Re ´mi Louf

Camille Roth

Marc Barthelemy
email: marc.barthelemy@cea.fr

Scaling in Transportation Networks

Subway systems span most large cities, and railway networks most countries in the world. These networks are fundamental in the development of countries and their cities, and it is therefore crucial to understand their formation and evolution. However, if the topological properties of these networks are fairly well understood, how they relate to population and socio-economical properties remains an open question. We propose here a general coarse-grained approach, based on a cost-benefit analysis that accounts for the scaling properties of the main quantities characterizing these systems (the number of stations, the total length, and the ridership) with the substrate's population, area and wealth. More precisely, we show that the length, number of stations and ridership of subways and rail networks can be estimated knowing the area, population and wealth of the underlying region. These predictions are in good agreement with data gathered for about 140 subway systems and more than 50 railway networks in the world. We also show that train networks and subway systems can be described within the same framework, but with a fundamental difference: while the interstation distance seems to be constant and determined by the typical walking distance for subways, the interstation distance for railways scales with the number of stations.

Introduction

Almost 200 subway systems run through the largest agglomerations in the world and offer an efficient alternative to congested road networks in urban areas. Previous studies have explored the topological and geometrical static properties of these transit systems [START_REF] Benguigui | The fractal dimension of some railway networks[END_REF][START_REF] Benguigui | A fractal analysis of the public transportation system of paris[END_REF][START_REF] Derrible | Network analysis of world subway systems using updated graph theory[END_REF][START_REF] Sienkiewicz | Statistical analysis of 22 public transport networks in poland[END_REF][START_REF] Levinson | Network structure and city size[END_REF], as well as their evolution in time [START_REF] Ferber | Modeling metropolis public transport[END_REF][START_REF] Roth | A long-time limit for world subway networks[END_REF][START_REF] Leng | Evaluating the evolution of subway networks: Evidence from beijing subway network[END_REF]. However, subways are not mere geometrical structures growing in empty space: they are usually embedded in large, highly congested urban areas and it seems plausible that some properties of these systems find their origin in the interaction with the city they are in. Previous studies [START_REF] Levinson | Density and dispersion: the co-development of land use and rail in london[END_REF][START_REF] Xie | Topological evolution of surface transportation networks[END_REF] have indeed shown that the growth and properties of transportation networks are tightly linked to the characteristics of urban environment. Levinson [9] for instance, showed that rail development in London followed a logic of both 'induced supply' and 'induced demand'. In other words, while the development of rail systems within cities answers a need for transportation between different areas, this development also has an impact on the organisation of the city. Therefore, while the growth of transportation systems cannot be understood without considering the underlying city, the development of the city cannot be understood without considering the transportation networks that run through it. As a result, the subway system and the city can be thought as two systems exhibiting a symbiotic behaviour. Understanding this behaviour is crucial if we want to gain deeper insights into the growth of cities and how the mobility patterns organise themselves in urban environments.

At a different scale, railway networks answer a need for fast transportation between different urban centers, and we therefore expect their properties to be linked to the characteristics of the underlying country. A model of growth has been recently proposed [START_REF] Louf | Emergence of hierarchy in cost-driven growth of spatial networks[END_REF], and relates the existence of a given line to the economical and geographical features of the environment. An interesting question is thus to know whether subways and railway networks behave in the same way, but at different scales. In other words, we are interested to know whether subways are merely scaled down railway networks, or whether they are fundamentally different objects, following different growth mechanisms. Also, the existence of scaling between the system's output and its size is important as it suggests that very general processes are governing the growth of these networks [START_REF] Banavar | Size and form in efficient transportation networks[END_REF][START_REF] Louf | How congestion shapes cities: from mobility patterns to scaling[END_REF].

Although many studies [START_REF] Derrible | Network analysis of world subway systems using updated graph theory[END_REF][START_REF] Levinson | Network structure and city size[END_REF][START_REF] Kansky | Structure of transportation networks: relationships between network geometry and regional characteristics[END_REF] explore the interplay between regional characteristics and the structure of transportation networks, a simple picture relating the network's most basic quantities and the region's properties is still lacking. In the spirit of what has recently been done for cities [START_REF] Louf | How congestion shapes cities: from mobility patterns to scaling[END_REF] and for railway networks [START_REF] Louf | Emergence of hierarchy in cost-driven growth of spatial networks[END_REF][START_REF] Black | An iterative model for generating transportation networks[END_REF], we propose here a large-scale framework and try to understand how subways and railway networks scale with some of the substrates' most basic attributes: population, surface area and wealth. As a result, we are able to relate the total ridership, the number of stations, the length of the network to socio-economical features of the environment. We find that these relations are in good agreement with the data gathered for 138 subway systems and 58 railway networks accross the world. In particular, we show that even if the main mechanisms are the same, the fact that both systems operate at different scales is responsible for their different behaviors. We believe this should lay the foundations for more specific and involved discussions.

Results

Framework

A transportation network is at least characterized by its total number of nodes (which are here train or subway stations), its total length, and the total (yearly) ridership. On the other hand, a city (or a country in the railway case) is characterized by its area, its population and its Gross Domestic Product (GDP). Because transportation systems do not grow in empty space, but result from multiple interactions with the substrate, an important question is how network characteristics and socio-economical indicators relate to one another. Naturally, a cost-benefit analysis seems to be the appropriate theoretical framework. This approach has been developed in the context of the growth of railway networks [START_REF] Louf | Emergence of hierarchy in cost-driven growth of spatial networks[END_REF][START_REF] Black | An iterative model for generating transportation networks[END_REF], and in these studies an iterative growth was considered: at each step an edge e is built such that the cost function

Z e ~Be {C e ð1Þ
is maximum. The quantity B e is the expected benefit and C e the expected cost of edge e. In the following, we consider networks after they have been built, and we assume that they are in a 'steady-state' for which we can write a cost function of the form

Z~X e Z e ~B{C ð2Þ
where B is the total expected benefits and C the total expected costs, mainly due to maintenance (in the steady state regime). We further assume that, during this steady-state, operating costs are balanced by benefits. In other words Z&0

Indeed, because lines and stations cost money to be maintained, we expect the network to adapt to the way it is being used. Therefore we can reasonably expect that at first order the cost of operating the system is compensated by the benefits gained from its use. In the following we will apply this general framework to subway and railway networks in order to determine the behavior of various quantities with respect to population and GDP.

Subways

In the case of subways, the total benefits in the steady-state are simply connected to the total ridership R and the ticket price f over a given period of time. The costs, on the other hand, are due to the maintenance costs of the lines and stations, so that we can write (for a given period of time)

Z sub ~Rf {E L L{E S N s ð4Þ
where L is the total length of the network, E L the maintenance cost of a line per unit of length, N S the total number of stations and E S the maintenance cost of a station (for a given period time). It is usually difficult to estimate the ridership of a system given its characteristics and those of the underlying city. Due to the importance of such estimates for planning purposes, the problem of estimating the number of boardings per station given the properties of the area surrounding the stations has been the subject of numerous studies [START_REF] Matsunaka | International comparison of the relationship between urban structure and the service level of urban public transportation: A comprehensive analysis in local cities in japan, france and germany[END_REF][START_REF] Kuby | Factors influencing light-rail station boardings in the united states[END_REF]. Here we are interested in the dependence of global, average behavior of the ridership on the network and the underlying city. Very generally, we write that the number R i of people using the station i will be a function of the area C i serviced by this station-the 'coverage' [START_REF] Derrible | Network analysis of world subway systems using updated graph theory[END_REF]-and of the population density r~P A in the city

R i ~ji C i r ð5Þ
where j i is a random number of order one representing the fraction of people that are in the area serviced by the station and who use the subway. The main difficulty is in finding the expression of the coverage. It depends, a priori, on local particularities such as the accessibility of the station, and should thus vary from one station to another. We take here a simple approach and assume that on average

C i *pd 2 0 ð6Þ
where d 0 is the typical size of the attraction basin of a given station.

If we assume that it is constant, the total ridership can be written as

R~X i R i *jpd 2 0 r N s ð7Þ
where j~1 Ns P i j i is of the order of 1. We gathered the relevant data for 138 metro systems across the world (see Materials and Methods), which we cross-verified when possible with the data given by network operators. We plot the ridership R as a function of N s r on Fig. 1 (left) and observe that the data is consistent with a linear behavior. We measure a slope of 800 km 2 =year which gives an estimate for d 0

d 0 &500 m ð8Þ
We illustrate this result on Fig. 1 (right) by representing each subway stations of Paris with a circle of radius 500 m. So far, the distance d 0 appears here an intrinsic feature of user's behaviors: it is the maximal distance that an individual would walk to go to a subway station.

