L Lahousse

S Boripatkosol2

S Leleu

J David

S Ducourtieux

T Coorevits

O Gibaru

LNE Activies in Nanometrology: flatness reference calibration algorithm

Introduction

The Laboratoire National de Métrologie et d'Essais (LNE) has developed an innovative ultra precision coordinate measuring machine [START_REF] Lahousse | Z calibration of the LNE ultra precision coordinate measuring machine[END_REF] traceable to the national length standard to measure three-dimensional objects with nanometric uncertainties (figure 1). The measuring range is 300 mm x 300 mm x 50 µm. The objective in term of uncertainty is to reach 30 nm in X and Y directions for a displacement of 300 mm and about few nanometers for a vertical displacement of 50 µm. On this machine, we use four capacitive sensors to measure the position along z direction. These sensors target the flat surface of cylinders (300 mm diameter) used as flatness references. To measure the shape of these aluminum references with nanometric uncertainties, we propose a measurement method based on a propagation process in which we introduce an angular measurement to compensate the curvature error inherent in this method.

The measurement process uses the same sensor technology (capacitive sensor) we use on the machine. This paper presents the measurement method, its validation and the first results.

Propagation process with 4 capacitive sensors and an electronic level.

The measurement principle is based on a propagation process. We displace a matrix of sensors along the profile to measure [START_REF] Gao | High accuracy profile measurement of a machined surface by the combined method[END_REF]. To introduce measurement redundancy, we use a matrix with 4 capacitive sensors. The distance d between each sensor is 20 mm (see figure 2). For each position j on the profile, we measure the distance (m 1j ,…,m 4j ) which separates the capacitive sensor from the flatness reference.

The angular motion of the sensor matrix is measured thanks to an electronic level. The unknown parameters are the position of the electronic level origin compared to the origin of the capacitive sensors, the relative positions of the capacitive sensors and the profile height called f. For each position j, we have to determine the motion errors of As we use 4 sensors, only two parameters are necessary to characterize their relative positions. We call em2 and em3 the gap between the central sensor and the peripheral ones (figure 2a). The same way, we fix the profile extremities equal to zero (figure 2b). The general equations that link all the parameters are given in the reference [START_REF] Elster | Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors[END_REF] and will not be presented here. Based on those equations, the system to be solved is:

For each position of the sensor matrix corresponds five lines in the system. The first four equations concern the distance measurement given by the capacitive sensor.

The 5th equation concerns the angular measurement given by the electronic level. We use three methods to solve the system.

In the first method, we solve subsystems in which we favor capacitive sensor measurements. In this method, we proceed in two steps. First, the capacitive sensor information is used to start the resolution of the system. The resulting profile contains

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ -
1 0 0 1 0 .. 0 0 0 0 0 .. 0 0 0 0 0 2 3 1 .. 0 0 0 0 0 .. 0 0 0 1 0 2 1 .. 0 0 0 0 1 .. 0 0 0 0 1 2 1 .. 0 0 0 0 0 .. 0 0 0 0 0 2 3 1 .. 0 0 0 0 0 .. 0 0 .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 0 0 0 0 .. 1 0 0 0 0 .. 0 0 0 0 0 0 0 .. 2 3 1 0 0 0 .. 0 0 0 1 0 0 0 .. 2 1 0 0 0 .. 0 0 0 0 1 0 0 .. 2 1 0 0 0 .. 1 0 0 0 0 0 0 .. 2 3 1 0 0 0 .. 0 1 1 0 0 0 0 .. 0 0 1 0 0 .. 0 0 0 0 0 0 0 .. 0 0 2 3 1 0 .. 0 0 0 1 0 0 0 .. 0 0 2 1 0 .. 1 0 0 0 1 0 0 .. 0 0 2 1 0 .. 0 1 0 0 0 0 0 .. 0 0 2 3 1 0 .. 0 0
a curvature error coming from the not perfect relative position of the sensors inside the matrix [START_REF] Gao | Precision measurement of cylinder straightness using a scanning multi-probe system[END_REF]. The second step consists in using the level information to eliminate the curvature error without local modification of the profile.

