer and Jia (1997) [3]). In the Supplementary Information Part 2, we explore ourresults when the metric for the evaluation of the effect of uncertainty considers alsothe distance to the various alternatives.
Dependence on the number and distribution of alternatives.
The results shown inFigure 2 provide an example for a specific configuration of six products. To assessthe generalizability of this example we examined the effect of the number and
ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 9 distribution of the alternatives on the match probability. The slope of (cid:104) P ρ (cid:105) where ρ = 0 serves as a useful metric, since it can be calculated directly from the lengthof boundaries. Higher slope indicates a stronger effect of the uncertainty on thematch probability. Increasing the number of alternatives increases the slope - whenmore alternatives are available, the probability for a correct match decreases (seeFigure 3a). This might seem counter-intuitive, as one would expect that morealternatives to choose from imply greater overall possibilities for a match. However,at the same time, more options mean more probability for a mismatch - as anindividual is surrounded by more alternatives, he is less likely to choose the optimalone . In our terminology, we say that the ball of uncertainty intercepts with alarger number of Voronoi cells. Note, that there is a body of literature on therelationship between the number of alternatives during choice process, and thelevel of satisfaction and regret. Having more choice alternatives to choose fromoften increases the difficulty of the task and reduces satisfaction (e.g. Schwartz2003 [29]; Haynes 2009 [30]).We further use the numerical simulations to explore how the distribution of thealternatives in the attribute space affects the match probability. Assume the loca-tion of the alternatives is drawn from a trimmed Gaussian distribution with width σ . Figure 3b presents the slope − d (cid:104) P ρ (cid:105) /dρ (cid:12)(cid:12)(cid:12) ρ =0 vs. σ for a market with 6 alter-natives. Increasing the width of the distribution increases the slope, thus reducingthe probability for a correct match. The limiting case of uniform distribution hasthe lowest probability for a match (see Figure 3b). The intuition behind this is thatthe more dense the alternatives are, they are more similar to each other, meaningthat the effective number of real alternatives is small, which, as illustrated in panela, implies a higher match probability. Allocating service representatives.
The results described above indicate thatunder uncertainty in preferences, mismatches are very likely to occur and can affecta considerable portion of the population, which creates a challenge for the centralplanner. As explained above, the authorities often employ service representatives(reps, hereafter), which assist individuals in understanding their true needs throughpersonal meetings. Thus, the central planner wishes to improve (cid:104) P ρ (cid:105) by introduc-ing meetings with reps, which once having met with an individual, improve theindividual’s uncertainty from ρ to a lower value ρ l < ρ . Same as with the originaluncertainty ball, our formulation practically allows heterogeneity in the amount ofimprovement: after the meeting with the service rep, a new perceived location isdrawn, within a smaller radius ρ l . The actual amount of improvement will naturallyvary for each individual and each dimension.Due to budget constraints these reps meet only a fraction b of the total populationof individuals. We therefore ask who are the individuals which, within a givenbudget, should receive assistance from a rep in a way that will maximize the numberof individuals who find their best matching alternative.When the reps are allocated randomly, the new expected probability of correctassignment is (1 − b ) (cid:104) P ρ (cid:105) + b (cid:104) P ρ l (cid:105) Therefore, if we fix ρ l and ρ , and assuming that reps are randomly assigned to thepopulation, increasing the proportion b of reps results in a linear increase of theexpected probability of correct assignment. We now check whether the central planner can improve the effectiveness of thereps by assigning them to specific individuals. To find the optimal assignmentof service reps we define the local increase in match probability obtained fromassigning a service rep to location x to be ∆( x, ρ, ρ l ) = P ρ l ( x ) − P ρ ( x ). Next, wechoose bN grid points, where N is the total number of points on the grid, that havethe maximal value of ∆( x, ρ, ρ l ), and reduce the uncertainty at these points to be ρ l .Finally, to calculate the improvement in the match probability obtained from thisprocess, we average P ρ ( x ) over the entire grid. We note that this optimal allocationscheme uses the true location x of each individual, since we want to find the optimalallocation and spot the individuals who will have the maximum benefit from theservice reps. In practice, as we stated above, x is not known to the central planner,and thus, the central planner’s implementation will be approximate, having its ownerror. We do not deal with such implementation error, but rather find the allocationwhich sets an upper limit to the benefit of the use of service reps.Figure 4 describes the overall improvement in (cid:104) P ρ (cid:105) for various budget values b , where a budget is measured as the overall proportion of available rep meetingsfor the entire population. Panel a illustrates the areas which found to be optimalfor receiving a meeting with the rep, within a budget b = 0 .