The average interstation distance ' 1 is another distance characteristic of the subway system. Rigorously, this distance depends on the average degree vkw of the network so that

' 1 ~2 L N s vkw
. It has however been found [START_REF] Roth | A long-time limit for world subway networks[END_REF] that for the 13 largest subway systems in the world, vkw[ 2:1,2:4 ½ , so that we can reasonably take vkw=2&1 and thus

' 1 ^L N s ð9Þ
The interstation distance depends in general on many technological and economical parameters, but we expect that for a properly designed system it will match human constraints. Indeed, if d 0 %' 1 , the network is not dense enough and in the opposite case d 0 &' 1 , the system is not economically interesting. We can thus reasonably expect that the interstation distance fluctuates slightly around an average value given by twice the typical station attraction distance d 0

d 0 ~'1 2 ~L 2 N s ð10Þ
It follows from this assumption that the interstation distance is constant and independent from the population size. In order to test our assumption, we plot on Fig. 2 (left) the total length of subway networks as a function of the number of stations. The data agrees well with a linear fit L*1:13 N S (r 2 ~0:93). We also plot on Fig. 2 (right) the normalized histogram of the inter-station length, showing that the interstation distance is indeed narrowly distributed around an average value ' 1 &1:2 km with a variance s&400 m, consistently with the value found above for d 0 &500 m. The outliers are San Francisco, whose subway system is more of a suburban rail service and Dalian, a very large chinese city whose metro system is very young and still under development.

As a result of the previous argument, we can express ' 1 in terms of the systems characteristics. Indeed, the total ridership now reads

R*jpr L 2 N s ð11Þ
If we assume to be in the steady-state Z sub &0, using the results from Eqs. [START_REF] Sienkiewicz | Statistical analysis of 22 public transport networks in poland[END_REF][START_REF] Louf | Emergence of hierarchy in cost-driven growth of spatial networks[END_REF], we find that the total length of the network and the number of stations are linked at first order in E s =E L by and that the interstation distance reads

L* 4E L pjf r z E s E L N s ð12Þ
' 1 ~4E L pjf r z E s E L ð13Þ
This relation implies that the interstation distance increases with the station maintenance cost, and decreases with increasing line maintenance costs, density and fare. We thus see that the adjustment of ' 1 to match 2 d 0 can be made through the fare price (or subsidies by the local authorities or national government).

At this point, it would be interesting to get reliable data about the maintenance costs and fares for subway systems in order to pursue in this direction and to test the accuracy of this prediction.

So far, we have a relation between the total length and the number of stations, but we need another equation in order to compute their value. Intuitively, it is clear that the number of stations -or equivalently the total length -of a subway system is an increasing function of the wealth of the city. We assume a simple, linear relation of the form

N s ~b G E s ð14Þ
where G is the city's Gross Metropolitan Product (GMP), and b the fraction of the city's wealth invested in public transportation. This relation can equivalently be interpreted as the proportional relation between the number of station per person and the city's development, as measured by its GMP per capita. On Fig. 3 (left) we plot the number of stations of different metro systems around the world as a function of the Gross Metropolitan Product of the corresponding city. A linear fit agrees relatively well with the data (R 2 ~0:73, dashed line), and gives E s b &10 10 dollars=station.

However, the dispersion around the linear average behaviour is important: more specific data is needed in order to investigate whether differences in the construction costs and investments (or the age of the system) can explain the dispersion, or if other important parameters need to be taken into account. Incidentally, another possibility would be to assume that the size of the system depends on the age of the system or the development of the city (measured by the GMP per capita). However, in both cases, we found poor correlations. At this stage, we thus conclude that the number of stations (respectively the density of stations) mostly depends on the total GMP (respectively the GMP per capita). Finally, we consider the number of different lines with distinct tracks. A natural question is how the number of lines N lines scales with the number stations N s , that is to say whether lines get proportionally smaller, larger or the same with the size of the whole system. We plot the number of lines as a number of stations on Fig. 3 (right) and find that the data agree with a linear relationship between both quantities (R 2 ~0:93). In other words, the number of stations per line is distributed around a typical value of 19, whatever the size of the system.

Railway networks

We first discuss an important difference between railway and subway networks. In the subway case, the interstation distance ' 1 is such that it matches human constraints: ' 1 *2 d 0 where d 0 is the typical distance that one would walk to reach a subway station. For the railway network, the logic is however different: while subways are built to allow people to move within a dense urban environment, the purpose of building a railway is to connect different cities in a country. In addition, due to the long distance and hence high costs, it seems reasonable to assume that each city is connected to its closest neighbouring city. In this respect, the railway network appears as a planar graph connecting in an economical way, randomly distributed nodes (cities) in the plane. If we assume that a country has an area A and N s train stations, the typical distance between nearest stations is The total length L*N s ' N is then given by

' N ~ffiffiffiffiffi ffi A N s s ð15Þ
L* ffiffiffiffiffiffiffiffiffi ffi AN s p ð16Þ
In order to test this relation for different countries, we plot the adimensional quantity L ffiffiffi ffi A p as a function of the number of stations N s on Fig. 4. A power law fit gives an exponent 0:50+0:08 (R 2 ~0:87), which is consistent with the previous argument.

At this point, we have a relation between L and N s , but we need to find expressions for the other quantities. In contrast with subway systems, due to distances involved, the ticket price usually depends on the distance travelled and we denote by f L the ticket price per unit distance. The relevant quantity for benefits is therefore not the raw number of passengers -as in subways -but rather the total distance travelled on the network T. Also, again due to the long distances spanned by the network, the costs of stations can be neglected as a first approximation, and we get for the budget the following expression

Z train ^T f L {E L L ð17Þ
In the steady-state regime Z train &0, or in other words the revenue generated by the network use must be of the order of the total maintenance costs [START_REF] Louf | Emergence of hierarchy in cost-driven growth of spatial networks[END_REF], which leads to

T* E L f L L ð18Þ
In addition, if we assume that the order of magnitude of a trip is given by ' N , the total travelled length is simply proportional to the ridership T*' N R leading to

R* E L N s f L ð19Þ
We thus plot the total daily ridership R as a function of the total number of stations N s (figure 5), and despite the small number of available data points, a linear relationship between these both quantities seems to agree with empirical data on average (R 2 ~0:86). This result should be taken with caution, however, due to the important dispersion that is observed around the average behaviour, and the small number of observations.

According to the previous result, the total length and the number of stations are related to each other. We now would like to understand what property of the underlying country determines the total length of the network. That is to say, why networks are longer in some countries than in others. As in subway systems, economical reasons seem appealing. Indeed, the railway networks of some large african countries such as Nigeria are way smaller than that of countries such as France or the UK of similar surface areas. A priori, when estimating the cost of a railway network, one should take into account both the costs of building lines and the stations. However, as stated above, considering the distances involved, the cost of building a station is negligible compared to that of building the actual lines. We thus can reasonably expect to have

L* a G E L ð20Þ
where G is here the country's Gross Domestic Product (GDP) used as an indicator of the country's wealth, and av1 the ratio of the GDP invested in railway transportation. We plot L as a function of G on Fig. 6 and the data agree well (R 2 ~0:91) with a linear dependence between L and G (note that we have more points here due to the fact that the data about the total length of a railway network is easier to get). Again, the dispersion indicates that the linear trend should only be understood as an average behaviour and that local particularities can have a strong impact on the important deviations observed. For instance, the United Arab Emirates are far from the average behaviour, with a 52 km network and a GDP of roughly 3|10 5 million dollars. Yet, the construction of a 1,200 km railway network has been decided in 2010, which would bring the country closer to the average behaviour. As in the case of subways, we also tried to see whether L could better be explained by the development of the country, as measured by its GDP per capita, but we didn't find any significant correlations.

Discussion

We observed scaling relations for global properties of railways and subways and the existence of such relations suggests that basic, common mechanisms are at play during their evolution. A probable reason for the presence of these systems is the mobility demand and their structure is driven by economical mechanisms that seem to be the same for all countries, independently from any cultural, or historical considerations. The fact that macroscopic properties seem to be independent from specific details opens the possibility for simple modelling, and in this spirit, we have proposed a general framework to connect the properties of railway and subway systems (ridership, total length and number of stations) to the socio-economic and spatial characteristics (population, area, GDP) of the country or city where they are built. Despite their simplicity, our arguments agree satisfactorily with data we gathered for almost 140 subway systems and 50 railway networks accross the world. As a result, and maybe surprisingly, the knowledge of simple characteristics of a country or a city are enough to give an estimate of the size and use of its transportation system.

It should be noted that the noise associated with the data (and sometimes their definition, see Material and Methods) makes it difficult to infer behaviours from the empirical analysis alone. Therefore, the most appropriate way to proceed, we believe, is to make assumptions about the systems and build a model whose predictions can then be tested against data.

This study suggests that the fundamental difference between railways and subways comes from the determination of the interstation distance. While it is imposed by human constraints in the subway case, the railway network has to adapt to the spatial distribution of cities in a country. This remark is at the heart of the different behaviors observed for railways and subways (see Table 1 for a summary of these differences).

The previous arguments are able to explain the average behaviour of various quantities. Nevertheless, it would be interesting to identify deviations from these behaviours, and see as suggested in [START_REF] Derrible | Network analysis of world subway systems using updated graph theory[END_REF] whether they are correlated with topological properties of the system, or other properties of the network and the region. We think that the relations presented here provide however a simple framework within which local particularities can be discussed and understood. We also think that this framework could serve as a useful null-model to quantify the efficiency of individual transportation networks, and compare them to each other. This would however require more specific data than those that were available to us.

