Xiangjun Qian

Arnaud De

La Fortelle

Fabien Moutarde

A Hierarchical Model Predictive Control Framework for On-road Formation Control of Autonomous Vehicles

This paper presents an approach for the formation control of autonomous vehicles traversing along a multi-lane road with obstacles and traffic. A major challenge in this problem is a requirement for integrating individual vehicle behaviors such as lane-keeping and collision avoidance with a global formation maintenance behavior. We propose a hierarchical Model Predictive Control (MPC) approach. The desired formation is modeled as a virtual structure evolving curvilinearly along a centerline, and vehicle configurations are expressed as curvilinear relative longitudinal and lateral offsets from the virtual center. At high-level, the trajectory generation of the virtual center is achieved through an MPC framework, which allows various on-road driving constraints to be considered in the optimization. At low-level, a local MPC controller computes the vehicle inputs in order to track the desired trajectory, taking into account more personalized driving constraints. High-fidelity simulations show that the proposed approach drives vehicles to the desired formation while retains some freedom for individual vehicle behaviors.

I. INTRODUCTION

Cooperative strategies for groups of autonomous vehicles start to attract attentions from both automotive industry and research institutions, due to their potential in improving traffic efficiency and reducing road accidents. Previous projects (PATH [START_REF] Shladover | Automated vehicle control developments in the PATH program[END_REF], CHAUFFER I & II [START_REF] Harker | PROMOTE-CHAUFFEUR II & 5.8 GHz vehicle to vehicle communications system[END_REF] and SARTRE [START_REF] Chan | Cooperative control of sartre automated platoon vehicles[END_REF]) have extensively studied a special form of cooperative driving: platoon, in which a group of vehicles forms a linear formation. It is shown that platooning vehicles brings a 15% to 30% fuel consumption reduction and a 3 to 5 times increase of road throughput [START_REF] Marjovi | Distributed graph-based convoy control for networked intelligent vehicles[END_REF].

This work extends the notion of platoon into a broader concept: convoy, defined as multiple cooperative vehicles spreading over multiple lanes maintaining a pre-designed formation. As an extension of platoon, convoy keeps the benefits of platoon. Moreover, it may find applications in lane-change assistances or the protection of VIP vehicles.

In the robotic and control community, generic formation control problem for multiple robots has been an active research area for decades. Roughly speaking, there are three approaches to tackle this problem, namely leaderfollowing, virtual structure and behavioral approaches. In leader-following approach [START_REF] Shao | Leader-following formation control of multiple mobile vehicles[END_REF], [START_REF] Consolini | Leaderfollower formation control of nonholonomic mobile robots with input constraints[END_REF], a leader is selected to track a reference trajectory, while other robots maintain Xiangjun Qian, Arnaud de La Fortelle and Fabien Moutarde are with MINES ParisTech, PSL -Research University, Centre for Robotics, 60 Bd St Michel, 75006 Paris, France {xiangjun.qian, arnaud.de la fortelle, fabien.moutarde}@mines-paristech.fr relative orientations/distances from the leader. The virtual structure approach [START_REF] Ren | Virtual structure based spacecraft formation control with formation feedback[END_REF], [START_REF] Ren | Decentralized scheme for spacecraft formation flying via the virtual structure approach[END_REF] considers the formation as a rigid structure. Robots are regarded as nodes in the structure in which the trajectory of the structure is first calculated and then transformed to the reference trajectories of individual robots. In behavioral approach [START_REF] Balch | Behavior-based formation control for multirobot teams[END_REF], robots are prescribed with several behaviors, notably goal seeking, local formation keeping, collision avoidance, etc.. The control of each individual robot is a weighted average of the control for each behavior. The global group behavior emerges from the behaviors of individual vehicles.

The literature in robot formation control lays a solid foundation for the convoy control problem of multiple autonomous vehicles. However, unique challenges exist for our problem. Firstly, vehicles are constrained to move in a highly structured environment (multi-lane road with lane markings). Thus the formation must adapt to the road shape and vehicles must avoid departing from their lanes. Secondly, each individual vehicle as well as the entire convoy must respect traffic rules and interact with other traffic participants accordingly. Thirdly, each vehicle must have a complete controller for autonomous driving, ruling out completely centralized approaches. This requirement ensures that a vehicle can still autonomously drive even if it loses communication with the convoy.

There exists some previous work that tackles the convoy control problem of autonomous vehicles on the road. Kato et al. [START_REF] Kato | Vehicle Control Algorithms for Cooperative Driving with Automated Vehicles and Intervehicle Communications[END_REF] consider a specific convoy problem with 5 vehicles spreading over two lanes using a leader-following approach. In [START_REF] Marjovi | Distributed graph-based convoy control for networked intelligent vehicles[END_REF], a distributed graph-based convoy control algorithm is proposed. Each vehicle memorizes and tracks its neighborhood. The local control input is calculated using the Laplacian graph method. The advantage of this method is that it is fully decentralized. However, the formation is limited to a rectangle shape and the vehicle controller is in its simplest form.

This paper proposes and validates a novel on-road convoy control algorithm. We adopt a hierarchical approach to separate the convoy control problem into a high-level virtual structure control problem and a low-level vehicle control problem. At high-level, we consider the convoy formation as a rigid virtual structure. We assume that the virtual structure is uni-dimensionally evolving along a reference path, taking into account speed limits, road curvatures and other traffic participants. A MPC controller is adopted to generate the reference trajectory for the virtual structure. This trajectory is then transformed into the reference trajectories of individual vehicles considering longitudinal and lateral offsets of vehi- cles. At low-level, we propose a complete non-linear MPC controller for autonomous driving of individual vehicles. Behaviors such as lane-keeping and collision avoidance can be readily integrated into the MPC. The cost function of the MPC is designed to penalize the deviation from a formation keeping reference trajectory.

The paper is structured as follows. In Section II, we present the modeling of the convoy control problem. Section III and Section IV present the formulations of the trajectory planning and the trajectory tracking problems, respectively. Section V validates our approach through simulations and Section VI closes the paper with conclusion.

II. MODELING

Next we introduce the modeling of the convoy control problem. In particular, Section II-A uses a kinematic bicycle model in the Frenet frame to describe the vehicle dynamics. Section II-B presents the modeling of multi-lane road as a set of parallel curves. Section II-C describes the virtualstructure-based convoy design. Section II-D classifies onroad objects and describes the handling strategies.

A. Vehicle Model

The car model is developed based on the kinematic bicycle model [START_REF] Kong | Kinematic and Dynamic Vehicle Models for Autonomous Driving Control Design[END_REF], which strikes a good balance between completeness and complexity. As we focus on on-road autonomous driving, in which vehicle motion is highly constrained by the road geometry, we re-express the kinematic bicycle model in the Frenet frame with respect to the centerline of the lane (Fig. 1). This Frenet frame approach is also discussed and applied to different vehicle dynamics models in [START_REF] Werling | Optimal trajectories for time-critical street scenarios using discretized terminal manifolds[END_REF], [START_REF] Weiskircher | Predictive ADAS : A Predictive Trajectory Guidance Scheme for Advanced Driver Assistance in Public Traffic[END_REF].

The model is given as:

ṡ = v cos(e ψ ) 1 1 -rκ(s) , (1a)
ṙ = v sin(e ψ ), (1b)
ėψ = β + v l r sin(β) -v cos(e ψ ) κ(s) 1 -rκ(s) , (1c) v = a, (1d) δ = ω, (1e)
where β = tan -1 lr l f +lr tan(δ) . s and r are the coordinates of the Center of Mass (CoM) in the road-following Frenet frame. s represents the curvilinear coordinate of the vehicle along the centerline and r is the lateral offset. e ψ is the heading alignment error of the with regard to the CoM centerline and v is the vehicle speed. l r and l f represent the distances from the CoM to the rear and the front axles. β is the difference between the angle of the CoM and the heading. κ(s) is the curvature of the centerline expressed as a function of s. δ is the steering angle. a and ω are two control inputs: acceleration and angular velocity.

B. Road Model

We give a formal description of multi-lane road. A road with L lanes is modelled as a collection of parallel curves Γ j∈{1,...,L} . We express the curve Γ j in parametric form as a function of its curvilinear coordinate s j in a 2D Cartesian plane: Γ j (s j ) ∈ R 2 . Two curves Γ j+1 and Γ j are said to be parallel as they satisfy

Γ j+1 (s j ) = Γ i (s j ) + h n(s j ), ( 2
)
where h is the signed distance of two curves and n is a unitary vector orthogonal to Γj (s j ) such that the determinant of two vectors are positive: det( Γj (s j ), n(s j )) > 0.

In what follows, we mention two properties of our road model, which will be useful in the rest of the paper.

1) Curvature: The curvature of the curve Γ j at coordinate s j reads κ j (s j ) = ẋ(s j )ÿ(s j ) -ẏ(s j )ẍ(s j ). The curvature of the parallel curve Γ j+1 parametrized by coordinate s j of curve Γ j reads

κ j+1 (s j ) = κ j (s j ) 1 -hκ j (s j ) . (3)
Note that a singularity exists for Eq. ( 3) when 1hκ j (s j ) = 0. We assume that hκ j (s j ) < 1, conforming to the normal road geometry.