2, for ρ l = 0 .
05 and ρ = 0 .
3. We see that the places for optimal allocation (in blue), are those that areclose to the boundaries between the Voronoi cells (white), but are not directly onthe boundaries. When the distance from the boundary is smaller than ρ l , meetinga rep will not significantly increase (cid:104) P ρ (cid:105) . Panel b presents (cid:104) P ρ (cid:105) as a function of thebudget b . While with random allocation, the improvement is linear with the budget,with the optimal allocation the curve shows a diminishing return and saturationat b ≈ .
7, meaning that the gain from allocating a service rep decreases as thenumber of allocated reps increases.To further demonstrate the effectiveness of service reps, we compare two waysto increase the match probability: the first is allocation reps as discussed, and thesecond is reducing the overall uncertainty of the population through means such aseducational or citizen involvement programs. Panel c shows, for each budget, whatis the uncertainty ρ that is equivalent to b percentage of the population meetingswith reps. A budget that allows meeting reps for 20% of the population increases (cid:104) P ρ (cid:105) from 0.8 to 0.88, which is equivalent to reducing ρ for the entire populationfrom 0.3 to 0.18. While in practice such a change in the entire population mightrequire long term educational and citizen involvement programs, the same resultcould be obtained by providing a relatively simple, easy to operate, front-deskservice to a pre-targeted population. Discussion
This paper deals with the overall impact of decisions, when individuals choosebetween alternatives, but have uncertainty as to the level of attributes that matchtheir preferences.We add to previous literature by suggesting a continuous measure for the prob-ability of a correct match, in a modeling framework that considers the entire set ofalternatives, attributes, and individuals, and can help central planners in designingtheir policies. We describe the attribute space as a Voronoi tessellation and userigorous analysis and numerical simulations to describe the probability for correct
ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 11 match in space as a function of the uncertainty, and to calculate the average per-centage of matches. We find that the overall mismatch can be considerable even forlow levels of uncertainty, and thus can be a concern for policy makers. We furtherexplore a practice often used by central planner - allocating service representativesto help individuals obtain the correct decisions. We use numerical simulations toshow that within a given budget, the allocation is most effective for individualswhose preferences are at a certain distance from the boundaries of a Voronoi cell -not too deep in the cell, but yet not too close to a boundary.This paper suggests several avenues for future research. First, one could re-examine our assumption on a uniform distribution of the population in the attributespace. Other distributions, such as bell-shaped distribution around a central valuemight diminish the impact of uncertainty (if, for example, there are several clustersof individuals and a single alternative is placed in the middle of each cluster), oralternatively enhance it (if preferences are centered around certain values, but thealternatives are scattered in space). An additional extension could be exploringthe issue of capacity constraints - the scenario in which a mismatch could prevent another individual from being correctly matched. A third topic of interest wouldbe endogenous sources of information, beside the reps, such as word-of-mouth fromother users. Since this additional information also has uncertainty, it can hypo-thetically work in both directions and its influence on the reps allocation is nottrivial.
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Acknowledgements
T.D. is grateful to the Azrieli Foundation for Azrieli Fellowships and is supportedby a quantum science and technologies fellowship given by the Israeli council forhigher education. R.P was supported by the Israeli Science Foundation and by theKMart foundation of the Hebrew University. Z.R. was supported by an AdvancedGrant from the European Research Council under the European Union’s Horizon2020 research and innovation programme/ERC grant agreement n o Author contributions statement
T.D., R.P., and Z.R. worked jointly and contributed equally to the paper. Allauthors reviewed the manuscript.
Additional information
Competing interests
The authors declare no competing interests.
ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 13
Supplementary Figure 1.
Voronoi tessellation. a.