While we have focused on an average, static description of metro systems, we believe that our study provides a better understanding of how these systems interact with the region they serve. This new insight is a necessary step towards a model for the growth of subway systems that takes the characteristics of the city into account. Indeed, although models of network growth exist, the length of networks and nodes at a given time is usually imposed exogeneously, instead of being linked to the socio-economic properties of the substrate. This study provides a simple approach to these complex problems and could help in building more realistic models, with less exogeneous parameters.

It would also be interesting to gather data about the exact structure of all the networks, to study whether there is a relationship between their topology (degree distribution, detour index, etc.) and properties of the substrate, as was done for the road network in [START_REF] Levinson | Network structure and city size[END_REF].

Finally, gathering historical data should allow to address the problem of the conditions for the appearance of a subway in a city. Indeed, we observe empirically that the GDP of the cities that have a subway system is always larger than about 10 10 dollars, a fact that calls for a theoretical explanation.

Materials and Methods

Data for 138 subways accross the world were mainly collected on Wikipedia [START_REF]Data about subway length, number of stations[END_REF], and cross-referenced with the operators' data when possible. The cities' GDP per capita was retrieved for 114 cities from Brooking's Global MetroMonitor [START_REF]GMP per capita data for different cities accross the world are available[END_REF]. The choice of population and city area was more subtle. Indeed, most subway systems span an area greater than the city core, and the relevant area therefore lies somewhere between the city core's area and the total urbanized area. We chose to use the population and surface area data for urbanized areas provided by Demographia [START_REF]Surface area and population data for urbanized areas accross the world are available on the Demographia[END_REF].

While data about ridership, network length were easily retrievable for more than 100 countries from the UIC Railisa 2011 database [START_REF]The Railisa database is[END_REF], data about the number of stations were more difficult to find. We had to use various data sources, mainly scrapping the operators' ticket booking websites. Data about the GDP, population and surface areas of different countries were obtained from the World Bank [START_REF]Data about the gdp of countries were taken from the World[END_REF], and the United Nations Statistics Division [START_REF]Population and surface areas of countries were taken from the Demographic Yearbook, available on the United Nation Statistical Divisions[END_REF].

All the data used for this study are publicly available in tsv format at [24].

G N s L

We summarize the difference of behaviour between subways and railways. The scaling of the average interstation length L=N s of the network with the number of stations N s reveals the different logics behind the growth of these systems. Another difference lies in the total ridership R: while it depends also on the population density P=A for subways, it only depends on the number of stations N s for train networks. Finally, the size of both types of networks can be expressed as a function of the wealth of the region, represented here by the GDP G. However, because the interstation length is constant for subways, the size can be expressed in terms of the number of stations N s or the length. In the railway networks case, the cost of stations is negligible compared to the building cost of lines, and the size is expressed in terms of the total length L. doi:10.1371/journal.pone.0102007.t001

Figure 1 .

 1 Figure 1. (Subway) Relationship between ridership and coverage. (Left) We plot the total yearly ridership R as a function of r N s . A linear fit on the 138 data points gives R&800 rN s (R 2 ~0:76) which leads to a typical effective length of attraction d 0 &500 m per station. (Right) Map of Paris (France) with each subway station represented by a red circle of radius 500 m. doi:10.1371/journal.pone.0102007.g001

Figure 3 .

 3 Figure 3. (Subway) Size of the subway system and city's wealth. (Left) We plot the number of stations for the different subway systems in the dataset as a function of the Gross Metropolitan Product of the corresponding cities (obtained for 106 subway systems). A linear fit (dashed line) gives N s ~2:5110 {10 G (R 2 ~0:73). (Subway) Number of lines and number of stations (Right) We plot the number of metro lines N lines as a function of the number of stations N s . A linear fit on the 138 data points gives N lines &0:053N s (R 2 ~0:93), or, in other words, metro lines comprise on average 19 stations. doi:10.1371/journal.pone.0102007.g003

Figure 4 .

 4 Figure 4. (Train) Total length and number of stations. Total length of the national railway network L rescaled by the typical size of the country ffiffiffi ffi A p as a function of the number of stations N s . The dashed line shows the best power-law fit on the 50 data points with an exponent 0:50+0:08 (R 2 ~0:87). doi:10.1371/journal.pone.0102007.g004

Figure 5 .

 5 Figure 5. (Train) Ridership and number of stations. The total yearly ridership R of the railway networks as a function of the number of stations. A linear fit on the 47 data points gives R*7:010 8 N s (R 2 ~0:86) doi:10.1371/journal.pone.0102007.g005

Figure 6 .

 6 Figure 6. (Train) Total length of the network and wealth. Total length of the railway network L as a function of the country GDP G. The dashed line shows the linear fit on the 138 data points which gives E L =a&10 4 dollars:km {1 (R 2 ~0:91). doi:10.1371/journal.pone.0102007.g006

Table 1 .

 1 Summary of the differences between subways and railways.

		Subway	Train
	L=N s	cste.	s	ffiffiffiffiffi ffi A
					N s
	R	P A	N s	N s

PLOS ONE | www.plosone.org

July 2014 | Volume 9 | Issue 7 | e102007

Acknowledgments

We thank Giulia Ajmone-Marsan and Ru ¨diger Ahrend for interesting discussions at an early stage of this project.

The authors have no support or funding to report.

The authors confirm that all data underlying the findings are fully available without restriction. All data files available from the URL http:// github.com/rlouf/data/tree/master/scaling_transportation

Author Contributions

Analyzed the data: RL CR MB. Contributed to the writing of the manuscript: RL CR MB.


Arthur E Attema
email: attema@eshpm.eur.nl

Olivier L'haridon

Gijs Van De Kuilen

Measuring Multivariate Risk Preferences in the Health Domain October 2018

Keywords: multivariate risk attitudes, health, prudence, temperance

Highlights

 Multivariate risk preferences are theoretically important to explain healthrelated behavior.

 However, empirical evidence on multivariate risk preferences is very scarce for health outcomes.

 We measure higher order risk preferences for health and wealth by means of the risk apportionment technique.

 We observe strong evidence for risk aversion and prudence for health and wealth in the gain domain, but more neutrality in the loss domain.

 In the multivariate setting, we find correlation aversion and cross-prudence for gains, and correlation seeking for losses.

 We conclude that there is clear evidence that multivariate risk preferences are sign-dependent.

Introduction

Health and health care are surrounded by a lot of uncertainty, both in terms of diagnostic and treatment risk, risk of future diseases, and financial risks. As a result, it is widely recognized that risk preferences are an important determinant of health-related decisions. The recent theoretical literature has shown that many health-related behaviors in unidimensional settings should not only be governed by risk aversion, but that the higher order risk preferences prudence and temperance complement risk aversion in important ways [START_REF] Courbage | Prudence and optimal prevention for health risks[END_REF][START_REF] Felder | Risk preferences: consequences for test and treatment thresholds and optimal cutoffs[END_REF][START_REF] Pauker | Moments When Utilities Are Functional[END_REF]. 1 Intuitively, prudence governs how optimal behavior changes in the presence of a background risk, while temperance determines whether independent zero-mean risks should be aggregated or not (see Section 2 for details). For example, in in the wealth domain, agents who are prudent and temperate should save more, but should invest fewer of these savings in risky assets, than those who are neither prudent nor temperate (e.g. [START_REF] Eeckhoudt | Putting Risk in Its Proper Place[END_REF][START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF][START_REF] Kimball | Precautionary Saving in the Small and in the Large[END_REF]. In the case of health, [START_REF] Courbage | Prudence and optimal prevention for health risks[END_REF] observe that the level of prudence is a main determinant of the optimal level of prevention for health risks, while [START_REF] Felder | Risk preferences: consequences for test and treatment thresholds and optimal cutoffs[END_REF] show that prudence and temperance affect optimal medical test and treatment decisions. As a result, [START_REF] Pauker | Moments When Utilities Are Functional[END_REF] advocates that the study of higher order risk attitudes should receive priority on the research agenda of medical decision making.

Many health-related decisions are, however, not unidimensional; they are taken in the presence of other -often uncontrollable and undiversifiable -risks. For instance, 1 Under expected utility (EU) maximization, risk aversion implies that the utility function is concave, prudence implies convex marginal utility, while temperance implies a negative fourth derivate of the utility function (e.g. [START_REF] Eeckhoudt | Putting Risk in Its Proper Place[END_REF].

A C C E P T E D M

A N U S C R I P T the decision to purchase health insurance typically involves both health and wealth related risks, and the treatment of a particular disease may involve multiple independent health risks (comorbidities). Recent theoretical literature has shown that decisions in these multidimensional settings are not only driven by univariate risk preferences, but also by the multivariate risk preferences correlation aversion, crossprudence and cross-temperance by governing the interaction of univariate risk preferences for different attributes (Bleichrodt et al., 2003a;[START_REF] Dardanoni | Uncertainty and the demand for medical care[END_REF][START_REF] Ebert | Measuring multivariate risk preferences[END_REF][START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF][START_REF] Rey | Health and wealth: How do they affect individual preferences? Geneva Pap[END_REF].

Taking health status and wealth as attributes, correlation aversion implies that the individual prefers to disaggregate fixed reductions in health and wealth, suggesting that, under expected utility (EU), the marginal utility of wealth increases as individuals become less healthy. If given a choice, a correlation averse individual would thus rather be rich in the state of the world in which she is sick than in the state in which she is healthy. Consequently, [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF] show that a correlation averse individual whose health is declining over time will be inclined to save more in a simple two-period saving model. This is because the higher future wealth level allows her to better cope with the decline in health state.