2) Change of reference frame: In a multi-lane road setting, the curvilinear coordinate s j of a vehicle with regard to curve Γ j differs from its curvilinear coordinate s j+1 with regard to curve Γ j+1 . The two coordinates satisfies the following relation:

s j+1 = sj 0 |1 -hκ j (ε j )|dε j (4)

C. Convoy Model

In the virtual-structure-based approach, the convoy is regarded as a rigid body with place-holders fixed in the formation to represent the desired positions [START_REF] Ren | Virtual structure based spacecraft formation control with formation feedback[END_REF]. Let V denote the number of member vehicles in the formation. We use i ∈ {1, ..., V} to label member vehicles and r i to label the desired lane of vehicle i, for example, r i = j if vehicle i is assigned to lane j. To determine the shape of the virtual structure, we introduce a reference point called the virtual center that evolves uni-dimensionally on a centerline Γ c . It is appropriate to consider the dynamics of the virtual center using a simple double integrator model:

ṡc = v c , (5a) vc = a c , (5b)
where s c and v c are curvilinear coordinate and speed on the curve Γ c . a c is the acceleration. Knowing the current position of the virtual center, the desired position of a vehicle i can then be defined by offseting the position of the virtual center, formally described by the tuple (∆s i,c , r i ) ∈ R×N + , where ∆s i,c is the desired longitudinal offset of the vehicle i with regard to the virtual center in the Γ c frame, r i is the lane label of the vehicle's desired lane. Fig. 2 provides an illustrative example defining a diamond-shaped convoy. The virtual center is assigned to lane Γ 2 . Vehicles V1 and V3 are assigned to lane Γ 2 with respectively 10 m and -10 m offsets from the virtual center. Vehicles V2 and V4 are assigned respectively to Γ 3 and Γ 1 with 0 longitudinal offset.

However, in local vehicle control, we consider the motion with regard to the vehicle's own reference frame Γ ri . The longitudinal coordinate in the Γ c frame then needs to be transformed to the coordinate in the Γ ri frame using the Eq. ( 4).

With the above formulation, a virtual-structure-based vehicle convoy can be uniquely defined by giving the position of the virtual center and the tuple (∆s i,c , r i ) for each member vehicle.

D. Object Model

Vehicle convoys must respect traffic rules and interact with other traffic participants and obstacles accordingly. This requirement demands a proper modeling of on-road objects. We propose to classify on-road objects into two categories: A Fig. 3: Illustration of on-road objects.

• Lane-Blocking Obstacles (LBOs). Static or dynamic obstacles that block one or more lanes. An example is given in Fig. 3 where the left lane is blocked by vehicle A.

• Non-Blocking Obstacles (NBOs). Static or dynamic obstacles that only block a small part of a lane. They can be dealt with by swerve maneuvers of member vehicles. The object B in Fig. 3 illustrates an avoidable obstacle on the border of the road. LBOs and NBOs need to be handled differently. A vehicle convoy cannot overtake LBOs unless it changes its formation. Thus LBOs must be handled in convoy trajectory planning level. An adequate convoy speed should be planned to keep safe distances from LBOs. On the other hand, NBOs should be ignored at high level and left for individual vehicles to handle, as local swerve maneuvers are sufficient.

Remark that LBOs may become NBOs and vice versa depending on the trajectory of obstacles and the changes of convoy formation. These problems are not considered in this paper.

III. MPC BASED TRAJECTORY GENERATION FOR THE VIRTUAL CENTER

Next we present how the trajectory of the virtual center can be generated using MPC. Let ξ c = [s c , v c ] ∈ Ξ c be the state of the virtual center and u c = [a c ] ∈ U c be the input. We assume that at a sampling time t = t 0 a desired trajectory for the interval t ∈ [t 0 , t 0 +T c ] should be provided. Consider the following cost function:

J c (ξ c ,u c ) = t0+Tc t0 (w 1 (v c (t) -v c,desire ) 2 + w 2 a 2 c (t))dt, (6)
where the first term penalizes the deviation of convoy speed from the desired speed v c,desire and the second term penalizes the control effort. These two terms are weighted through the tuning parameters w 1 and w 2 .

At each sampling time t = t 0 , we solve the following constrained optimization problem:

min uc J c (ξ c , u c ), subj. to ∀t ∈ [t 0 , t 0 + T c ], ξc (t) = f c (ξ c (t), u c (t)), (7a) ξ c (t) ∈ Ξ c , u c (t) ∈ U c , (7b) v 2 c (t)κ c (s c (t)) ≤ āc,lat , (7c) f c (ξ c (t), ξ o,c (t)) ≤ 0, ∀o, (7d)
Eq. (7a) is the compactly written form of the virtual center model [START_REF] Shao | Leader-following formation control of multiple mobile vehicles[END_REF]. (7b) enforces bound constraints on state and input, such as speed limit and acceleration bounds. The lateral acceleration of convoy is limited through (7c). Constraint (7d) is used to avoid collision with LBOs where ξ o,c is the state of a LBO o in Γ c frame. A possible design of this constraint can be:

s c (t) + v c (t)T desire + ≤ s o,c (t) (8)
where s o,c (t) is the estimated longitudinal trajectory of LBO o in Γ c frame, T desire is the desired constant time gap and is the minimal standstill distance. This constraint forces the virtual center to keep a constant time gap from the LBO. Solving the optimization problem results in an optimal trajectory ξ * c (t) for the horizon [t 0 , t 0 + T c ]. This trajectory can then be transformed to reference trajectories to be tracked by member vehicles. Problem [START_REF] Ren | Virtual structure based spacecraft formation control with formation feedback[END_REF] is solved repeatedly at a sampling rate of 1/∆T c , incorporating new measurements of the environment.

IV. MPC BASED TRAJECTORY TRACKING CONTROL FOR VEHICLES

In this section, we consider the tracking control design for an arbitrary member vehicle i of the convoy. As the tracking control problem is always considered in the reference frame Γ ri , for the sake of simplicity, we drop out the subscript r i so that s i,ri becomes s i , and apply similar rules to other state variables. We denote the vehicle state

ξ i = [s i , e l,i , e ψ,i , v i , δ i ] ∈ Ξ i and the input u i = [a i , ω i ] ∈ U i .
Consider the following cost function in least-square form:

J i (ξ i , u i ) = Ti 0 (||ξ i (t) -ξ i,ref (t)|| 2 Q + ||u i (t) -u i,ref (t)|| 2 R ), (9)
where T i is the prediction horizon. ξ i,ref (t) = [s i,ref (t), 0, 0, 0, 0] is the formation keeping reference trajectory from the high level MPC and u i,ref (t) is the reference input usually set to 0. Q = diag(q 1 , ...q 5 ) and R = diag(r 1 , r 2 ) are two positive diagonal weighting matrices. It is shown in [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF] that the stability of MPC can be guaranteed by properly choosing a final state constraint and a final cost. However, in our problem it is difficult to fix a final state or a final state set due to the existence of obstacles. An empirical trial-and-error approach is used to find a long-enough horizon T i that is stabilizing.

At each sampling time t = t 0 , we solve the following constrained optimization problem: Here we provide some details on the constraint design for NBOs. Consider an obstacle described as a polygon in Fig 4 . If directly integrated into the optimization problem, the resulting constraint is not differentiable and it creates two homotopy classes of feasible trajectories. We choose to assign the obstacle to its nearest boundary (in this case the lower one), augment the obstacle region to a triangle and use a parabolic constraint in the form of r ≥ as 2 + bs + c to approximate the obstacle region. The unknown coefficients a, b, c can be calculated by solving the following linear system.

min ui J i (ξ i , u i ), subj. to ∀t ∈ [t 0 , t 0 + T c ], ξi (t) = f i (ξ i (t), u i (t)), (10a) ξ i (t) ∈ Ξ i , u i (t) ∈ U i , (10b)
v i (t) βi (t) + v i (t) l r sin(β i (t)) ∈ [-ā i,lat , āi,lat ], ( 10c
)
f i (ξ i (t), ξ o (t)) ≤ 0, ∀o. (10d)
  s 2 0 s 0 1 s 2 1 s 1 1 s 2 2 s 2 1     a b c   =   r 0 r 1 r 2   . (11)
The solution of optimization problem [START_REF] Kato | Vehicle Control Algorithms for Cooperative Driving with Automated Vehicles and Intervehicle Communications[END_REF] is the optimal control input u * i (t), t ∈ [t 0 , t 0 + T i ] as well as the associated optimal trajectory ξ * i . Let ∆T i be the replanning interval. Then only the fraction [t 0 , t 0 +∆T i ] of u * i (t) will be fed into the vehicle low-level controller and the problem is solved again at t = t 0 + ∆T i .