In the second method, we use the information of the electronic level to calculate the rotation of the sensor matrix instead of the capacitive sensor measurement. In this method, we favor electronic level measurement In the third method, we solve the whole system using the weighted least squares method that consist in dividing the system of equations by an estimation of each measurement uncertainty. To solve the system, we minimise

( ) 2 N 1 i i i i b ax y ∑ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - σ
. As the weight of each measurement is inversely proportional to its uncertainty, the result is physically better. However, it is necessary to know the sensor measurement uncertainty and the level measurement uncertainty.

Experimental results

At the moment, to test the method, a dedicated bench was developed (figure 3). above the flatness reference. The resolution of the inclination sensor is one microradian that is not sufficient to be comparable with the resolution of the capacitive sensors. Nevertheless, the repeatability of the inclination sensor is better than its resolution so it is possible to increase the resolution making the sensor oscillate. For that purpose, two piezoelectric actuators are introduced in the developed bench: one to rotate the sensor matrix and another one to make the reference oscillate. We suppose that the capacitive sensors and electronic level are not yet calibrated and that we do not inclination sensor flatness reference 4x4 capacitive sensor matrix and integrated inclination sensor surface plate have a reliable estimation of their uncertainty. To compare the three solving methods, we introduce a parameter "L" which characterizes the ratio of the capacitive sensor measurement uncertainty with the electronic level measurement uncertainty

( lev sen L σ σ =
). The variation of this parameter corresponds to a change of the information weighting. We present on the figure 4, the deviation to the mean value of flatness reference straightness profiles obtained using the weighted least squares method with "L" equal to one millimeter

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = µrad x nm x lev sen σ σ
. Each position corresponds to an average over one oscillation of the inclination sensor. The oscillation amplitude is 10 µrad. On this graph, we estimate the repeatability to +/-4nm. This repeatability confirms the stability and the efficiency of the measurement. We notice in Fig. 5 that the shape of the profile evolves between two extreme profiles corresponding to the two extreme methods. For high value of L (L= 100 mm), the profile converges to the one we obtain when we favor the level information (second method). When L is small (L= 0.1 mm), the situation is reversed and the profile converges to the one we obtain when we favor the capacitive sensor information (first method).

The 60 nm deviation between the different methods is significant in regard of our objectives. This is due to the inconsistency of the capacitive sensor and electronic level information. The calibration of the two sensors will significantly decrease this inconsistency. The experimental evaluation of the sensor measurement uncertainty will allow fixing L and consequently to find the optimal weighting for the weighted least squares method.

Conclusion

The few nanometer repeatability of the profile measurement shows the efficiency of the method in which we increase the level resolution by averaging the measurement during level oscillations. The several methods used to solve the system show a deviation of 60 nm that corresponds to the inconsistency of the information coming from the electronic level and the capacitive sensors. When these sensors will be calibrated, we will use the weighted least squares method to improve the presented straightness measurement method.

Figure

  Figure 1: photograph of the machine

Figure 2 :

 2 Figure 2: straightness measurement using 4 capacitive sensors (left), convention used to locate the relative position of the sensor (a) the points of the profile (b)

Figure 3

 3 Figure 3: developed bench Two inclination sensors are integrated: one inside the capacitive sensor matrix and one

Figure 4 :Figure 5 :

 45 Figure 4: deviation to the mean value of straightness profiles obtained using the weighted least squares On figure 5, we present the evolution of the straightness profile when the value of "L" changes.