Cross-prudence for health and wealth implies that the individual can better cope with the "pain" of facing a health risk when she has more wealth (and vice versa). As a result, an individual who is cross-prudent in health will spend more resources on preventive tertiary care if her future wealth risk increases [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF].

Finally, an individual who is cross-temperate for health and wealth prefers to disaggregate health and wealth related risks over different states of nature (see Section 2 for details). Despite the importance of multivariate risk preferences for health-related behaviors from a theoretical perspective, there is surprisingly little empirical evidence regarding the prevalence of multivariate risk preferences in the health domain. In this study, we report the results of an experiment designed to elicit multi-and univariate higher order risk attitudes for health (longevity) and wealth, using the risk apportionment method introduced by [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF]. The main strength of this method is that it identifies multivariate risk preferences in a model-free way, i.e., without having to assume that individuals maximize a particular preference model such as EU. This is especially important given the fact that EU has often been found to fail as an accurate descriptive model of decision making in the health domain (e.g. [START_REF] Bleichrodt | A Parameter-Free Elicitation of the Probability Weighting Function in Medical Decision Analysis[END_REF][START_REF] Bleichrodt | Making descriptive use of prospect theory to improve the prescriptive Use of expected utility[END_REF][START_REF] Loomes | The use of QALYs in health care decision making[END_REF]. Another strength of the measurement method is that is based on direct choices between two lotteries, making it well-suited for experimentation.

We make the following contributions to the literature on risk preferences in the health domain. First, we contribute to the fast-emerging literature on higher order risk preferences ( e.g. Deck andSchlesinger, 2014, 2018;[START_REF] Ebert | Measuring multivariate risk preferences[END_REF][START_REF] Ebert | Joint measurement of risk aversion, prudence, and temperance[END_REF][START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF]see Trautmann and van de Kuilen, 2018 for an overview). Only [START_REF] Krieger | Patient Preferences and Treatment Thresholds under Diagnostic Risk -An Economic Laboratory Experiment[END_REF] have elicited higher order risk attitudes in a health context, where they observed risk aversion and prudence. [START_REF] Krieger | Patient Preferences and Treatment Thresholds under Diagnostic Risk -An Economic Laboratory Experiment[END_REF] used monetary outcomes to proxy suffering from a disease. The advantage of this approach is that real incentives can be used, but the downside of the approach is that it is invalid if subjects value being ill and receiving less money differently. For this reason -and to avoid combining real incentives for one attribute with hypothetical incentives for another -we use hypothetical outcomes for wealth and longevity in this study. An advantage of this approach is that we are able to investigate

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higher order risk preferences in the health domain for significant outcomes. Although it is common practice in experimental economics to adopt a real incentive scheme, there is evidence suggesting there are no systematic differences in the results obtained with hypothetical incentives compared to real incentives for monetary outcomes (e.g., [START_REF] Abdellaoui | The rich domain of uncertainty: Source functions and their experimental implementation[END_REF][START_REF] Bardsley | Experimental Economics: Rethinking the Rules[END_REF][START_REF] Cohen | Measuring time preferences[END_REF].

Second, we provide model-free direct evidence on whether individuals exhibit a nonlinear preference toward correlation for wealth and health (longevity). Several papers have investigated attitudes toward correlation for wealth and health indirectly by estimating the interaction between the marginal utility of wealth and health status, finding mixed evidence [START_REF] Brown | Heterogeneity in State-Dependent Utility: Evidence from Strategic Surveys[END_REF][START_REF] Domeij | Consumption and Health[END_REF][START_REF] Edwards | Health risk and portfolio choice[END_REF][START_REF] Evans | Estimation of state-dependent utility functions using survey data[END_REF][START_REF] Finkelstein | WHAT GOOD IS WEALTH WITHOUT HEALTH? THE EFFECT OF HEALTH ON THE MARGINAL UTILITY OF CONSUMPTION[END_REF][START_REF] Gyrd-Hansen | A Stated Preference Approach to Assess whether Health Status Impacts on Marginal Utility of Consumption[END_REF][START_REF] Lillard | Uncertain health and survival: Effects on end-of-life consumption[END_REF]Sloan et al., 1998;Tengstam, 2014;Viscusi and Evans, 1990). Other papers have investigated attitudes toward correlation between longevity and health states, generally finding correlation seeking for chronic health states, e.g., people typically prefer to live either long without pain or short with pain rather than living long with pain or short without pain (e.g. [START_REF] Pliskin | Utility functions for life years and health status[END_REF]Wakker, 2010). We take longevity and wealth as attributes and use a model-free approach, i.e., without assuming that individual maximize a particular preference model such as EU.

Third, we provide the first empirical measurements of the prevalence of crossprudence and cross-temperance for longevity and wealth in our sample. As shown by [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF], saving behavior in the presence of health risks is determined by these risk traits. In a survey of retired individuals, [START_REF] Edwards | Health risk and portfolio choice[END_REF] found indirect evidence for cross-temperance for wealth and health by observing that the proportion of risky financial assets held by respondents' decreases with increases in perceived

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future health risk. We are the first to test this conjecture directly in a controlled environment without making the assumption that individuals maximize EU. Fourth, we investigate whether higher order risk preferences for longevity and multivariate risk preferences for longevity and wealth are sign-dependent. Previous literature has shown that attitudes towards risk depend crucially on whether outcomes are in the gain or in the loss domain (Tversky and Kahneman, 1992). Several recent papers have studied the robustness of higher order risk preferences toward framing of the outcomes [START_REF] Deck | Exploring higher order risk effects[END_REF][START_REF] Ebert | Measuring multivariate risk preferences[END_REF][START_REF] Ebert | Testing for Prudence and Skewness Seeking[END_REF][START_REF] Maier | Experimental evidence on higher order risk preferences with real monetary losses[END_REF]. The common finding of prudence and temperance for wealth appears to be robust to such framing effects. In most of these studies however, subjects are asked to apportion either improvements (gains) or reductions (losses) across equally likely states of nature. The final outcomes of the lotteries are kept constant by manipulating the initial endowment from/to which these reductions or improvements are added/subtracted. As noticed by [START_REF] Bleichrodt | Higher order risk preferences for gains and losses[END_REF], this manipulation does not affect the reference point of subjects if one uses the expected value of a lottery, the maximum of the minimum outcomes of a lottery, or the status quo as a reference point. The fact that these studies did not replicate the commonly found risk aversion for gains and risk seeking for losses for moderate probabilities, indeed suggests that these studies did not induce a shift in the reference point. [START_REF] Bleichrodt | Higher order risk preferences for gains and losses[END_REF] do observe sign-dependence for prudence preferences in a setting where outcomes are framed as reductions or improvements compared to a (salient) initial endowment. We investigate whether multivariate risk preferences toward wealth and longevity are sign-dependent by considering the apportionment of reductions and improvements of wealth and We observe pervasive univariate risk aversion and prudence for monetary gains combined with risk aversion and prudence for gains in longevity. In the loss domain, however, we observe a shift toward more neutrality for risk and prudence for outcomes in euros and outcomes in life years. More importantly, individuals are correlation averse for wealth and longevity for gains, but correlation seeking for wealth and longevity for losses, suggesting that attitudes toward correlation might be sign-dependent. Finally, in the gain domain we observe cross-prudence for wealth, which implies part of precautionary savings by households might be driven by the desire to protect oneself against future longevity risks with higher wealth. More generally, our findings provide evidence on the validity of assumptions underlying models on demand for health insurance, prevention behavior and demand for medical care.

The remainder of this paper is organized as follows. Section 2 introduces notation and describes the risk apportionment technique. Section 3 presents the experimental design and Section 4 gives the results. We end the paper with a discussion in Section 5.

Background

Univariate Risk preferences

Suppose an individual is endowed with an initial wealth 𝑊 and an initial longevity 𝐻.

We assume that the individual always prefers more wealth to less wealth and more longevity to less longevity. In what follows we assume that all variables are chosen such that final wealth and final longevity levels always remain strictly positive. Let 𝑤 1 and 𝑤 2 2 A constant initial endowment induces a change in reference point if subjects use the expected value or the maximum of the minimum outcomes of a lottery as a point of reference, as has recently been found; in particular, [START_REF] Baillon | Searching for the Reference Point[END_REF] observe that the majority of subjects use the status quo or the maximum of the minimum outcomes as their reference points in a high stakes experiment. As can be seen in Figure 1, the choice between the prospect A and B amounts to whether the individual prefers to aggregate or disaggregate the fixed wealth reductions across equally likely states of nature. If the individual prefers to disaggregate these fixed wealth reductions, she prefers prospect A and the individual is said to be risk averse for wealth.