V. EXPERIMENTS AND RESULTS

Experiments are carried out using the high-fidelity robotic simulator Webots [START_REF] Michel | Webots: Professional Mobile Robot Simulation[END_REF]. Webots provides a realistic vehicle model with steering dynamic response and friction of tires. The proposed convoy algorithm is implemented in Webots using C++ language and optimization problems are solved using ACADO toolkit [START_REF] Houska | ACADO Toolkit -An Open Source Framework for Automatic Control and Dynamic Optimization[END_REF]. The integrated simulation runs on an Ubuntu machine with Intel Core i7 CPU clocked at 3.40GHz and 32GB RAM.

In our experiments, we consider a diamond formation as in Fig. 2. The trajectory generation module is implemented in the foremost vehicle (V1). Note that it can also be implemented in other vehicles or even in a remote server. Vehicles are mounted with localization system and communication devices. Vehicle localization has a 0.2 m precision and communication devices are assumed to be delay-free for this experiment (delays will be considered in the future research). The trajectory of the virtual center is disseminated to all vehicles through communication. The trajectory tracking module is implemented in each vehicle. Here we only consider homogeneous convoys, in which all vehicles have the same characteristics. We consider three sets of experiments, respectively:

• convoy control while driving in a road with straight and curvy segments, • convoy control while driving in a road with a speedvarying LBO in front, • convoy control while driving in a road with multiple NBOs.

In the first experiment, we aim to demonstrate the formation keeping capability of our approach under localization noise. Vehicle convoy starts from zero speed and non-zero errors in formation. Fig. 5a presents the trajectories of convoy vehicles. Fig. 5b provides the evolution of formation errors defined as the difference between the actual vehicle position and the desired position. We observe that at the beginning, vehicles quickly converge to the desired formation. The formation errors of all vehicles decrease to the level of localization uncertainty in less than 5 s. Formation errors increase slightly (while still less than 1 m) when convoy enters and exits the curvy segment. Fig. 5c shows the evolution of speed of the virtual center and individual vehicles. We remark two points: 1) convoy speed is adjusted in the curvy segment to reduce lateral acceleration. 2) Individual vehicle speeds track closely the convoy speed except in the curvy segment, where the vehicle in the inner lane has a smaller speed while the vehicle in the outer lane has a higher speed.

The second experiment aims to verify if the convoy can safely handle speed-varying LBOs. The desired time space is selected as 2 s. Fig. 7 presents the speed curves of the convoy and the LBO in the front and the distance between the virtual center and the LBO. The LBO first decelerates and then accelerates to recover its original speed. We observe that the convoy adapts its speed accordingly. We also observe that a safe distance is always kept.

The third experiment demonstrates the capability of our approach to avoid on-road NBOs. Three NBOs are distributed over two lanes. Vehicle convoy traverses the road segment at a constant speed of 12 m/s. Trajectories are illustrated in Fig. 8, we observe that the vehicles successfully avoid NBOs while maintaining the formation.

The average computation time for the trajectory generation of the virtual center is 0.033 s, and the standard deviation is 0.014 s. For the computation time of trajectory tracking, the average is 0.071 s, and the standard deviation is 0.020 s.

Videos of the three experiments are available on-line1 and can also be found in the supplementary material of the submission.

VI. CONCLUSION

We have presented a hierarchical MPC scheme for on-road convoy control of autonomous vehicles. The high level MPC generates the reference trajectory of the convoy, considering various constraints. The reference trajectory is then communicated to member vehicles with different formation offsets. At low level, a local MPC tracking controller is used to drive each vehicle towards its desired trajectory. Experiments have demonstrated the effectiveness of the approach in handling realistic on-road conditions such as curvy segments, other traffic participants and obstacles. The current paper focuses on the modeling of convoy and the surrounding enviroment and the presentation of the components of MPC based scheme. In the future, we will consider more advanced issues such as the incorporation of uncertainties from communication and perception and the dynamic formation modification in merging scenario. Moreover, we plan to implement and demonstrate our scheme in real vehicles under the framework of the European project AutoNET2030 [START_REF] De La Fortelle | Network of Automated Vehicles: the AutoNet2030 vision[END_REF].

Fig. 1 :

 1 Fig. 1: Kinematic bicycle model in Frenet frame.

Fig. 2 :

 2 Fig. 2: Convoy formation defined by curvilinear longitudinal offset and lane number

Fig. 4 :

 4 Fig. 4: Approximation of obstacle region by a parabola.

  Convoy parameters are given as: 0 ≤ v c ≤ 15 m/s, v c,desire = 12 m/s, |a c | ≤ 1.5 m/s 2 and āc,lat = 1 m/s 2 . The parameters used for trajectory generation of the virtual center is w 1 = 1, w 2 = 4, T c = 10 s and ∆T c = 0.256 s. Vehicle parameters are given as: 0 ≤ v i ≤ 20 m/s, |a i | ≤ 2.5 m/s 2 , |ā i,lat | = 2.5 m/s 2 , |δ| ≤ 0.64 rad, |ω| ≤ 0.05 rad/s, l r = 1.30 m and l f = 1.70 m. The parameters used for trajectory tracking are Q = diag(15, 8, 1000, 0, 20), R = diag(1, 600), T i = 5 s and ∆T i = 0.128 s.

Fig. 5 :

 5 Fig. 5: First experiment: (a) trajectory of the convoy, (b) formation error and (c) speed.

  Fig.6

Fig. 7 :

 7 Fig.7: Second experiment: speed of the virtual center and LBO, distance from the virtual center to LBO.

Fig. 8 :

 8 Fig. 8: Third experiment: trajectory of the convoy. Black boxes mark obstacles.

https://youtu.be/70wMhrYuCTo

ACKNOWLEDGMENT

We thank Florent Altché and Philip Polack for their insightful comments on the paper.

This research was supported by the European FP7 project AutoNet2030 (Grant Agreement NO. 610542).


Emanuele Ferragina

Martin Seeleib-Kaiser

 has transformed and inspired social policy research over the past two decades. Various contributions have confirmed his typology, while others have challenged, and expanded, it from substantive and methodological perspectives. This article contributes to this debate in two ways. First, it provides a comprehensive analysis of the different typologies proposed in the literature, employing the concept of 'ideal types'. Second, it elaborates new directions for research along three dimensions: (1) improving measurement validity by linking macro and micro data to overcome assumptions, largely based on the average (production) worker;

(2) assessing the reliability of typologies over time; (3) systematically integrating both the workwelfare as well as the care-welfare dimensions.

; however these reviews have been neither exhaustive nor designed to synthesise the empirical evidence presented in the literature and propose new avenues for research.

Introduction

The 'three worlds of welfare capitalism' (TWWC) is part of a long sociological tradition (Esping- [START_REF] Esping-Andersen | The three worlds of welfare capitalism[END_REF] rooted in deductive reasoning and the use of ideal types. As Max [START_REF] Weber | On the methodology of the social sciences[END_REF][START_REF] Weber | On the methodology of the social sciences[END_REF]) highlighted, 'the construction of a system of abstract and therefore purely formal propositions ..., is the only means of analysing and intellectually mastering the complexity of social life ' (1904 [1949]: 87). In this vein, Esping- [START_REF] Esping-Andersen | The three worlds of welfare capitalism[END_REF] constructed the welfare regime typology acknowledging the ideational importance and power of the three dominant political movements of the long 20th century in Western Europe and North America, that is, social democracy, Christian democracy (conservatism) and liberalism 1 .

The ideal social-democratic welfare state is based on the principle of universalism, granting access to benefits and services based on citizenship. Such a welfare state is said to provide a relatively high degree of autonomy, limiting the reliance on family and market. In order to achieve autonomy, social-democratic welfare states are characterised by a high level of decommodification and a low degree of stratification. Social policies are perceived as 'politics against the market' (Esping- [START_REF] Andersen | Politics against markets[END_REF]. Christian-democratic welfare states are based on the principle of subsidiarity and the dominance of social insurance schemes, offering a medium level of decommodification and a high degree of social stratification. The liberal regime is based on the notion of market dominance and private provision; ideally, the state only interferes to ameliorate poverty and provide for basic needs, largely on a means-tested basis. Hence, the decommodification potential of state benefits is assumed to be low and social stratification high.

Various contributions have confirmed his typology, while others have challenged, and expanded it, from substantive and methodological perspectives. A number of scholars have criticised Esping-Andersen's typology for the 'misclassification' of specific countries. [START_REF] Leibfried | Towards a European welfare state?[END_REF] and [START_REF] Ferrera | Modelli di solidarietà[END_REF] have argued that Mediterranean countries constitute a separate regime with limited social insurance coverage, which is (at least partially) rooted in clientelism. [START_REF] Jones | The pacific challenge[END_REF] and [START_REF] Goodman | The East Asian welfare state[END_REF] have argued that East-Asian welfare states also form a separate regime. Using the concept 'family of nations', [START_REF] Castles | Families of nations[END_REF] identified the Antipodean countries as radical welfare states. Others have argued that Esping-Andersen's regime typology is quite limited to the work-welfare nexus and not applicable to other welfare state domains, such as the care-welfare nexus [START_REF] Lewis | Gender and the development of welfare regimes[END_REF]), 2 education [START_REF] Busemeyer | Education[END_REF] or healthcare (Bambra, 2005;[START_REF] Wendt | Mapping European healthcare systems[END_REF][START_REF] Reibling | Healthcare systems in Europe[END_REF].