Catherine Bouteiller

Bruno Faivre D'arcier

From "distance-based" price discrimination to "use-based" price discrimination: tracks to improve fare box revenue in urban public transport

The aim of this paper is to initiate a discussion on how to improve urban public transport fare box revenue by increasing quality of revenue and especially adequacy of "distance-based" fares to the public transport network and the urban form of the city. What are the determining factors of an appropriate fare policy? Are customers' expectations fully taken into considerations into fare grids? We consider that "use-based" fare policy can respond to the search of friendly use network, seamless strategy for a better mobility in the urban area and equity of customers regarding their social conditions and their use of the network. Public transport business model must take into account fare policy as a variable of the economic performance of the system.

We will analyse the public transport network and fare grid structure in 10 cities corresponding to different types of urban morphologies: mono centric (Tokyo, Singapore), composite (Seoul, Paris, London, Madrid, Toronto) or poly-centric (Amsterdam, Copenhagen, Berlin). Our study consists in comparing "distance-based" fare system chosen by these cities, through 3 scores. By studying "distance-based" price discrimination such as distance based, zone based or cells comb based, we will have elements to understand how delivering the appropriate pricing system contributes to a better use of the public network in a given spatial form of the city. Then we will focus on "use-based" price discrimination through combination of 4 scores relying on the analysis of 4 criteria: time period, number of trips, mean of payment, mode use.

Fare grids can be a tool to respond more accurately to the changing needs of the different urban schemes and to customers' expectations. But the study of fare grids enhances the complexity of combination of fare possibilities offered by fare management tools to implement "use-based" fare discrimination. We will go further with the idea that not only fare grids are a lever to improve fare box but also a determinant element in network usability and orientation of demand.

Fare grid, fare box, urban planning, monocentric cities, composite cities, poly centric cities, price discrimination

From "distance-based" price discrimination to "use based" price discrimination (BOUTEILLER, Catherine; FAIVRE D'ARCIER, Bruno) 13

INTRODUCTION

For 10 years, fare box ratios have been constantly decreasing in France (-31% to -36% according to network sizes) and public transport commercial revenue covers operating costs by 35% only (GART, 2012). In April 2012, a few experts set off reflexions on public transport business model and debts (Bonnafous and al, 2011). Public transport business model has to explore all tracks to improve revenue. Fare discrimination strategies deserve to be studied among other solutions. The observation of French trends (GART, 2012) over the last decade shows a constant raise in terms of kilometric supply (+13% to +37%). But raise of patronage do not follow the trend (+8% to +15%) and surprisingly average price for day tickets fell by 11% in real terms, and 18% for annual season ticket. Monthly season ticket is the more common fare and its decrease varies with the size of considered network. At the same time fare recovered per trip is always getting lower. This trend should be reversed. French public authorities, but Paris, have been implementing massively flat fares that they consider as a friendly use solution.

Looking at what's happening in Europe or in a few cities around the world, practices appeared to be different: flat fare is not the norm. Fare discrimination has been introduced through fare management systems that allow combination of fares. Simplicity of the tools (smart cards) and marketing; offer seamless use of the networks This paper analyses price discrimination in 10 cities around the word. Fare grids have been studied through a methodology that split up price into three components: "distance-based" fare, "use-based" fares and "passenger-based" fares. The two first dimensions will be studied in this paper:

Distance-based price discrimination reflects a situation where price depends on characteristics of the trips and network capability to respond to spatial demand of trips (zonal fares, kilometric, cells zonal). Questioning distance-based score will lead to analyze pertinence of the distance-based grid chosen by the PTA. These relationships are not often studied as mentioned: "despite their importance to the public user experience the literature on the relationship between fare structures and public transport network design is surprisingly sparse" [START_REF] Dodson | Principles of Public Transport Network Planning: A review of the emerging literature with select examples, Griffith Urban Research Program[END_REF].

Second component of fare studied in this paper is "use-based" price discrimination. Use-based price discrimination is a terminology that associates ticket based strategies and use based practices in relation to the willingness of the customers to travel in special periods, using combination of specific modes, paying with their preferred supports. Use-based price discrimination will be approached through four dimensions of usage: time of use, number of trip use, mode use and mean of payment use.