An ex-ample for a two dimensional square [0 , of side length 1, where6 alternatives (yellow) divide the area to distinct Voronoi cells. b. In this example, ρ = 0 .
1. The probability P ρ ( x ) that an individual x (marked by the black dot) chose the correct Voronoi cell is therelative area of the part of the ball of radius ρ around x which liesin the same Voronoi cell as x . Supplementary Figure 2.
Match probabilities. a.
The lo-cal probability for a match, P ρ ( x ), is plotted as a color map forthe example shown in Figure. 1. b. Average probability for amatch (cid:104) P ρ (cid:105) as a function of ρ for this configuration. Displayed isa comparison between the small ρ linear approximation and thenumerical calculation. The probability for correct match rapidlydecreases with ρ and the decrease is attenuated for large values of ρ , until saturation. Appendix A. Proof of the formula for the first variation of theexpected probability of correct assignment
We start with a space of attributes A which is a K -dimensional box, A = [ a , b ] × [ a , b ] × [ a K , b K ] with side-lengths b i − a i . We are given partition of the space of Supplementary Figure 3.
Effects of the distribution andthe number of alternatives on the probability for a matcha. − d (cid:104) P ( ρ = 0) (cid:105) /dρ vs. the number of alternatives. For eachnumber of alternatives we generated 100 market configurationssampled from a uniform distribution. For each configuration wecalculated the length of the boundaries between the resultingVoronoi cells and used equation 5 to compute d (cid:104) P ( ρ = 0) (cid:105) /dρ .We present the average value of the different configurations. Theerror bar shows the standard deviation. b. Dependence on the dis-tribution of alternatives: − d (cid:104) P ( ρ = 0) (cid:105) /dρ vs. σ , where σ is thewidth of a trimmed Gaussian distribution, from which the locationof alternatives is sampled. The simulation procedure is similar topanel (a). We vary σ for the case of 6 alternatives (panel b)attributes A into Voronoi cells with disjoint interiors: A = (cid:96) Jj =1 D j . The cellsare convex polytopes, so that the boundary of each cell is covered by finitely manyhyperplanes.We fix an uncertainty factor, a ball of radius ρ >
0, and for a point x ∈ A weask what is the probability P ρ ( x ) that we assign the correct Voronoi cell, given thisuncertainty factor ρ ? That is, given that x ∈ D j , what is the probability that weassign D j as the basin of attraction, using an error bar of ρ ? Note that the problemonly makes sense for small ρ , because once ρ is sufficiently large so that the ball B ( x, ρ ) covers all of A , say ρ > ρ max , then the question is independent of ρ .We can write a formula for the expected value (cid:104) P ρ (cid:105) of P ρ ( x ) (as we average over x ): Proposition 1. (7) (cid:104) P ρ (cid:105) = 1vol A (cid:88) j (cid:90) D j vol (cid:16) D j ∩ B ( x, ρ ) (cid:17) vol B ( x, ρ ) dx where B ( x, ρ ) is the ball around x of radius ρ .If ρ > ρ max , then (cid:104) P ρ (cid:105) saturates at (cid:104) P ρ (cid:105) = (cid:80) Jj =1 (vol D j ) / (vol A ) .Proof. To see (7), recall that given that x ∈ D j , the probability P ρ ( x ) that weselect D j as the basin of attraction is the relative area of the intersection of the ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 15
Supplementary Figure 4.
Improving match probabilitiesusing service reps a.
Areas which maximize the effectiveness ofservice reps (blue), within a budget b = 0 .
2, for ρ l = 0 .
025 and ρ = 0 . b. Average probability for a match (cid:104) P ρ (cid:105) as a function ofthe budget b , where ρ l = 0 .
025 and ρ = 0 .