Although valuable, many of these criticisms [START_REF] Kasza | The illusion of welfare regimes[END_REF]Bambra, 2006;[START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF] have not addressed the major methodological and theoretical shortcomings of Esping-Andersen's approach; 3 moreover, some have confused the concept of 'ideal types' with 'real types'. In this article, we highlight the importance of ideal types and review 23 quantitative studies analysing welfare regimes and demonstrate the validity of the TWWC, before suggesting a number of methodological improvements, especially the need to take a more dynamic approach and to integrate micro-and macro-level data in order to obtain a more nuanced perspective on welfare state regimes.

The sociological foundations of typologies

The tradition of historical nominalism and the anthropological interpretation of culture and society [START_REF] Benedict | Patterns of culture[END_REF][START_REF] Benedict | Patterns of culture[END_REF]) have been always hostile to classifications, generalisations and ultimately comparison. From this point of view, ideal types are only a vain sport, a meaningless academic game. Hence, typologies have to be avoided, as they constitute artificial constructions. Moreover, societies have to be analysed without presupposition through a fine-grained detailed case analysis (for a critique, see Smelser, 2003: 648). Although historically much of the academic work on social policy was dominated by idiosyncratic case study analyses and this tradition still seems to dominate much of the textbook market for social policy (for a review, see [START_REF] Seeleib-Kaiser | Learning from Britain?[END_REF], the comparative welfare state research community has largely accepted the benefits of using ideal types and typologies as heuristic devices, as evidenced by the high number of references to Esping-Andersen's seminal work. The longstanding debate about the appropriateness of Esping-Andersen's regime typology can be interpreted as a symptom of the 'adolescence' [START_REF] Weber | On the methodology of the social sciences[END_REF][START_REF] Weber | On the methodology of the social sciences[END_REF]: 104) of comparative social policy research, as scholars have not yet learned 'to navigate safely in the vast area of empirical facts' [START_REF] Weber | On the methodology of the social sciences[END_REF][START_REF] Weber | On the methodology of the social sciences[END_REF]: 104) without using the TWWC ideal typical construction.

At times the concept of 'ideal types' was confused with 'real types', although Esping-Andersen (1990: 28-29) himself was keenly aware of the impurity of real types:

We show that welfare states cluster, but we must recognize that there is no single pure case. The Scandinavian countries may be predominantly social democratic, but they are not free of crucial liberal elements. Neither are the liberal regimes pure types. The American socialsecurity system is redistributive, compulsory and far from actuarial. At least in its early formulation, the New Deal was as social democratic as was contemporary Scandinavian social democracy. And European conservative regimes have incorporated both liberal and social democratic impulses. Over the decades, they have become less corporatist and less authoritarian.

Nevertheless, some countries can be considered as prototypes, that is, countries with welfare state arrangements that very closely relate to the structures of the identified ideal types. Based on Esping-Andersen's decommodification and social stratification index (see Table 1) (see also the Appendix for a detailed explanation of the concepts and the construction of the scores), Sweden (39.1), Norway (38.3) and Denmark (38.1) are the only countries placed one standard deviation above the mean decommodification score (34.9) and the highest ranked in terms of socialist attributes, therefore they seem to clearly fit in the social-democratic regime, whereas France and Germany are very close to the average (27.2), constituting the core of the Christian-democratic regime (also for their high level of conservatism). The United States (US) (13.8) has the lowest decommodification score, if we base our analysis on the decommodification score suggested by Scruggs and Allan (2006: 61), and also has the highest score of liberal attributes. Esping- Andersen's (1990: 54) lower scores for Australia and New Zealand seem to be the result of an arbitrary score attributed to the contributory period and population covered (Esping- Andersen, 1990: 54). 4 All other countries can be understood as approximations, more or less closely fitting the ideal types. * The low scores for Australia and New Zealand seem to be the result of an arbitrary score attributed to the contributory period and the population covered (Esping- Andersen, 1990: 54;[START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF]. ** Standard deviation Source: Authors' elaboration based on Esping- Andersen (1990: 52;1999: 74) and [START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF] The empirical work on the 'typology business'

The selection of studies for our review of the regime literature is based on the following criteria: (1) the use of quantitative methods; (2) the analysis of a minimum of 10 countries of the Organisation for Economic Co-operation and Development (OECD); (3) and the presence of a final classification.

These criteria led us to select 23 studies. 5 Based on the six types introduced in the comparative welfare state literature, that is, the social-democratic, Christian-democratic, liberal, Mediterranean, radical and hybrid, 6 we created a double entry table (Table 2), which establishes how often a country has been classified in one of the six types suggested in the literature (Table 3).

Among these quantitative analyses, most scholars -largely with the exception of Castles (1993) -did not attempt to propose new theoretically rooted typologies (or ideal types), but focused their attention on the inclusion of a wide range of indicators 7 or on the use of more sophisticated statistical techniques. 8 They certainly improved our understanding of welfare typologies from different perspectives, but the result is a confused debate, in which the only reference point seems to be Esping-Andersen's contribution.

The aim of this review is to bring all these studies together in order to verify whether Esping-Andersen's typologies constitute a good guidance for comparative social policy analysis (in particular for the analysis of the 'ideal-typical' cases). The comparison of these different studies is fairly complicated for two reasons: first, typologies are based on different policy domains/indicators; and second, typologies are based on data collected at different points in time.

The use of different policy domains (ie, pensions, unemployment, work injuries, sickness, disability, family, health and education policy, as shown in detail in Table A of the Appendix) or indicators (ie, social expenditure, benefit equality, fairness of the taxation system, social expenditure, institutionalism and corporatism) proposes two challenges for any systematic classification: first, the selection of studies to be included in the final classification; and second, the exclusion of analyses limited to certain policy domains.

The final classification is based on all 23 studies because the 'purity' of Esping-Andersen's ideal types should be verified by considering all sorts of indicators and policy domains. However, when discussing the 'divergent typologies' from Esping-Andersen's measurement, we analyse only studies based on cash transfers and those that included a mix of indicators (including indicators for social transfers) because they relate closely to the concepts of 'decommodification', 'social stratification' and 'defamilialisation' (used by Esping-Andersen's to classify countries and described in the Appendix). Furthermore, education can be characterised as 'market making' and not 'market breaking' (cf [START_REF] Streeck | From market making to state building? Reflections on the political economy of European social policy[END_REF], thereby following a distinctive, different logic from decommodification, social stratification and defamilialisation. Similarly, we exclude healthcare, as the provision of healthcare is not primarily about decommodifying or destratifying workers; in addition, the introduction of healthcare systems was not closely associated with different political movements, as argued by Esping-Andersen regarding the various transfer programmes, but mainly with the strength of service providers and the professional organisations of physicians (Heidenheimer, 1973: 323-4). As argued by Moran (2000: 139), 'health care systems are ... shaped by dynamics of their own'. Based on this rationale, we exclude education and healthcare policies. Many deviations from Esping-Andersen's typology are simply explained by the inclusion of indicators related to the healthcare and education system (Obinger andWagschal, 1998, 2001;[START_REF] Saint-Arnaud | Convergence or resilience?[END_REF][START_REF] Powell | Welfare regimes and the welfare mix[END_REF]Bambra, 2005;[START_REF] Castles | Worlds, families, regimes[END_REF]Schröder, 2008).

None of the models based on education and health care indicators is included in this section, but is taken into account in the summary tables (Tables 2 and3) and the final classification (Figure 1). Another constraint to the comparability of different typologies is the use of data collected at different points in time (from 1950s onwards). At a theoretical level, the comparison between typologies based on data from different points in time might seem less problematic because, according to Esping- Andersen (1990: 9-12), regimes are the result of the influence of dominant political movements, suggesting stability over time. 9 According to the criteria illustrated above and on the basis of the consistency of the ranking for each country in the various studies, the quantitative data collected in Table 2 are reorganised along a continuum, ranging from the most social-democratic to the most liberal welfare state, in order to test Esping-Andersen's ideal-typical construction. We identify the following groups of countries based on the consistency of the classifications:

• 'pure countries': classified more than 80% of the time in the same regime type (Sweden, Norway and Denmark -social-democratic; France, Germany and Austria -Christian-democratic; and the US -liberal);

• 'medium-high internal consistency countries': classified between 61% and 80%of the time in the same regime type (Finland -social-democratic; Belgium and Italy -Christian-democratic; and Canada, the UK, Australia and Japan -liberal);

• 'medium internal consistency countries': classified between 51% and 60% of the time in the same regime type (New Zealand and Ireland -liberal).

Among the countries included in the various analyses, only Switzerland and the Netherlands are not classified in the same regime type in more than 50% of the studies. Both countries are hybrids, classified in the literature in four distinct 'worlds' (Table 3), and have therefore been excluded from our further systematic literature review. Goodin (2001: 15) describes the Netherlands as a postproductivist model, which does not follow the decommodification logic used to build welfare typologies, and Obinger (1998) argues that the Swiss welfare state cannot be clearly categorised.