Passenger-based price discrimination will not be studied as it is very commonly practiced and well known by public transport authorities: it is a segmentation relying on demographic groups: senior, student, unemployed…It refers to social rebates. Public transport authority's challenge as far as fare strategy is concerned "is to determine a tariff structure that reconciles the user's need for an affordable public service with the commercial interests of the operators, while at the same time pursuing the authority's social and spatial objectives" [START_REF] Mezghani | Study on e-ticketing in public transport[END_REF]. First part will explore conformity between city morphology and public transport organization, showing that passengers flow characteristics will be key information to choose the appropriate distance-based fare grid structure. Second part makes an evaluation of distance-based fare grid by studying relations between fare and distance to the center for a monthly season ticket. Public transport authority may use distance-based grid to give priority of development to specific areas thanks to specific profile of fares. Third part offers a methodology to estimate capability of Public Transport Authorities to make price discrimination on "distance-based" practice or "use-based" practice of the network and gives a mapping of the cities.

Polycentric structures

These cities have a structure with no dominant central city but several interacting main cores generating different travel behaviours [START_REF] Cervero | Travel demand and the 3Ds: density, diversity, and design[END_REF]. Bertaud characterizes polycentric model describing cities where "functions were very much in the same way as a monocentric city, jobs, wherever they are, attract people from all over the city. The pattern of trips is different, however. In a polycentric city each sub-centre generates trips from all over the built-up area of the cities. Trips tend to show a wide dispersion of origin and destination, appearing almost random". In these cities, main passenger's flows are very specific [START_REF] Levine | Decentralization of Jobs and Emerging Suburban Commute[END_REF]: -Short distance trip in suburbs and cross commuting (Van der Laan, 1998).

-Very strong need of interaction between modes to connect all suburbs: bike/train / park and ride.

-Strong integration between sub networks of the metropolitan area as demonstrated in the Ramstad area [START_REF] Clark | Commuting in restructuring urban regions[END_REF].

-Place of car use must be considered in suburbs.

Composite model

"Poly-centric" cities defined above, corresponds also to cities called "polynuclear" [START_REF] Lemoine | Système tarifaire des transports collectifs: Éléments de réflexion[END_REF]. Between "polycentric" cities and "monocentric "cities, there are cities undergoing changes. Bertaud calls them "composite" cities and gives the following definition: "The "composite model" is the most common type of urban spatial structure. It contains a dominant centre but a large number of jobs are also located in the suburbs. In this type of cities most trips from the suburbs to the CBD will be made by mass transit while trips from suburb to suburb will use individual cars, motorcycles, collective taxis or minibuses. The "composite model" is in fact an intermediary stage in the progressive transformation of a monocentric city into a polycentric one. As a city population grows and the built-up area expands, the city centre becomes more congested and progressively loses its main attraction. The original raison d'être of the CBD was based on its easy accessibility by all the workers and the easy communication within the centre itself because of its spatial concentration." [START_REF] Bertaud | GHG Emissions Urban mobility and efficiency of urban morphology[END_REF]. In composite cities, characteristics of monocentric and polycentric cities are mixed:

-Fewer trips from central to peripheral area.

-New trips from suburb to suburb.

-Long distance trips (depending of area size) for workers commuting.

-Urbanization along urban corridors. Urban network should change from radial scheme to dispersed schemes. Otherwise car use might be considered as the only alternative to public transport when network infrastructure doesn't follow commuting patterns changes. [START_REF] Bertaud | The spatial organization of cities: deliberate outcome or unforeseen consequence? working paper[END_REF] gives the following summary of evolution of the forms: "No city is ever 100% monocentric, and it is seldom 100% polycentric (i.e., with no discernible "downtown"). Some cities are dominantly monocentric, others are dominantly polycentric and many are in between. Some circumstances tend to accelerate the mutation toward polycentricism-a historical business centre with a low level of amenities, high private-car ownership, cheap land, flat topography, grid street design-and others tend to retard it-a historical centre with a high level of amenities, rail-based public transport, radial primary road network, and difficult topography preventing communication between suburbs."