15. The orange line showsa linear improvement when the reps are assigned randomly. Theblue dots show maximal improvement when the reps are allocatedoptimally. c. The equivalent overall ρ which results in the same (cid:104) P ρ (cid:105) as an optimal allocation of reps within a given budget b .ball of radius ρ with the cell D j : P ρ ( x ) = vol( D j ∩ B ( x,ρ ))vol B ( x,ρ ) , x ∈ D j , x / ∈ D j . We want to compute the expected value of P ρ ( x ) (we average over x ): (cid:104) P ρ (cid:105) = 1vol A (cid:88) j (cid:90) D j P ρ ( x ) dx = (cid:88) j A (cid:90) D j vol (cid:16) D j ∩ B ( x, ρ ) (cid:17) vol B ( x, ρ ) dx. To see the saturation value, take ρ to be larger than the diameter of the spaceof attributes A , so that for each point x , the ball B ( x, ρ ) coincides with all of A .Then for each x ∈ A , the intersection B ( x, ρ ) ∩ D j = D j , and we can compute (cid:104) P ρ (cid:105) simply as a conditional expectation, by writing P ρ ( x ) = J (cid:88) j =1 D j ( x ) vol D j vol A and then (cid:104) P ρ (cid:105) = (cid:82) A P ρ ( x ) dx vol A = 1vol A J (cid:88) j =1 (cid:90) A D j ( x ) vol D j vol A dx = J (cid:88) j =1 (cid:16) vol D j vol A (cid:17) as claimed. (cid:3) A.1.
The one dimensional case.
Equation (7) makes sense in any dimension,but it is only in dimension K = 1, when the space of attributes is an interval A = [0 , L ], that we know how to extract an exact expression from it for small ρ . Proposition 2.
For K = 1 , and ρ < min j ( a j +1 − a j ) , (cid:104) P ρ (cid:105) = 1 − J − ρ length A . Proof.
We use equation (7): In this one-dimensional case, the Voronoi cells areintervals D j = [ a j , a j +1 ], with 0 = a < a < · · · < a J +1 = L . We assume that2 ρ < min j length D j = min j ( a j +1 − a j ) . To compute the contribution of each cell D j = [ a j , a j +1 ], divide the region ofintegration into an interior region D int j := [ a j + ρ, a j +1 − ρ ] and two boundaryregions [ a j , a j + ρ ] and [ a j +1 − ρ, a j +1 ].For x in the interior region, we have B ( x, ρ ) ⊂ D j so that D j ∩ B ( x, ρ ) = B ( x, ρ )and hence (cid:90) D int j length (cid:16) D j ∩ B ( x, ρ ) (cid:17) length B ( x, ρ ) dx = (cid:90) D int j dx = length( D int j )= ( a j +1 − ρ ) − ( a j + ρ )= a j +1 − a j − ρ = length( D j ) − ρ. To compute the contribution of the boundary components, note that there aretwo types, corresponding if they coincide with the boundary of the interval, namely j = 1 , J + 1, or not ( j = 2 , . . . , J ).For components which do not intersect the boundary, namely if j (cid:54) = 1 , J + 1 thenfor all x ∈ [ a j , a j + ρ ] ∪ [ a j +1 − ρ, a j +1 ], the “ball” B ( x, ρ ) = [ x − ρ, x + ρ ] has length2 ρ , but B ( x, ρ ) ∩ D j = [ x − ρ, x + ρ ] ∩ [ a j , a j +1 ] = (cid:40) [ a j , x + ρ ] , a j ≤ x ≤ a j + ρ [ x − ρ, a j +1 ] , a j +1 − ρ ≤ x ≤ a j +1 (see Supplementary Figure 5) andlength B ( x, ρ ) ∩ D j = (cid:40) x + ρ − a j , a j ≤ x ≤ a j + ρa j +1 + ρ − x, a j +1 − ρ ≤ x ≤ a j +1ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 17 Supplementary Figure 5.