On the other hand, if the individual prefers to aggregate these fixed wealth reductions, she prefers prospect B and the individual is said to be risk seeking for wealth. [START_REF] Eeckhoudt | Putting Risk in Its Proper Place[END_REF] show that the choice for prospect A (B) implies that the utility function for wealth is concave (convex). Risk aversion for longevity can be defined in an analogous way by considering the apportionment of fixed reductions in longevity ℎ 1 and ℎ 2 instead of fixed reductions in wealth, as depicted in Figure 2 below. The individual is said to be risk seeking (averse) for longevity if she prefers to (dis)aggregate ℎ 1 and ℎ 2 across equally likely states of nature, i.e., if she prefers prospect A to prospect B in Figure 2. [START_REF] Eeckhoudt | Putting Risk in Its Proper Place[END_REF] show that any nth-order risk attitude can be defined in a more general framework, i.e., via risk apportionment, by considering the apportionment of zero-mean health and wealth risks instead of considering the apportionment of fixed reductions in health and wealth. For example, to define univariate prudence, the fixed reductions 𝑤 2 and ℎ 2 are replaced by a zero-mean risk wealth 𝑤 ̃2 and zero-mean longevity risk ℎ ̃2 in Figure 1 and Figure 2, respectively. A prudent individual prefers to disaggregate the sure loss in the first outcome and the zero-mean risk of the second outcome in two different states of nature, rather than aggregating them. For example, an individual exhibiting univariate prudence for wealth will always prefer to attach the zero-mean risk 𝑤 ̃2 to the state with the higher wealth (𝑊, in prospect A) than to the state with the lower wealth (𝑊 -𝑤 1 , in prospect B).

𝑊 -𝑤 1 -𝑤 2 , 𝐻 𝑊, 𝐻 ½ ½ 𝑊 -𝑤 2 , 𝐻 ½ ½ A B 𝑊 -𝑤 1 , 𝐻 𝑊, 𝐻 -ℎ 1 -ℎ 2 𝑊, 𝐻 ½ ½ 𝑊, 𝐻 -ℎ 2 ½ ½ A B 𝑊, 𝐻 -ℎ 1 A C C E P T E D M A N U S C R I P T
Similarly, an individual exhibiting univariate prudence for longevity will always prefer to apportion the zero-mean risk ℎ ̃2 to the state with the higher longevity (𝐻) than to the state with the lower longevity (𝐻 -ℎ 1 ). Figure 2 illustrates univariate prudence for longevity. temperate decision maker dislikes aggregating zero-mean risks in the same state of nature, i.e., the zero-mean risks are mutually aggravating. Under EU, univariate risk aversion, prudence and temperance for attribute x imply uxx < 0, uxxx > 0, and uxxxx < 0, respectively. 3

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Multivariate risk preferences

3 Throughout this paper, we let u(x, y) denote a two-attribute, four-times differentiable utility function. uxx denotes the second derivative of the utility function with respect to attribute x. We follow the same convention for higher order and cross-derivatives, e.g., uxy denotes the cross derivative of the utility function with respect to attribute x and y.

𝑊, 𝐻 -ℎ

1 + ℎ ̃2 𝑊, 𝐻 ½ ½ 𝑊, 𝐻 + ℎ ̃2 ½ ½ A B 𝑊, 𝐻 -ℎ 1 A C C E P T E D M A N U S C R I P T
As shown by [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF] the risk apportionment approach can also be used to define multivariate risk attitudes. A behavioral definition of correlation aversion, also called multivariate risk aversion, was first given by de Finetti (1952), [START_REF] Richard | Multivariate risk aversion, utility independence and separable utility functions[END_REF], and [START_REF] Epstein | Increasing generalized correlation: a definition and some economic consequences[END_REF]. Correlation aversion between longevity and wealth is defined as the preference of prospect A over B depicted in Figure 4. As can be seen in Figure 4, the individual faces two fixed reductions in prospect A, whatever the state of nature. The realization of uncertainty determines the nature of the reductions: a reduction in wealth 𝑤 1 or a reduction in longevity ℎ 2 . In prospect B both reductions are aggregated in the same state of nature. Thus, the choice between prospect A and B amounts to whether the individual prefers to aggregate the fixed reductions in wealth and longevity or not. Correlation aversion (seeking) implies that an individual prefers a higher level of wealth in the case of lower (higher) longevity. [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF] show that the choice for A implies that the cross-derivative of the utility function is negative, i.e., uxy ≤ 0.

𝑊 -𝑤 1 , 𝐻 -ℎ 2 𝑊, 𝐻 ½ ½ 𝑊, 𝐻 -ℎ 2 ½ ½ A B 𝑊 -𝑤 1 , 𝐻 A C C E P T E D M A N U S C R I P T
sure reduction of longevity and a zero-mean wealth risk in two different states of nature, rather than pooling them together in the same state of nature. Thus, cross-prudence for longevity implies that the individual can better cope with a wealth risk if her longevity is higher. Similarly, cross-prudence for wealth is defined as the preference for the prospect Finally, cross-temperance for wealth and longevity is defined as the preference for prospect A over prospect B depicted in Figure 5. As can be seen in the figure, the choice between A and B amounts to whether or not the individual prefers to aggregate the independent zero-mean risks 𝑤 ̃1 and ℎ ̃2. In the words of [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF] and [START_REF] Kimball | Standard risk aversion[END_REF], for a cross-temperate individual the independent wealth and longevity risk are "mutually aggravating", and, hence, she prefers to disaggregate them and prefers prospect A.

Experiment

𝑊 + 𝑤 ̃1, 𝐻 + h ̃2 𝑊, 𝐻 ½ ½ 𝑊, 𝐻 + ℎ ̃2 ½ ½ A B 𝑊 + 𝑤 ̃1, 𝐻 A C C E P T E D M A N U S C R I P T

Subjects

The experiment was conducted at Erasmus University Rotterdam, the Netherlands, and was programmed in z-Tree [START_REF] Fischbacher | z-Tree: Zurich toolbox for ready-made economic experiments[END_REF]. In total, 98 subjects (43.9% female) took part in sessions of up to four participants. Participants were remunerated in course credits. Upon arrival at the experiment, subjects we given oral instructions and were informed that there were no right or wrong answers. The experiment consisted of two separate parts. The first part concerned the elicitation of univariate risk preferences; the second part concerned the elicitation of multivariate risk preferences.

At the start of each part, subjects read the written instructions and were required to answer several practice questions to check for understanding of the instructions. Online

Appendix 2 details the instructions for both parts of the experiment. An elaborate description of all traits we used in the experiment is provided in Appendix A.

Procedure and stimuli

Subjects were asked to make a total of 69 choices between prospects yielding outcomes in terms of wealth and life expectancy. Within each part of the experiment (univariate vs. multivariate), the order of the choice tasks and the position of each lottery on the screen (left vs. right) was randomized. Subjects were endowed with €240,000 and 40 years of longevity. The 40 years are about 15 to 20 years less than the life-expectancy of the average subject in our sample. Because our experiment included improvements and reductions in life years, it was important to have a baseline value that was compatible with a reasonable final longevity in the gain domain. Regarding wealth, subjects were given a baseline endowment of €240,000. We stressed the fact that choices involving increments and decrements of wealth would be proposed during the experiment. It was

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also underlined that those increments and decrements could be safe (realized for sure) or risky. Lotteries were presented in a non-reduced form in order to stay as close as possible to the existing literature on higher order risk preferences.

Table 1 lists the choice tasks that were used in the experiment in the gain domain.

[(€240 ± €20, 40𝑦); (€240 ± €20, 40𝑦)] [(€240 ± €20 ± €20, 40𝑦); (€240, 40𝑦)] 50 A C C E P T E D M A N U S C R I P T
As can be seen in Table 1, the first 9 choices (3 for each trait) were designed to elicit univariate attitudes towards risk and correlation for wealth and longevity for gains.

Attitudes towards (cross-) prudence for gains were measured in the next set of 12 choices. The final set of 9 choices was designed to elicit subjects' attitudes towards (cross-) temperance. For gains, 7 choices were repeated to probe the degree of consistency of responses.

In addition to these choice tasks, individuals were given 32 choice tasks involving the apportionment of losses, summarized in Table 2.

Analyses

In our analyses, we use the amount of choices (out of 3) that are compatible with a given risk trait as our measurement of the strength of multi-and univariate risk preferences.

One advantage of using the amount of choices as a measure of the strength of preference is that it allows us to directly compare our results with the existing literature where this measure is used most (Deck andSchlesinger, 2010, 2014;[START_REF] Ebert | Measuring multivariate risk preferences[END_REF][START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF]. More importantly, the measure provides a ranking of subjects in terms of these preferences without having to assume that subjects maximize a particular preference model, such as EU. As a result, our analysis is modelfree in the sense that it retains validity if EU theory fails to accurately describe choice behavior (Starmer, 2000).

At the aggregate level, we report the mean number of choices and the average percentage of choices over tasks. At the individual level, we classify subjects depending on their choices, using the following criterion. A subject is classified according to a risk trait if the majority of her choices is consistent with that particular trait. Thus, for example, an individual is classified as being correlation averse (seeking) if the majority of her choices are compatible with correlation aversion (seeking). We perform a series of chi-squared tests to check whether the observed distribution of preferences deviates from the distribution that would be observed if subjects choose randomly. All tests show that choices were not made at random, with the notable exception of prudence for losses in longevity (see Online Appendix 1 for details).

Consistency checks

Subjects made the same choice in 77.44 percent of the repeated choices. This rate is consistent with the usually observed consistency rates in experiments (Stott, 2006).