The pure countries

Austria, Denmark, France, Germany, Norway, Sweden and the US are 'pure models'. Sweden and the US are classified in the same world in every analysis; their internal consistency places their social security systems at the opposite ends of the continuum from social-democracy to liberalism. The other countries are in an intermediate position (see the next section). Like Sweden, Denmark and Norway form part of the social-democratic regime. Only Goodin (2001) classified Norway as liberal (Table 3), based on its high labour market participation and low social security expenditure. However, using these criteria is somewhat problematic, as the Scandinavian model is based on high labour market participation [START_REF] Huber | Development and crisis of the welfare state[END_REF], which can in effect lead to lower spending levels, as is demonstrated by the fact that according to the most recent OECD spending figures, France has the highest social expenditure (OECD, 2010).

Hence, we consider Goodin's classification as an outlier, which does not undermine the overall classification of Norway as a social-democratic welfare state.

The Danish welfare state was classified by Bambra (2006) as Christian-democratic and as liberal by [START_REF] Korpi | The paradox of redistribution and strategies of equality[END_REF]. Whereas Bambra's decommodification score seems to be out of sync with all other decommodification scores, 10 without providing any explanation, [START_REF] Korpi | The paradox of redistribution and strategies of equality[END_REF] included the Danish case en passant in their analysis of pension models. 11 The latter classified countries along the dimensions of targeted versus universal benefits and redistribution. According to 9 In case of more than one classification presented in the study we have included the most recent typology in our analysis. 10 ll other models and in particular the recalculation proposed by [START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF] confirm Esping-Andersen rather than Bambra. 11 The country appears only in one table and not in the remaining empirical discussion.

Korpi and Palme's classification, Denmark belongs to the 'basic security model', because the public pension system is based primarily on flat-rate universal benefits, whereas the other Scandinavian/social-democratic welfare states have built their pension systems on the 'encompassing security model', which includes the provision of earnings-related benefits on top of the universal basic provision (Korpi and Palme, 1998: 668-9). Neither Bambra (2006) nor [START_REF] Korpi | The paradox of redistribution and strategies of equality[END_REF] seem(s) to invalidate the overall classification of Denmark as belonging to the socialdemocratic regime.

Austria, France and Germany represent the 'pure' Christian-democratic cases (Table 3). Only Bambra (2006) classified France in the social-democratic group. 12 Germany, on the other hand, has been classified as a hybrid by [START_REF] Ragin | A qualitative comparative analysis of pension systems[END_REF], based on its structure of the pension system, 13 and by [START_REF] Obinger | Drei Wohlfahrtsstaates?[END_REF] using Esping-Andersen's social stratification index. 14 [START_REF] Ragin | A qualitative comparative analysis of pension systems[END_REF] defined Germany a spare case close to the Christian-democratic group, and [START_REF] Obinger | Families of nations and public policy[END_REF] in a more recent article classified Germany as Christian-democratic, in line with mainstream literature. Both instances lead us not to question the purity of Germany's classification.

While Austria is usually identified as belonging to the Christian-democratic welfare regime, Esping- [START_REF] Esping-Andersen | The three worlds of welfare capitalism[END_REF] and Bambra (2006) have classified the Alpine republic as belonging to the social-democratic model, largely as a result of its generous pension system. [START_REF] Powell | Welfare regimes and the welfare mix[END_REF] have classified Austria as liberal, mainly as a result of including education-related indicators in their analysis. Based on our systematic literature review of studies, we have no doubt that Austria can be considered as a 'pure' Christian-democratic welfare state, especially if one excludes employment policy, which longer than in many other European countries had been based on the concept of 'Keynesianism' [START_REF] Scharpf | Crisis and choice in European social democracy[END_REF].

Medium-high internal consistency

Finland (social-democratic), Belgium and Italy (Christian-democratic), Canada, the UK, Australia and Japan (liberal) are part of the medium-high internal consistency group (Table 2). While Finland is a social-democratic country with a strong secondary Christian-democratic component (Tables 2 and3), largely based on its pension system (Esping- [START_REF] Esping-Andersen | The three worlds of welfare capitalism[END_REF][START_REF] Ragin | A qualitative comparative analysis of pension systems[END_REF][START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF], Belgium can be categorised as a Christian-democratic country with a secondary social-democratic component (Table 1), captured by various decommodification indices (Esping- [START_REF] Esping-Andersen | The three worlds of welfare capitalism[END_REF]Bambra, 2006;[START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF] and some other indicators [START_REF] Castles | Identifying welfare state regimes[END_REF]. 15 At the political level, the Christian-democratic nature of the Belgian welfare state regime and its secondary social-democratic component is explained by the dominance of Christian Democrats in government coalitions, in which socialist parties (especially from Wallonia) were the junior partners (Seiler, 1977: 460).

Italy clearly belongs to the Christian-democratic regime type (Tables 2 and3) whenever the analysis is limited to three ideal types. If, however, a fourth ideal type, the Mediterranean type, 16 is added, Italy has always been classified as Mediterranean [START_REF] Bonoli | Classifying welfare states[END_REF][START_REF] Gallie | Welfare regimes and the experience of unemployment in Europe[END_REF][START_REF] Obinger | Families of nations and public policy[END_REF][START_REF] Saint-Arnaud | Convergence or resilience?[END_REF][START_REF] Soede | Unequal welfare states, The Hague, The Netherlands: Social Cultural Planning Office[END_REF][START_REF] Castles | Worlds, families, regimes[END_REF]. This raises a methodological question about the number of ideal types to be included in an analysis of rich democracies. 17 Nevertheless, based on the principle of parsimony and in accordance with the overwhelming number of other analyses, we maintain that Italy should be considered as belonging to the Christian-democratic type as suggested by Esping-Andersen (Table 3).

An overwhelming majority of scholars have grouped Canada into the liberal category (Table 3) and only Bambra (2004Bambra ( , 2006) ) and [START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF] have placed the country into the Christian-democratic group. In order to overcome the lack of data for female labour force participation rates (one of the four indicators used by her to measure defamilialisation), Bambra (2004: 205) attributed to Canada a score on the basis of the average values of all the other countries included in her analysis (Bambra, 2004: 205). This procedure, although acknowledged by Bambra, seems arbitrary and questions the overall validity and reliability of the classification. The other two models that include Canada in the Christian-democratic cluster (Bambra, 2006;[START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF] have used higher decommodification scores than Esping-Andersen for the Canadian pension system. 18 Similar to Canada, the UK also belongs to the group of countries with medium-high internal consistency (Table 2), but with a slightly higher fragmentation (Table 3). The hybrid classification is partly explained by the transitional state of the pension system. 19 Using the concept of 'defamilialisation', Bambra (2004) groups the UK into the Christian-democratic regime. In her analysis she uses a wage replacement rate of 50% for maternity leave for a duration of 18 weeks (Bambra, 2004: 205), which seems a too-generous estimate, as in 1996 maternity leave provided a wage replacement rate of 90% for the first six weeks and a flat-rate benefit of £55 per week for the remaining 12 weeks [START_REF] Gauthier | Comparative family policy database[END_REF]. Hence, the average replacement rate only reached the level of 50% for mothers with an income below the average wage. [START_REF] Castles | Identifying welfare state regimes[END_REF], [START_REF] Korpi | The paradox of redistribution and strategies of equality[END_REF] and Shalev (2007) have classified Australia as radical and Gal (2004) as Christian-democratic (Table 3). Gal's typology does not seem to be representative of the generosity of the welfare state systems as a whole, as his analysis is limited to work injuries programmes, which represent less than 1% of the total public social expenditure (OECD, 2010). The introduction of a radical type encounters the same issues raised above relating to the Mediterranean type, that is, the number of ideal types to be included in the analysis. Only when the radical type is included in the analysis, and with the exception of Gal's analysis, Australia is not classified as liberal.

Japan does not seem to clearly fit any of the typologies and has been characterised as belonging to a distinct East-Asian or Confucian welfare state [START_REF] Goodman | The East Asian welfare state[END_REF]. Esping- [START_REF] Esping-Andersen | Hybrid or unique?[END_REF] has rejected this criticism, and has labelled Japan as a 'hybrid'. However, we suggest excluding Japan from our analysis in this article altogether for two reasons: (1) it is impossible to assess quantitatively whether Japan belongs to a distinct East-Asian or Confucian regime, as robust comparable data for the time period in other East-Asian or Confucian countries are unavailable; and (2) as Japan's political and societal system is rooted in the tradition of neither liberalism, nor Catholicism nor social democracy, it seems inappropriate to characterise the Japanese welfare state as belonging to either of the regimes, even if some of the variables are closely related to certain regime characteristics. Obviously, this does not mean that Japan per se should be excluded from comparative welfare state analysis of rich democracies, but that any such analysis should include 17 Studies that group countries using descriptive statistics (employing the decommodification index [see Esping- [START_REF] Esping-Andersen | The three worlds of welfare capitalism[END_REF] or double entry tables [START_REF] Castles | Identifying welfare state regimes[END_REF]) force the number of typologies. They determine arbitrarily the existence of three or four worlds; hence, countries with a strong secondary component (ie, Italy, Australia, New Zealand, Ireland and Finland) might be part of a regime only because of the method used. 18 This is partially due to the different assumptions used to calculate pension replacement rates by Esping- Andersen (1990: 54) and Scruggs and Allan (2006: 64). 19 During the 1980s, the pension system underwent a series of incremental reforms. [START_REF] Ragin | A qualitative comparative analysis of pension systems[END_REF] analysis, based on 1986 data, reflects this transitional phase, which consequently led to the classification of the country as hybrid.

functional equivalents of public social policy to make the comparison meaningful (see [START_REF] Seeleib-Kaiser | Globalisierung und Sozialpolitik[END_REF][START_REF] Estevez-Abe | Welfare and capitalism in postwar Japan[END_REF].