In this context, the shape of the public transport network implemented must be considered. It reflects needs of mobility at one given time of the development of the city.

I-B FROM URBAN PLANNING TO TRANSPORTATION NETWORK

Several studies explain the challenge of deploying a public transport network in the different forms of cities to respond to the demand of the different commuting patterns. Several network strategies have been described: radial network [START_REF] Thompson | Planning Considerations for alternative transit route structures[END_REF], [START_REF] Dodson | Principles of Public Transport Network Planning: A review of the emerging literature with select examples, Griffith Urban Research Program[END_REF], corresponds to mono centric cities, dispersed network strategy [START_REF] Thompson | Planning Considerations for alternative transit route structures[END_REF] is more common in composite cities, and radial network strategy can be found in polycentric cities with specific grid [START_REF] Newman | Sustainability and Cities: Overcoming Automobile Dependence[END_REF]. Routes and nodes described respond to some of the needs of the passengers depending on the structure of the cities. But as cities change forms quickly as well as passenger commuting patterns, public transport infrastructure have difficulties in responding to new demands… The phenomenon of dispersion in the metropolitan area leads to new schemes of mobility. "Individual behaviour reshapes urban form in the long terms [START_REF] Du | Exploring Reciprocal Influence between Individual Shopping Travel and Urban Form: Agent-Based Modeling Approach[END_REF]. Public transport networks tend to reflect the morphology of the cities that evolves from one type to another, for example from mono-centric shape to poly-nuclear or polycentric shape. Public transport structures need to adapt. Nevertheless, as infrastructure costs are high, discrepancies appear. More agile systems such as smartcards can allow seamless use of all modes. The result is a better use of the network. Urban forms of the city and the public transport network shape characteristics have a strong impact on cost structure of the

II -CHARACTERISTICS OF "DISTANCE-BASED" FARE SYSTEMS

Can fare grids be a response to customer's need of mobility and influence public transport use at the same time? Fare policy and fare grid reflect organization of the network. But, whereas public transport network infrastructure is not flexible and need investments, fare grid can introduce flexibility to give a better orientation of use of the network. Fare can be an incentive to use a mode, a route.

II-A THE MAIN CHALLENGES FOR A TRANSPORT PRICING POLICY

EMTA (European Metropolitan Transport Authorities) study on electronic ticketing in public transport gave the following definition of public transport pricing policy [START_REF] Mezghani | Study on e-ticketing in public transport[END_REF]: "The greatest challenge for a pricing policy is to determine a tariff structure that reconciles the user's need for an affordable public service with the commercial interests of the operators, while at the same time pursuing the authority's social and spatial objectives ".

Pricing strategy must reconcile targets of the 3 actors of the public transport:

For public authorities, fare should simultaneously increase public transport modal share, attract passengers, set low prices and simple tariffs, balance prices and limit exclusion, increase multi-modality and inter-modality, minimize public subsidies or financial compensation clearing and share revenues.

As far as the operator is concerned, fare strategy should cover costs and maximize profit, build an attractive public transport system (image & loyalty card), ease control routes, reduce fraud, simplify revenue collection.

For the passenger, main expectations are minimizing transport cost, facilitating use of PT, multi-modal use and equity. Pricing policy on the other hand has to take into account two main dimensions: spatial and social organization is an input: extensive routes and highly density networks are not priced the same way. Price should be fair for long distance commuter especially when low revenue customers live far from central business district where they work… A second dimension is the demographic profile of the users of the network: are they mainly students, workers, or senior citizen? [START_REF] Fatisson | Principes de tarification sociale dans les transports publics urbains en Europe, mémoire Master2 Transport, Espace, Réseaux[END_REF] Price policy is built on three pillars

Mezghani explains in EMTA study [START_REF] Mezghani | Study on e-ticketing in public transport[END_REF] that "journey-based" price discrimination reflects characteristics of the trips. As it is directly linked to the spatial dimension, it is suggested to be called "distance-based" fare strategy in this paper.