Given that the position of lotteries on the screen (left vs right) was randomly determined for every choice, consistency would not be observed to such a high degree of subjects would choose according to a heuristic such as always choosing left or right.

We found some variability in consistency between the different tasks, with a (betweentask) standard deviation of 9.26 percent. 4

Univariate risk preferences

Table 1 shows the aggregate results on univariate risk preferences. It shows both the mean number of choices and the standard deviation computed over the three choices.

For gains, we found risk aversion to be the predominant pattern for both longevity and wealth, with a large majority of the choices compatible with risk aversion in both cases.

Prudence is also a dominant feature for gains, for both attributes. Interestingly, Table 3 shows that we do not observe a significant deviation from neutrality for temperance for both wealth and longevity. This is in line with the mixed findings in the literature, where some papers have observed a preference for intemperance (e.g. Baillon et al., 2018; 4 A closer look at the different repeated tasks shows that consistency rates were significantly higher for gains than for losses (Wilcoxon, p-value 0.017). We also found consistency to be higher on measures of risk aversion (univariate risk aversion and correlation aversion) than on measures of prudence (Wilcoxon, p-value <0.001) or temperance (Wilcoxon, p-value 0.001). We however found no difference in consistency between univariate tasks and multivariate tasks (Wilcoxon, p-value 0.931). For losses, we do not observe significant deviations form linearity; the mean preference is indicative of a neutral attitude toward risk and prudence for wealth and longevity. This finding corroborates recent results obtained by [START_REF] Bleichrodt | Higher order risk preferences for gains and losses[END_REF], who observe a shift from risk aversion and modest prudence for gains to risk neutrality and imprudence for losses for monetary outcomes. We performed a series of chi-squared tests to check whether the observed distribution of preferences deviated from the distribution that would be observed if subjects choose randomly. All tests show that choices were not made at random, with the notable exception of prudence for losses in longevity. Table 4 shows the classification of individuals, based on their univariate risk preferences, together with the p-value of a one-sided binomial test for comparison between the percentage of individuals and 50%. Table 4 confirms the aggregate findings. A large majority of individuals are classified as being risk averse for both longevity and wealth. We also find evidence for prudence for longevity and (marginally)

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for wealth. Temperance for longevity is marginally significant. For losses, we observe that there are more risk seekers than risk averters, but the proportion of risk seekers is not significantly different from 50% in either case. Imprudence for longevity and prudence for wealth are also not significant in the loss domain. The aggregate results indicate that risk attitudes shift from risk aversion and prudence for gains toward risk-and prudence neutrality for losses. Such systematic sign-dependence of risk attitudes is not corroborated by the analysis at the individual level. A Fisher test shows no significant association between risk attitudes in the gain domain and risk attitudes in the loss domain (p-value=0.51 for longevity and 0.59 for wealth). We also find no association in prudent behavior between gains and losses.

Individual results show that prudence is not systematically associated with risk aversion. The absence of relationship between risk attitudes and prudence holds for longevity and for wealth, for both gains and losses (Fisher tests, all p-values>0.31).

Temperance (intemperance) is clearly associated with risk aversion (seeking), except for monetary losses (Fisher tests, all p-values <0.03, except for wealth and losses, p-value=0.23). Likewise, no systematic association between prudence and temperance is observed (Fisher tests, all p-values >0.09).

For gains, the combination of risk aversion, prudence and temperance corresponds to the modal univariate risk preference for both longevity and wealth (for 31.6 and 27.5 % of subjects, respectively). Combinations of risk aversion, prudence and intemperance, and risk aversion, imprudence and intemperance are also common. For losses, the combination of risk aversion, prudence and temperance corresponds to the modal univariate risk preference for wealth only (for 19.4% of subjects). For longevity, such risk preferences were not uncommon (17.3% of subjects), but the modal preference implies a combination of risk seeking, imprudence and intemperance preferences (for 19.4% of subjects).

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Multivariate risk preferences

Table 5 shows the aggregate results concerning multivariate risk preferences. For gains, the modal choice pattern is indicative of a preference for correlation aversion for longevity and wealth; more than 76% of the choices are compatible with correlation aversion. Interestingly, for losses, we observe the opposite preference; 68% of choice are consistent with correlation seeking behavior. We thus extend the often observed sign-dependence for univariate risk attitudes to multivariate risk attitudes. Subjects prefer to disaggregate improvements in wealth and longevity, while they prefer to aggregate fixed reductions in these attributes.

Interestingly, we observe cross-prudence for wealth, but not so for longevity. Thus, people can better cope with a health risk in the future, if they have more wealth, which induced people to save precautionary to protect themselves against future health risk, according to theory [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF]. For losses, we observe cross-prudence neutrality. Finally, we observe cross-temperance neutrality; people do not exhibit a nonlinear attitude toward aggregating or disaggregating wealth and longevity risks.

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Table 4 shows the classification of individuals, based on their multivariate risk preferences, together with the p-value of a one-sided binomial test for the comparison between the percentage of individuals and 50%. Table 6 confirms the aggregate results.

A large majority of individuals is classified as correlation averse for gains and correlation seeking for losses. While a majority of subjects (53 out of 98) exhibited correlation aversion in the gain domain and correlation seeking in the loss domain, we could not reject the hypothesis that both behaviors were unrelated (Fisher test, p-value 0.414). We also found evidence for cross-prudence for wealth in the gain domain. We only found marginal significance for cross-imprudence for longevity in the loss domain. For both gains and losses, we found significant associations between cross-prudence for longevity and crossprudence for wealth (Fisher tests, p-values 0.006 for gains and 0.002 for losses). Table 6 also shows we found no clear evidence against cross-temperance for longevity and wealth. Cross-prudence is related to univariate prudence, but for gains only. Cross-prudence in longevity for related to prudence for wealth (Fisher test, p-value 0.004) and crossprudence for wealth is related to prudence for longevity (Fisher test, p-value 0). For

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We also found a positive association between cross-temperance and temperance in the univariate context. This holds for longevity (Fisher test, p-value 0.024) and for wealth (Fisher test, p-value 0.002).

For gains, the combination of correlation aversion, cross-prudence and crossintemperance corresponded to the modal multivariate risk preference when crossprudence for wealth was considered (for 27.55 % of subjects). When cross-prudence for longevity was considered, this combination, although shared by a minority of subjects (20.41 % of subjects) was not the modal preference. Indeed, the combination of correlation aversion, cross-imprudence in longevity and cross-intemperance was shared by 23.47 % of subjects. For losses, whatever the definition of cross-prudence, the modal multivariate risk preference revealed correlation seeking, cross-imprudence and cross-intemperance (28.57 % of subjects for cross-prudence in wealth, 28.57 % of subjects for cross-prudence in longevity).

Discussion

In this study we have shown how univariate and multivariate risk attitudes for longevity and wealth can be measured in a direct way, without the need for any modeling assumptions. We have also been the first to measure cross-prudence and cross-temperance for wealth and longevity, and, more generally, the first to measure these traits in a sign-dependent framework.

Regarding univariate risk preferences, for monetary gains we confirm the common finding of risk aversion and prudence (see Trautmann and van de Kuilen, 2018).

Moreover, we do not observe a strong preference toward temperance, which is in line with the mixed findings in the literature where some papers have observed a preference

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for intemperance (e.g., [START_REF] Deck | Exploring higher order risk effects[END_REF][START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF], while other studies have found the opposite (e.g., [START_REF] Deck | Consistency of higher order risk preferences[END_REF][START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF].

The fact that our findings in the wealth domain with hypothetical incentives confirm the common findings obtained under real incentives strengthens the recent observation by

Trautmann and van de Kuilen ( 2018) that the provision of monetary rewards itself does not appear to affect preferences toward higher order risk. 5

The pattern of risk aversion and prudence toward monetary gains extends toward higher order risk preferences regarding gains in longevity. People are not only risk averse, but also prudent for longevity. We replicate the shift toward neutrality when considering losses; in the loss domain, the modal choice pattern is indicative of neutrality toward risk and prudence. This finding corroborates recent results obtained by [START_REF] Bleichrodt | Higher order risk preferences for gains and losses[END_REF], who observe a shift from risk aversion and modest prudence for gains to risk neutrality and imprudence for losses for monetary outcomes. Similar results were also reported by Smith et al. (2002), who found 73% of choices for gains to be risk averse and 55.5% of choices to be risk seeking for losses, and others (e.g. [START_REF] Abdellaoui | Parameter-Free Elicitation of Utility and Probability Weighting Functions[END_REF][START_REF] Abdellaoui | A tractable method to measure utility and loss aversion under prospect theory[END_REF][START_REF] Fehr-Duda | Probability and Risk: Foundations and Economic Implications of Probability-Dependent Risk Preferences[END_REF]. We extend these findings toward longevity risks.

Regarding multivariate risk preferences, we observe correlation aversion for wealth and longevity in the gain domain. Thus, people prefer to separate fixed improvements in wealth and longevity. This pattern is reversed when fixed deteriorations in wealth and longevity are considered, i.e., we observe a preference for correlation seeking, as predicted by Bleichrodt et al. (2003b) based on the finding of some studies that the 5 This observation is mainly based on the findings by [START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF], who do not observe any differences in the degree of risk aversion, prudence, and temperance between a real and a hypothetical payoff condition in a representative sample of the Dutch population. Viscusi and Evans, 1990). This marked difference between gains and losses may be considered striking, and is perhaps due to the specific design used in the current study.