Medium internal consistency

The medium internal consistency group includes only two liberal countries: New Zealand and Ireland (Table 2). Similar to Australia, New Zealand has been classified as radical [START_REF] Castles | Identifying welfare state regimes[END_REF]Shalev, 2007) and hybrid [START_REF] Ragin | A qualitative comparative analysis of pension systems[END_REF]. Although Ireland has overwhelmingly been categorised as liberal, [START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF], Bambra (2004Bambra ( , 2006) ) and [START_REF] Shalev | Introduction[END_REF]Shalev ( , 2007) ) have categorised Ireland as Christian-democratic, using decommodification and defamilialisation (Bambra, 2004) indices and a range of other indicators. Based on the strong influence of the Catholic Church (especially with regard to family policies; see [START_REF] Daly | The functioning family: Catholicism and social policy in Germany and the Republic of Ireland[END_REF], it seems plausible that the Irish welfare state has a number of elements that place the country close to the Christian-democratic regime cluster.

Discussion

The results of the previous section grosso modo confirm the existence of three worlds of welfare capitalism; all countries (with the exception of the Netherlands and Switzerland) are classified in more than 50% of the studies in one of the three original worlds defined by Esping-Andersen (Table 2). 20 Furthermore, seven countries are 'pure' (Austria, Denmark, France, Germany, Norway, Sweden and the US), six are medium-high consistent (Australia, Belgium, Canada, Finland, Italy and the UK), and two are medium-consistent (Ireland and New Zealand). The social-democratic and the Christiandemocratic worlds are smaller and more cohesive than the liberal one (Table 2).

The countries analysed are placed on a continuum based on three criteria (Figure1): ( 1) the number of times each country has been classified in the same group; (2) the positioning of the seven pure countries; and (3) the presence of countries with a high secondary component. 21 Socialdemocratic countries are positioned to the left-hand side, the Christian-democratic in the middle and the liberal on the right-hand side.

Sweden, Norway and Denmark are the pure social-democratic cases. Finland belongs to the same group, however, differently from the other three social-democratic countries has a large secondary Christian-democratic component (see Table 2 and Figure 1). Austria, France and Germany are 'pure' Christian-democratic welfare states. Although Belgium also belongs to the Christiandemocratic regime type, it has a number of social-democratic elements, which, however, fall short of our definition of 'secondary component'. Italy, on the other hand, has a strong secondary Mediterranean component (Table 2 andFigure 1). 20 The classification is even closer to Esping-Andersen's regimes if the mixed typologies (those that include health and education) are excluded (Table 3). 21 Attributed to the countries classified in more than 25% (because this score represents half membership) of the studies into a category different from the main classification. As already emphasised, the liberal regime is the least cohesive and occupies 40% of the total spectrum (six countries out of 15). The US is the only 'true' liberal country, since Canada incorporates some Christian-democratic elements (Table 2), and the UK has been classified in five different regimes (Table 2). Australia and New Zealand have been classified into a separate radical category by about 20% of the studies reviewed. Finally, Ireland completes this world with its strong secondary Christian-democratic component (Table 2 and Figure 1). The continuum elaborated to review the quantitative models confirms Esping-Andersen's theory and affirms its connection to Weberian ideal types. In the next section, attention turns to measurement validity, and the proposal of an innovative range of ways to improve the 'typology business'.

Regime futures?

The results reported in the previous section confirm the overall validity of Esping-Andersen's typology. However, many analyses aiming to test the validity of TWWC insufficiently deal with the issue of measurement validity, are largely based on a static measurement of welfare regimes and lack a systematic integration of the work-welfare and the care-welfare nexi. In order to connect the evolution of welfare state systems over time to Esping-Andersen's theoretical framework, we suggest:

(1) using latent class analysis (LCA) or fuzzy sets, as suggested by [START_REF] Kvist | Exploring diversity: measuring welfare state change with fuzzy-set methodology[END_REF], to link micro and macro data to improve measurement validity; 22 (2) investigating welfare state evolution without assuming path dependence, by exploiting panel data; (3) analysing the work-welfare and care-welfare nexi in an integrated manner. 23 22 A first attempt to use micro data to investigate regime theory has been performed by [START_REF] Goodin | The real worlds of welfare capitalism[END_REF]; however, they did not systematically integrate micro and macro data, as they assumed the macro categorisation to be fixed. First, measurement validity can be improved by using LCA to link micro and macro data. The three quantitative methods (descriptive statistics, 24 cluster analysis 25 and principal component analysis [PCA] 26 ) used to class countries in different regimes do not sufficiently link welfare regime theory and empirical measurement. The use of descriptive statistics imposes a fixed number of typologies and requires the establishment of thresholds and cut-off points; cluster analysis cannot sufficiently connect theoretical hypotheses and measurement; 27 and PCA does not deal with the possibility that the co-variation of the observed variables can potentially be dependent on the existence of one or more latent variables (ie, welfare generosity), which exert a causal inference on the observed variables.

SE NO DK

Conversely, LCA has a confirmatory character because concepts and causal models have to be derived from theoretical and deductive perspectives and then tested with the available empirical data. Those concepts and causal models are never purely derived from the existing data, as is the case for PCA, but they need to have a strong theoretical background. LCA creates latent variables from the indicators, but differently from PCA handles measurement errors, resulting in more stable measures of unobserved variables. LCA can be proficiently exploited to verify the robustness of the Esping-Andersen regimes and statistically generate more comprehensive and theoretical sophisticated typologies.

Furthermore, welfare generosity is usually calculated on the basis of entitlements for the average (production) worker (APW). 28 However, if the past three decades in many OECD countries can be accurately characterised as an 'age of dualization', largely maintaining previous levels of social protection for labour market insiders, while retrenching policies for an increasing number of outsiders, as suggested by Emmenegger et al (2012: forthcoming), we need to move beyond the 23 For other innovative approaches, see [START_REF] Kvist | Exploring diversity: measuring welfare state change with fuzzy-set methodology[END_REF] and [START_REF] Scruggs | Welfare state generosity across space and time[END_REF]. 24 [START_REF] Castles | Identifying welfare state regimes[END_REF], [START_REF] Bonoli | Classifying welfare states[END_REF], [START_REF] Korpi | The paradox of redistribution and strategies of equality[END_REF], [START_REF] Gallie | Welfare regimes and the experience of unemployment in Europe[END_REF], Goodin (2001), Bambra (2004Bambra ( , 2005Bambra ( , 2006) ) and [START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF]. 25 [START_REF] Kangas | The politics of social security[END_REF], Obinger andWagschal (1998, 2001), [START_REF] Gough | Social assistance regimes[END_REF], [START_REF] Kautto | Investing in services in West European welfare states[END_REF], [START_REF] Saint-Arnaud | Convergence or resilience?[END_REF], [START_REF] Powell | Welfare regimes and the welfare mix[END_REF], Bambra (2007b) and Castles and Obineger (2008). 26 [START_REF] Shalev | Introduction[END_REF]Shalev ( , 2007)), Wildeboer Shut et al (2004), Schröder (2008) and Vrooman (2009). 27 Three types of cluster analysis have been applied to welfare regime theory: (1) hierarchical cluster analysis (HCA), which finds the closest pairs of cases and then by amalgamation combines other cases to form clusters;