The overlap of the “balls” B ( x, ρ ) =[ x − ρ, x + ρ ] with the Voronoi intervals D j = [ a j , a j +1 ].so that (cid:90) a j + ρa j length (cid:16) D j ∩ B ( x, ρ ) (cid:17) length B ( x, ρ ) dx = (cid:90) a j + ρa j x + ρ − a j ρ dx = 34 ρ and (cid:90) a j +1 a j +1 − ρ length (cid:16) D j ∩ B ( x, ρ ) (cid:17) length B ( x, ρ ) dx = (cid:90) a j +1 a j +1 − ρ a j +1 + ρ − x ρ dx = 34 ρ. Altogether, we obtain for j (cid:54) = 1 , J that (cid:90) D j p ρ ( D j )( x ) dx = length( D j ) − ρ + 34 ρ + 34 ρ = length D j − ρ. For components which do intersect the boundary, that is for D = [0 , a ] or D J = [ a J , L ], we have B ( x, ρ ) = B ( x, ρ ) ∩ D = [0 , x + ρ ] , x ∈ D , and B ( x, ρ ) = B ( x, ρ ) ∩ D J = [ x − ρ, L ] , x ∈ D J so that length B ( x, ρ ) ∩ D j length B ( x, ρ ) = 1 , x ∈ D ∪ D J and we get a contribution of (cid:90) ρ dx = ρ = (cid:90) LL − ρ dx. Therefore, (cid:90) D j P ρ ( x ) dx = length D j − ρ + 34 ρ + ρ = length D j − ρ, j = 1 , J. Altogether we find (cid:104) P ρ (cid:105) = 1length A J (cid:88) j =1 (cid:90) D j P ρ ( x ) dx = 1length A (cid:16) length D − ρ + J − (cid:88) j =2 (cid:18) length D j − ρ (cid:19) + length D J − ρ (cid:17) = 1length A (cid:16) J (cid:88) j =1 length D j − J − ρ (cid:17) = 1 − J − ρ length A as claimed. (cid:3) A.2.
Higher dimensions K ≥ . We now pass to the higher dimensional case K ≥
2. Our goal in this section is to obtain an exact formula for the first variation of (cid:104) P ρ (cid:105) , that is for the slope at ρ = 0.For each Voronoi cell D j , we denote by ∂ int D j the part of the boundary of D j which does not lie on the boundary of the box (the space of attributes) A . Proposition 3.
In dimension K ≥ , the mean probability for correct assignment (cid:104) P ρ (cid:105) for ρ small is (cid:104) P ρ (cid:105) ∼ − c K vol A (cid:88) j vol K − ( ∂ int D j ) · ρ, ρ (cid:38) where (8) c K = 12 Γ (cid:0) K + 1 (cid:1) √ π Γ (cid:0) K +32 (cid:1) = π m ( m +1) ( m +1 m ) , K = 2 m even m +2 (cid:0) m +1 m (cid:1) , K = 2 m + 1 odd . Proof.
We start by using equation (7)(9) (cid:104) P ρ (cid:105) = 1vol( A ) (cid:88) j (cid:90) D j vol ( B ( x, ρ ) ∩ D j )vol B ( x, ρ ) dx . There are two types of points x ∈ D j : Type I, those x ∈ D j so that the ball B ( x, ρ )is entirely contained in D j , and type II are the rest (Supplementary Figure 6).Note that if x is close to the boundary of A : dist( x, ∂ A ) < ρ , but far from theinterior boundary of the cell, that is dist( x, ∂ int D j ) > ρ , then B ( x, ρ ) ⊆ D j isentirely contained in the cell, even though it is only a truncated ball (SupplementaryFigure 7). This means that these points are type I. Thus type II points are preciselythose x ∈ D j so that dist( x, ∂ int D j ) < ρ .For type I points, we have B ( x, ρ ) ∩ D j = B ( x, ρ ) so that the quotient of volumesequals unity: vol ( B ( x, ρ ) ∩ D j )vol B ( x, ρ ) = 1 , x of type I . Thus the type I points contribute(10) 1vol( A ) (cid:88) j (cid:90) x ∈ D j x type I vol ( B ( x, ρ ) ∩ D j )vol B ( x, ρ ) dx = (cid:88) j vol { x ∈ D j type I } vol( A ) . ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 19
Supplementary Figure 6.
Type I region, type II region(shaded) and the excised points near the boundary (shaded regionwith stripes).The type II points are contained in a “strip” around the interior boundary of“width” 2 ρ . We excise the contribution of points which are also ρ -close to ∂ A orto more than one interior face (Supplementary Figure 6). The volume of thesepoints is bounded by O ( ρ ), since they are at distance ≤ ρ from the intersec-tion of two faces or the intersection of a face with ∂ A , which has codimension 2.Since vol ( B ( x, ρ ) ∩ D j ) / vol B ( x, ρ ) ≤ O ( ρ ), which is negligible. Thus we need only consider points x withdist( x, ∂ int D j ) < ρ and in addition that B ( x, ρ ) is an actual Euclidean ball, not atruncated one. Proposition 4.