Future research could therefore establish whether attitudes toward correlation between health and wealth are genuinely domain-specific or whether attitudes toward correlation depend on stimuli. The method introduced in this paper can be used to test these two explanations in any setting in a model-free way.

Overall, our results show that the elicitation of higher order risk attitudes is subject to sign-dependence. In our experiment, the signs of the outcomes had a large impact on decisions, with more noise for losses than for gains, and a reversal from correlation averse behavior for gains to correlation seeking behavior for losses. This implies that framing and sign-dependence are more important in the study of higher order risk attitudes than previously thought, since most theoretical studies on this topic do not account for the agents' initial endowment (Bleichrodt et al., 2003a;[START_REF] Courbage | Prudence and optimal prevention for health risks[END_REF][START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF]. Therefore, one should be cautious when applying the risk apportionment technique to elicit preferences for health, especially when they involve losses. Another implication of our findings of sign-dependence lies in the physicianpatient interaction. If the heterogeneity of patients' preferences is measured by a risk apportionment technique, it is important to keep in mind the framing effects we found.

If patients reveal different behavior for gains than for losses, it is hard to tell which preferences should be considered in the physician's treatment recommendation. This also raises the question as to whether such recommendations should be descriptive or normative. Because our results are descriptive, and we do not have separate information on the extent to which this behavior was caused by non-rational components (cf., for example, a study that disentangles risk aversion in a utility normative conclusions from our study. However, the findings of sign-dependence suggest that, for normative purposes, it would be better to express the potential health outcomes in terms of final outcomes, instead of framing them as gains or loss relative to some reference points. It should be mentioned that a weakness of using a fixed reference point is that our results cannot be generalized to other reference points. This highlights the need for further research into this topic using different reference points.

Because this study comprises a new elicitation method, we used a convenience sample to test its feasibility and reliability. Future work could implement this methodology in a sample more representative of the general public, increasing external validity. Such work could also aim to verify some predictions that follow from our results, such as if differences exist in higher order risk attitudes between lower and higher incomes. This kind of findings would have large consequences for policy making.

For instance, our finding of cross-prudence for wealth gains would imply that people who expect a health shock in the future will save more now than people who do not expect this, or than those who currently experience a health shock. In addition, the observation of correlation seeking for losses will lead retirees to anticipate that health deteriorations will boost their marginal utility of income due to impeded home production. The presence or not of spouses is also of special importance in this regard, since individuals with a spouse will have more possibilities for free informal care. As a result, they are less likely to have to hire a caregiver if they get sick, so they are less likely to be correlation seeking (Edwards, 1998).

Our results likewise have important implications for health insurance. Expectations about future health risks are related to expected future medical expenses and will therefore influence current health insurance decisions (e.g. amount of deductible,

A C C E P T E D M A N U S C R I P T
supplementary insurance) and savings. The degree of prudence for wealth will affect the reaction of current health insurance choice and savings to the increased future medical expenses risk; whereas, the degree of cross-prudence for wealth will affect the response of health insurance uptake to the future health risk. Moreover, the degree of correlation aversion is important here if one anticipates an adverse health shock;

correlation averse [seeking] individuals will then spend [save] more now. For example, Rosen and Wu (2004) observed that poor health is associated with safer investment portfolios regardless of health insurance status or other medical expenditures. This implies the individuals studied by Rosen and Wu ( 2004) were cross-prudent for health status on the aggregate.

In the main analysis, we used the number of choices as a measure of the strength of preferences. As previously mentioned, this allows us to directly compare our results

with the existing literature where this measure is used most (Deck andSchlesinger, 2014, 2010;[START_REF] Ebert | Measuring multivariate risk preferences[END_REF][START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF], and it provides a ranking of subjects in terms of these preferences without having to assume that subjects maximize a particular preference model, such as EU. A drawback is that our information about the amount of (higher order) risk aversion is admittedly limited. Future research should therefore strive toward more precise estimates of these risk attitudes, e.g. by eliciting the intensity of risk attitudes instead of only its sign, as for example performed by [START_REF] Ebert | Joint measurement of risk aversion, prudence, and temperance[END_REF] for univariate risk attitudes. bring more clarity on the sign of the cross-derivative of these two attributes, since, at present, there is mixed empirical evidence of the relation between marginal utility of wealth and health status. Some studies reported the marginal utility to increase with improved health status [START_REF] Brown | Heterogeneity in State-Dependent Utility: Evidence from Strategic Surveys[END_REF][START_REF] Domeij | Consumption and Health[END_REF][START_REF] Finkelstein | WHAT GOOD IS WEALTH WITHOUT HEALTH? THE EFFECT OF HEALTH ON THE MARGINAL UTILITY OF CONSUMPTION[END_REF]Gyrd-Hansen, 2017;Sloan et al., 1998;Viscusi and Evans, 1990), whereas others report evidence for the opposite relation [START_REF] Edwards | Health risk and portfolio choice[END_REF][START_REF] Evans | Estimation of state-dependent utility functions using survey data[END_REF][START_REF] Lillard | Uncertain health and survival: Effects on end-of-life consumption[END_REF]Tengstam, 2014).

Other interesting research topics would be the elicitation of higher order (multivariate) preferences for the health outcomes of different groups in society (i.e., equity attitudes, [START_REF] Courbage | Priority setting in health care and higher order degree change in risk[END_REF]) and the elicitation of self-versus-other preferences à la Ebert and van de Kuilen (2018) in a health context. The latter could shed some light on the current debate about the role of relative positions in health (e.g. Wouters et al., 2015).

It has been observed that the degree of prudence compared to the degree of risk aversion typically observed in experiments is too strong to be adequately modeled by widely-used utility functions such as CARA or CRRA utility under EU. Moreover, risk averse individuals should be temperate under these specifications (e.g. [START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF]Trautmann and van de Kuilen, 2018). As a result, several papers have discussed the implications of attitudes towards higher order risk under non-EU models such as prospect theory in the wealth domain (e.g. [START_REF] Bleichrodt | Saving under rank-dependent utility[END_REF][START_REF] Bleichrodt | Higher order risk preferences for gains and losses[END_REF]Deck andSchlesinger, 2010, 2014). Our finding of strong risk aversion and temperance neutrality for monetary gains combined with the fact that attitudes toward higher order risk are more pronounced for gains than for losses add to the notion that investigating the implications of higher order risk under non-EU models temperance neutrality and sign-dependence) suggests that this discussion is also highly relevant for the health domain; attitudes toward higher order risk for longevity might be better captured by non-EU models. The measurements presented in this paper could serve as an input for the calibration of such models.

Conclusion

In this study we have shown how the risk apportionment technique can be applied to measure multivariate, higher order, risk preferences for longevity and wealth. In addition, we implemented a sign-dependent framework in order to test for differences between gains and losses. We observe that individuals in both domains are risk averse and prudent for gains, while they are closer to risk and prudence neutrality for losses.

There is clear evidence that multivariate preferences are sign-dependent, with correlation aversion for gains and correlation seeking for losses. Therefore, we conclude that framing and sign effects are important factors in the experimental measurement of (higher order) multivariate risk preferences.

  keeping initial endowments constant. 2

  amounts and let [x; y] denote a prospect yielding either x or y with equal (50/50) chance. Risk aversion for wealth is defined as a preference between the two prospects depicted in Figure1.

Figure 1 :

 1 Figure 1: A behavioral condition of univariate risk aversion for wealth

Figure 2 :

 2 Figure 2: Behavioral condition of univariate risk aversion for longevity

Figure 3 :

 3 Figure 3: Behavioral condition of univariate prudence for longevity

Figure 4 :

 4 Figure 4: Behavioral definition of correlation aversion between wealth and longevity

Figure 5 :

 5 Figure 5: Behavioral definition of cross-temperance

  first 9 choices (3 for each trait) were designed to elicit univariate attitudes towards risk and correlation for wealth and longevity for losses. Attitudes towards (cross-) prudence for losses were measured in the next set of 12 choices. Since the apportionment concerned either a reduction or an improvement of the fixed outcome, the temperance questions were not different for gains and losses, and were only posed once. For losses, we repeated 2 choices to probe the degree of consistency of responses.

  [START_REF] Deck | Exploring higher order risk effects[END_REF], while others studies have found the opposite (e.g.,Deck and Schlesinger 2014, Noussair et al., 2014).