(2) k-means cluster analysis (KCA), which proceeds to the amalgamation of cases by starting from a fixed number of regimes determined by the researcher; and (3) HCA and KCA combined. HCA has been applied to different dimensions of social policy, that is, healthcare, education and social security [START_REF] Kangas | The politics of social security[END_REF]Obinger andWagschal, 1998, 2001;[START_REF] Kautto | Investing in services in West European welfare states[END_REF][START_REF] Saint-Arnaud | Convergence or resilience?[END_REF][START_REF] Powell | Welfare regimes and the welfare mix[END_REF], generating a wide range of welfare regime typologies. This widespread use of HCA has been criticised for its inductive foundations. According to this critique, HCA cannot support a theoretical framework thereby contravening Esping-Andersen's theoretical premises [START_REF] Gough | Social assistance regimes[END_REF]. In order to overcome this issue, Gough employed KCA and Bambra (2007b) and [START_REF] Castles | Worlds, families, regimes[END_REF] subsequently integrated HCA and KCA. Despite this methodological advance, the evaluation of welfare generosity with all types of cluster analyses is just as arbitrary as grouping countries based on descriptive statistics, because: (1) the selection of the number of clusters is completely subjective, one can decide to stop at different levels of amalgamation and there are no precise rules to know when to stop the analysis; (2) clusters are generated in every case, no matter what indicators are used; (3) the analyses are unstable: alternative algorithms produce totally different results starting from the same data; and most importantly (4) these types of analyses do not adequately connect deductive reasoning and empirical measurements because of the predetermination of the number of regimes (necessary to apply KCA), contrarily to what has been suggested in the literature [START_REF] Gough | Social assistance regimes[END_REF], is not a theoretically driven way to connect deductive reasoning and empirical measurement. Deductive reasoning can be formulated, instead, on the basis of typologies driven by the selection criteria (as in the case of latent class analysis) in accordance with a refined theoretical framework. 28 [START_REF] Scruggs | Welfare state generosity across space and time[END_REF] proposed to overcome this problem by using various household types. This would considerably increase the accuracy of the models; however, it would not completely resolve this issue, as the typologies would still be based on statutory entitlements, albeit for different socio-economic groups.

entitlements for the APW to assess the generosity of welfare regimes. A model based on survey data at the individual or household level 29 might help to overcome this problem, because it provides the possibility to calculate different 'real' decommodification indices (or other generosity indices), distinguishing between various socioeconomic groups, for instance along the dimensions of gender, education and skill levels. 30 Second, welfare regime typology analyses are often rather static: Esping-Andersen's (1990) original analysis related to data from around 1980 and most subsequent studies were also limited to a snapshot of a certain point in time. A cornerstone of the regime typology is its foundation in the three main political movements of Western Europe, that is, social democracy, Christian democracy and liberalism (cf Esping- [START_REF] Esping-Andersen | The three worlds of welfare capitalism[END_REF]). If, however, political parties have undergone significant transformations over the past decades, as some research suggests (see [START_REF] Pontusson | Explaining the decline of European social democracy[END_REF][START_REF] Seeleib-Kaiser | Neubeginn oder Ende der Sozialdemokratie?[END_REF][START_REF] Rueda | Social democracy inside out[END_REF][START_REF] Seeleib-Kaiser | Party politics and social welfare[END_REF], we should assess stability and change of welfare regimes over time. Using longitudinal micro data at the individual or household level would seem to be an appropriate approach to capture possible dynamics. However, comparable panel data are only available for a limited number of countries 31 and thus using micro simulations for various stylised workers, for example, APW, high-and low-general skilled workers in the service sector, at various points in time might be a suitable alternative (for pensions, cf [START_REF] Bridgen | Politically dominant but socially flawed[END_REF].

Third, the work-welfare nexus has been at the centre of the welfare regime typology debate, primarily focusing on 'old' social risks, such as old age and unemployment. However, research has pointed to the emergence of new social risks since the 1980s in many rich democracies [START_REF] Bonoli | Time matters: post-industrialization, new social risks, and welfare state adaptation in advanced industrial democracies[END_REF]. Although the work-welfare nexus still constitutes a core element of welfare regimes, defining social risks and the consequent generosity of entitlements to protect citizens only from this point of view seems outdated. The care-welfare nexus, as emphasised by [START_REF] Lewis | Gender and the development of welfare regimes[END_REF], constitutes the second core element of the welfare state and cannot be neglected. Demographic trends and the difficulty for parents to reconcile work and care further demonstrate the importance of this nexus. Many authors have argued that the future of welfare state systems will be dependent on the ability to balance work and family life [START_REF] Lewis | Women and social policies in Europe: Work, family and the state[END_REF][START_REF] Crompton | Employment and the family[END_REF][START_REF] Gilbert | A mother's work[END_REF][START_REF] Esping-Andersen | The incomplete revolution[END_REF]. Thus, to capture the dynamics of welfare state change it would seem appropriate to analyse welfare regimes along both the work-welfare and the care-welfare nexi, as retrenchment along one dimension might coincide with the expansion of policies in the other [START_REF] Bleses | The dual transformation of the German welfare state[END_REF][START_REF] Fleckenstein | The dual transformation of social protection and human capital: comparing Britain and Germany[END_REF]. Although [START_REF] Esping-Andersen | The social foundations of postindustrial economies[END_REF] took into account some of the criticisms highlighted by [START_REF] Lewis | Gender and the development of welfare regimes[END_REF] to integrate the work-welfare and the care-family welfare nexi, his empirical model was somewhat unbalanced, as he compared the decommodification index based on data from around 1980 with the defamilialisation index based on data from the 1990s. Once again we suggest a dynamic approach, integrating the decommodification and defamilialisation indices.

Conclusion

Our literature review confirms the existence of three worlds of welfare capitalism in the light of ideal types. We propose a classification of rich democracies on a continuum from the most purely socialdemocratic (Sweden) to the most liberal country (the US). Following [START_REF] Weber | On the methodology of the social sciences[END_REF][START_REF] Weber | On the methodology of the social sciences[END_REF]) and 29 Micro data should be used to calculate individual (or household level) generosity indices that can then be aggregated according to different demographic characteristics. 30 The Cornell Equivalent File (CNEF, 1970(CNEF, -2008) ) (ECHP, 1994(ECHP, -2001) ) and the longitudinal component of its successor, the European Union Statistics on Income and Living Conditions (EU-SILC, 2005 onwards); and the Luxembourg Income Study (LIS, 1980(LIS, -2004)). 31 See previous note. [START_REF] Durkheim | Les règles de la méthode sociologique[END_REF], Esping-Andersen emphasised the importance of deduction to develop sociological theory; his typology provides an excellent starting point and heuristic device for empirical research.

The critiques proposed in the field, from nominalist and technical perspectives, did not always seem to fully grasp the theoretical foundations of the three worlds of welfare capitalism. Despite all their criticisms, subsequent analyses overall have confirmed the consistency of the three worlds, as all countries3432 have been classified in more than 50% of the respective studies in the same group. This empirical evidence has been used in this article to construct an ideal-typical continuum. The continuum shows that different countries and regimes have dissimilar internal consistencies; in particular the liberal model appears to be the least homogeneous (Figure 1). Finally, we have emphasised the necessity to connect the deductive framework provided by Esping-Andersen with more refined empirical instruments to analyse welfare typologies using theoretically specified methods (like LCA), analysing the provision not only for the APW, considering the potential evolution of regimes over time and integrating systematically the work-care and care-welfare nexi.

Finally, we want to emphasise that although typologies are not set in stone, their existence is essential in providing theoretical orientation and meaning as well as in establishing new methodological principles. Too often, sophisticated data analyses become an end rather than constituting a means to understanding the real world. Ideal types can function as beacons in the twilight of complex empirical research.

All research in the cultural sciences in an age of specialisation, once it is oriented towards a given subject matter through particular settings of problems and has established its methodological principles, will consider the analysis of the data as an end in itself. It will discontinue assessing the value of the individual facts in terms of their relationships to ultimate value-ideas. Indeed, it will lose its awareness of its ultimate rootedness in the valueideas in general. And it is well that it should be so. But there comes a moment when the atmosphere changes. The significance of the unreflectively utilised viewpoints becomes uncertain and the road is lost in the twilight. The light of the great cultural problems moves on. Then science too prepares to change its standpoint and its analytical apparatus and to view the streams of events from the heights of thought. [START_REF] Weber | On the methodology of the social sciences[END_REF][START_REF] Weber | On the methodology of the social sciences[END_REF]: 112) To conclude, typologies are a fundamental heuristic tool for welfare state scholars, even for those who claim that in-depth analysis of a single case is more suited to capture the complexity of different social policy arrangements. Welfare typologies have the function to provide a comparative lens and place even the single case into a comparative perspective -to paraphrase Goethe, 'those who know nothing of foreign languages [read other welfare regimes] know nothing of their own'; wanderlust educates. maximum of welfare obligations to the household'. Conversely he defines defamilialization as follows (Esping-Andersen, 1999: 45):

I shall use 'de-familialization', yet another admittedly awkward word, to capture policies that lessen individuals' reliance on the family; that maximize individuals' command of economic resources independently of familial or conjugal reciprocities. It is, like the concept of decommodification, empirically more a matter of degree than of an 'either-or'. Given that women's (or at least mothers') family responsibilities easily restrict their ability to gain full economic independence solely via work, their de-familialization, as many studies have shown, depends uniquely on the welfare state. The indicators included to measure defamilialisation are: the intensity of family welfare provision (measured as the percentage aged living with children, unemployed youth living with parents as a share of total, weekly unpaid hours of work for women [Esping-Andersen, 1999: 63]), the entity of welfare state services to families (measured as family service spending as a percentage of GDP, public child care coverage, home-help coverage [Esping- Andersen, 1999: 71]).

  AT = Austria, AU = Australia, BE = Belgium, CA = Canada, CH = Switzerland, DE = Germany, DK = Denmark, FI = Finland, FR = France, IE = Ireland, IT = Italy, JP = Japan, NL = Netherlands, NO = Norway, NZ = New Zealand, SE = Sweden, UK = United Kingdom, US = United States. Notes: Esping-Andersen miscalculated the decommodification score for the US. Therefore we provide the initial value [in brackets] and the corrected score.