For ρ sufficiently small, the contribution of type II points is (cid:88) j vol( x ∈ D j type II)vol( A ) − c K vol K − ∂ int D j vol( A ) ρ + O ( ρ ) . Putting together equation (10) and Proposition 4 gives Proposition 3. (cid:3)
A.3.
Proof of Proposition 4.
Fix a component H of the interior boundary ∂ int D j ; H is a hyperplane. After rotation, reflection and translation of the dia-gram, we may assume that the boundary component H is the coordinate hyperplane H = { x = ( x , . . . x K ) : x K = 0 } , and that the cell D j lies in the top half-space H + = { ( y , . . . , y K ) : y K ≥ } (Supplementary Figure 7). Then for every x ∈ H + ,we have dist( x, H ) = x K and we assume that 0 ≤ x K ≤ ρ . We need to compute (cid:90) x :( x ,...,x K − , ∈ H vol ( B ( x, ρ ) ∩ D j )vol B ( x, ρ ) dx . . . dx K = 1vol B (0 , ρ ) (cid:90) ≤ x K ≤ ρ ( x ,...,x K − , ∈ H vol ( B ( x, ρ ) ∩ D j ) dx . . . dx K . Supplementary Figure 7.
A truncated ballWe fix the first K − x = ( x , . . . , x K − ), and compute the integralover x K : Lemma 5.
Fix ( x , . . . , x K − ) so that ( x , . . . , x K − , ∈ H . Then B (0 , ρ ) (cid:90) ρx K =0 vol (cid:16) B (cid:16) ( x , . . . , x K − , x K ) , ρ (cid:17) ∩ H + (cid:17) dx K = 1 − c K · ρ + O ( ρ ) . Proof.
Since the integral is independent of the first K − x , . . . , x K − ) = (0 , . . . , (cid:90) ρx K =0 vol (cid:16) B (cid:16) (0 , . . . , , x K ) , ρ (cid:17) ∩ H + (cid:17) dx K . The set B (cid:16) (0 , . . . , , x K ) , ρ (cid:17) ∩ H + is the bigger half of the ball B (cid:16) (0 , . . . , , x K ) , ρ (cid:17) (see Supplementary Figure 7); we find it easier to compute the integral over thecomplementary, smaller half, which is a spherical cap (Supplementary Figure 8),and this in turns equals Supplementary Figure 8.
A spherical cap (cid:90) ρx K =0 vol (cid:16) B (cid:16) (0 , . . . , , x K ) , ρ (cid:17) ∩ H + (cid:17) dx K = vol B (0 , ρ ) − (cid:90) ρx K =0 vol (cid:16) B (cid:16) (0 , . . . , , − x K ) , ρ (cid:17) ∩ H + (cid:17) dx K . Dividing by vol B (0 , ρ ) gives1 − B (0 , ρ ) (cid:90) ρ A ( x K ) dx K ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 21 where A ( x K ) is the volume of the small spherical cap A ( x K ) := vol (cid:16) B (cid:16) (0 , . . . , , − x K ) , ρ (cid:17) ∩ H + (cid:17) = vol { ( (cid:126)y, z ) : (cid:126)y ∈ R K − , z ≥ , | (cid:126)y | + ( z + x K ) ≤ ρ } = (cid:90) (cid:126)y ∈ R K − | (cid:126)y | ≤ ρ − x K (cid:90) − x K + √ ρ −| (cid:126)y | z =0 dzd(cid:126)y = (cid:90) | (cid:126)y | ≤ ρ − x K (cid:16)(cid:112) ρ − | (cid:126)y | − x K (cid:17) d K − (cid:126)y = (cid:90) | (cid:126)y | ≤ ρ ( | (cid:126)y | + x K ≤ ρ ) (cid:16)(cid:112) ρ − | (cid:126)y | − x K (cid:17) d K − (cid:126)y. Now integrate over x K ∈ [0 , ρ ]: Switching order of integration gives (cid:90) ρ A ( x K ) dx K = (cid:90) ρ (cid:90) | (cid:126)y | ≤ ρ ( | (cid:126)y | + x K ≤ ρ ) (cid:16)(cid:112) ρ − | (cid:126)y | − x K (cid:17) d K − (cid:126)ydx K = (cid:90) | (cid:126)y | ≤ ρ (cid:90) ρx K =0 (cid:16)(cid:112) ρ − | (cid:126)y | − x K (cid:17) ( | (cid:126)y | + x K ≤ ρ ) dx K d K − (cid:126)y = (cid:90) | (cid:126)y | ≤ ρ (cid:90) √ ρ −| (cid:126)y | x K =0 (cid:16)(cid:112) ρ − | (cid:126)y | − x K (cid:17) dx K d K − (cid:126)y = (cid:90) | (cid:126)y | ≤ ρ ( ρ − | (cid:126)y | ) d K − (cid:126)y − (cid:90) | (cid:126)y | ≤ ρ