Figure 6

 6 Figure6shows the distributions of the number of correlation averse choices, in the

Figure 6 :

 6 Figure 6: Correlation aversion for gains and losses, distribution of the number of

  income increases with increases in health (e.g.Sloan et al., 1998;

  probability weighting component), we are not able to draw firm

  Another future research avenue is to investigate the combination of two different health risks, e.g. by considering two comorbidities, one affecting longevity, and one affecting quality of life.Bleichrodt et al. (2003a) have demonstrated the theoretical importance of these concepts, but thus far these theoretical predictions have not yet been empirically tested. The risk apportionment technique applied in this study may in future work combining wealth and quality of life. Such an approach may

  avenue for future research. The fact that we observe the same behavioral pattern for outcomes in terms of longevity (i.e., strong risk aversion combined with

Table 1 :

 1 Choice Tasks for Gains

	Task a	Trait	Prospect A	Prospect B	Choice % b
	1* 2 3	Risk aversion for wealth	[(€240 + €44, 40𝑦); (€240 + €20, 40𝑦)] [(€240 + €32, 40𝑦); (€240 + €44, 40𝑦)] [(€240 + €20, 40𝑦); (€240 + €32, 40𝑦)]	[(€240 + €44 + €20, 40𝑦); (€240, 40𝑦)] [(€240 + €32 + €44, 40𝑦); (€240, 40𝑦)] [(€240 + €20 + €32, 40𝑦); (€240, 40𝑦)]	81.63 / 80.61 81.63 79.59
	4 5* 6	Risk aversion for longevity	[(€240, 40𝑦 + 9𝑦); (€240, 40𝑦 + 5𝑦)] [(€240, 40𝑦 + 7𝑦); (€240, 40𝑦 + 9𝑦)] [(€240, 40𝑦 + 5𝑦); (€240, 40𝑦 + 7𝑦)]	[(€240, 40𝑦 + 9𝑦 + 5𝑦); (€240, 40𝑦)] [(€240, 40𝑦 + 7𝑦 + 9𝑦); (€240, 40𝑦)] [(€240, 40𝑦 + 5𝑦 + 7𝑦); (€240, 40𝑦)]	87.76 91.84 / 86.73 85.71
	7		[(€240 + €20, 40𝑦); (€240, 40𝑦 + 9𝑦)]	[(€240 + €20, 40𝑦 + 9𝑦); (€240, 40𝑦)]	74.49
	8*	Correlation aversion	[(€240 + €44, 40𝑦); (€240, 40𝑦 + 7𝑦)]	[(€240 + €44, 40𝑦 + 7𝑦); (€240, 40𝑦)]	75.51 / 72.45
	9		[(€240 + €32, 40𝑦); (€240, 40𝑦 + 5𝑦)]	[(€240 + €32, 40𝑦 + 5𝑦); (€240, 40𝑦)]	80.61
	10		[(€240 + €44, 40𝑦); (€240 ± €20, 40𝑦)]	[(€240 + €44 ± €20, 40𝑦); (€240, 40𝑦)]	65.31
	11	Prudence for wealth	[(€240 + €44, 40𝑦); (€240 + €(15, -5), 40𝑦)]	[(€240 + €44 + €(15, -5), , 40𝑦); (€240, 40𝑦)]	51.02
	12*		[(€240 + €20, 40𝑦); (€240 ± €5, 40𝑦)]	[(€240 + €20 ± €5, 40𝑦); (€240, 40𝑦)]	56.12 / 56.12
	13		[(€240, 40𝑦 + 9𝑦); (€240, 40𝑦 ± 6𝑦)]	[(€240,40𝑦 + 9𝑦 ± 6𝑦); (€240, 40𝑦)]	66.33
	14	Prudence for longevity	[(€240, 40𝑦 + 7𝑦); (€240, 40𝑦 + (3, -1)𝑦)]	[(€240,40𝑦 + 7𝑦 + (3, -1)𝑦); (€240, 40𝑦)]	56.12
	15		[(€240, 40𝑦 + 5𝑦); (€240, 40𝑦 ± 2𝑦)]	[(€240,40𝑦 + 5𝑦 ± 2𝑦); (€240, 40𝑦)]	64.29
	16 17 18	Cross-Prudence for longevity	[(€240 ± €20, 40𝑦); (€240, 40𝑦 + 9𝑦)] [(€240 + €(15, -5), 40𝑦); (€240, 40𝑦 + 7𝑦)] [(€240 ± €5, 40𝑦); (€240, 40𝑦 + 5𝑦)]	[(€240 ± €20, 40𝑦 + 9𝑦); (€240, 40𝑦)] [(€240 ± €(15, -5), , 40𝑦 + 7𝑦); (€240, 40𝑦)] [(€240 ± €5, 40𝑦 + 5𝑦); (€240, 40𝑦)]	50 39.8 43.88
	19* 20 21	Cross-Prudence for wealth	[(€240 + €44, 40𝑦); (€240, 40𝑦 ± 6𝑦)] [(€240 + €32, 40𝑦); (€240, 40𝑦 + (3, -1)𝑦)] [(€240 + €20, 40𝑦); (€240, 40𝑦 ± 2𝑦)]	[(€240 + €44, 40𝑦 ± 6𝑦); (€240, 40𝑦)] [(€240 + €32,40𝑦 + (3, -1)𝑦); (€240, 40𝑦)] [(€240 + €20, 40𝑦 ± 2𝑦); (€240, 40𝑦)]	45.92 / 35.71 35.71 37.76
	22	Temperance for wealth

Table 2 :

 2 Choice Tasks for Losses

	Task a	Trait	Prospect A	Prospect B	Choice % b
	1 2 3	Risk aversion for wealth	[(€240 -€6, 40𝑦); (€240 -€30, 40𝑦)] [(€240 -€18, 40𝑦); (€240 -€6, 40𝑦)] (€240 -€18, 40𝑦)] [(€240 -€30, 40𝑦);	[(€240 -€6 -€30, 40𝑦); (€240, 40𝑦)] [(€240 -€18 -€6, 40𝑦); (€240, 40𝑦)] (€240, 40𝑦)] [(€240 -€30 -€18, 40𝑦);	42.86 48.98 53.06
	4 5* 6	Risk aversion for longevity	[(€240, 40𝑦 -1𝑦); (€240, 40𝑦 -5𝑦)] [(€240, 40𝑦 -3𝑦); (€240, 40𝑦 -1𝑦)] (€240, 40𝑦 -3𝑦)] [(€240, 40𝑦 -5𝑦);	[(€240, 40𝑦 -1𝑦 -5𝑦); (€240, 40𝑦)] [(€240, 40𝑦 -3𝑦 -1𝑦); (€240, 40𝑦)] (€240, 40𝑦)] [(€240, 40𝑦 -5𝑦 -3𝑦);	42.86 47.96 / 45.92 51.02

Table 3 :

 3 Univariate Risk Preferences Across Domains

		Gains,	Gains,	Losses,	Losses,
		mean	sd	mean	sd
	Risk aversion, Longevity	2.65***	0.73	1.42	1.08
	Prudence, Longevity	1.87***	1.07	1.45	0.99
	Temperance, Longevity	1.71*	1.06
	Risk aversion, Wealth	2.43***	1.02	1.45	1.11
	Prudence, Wealth	1.72**	1.01	1.53	0.99
	Temperance, Wealth	1.51	0.99
	Notes: *** (**,*) indicates a significant difference from 1.5 at the 1% (5%, 10%) level (one-sample
	two sided Wilcoxon-signed rank tests).

Table 4 :

 4 Classification of individuals by context based on univariate risk preferences

	% of individuals	Gains p-value Losses p-value
	Risk averse, Longevity	90.82 <0.01	44.90	0.18
	Prudent, Longevity	62.24 0.01	45.92	0.24
	Temperate, Longevity	58.16 0.065
	Risk averse, Wealth	83.67 <0.01	48.98	0.46
	Prudent, Wealth	58.16 0.065	56.12	0.13
	Temperate, Wealth	54.08 0.24
	In order to test the consistency of risk preferences across domains, we tested for the
	association between univariate risk preferences for longevity and univariate risk
	preferences for wealth. We find a significant association between risk preferences for
	longevity and wealth, at any order. First, risk attitudes for gains and losses are
	correlated (Fisher test, both p-values <0.001), with risk aversion (seeking) in the
	longevity domain associated with risk aversion (seeking) in the monetary domain. The
	same applies to prudence: prudent (imprudent) individuals in one domain are also
	prudent (imprudent) in the other domain (Fisher test, both p-values <0.025 for gains

and losses). For temperance, we find a similar result (Fisher test, p-value <0.001).

Table 5 :

 5 Multivariate Risk Preferences for Longevity and Wealth

		Gains,	Gains,	Losses,	Losses,
		mean	sd	mean	sd
	Correlation aversion	2.31***	1.07	0.97***	1.09
	Cross-Prudence in Longevity	1.66	1.08	1.34	1.07
	Cross-Prudence in Wealth	1.81***	1.02	1.39	1.06
	Cross-Temperance	1.38	1.01

Notes: *** (**,*) indicates a significant difference from 1.5 at the 1% (5%, 10%) level (one-sample Wilcoxon-signed rank tests).

Table 6 :

 6 Classification of individuals by context based on multivariate risk preferences

	% of individuals	Gains	p-value Losses p-value
	Correlation aversion	78.57	<0.01	28.57	<0.01
	Cross-Prudence in Longevity	52.04	0.38	42.86	0.09
	Cross-Prudence in Wealth	63.27	<0.01	47.96	0.38
	Cross-Temperance	45.92	0.24

Acknowledgments

This paper has benefited from the comments and suggestions of two anonymous referees, Mathias Kifmann (the editor), participants of seminars at the University of Southern Denmark and Newcastle Business School, and participants of the FUR conference in York and the EUHEA conference in Maastricht.

losses, however, we observe no systematic association between multivariate crossprudence and univariate prudence (both p-values >0.10).