  = Austria, AU = Australia, BE = Belgium, CA = Canada, CH = Switzerland, DE = Germany, DK = Denmark, FI = Finland, FR = France, IE = Ireland, IT = Italy, JP = Japan, NL = Netherlands, NO = Norway, NZ = New Zealand, SE = Sweden, UK = United Kingdom, US = United States. Notes: 1. We only mention those authors who are not in accordance with our classification. 2. Studies indicated in bold are those that do not include indicators for health and education. Source: Authors' elaboration.

Figure 1 :

 1 Figure 1: Final classification

  provides a good source of comparable micro data on demographics, income and employment (including data on the UK [BHPS], Australia [HILDA], Korea [KLIPS], Switzerland [SHP], Canada [SLID], the US [PSID] and Germany [SOEP]). Other comparative sources of micro data are: the European Community Household Panel

Table 1 -Revisiting Esping-Andersen's decommodification and stratification scores to identify liberal, conservative and social-democratic 'ideal types'

 1

		Decomodification		Social stratification	Esping-Andersen
						prototypes
	Country	Scores		Country	Degrees of
					liberalism,
					conservatism and
					socialism
	AU	13*		US	High liberalism	US
	US	14.2 [13.8]		CA	High liberalism
	NZ	17.1*	Mean -one	CH	High liberalism
			standard
		deviation = 19.5
	CA	22		AU	Medium-high
					liberalism
	IE	23.3		JP	Medium-high
					liberalism
	UK	23.4		UK	Medium-high
					liberalism
	IT	24.1		IE	Low liberalism
	JP	27.1		NZ	Low liberalism
	FR	27.5	27.2 (Mean)	FR	High conservatism	FR
	DE	27.7		DE	High conservatism	DE
	FI	29.2		AT	High conservatism
	CH	29.8		BE	High conservatism
	AT	31.1		IT	High conservatism
	BE	32.4		NL	Medium-high
					socialism
	NL	32.4		FI	Medium-high
					socialism
	DK	38.1	Mean + one	DK	High socialism	DK
			standard
			deviation (34.9)
	NO	38.3		NO	High socialism	NO
	SE	39.1		SE	High socialism	SW
	Mean	27.

Table 2 : Classification in the same model as percentage of the total number of classifications

 2

	Country	Social-	Christian-	Liberal	Mediterranean	Radical	Hybrid	Total
		democratic	democratic					classifications

Table 3 : Classification proposed in the literature (in bold: typologies that exclude indicators for health and education)

 3

	Country	Social-democratic	Christian-democratic	Liberal	Mediterranean	Radical	Hybrid
	AU		Gal (2004), Powell and	15 classifications		Castles and
			Barrientos (2004)			Mitchell (1992),
						Korpi and Palme
						(1998), Obinger
						and Wagschal
						(1998), Shalev
						(2007)
	AT	Esping-Andersen	14 classifications	Powell and Barrientos
		(1990), Bambra (2006)		(2004)
	BE	Esping-Andersen	16 classifications	Powell and Barrientos			Obinger and Wagschal (1998)
		(1990), Castles and		(2004)
		Mitchell (1992),
		Bambra (2006),
		Scruggs and Allan
		(2006)
	CA	Bambra (2005)	Bambra (2004, 2006),	15 classifications
			Scruggs and Allan (2006)
	DK	20 classifications	Bambra (2006)	Korpi and Palme (1998)
	FI	12 classifications	Esping-Andersen (1990);				Obinger and Wagschal (1998)
			Ragin (1994), Powell and
			Barrientos (2004), Scruggs
			and Allan (2006), Castles
			and
			Obinger (2008)
	FR	Bambra (2006)	21 classifications
	DE		21 classifications				Ragin (1994), Obinger and
							Wagschal
							(1998)
	IE		Shalev (1996), Bambra	10 classifications	Obinger and		Ragin (1994), Obinger and
			(2004, 2006), Bambra		Wagschal		Wagschal
			(2005),		(2001)		(1998)
			Scruggs and Allan (2006),
			Shalev (2007)

Table A :

 A Typologies, policy domains and data

For a critique, see[START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF].

Accordingly, and in addition to the various qualitative studies, the following quantitative analyses were excluded:[START_REF] Kangas | The politics of social security[END_REF],[START_REF] Fawcett | Social exclusion, social citizenship and decommodification[END_REF],[START_REF] Gough | Social assistance regimes[END_REF],[START_REF] Lødemel | An offer you can't refuse[END_REF],[START_REF] Kautto | Investing in services in West European welfare states[END_REF];[START_REF] Mcmenamim | Is there an East-Central European variety of democratic capitalism?[END_REF];[START_REF] Ferreira | Welfare regimes in the EU 15 and in the enlarged Europe[END_REF]. Table A includes all studies reviewed.

Hybrid' is a residual category used by many authors (eg,[START_REF] Ragin | A qualitative comparative analysis of pension systems[END_REF] Obinger and Wagschal, 1998, 2001).

See Table A in the Appendix.

Mainly descriptive statistics, cluster analysis and principal component analysis.

As discussed for Denmark, also in this caseBambra (2006) presents the only discrepant classification[START_REF] Scruggs | Welfare state decommodification in 18 OECD countries[END_REF] using a similar indicator to Esping Andersen include the country in the Christian-democratic regime).

He based his conclusions on the private sector's share of total pension expenditure, the expenditure on pensions of public employees as a percentage of Gross Domestic Product (GDP), the number of occupational programmes and the pension decommodification index.

This typology includes indicators related to health care.

[START_REF] Castles | Identifying welfare state regimes[END_REF] included expenditure data, benefit equality and taxation.

[START_REF] Leibfried | Towards a European welfare state?[END_REF] was the first to theorise about a distinct Mediterranean regime.

With the exception of the Netherlands and Switzerland.

8 IT 12 classifications Scruggs and Allan (2006) Bonoli (1997), Gallie and Paugam (2000), Saint-Arnaud and Bernard (2003), Soede et al (2004), Castles and Obinger (2008) JP Esping-Andersen (1990), Korpi and Palme (1998), Bambra (2005) 9 classifications Castles and Obinger (2008), Schröeder (2008) NL Esping-Andersen (1990), Powell and Barrientos (2004); Scruggs and Allan (2006) 11 classifications Korpi and Palme (1998), Saint-Arnaud and Bernard (2003) Castles and Obinger (2008)

supported by the ESRC Grant

Appendix

Decommodification

According to Esping-Andersen 'decommodification refers to the degree of to which individuals, or families, can uphold a socially accepted standard of living independently of market participation' (Esping- Andersen, 1999: 35-47). Esping-Andersen measured decommodification by considering the eligibility rules and the restrictions of entitlements, levels of income replacement and the range of cash benefits provided to deal with traditional social risks: unemployment, sickness and old age (Esping- Andersen, 1990: 47). Based on these indicators, countries are divided into the following regimes (see Table A):

• liberal: Australia, Canada, Ireland, New Zealand, the UK and the US;

• conservative: Finland, France, Germany, Italy, Japan and Switzerland;

• social-democratic: Austria, Belgium, Denmark, the Netherlands, Norway and Sweden.

Social stratification

The social stratification concept attempts to measure through a wide range of indicators how key welfare state key institutions operate in structuring class and social order: '[t]he organizational features of the welfare state help determine the articulation of social solidarity, division of class, and status differentiation' (Esping- Andersen, 1990: 55). Welfare states participate in the process of social stratification according to their own historical legacy (conservative, liberal, socialist).

Social stratification is measured by using seven indicators (Esping-Andersen: 1990: 71):

• corporatism (conservative principle), 'measured as a number of occupationally distinct public pension schemes'; • etatism (conservative principle), 'measured as expenditure on pensions to government employees as % of the GDP'; • means-tested poor relief spending (liberal principle), 'measured as % of total public expenditure'; • private pensions spending (liberal principle), 'measured as % of total pension';

• private health spending (liberal principle), 'measured as % of total health spending';

• average universalism (socialist principle), 'measured as the average score for sickness, unemployment and pensions'; • average benefit equality (socialist principle), measured as 'average differential between basic and minimum social benefits for sickness, unemployment, and pensions (based on net, aftertax benefits). Benefit differentials are based on the ratio of guaranteed basic social benefit to the legal maximum benefit in the system'.

Based on these indicators, countries are divided into the following regimes:

• liberal: Australia, Canada, Japan, Switzerland and the US; conservative: Austria, Belgium, France, Germany and Italy; • social-democratic: Denmark, Finland, the Netherlands, Norway and Sweden • not clearly classified: Ireland, New Zealand and the UK.

Defamiliarisation

The defamilialisation concept captures the level of familism in each welfare state. With the term 'familistic welfare state', Esping- Andersen (1990: 45) Social expenditure (as a % of GDP), social expenditure financed through contributions (as a % of GDP)

OECD (1989,1992)