12 ( ρ − | (cid:126)y | ) d K − (cid:126)y = 12 (cid:90) | (cid:126)y | ≤ ρ ( ρ − | (cid:126)y | ) d K − (cid:126)y = ρ K +1 (cid:90) | (cid:126)y |≤ (1 − | (cid:126)y | ) d K − (cid:126)y. When K = 2, this equals ρ (cid:90) y = − (1 − y ) dy = 23 ρ and dividing by the area of B (0 , ρ ) = πρ gives1area B (0 , ρ ) (cid:90) ρ A ( x ) dx = 23 π ρ = c ρ. For K ≥
3, we will use polar coordinates: In G ≥ G = K − G = K ) x j = r cos( θ j ) j − (cid:89) k =1 sin θ k , j = 1 , . . . , G − , x G = r G − (cid:89) k =1 sin θ k with r ≥
0, 0 ≤ θ j ≤ π for j = 1 , . . . , G − ≤ θ G − ≤ π . The Jacobian ofthis transformation is J G ( r, θ ) = r G − G − (cid:89) j =1 (sin θ j ) G − − j . The volume of the ball B (0 , ρ ) ⊂ R K in dimension G = K is thusvol K B (0 , ρ ) = (cid:90) ρr =0 r K − dr K − (cid:89) j =1 (cid:90) πθ j =0 (sin θ j ) K − − j dθ j (cid:90) πθ K − =0 dθ K − = ρ K πK (cid:90) πθ =0 (sin θ ) K − dθ · K − (cid:89) j =2 (cid:90) π (sin θ j ) K − − j dθ j = 2 πK (cid:90) πθ =0 (sin θ ) K − dθ · K − (cid:89) i =1 (cid:90) π (sin θ j ) K − − i dθ i · ρ K = 2 πK √ π Γ (cid:0) K − (cid:1) Γ (cid:0) K (cid:1) · K − (cid:89) i =1 (cid:90) π (sin θ j ) K − − i dθ i · ρ K . Using polar coordinates in R K − , K ≥ G = K − (cid:90) | (cid:126)y |≤ (1 − | (cid:126)y | ) d K − (cid:126)y = (cid:90) (1 − r ) r K − dr K − (cid:89) j =1 (cid:90) π (sin θ j ) K − − j dθ j (cid:90) π dθ K − = 2 K − π K − (cid:89) j =1 (cid:90) π (sin θ j ) K − − j dθ j so that (cid:90) ρ A ( x K ) dx K = 1 K − π K − (cid:89) j =1 (cid:90) π (sin θ j ) K − − j dθ j · ρ K +1 . Dividing we find that1vol B (0 , ρ ) (cid:90) ρ A ( x K ) dx K = KK − (cid:0) K (cid:1) √ π Γ (cid:0) K − (cid:1) · ρ = Γ( K + 1)2 √ π Γ( K +32 ) · ρ which equals c K ρ . (cid:3) We can now complete the proof of Proposition 4: Until now, we have fixed the co-ordinates ( x , . . . , x K − ), where the particular face of the cell is ( x , . . . , x K − , ∈ H ∩ D j ; integrating over these coordinates, we obtain the ( K − O ( ρ ) , and summing over all interior faces ofthe cell D j and then over the various cells, we obtain1vol A (cid:88) j (cid:16) vol( x ∈ D j : of type II) − vol K − ( ∂ int D j ) c K ρ (cid:17) + O ( ρ )as asserted by Proposition 4. (cid:3) Appendix B. Distance Based Matching Metric
We explore the sensitivity of our results in Figure 2 of the main text to a matchingmeasure which is based on distance rather than a matching/non-matching binaryclassification. One can say, that a binary match/non-match classification doesnot provide information as to how much the chosen alternative is worse than theoptimal one, and thus it makes it harder to evaluate the overall dissatisfaction inthe population. A metric which measures the average distance between the possible
ODELLING ALLOCATION MATCH UNDER UNCERTAINTY 23 chosen alternatives and the true location, might provide additional information asto the level of satisfaction of the individual from the chosen alternative.To construct such metric, consider an individual at location x , within a singleVoronoi cell D j . If there is no uncertainty in the perceived location of that individ-ual ( ρ = 0),the individual is assigned to alternative j , at a distance d ( ρ = 0 , x ) = | x − j | to the alternative. If the individual mistakenly perceives his position as y , leading to choosing another alternative, i , then the distance between the truelocation and the chosen alternative is | x − i | . Assuming a uniformly distributederror ball of radius ρ around x , we obtain the average distance between the chosenalternative and the true position as:(11) d ( ρ, x ) = (cid:88) j vol (cid:16) D j ∩ B ( x, ρ ) (cid:17) vol B ( x, ρ ) | x − j | Supplementary Figure 9a shows the effect of uncertainty on the average distanceto the chosen alternative: d ( ρ, x ) − d ( ρ = 0 , x ). By integrating over the attributespace we obtain the average distance between all of the individuals and their chosenalternatives. To measure the elasticity of the overall match on the error ρ , we dividethe above average by the average distance obtained for ρ = 0:(12) (cid:104) d ( ρ ) (cid:105) = (cid:82) d ( ρ, x ) dx (cid:82) d ( ρ = 0 , x ) dx The metric (cid:104) d ( ρ ) (cid:105) represents the average incremental distance between the truelocation and all the possible chosen alternatives within the error ball, relative tothe no error case. The larger is its deviation from 1, the higher is the distancebetween the individuals and their chosen alternatives.Supplementary Figure 9a visualizes the effect of uncertainty for the case of theattribute space shown in Figures 1 and 2 of the main text. We plot d ( ρ, x ) − d ( ρ = 0 , x ) at each point of the attribute space. Just as in the case of the binarymetric, most of the effect of the uncertainty lies within a strip of radius ρ aroundthe boundaries. However, unlike the binary metric, where the boundaries are themost sensitive to the occurrence of a mismatch, for the distance-based metric, theboundaries are the regions which are the least sensitive to a mismatch, as thedistance to the alternatives on both sides of the boundary is of similar magnitude.Supplementary Figure 9b shows 1 / (cid:104) d ( ρ ) (cid:105) vs. ρ for the same attribute space.The value of 1 / (cid:104) d ( ρ ) (cid:105) decreases with ρ . Note, that unlike the linear decrease inthe binary metric, this decrease, for low values of ρ , can be fitted by a parabola (cid:104) d ( ρ ) (cid:105) ∝ ρ . The effect of the uncertainty is thus second order in ρ .To compare the effect of the uncertainty between the binary and the distancebased cases, consider ρ = 0 .
15. The mismatch probability in the binary case is 20%(as shown in Figure 2 of the main text), however the value of (cid:104) d ( ρ ) (cid:105) is 1.03. Thatis, although on average 20% of the population is expected to choose an alternativewhich is not optimal, the distance to their chosen alternatives is expected to increaseby 3%. Supplementary Figure 9.
Distance based matching. a.
The effect of the uncertainty on the average distance to the chosenalternative: d ( ρ, x ) − d ( ρ = 0 , x ) obtained using numerical simu-lations for the example shown in Figure 1 of the main text and ρ = 0 . b. / (cid:104) d ( ρ ) (cid:105) vs. ρρ