Zonal concentric fare, cell zones fare, flat fare and kilometric fare belong to "distance-based" strategy. The second pillar of price strategy described by Mezghani is "passenger based" price discrimination and concessional fare. These fares are linked to demographic segmentation.

The third pillar, that is not very often studied, will be called "use-based" price discrimination. [START_REF] Fearnley | Une décennie de tarification dans les réseaux de transport urbains[END_REF] describes what would be "inventive pricing". Innovation described by Ferney is time differentiation (off peak fare), membership fees, surcharge on high standard (airport fare), rebate and pre-sold ticket.

To increase fare box as [START_REF] Fearnley | Une décennie de tarification dans les réseaux de transport urbains[END_REF] explained, there is not much to expect from "passengerbased" price discrimination as it is mostly linked to social policy. Flexible fare systems can be the solution to maintain modal share and service levels. Flexible fare system is a notion very close to "use-based" price discrimination: for example giving possibility to use a public transport service and pay afterwards at the end of the month can be very convenient for those who do not want to care about tickets and season tickets. Flexible fare means that it will be charged different price for the same route (for example different prices for different kind of services: air con bus, express or standard bus).

In fact "inventive pricing" of Ferney is also a mix of "ticket-based" strategy and "use-based" strategy. Fearnley's inventive pricing belongs to "use-based" price discrimination. As fare management is always improving, these strategies of differentiation are getting more and more common. Their impact on fare box revenue has to be studied and part 3 suggests a method to evaluate the capability of public transport authorities to implement "use-based" price discrimination.

II-B "DISTANCE-BASED" FARE GRID DESIGN DEPENDS ON MOBILITY PATTERNS

"Distance-based" pricing is the most common used grid and reflects PT's network orientation.

Fare strategy must take into account public transport network forms to respond to commuting patterns. This idea has been developed by (Lemoine and Predali,2007) who made a correspondence between the morphology of the city and the distance based grid. They consider six case studies belonging either to composite (called polycentric by the authors) or multipolar models: Chicago, Los Angeles, London, Montreal, Munich and Berlin. "Pricing structure is related to the urban region structures and is seen as an answer to mobility needs: most polycentric areas have concentric zone structures, multipolar areas apply cells pricing".

The following table gives a definition of all "distance-based" strategies from flat fare to distance based fare and gives the impact of the grid on the spatial use of the network. [START_REF] Mezghani | Study on e-ticketing in public transport[END_REF] From "distance-based" price discrimination to "use based" price discrimination (BOUTEILLER, Catherine; FAIVRE D'ARCIER, Bruno) 13 th WCTR, July 15-17, 2013 -Rio, Brazil 8 Do public transport authorities implement these strategies systematically on their territories? Is there a link between implemented fare strategies and characteristics of the city: urban sprawl, need for motorized trip in the peripheral area?

III-ASSESSING "DISTANCE-BASED" AND "USE-BASED" FARE GRIDS

This part proposes a methodology to see if the fare strategies of ten cities are reflective of their urban development. As previously said, price discrimination can be described through three components: "distance-based" score, "use-based" score and "passenger-based" score. A focus will be made on "distance-based" score and "use-based" score as "passengerbased" score has already been studied by [START_REF] Hodson | Price differentiation and fare integration in urban public transport[END_REF]. On one hand he considers discrimination on journey characteristics such as: time period for traveling, travelled distance and mode choice. On another hand he considers differentiation on the basis of tickets characteristics: availability of tickets at cheaper price than single for example.

III A-METHODOLOGY AND DEFINITION OF THE STUDY

Distance-based, use-based and passenger-based price discriminations are three approaches to qualify a fare system in a given city. Most of public transport authorities, except those using flat fare, are making a combination of fare related to these notions. 8) on each metrics is expressed as a proportion of the maximum score on that metrics for the 10 cities studied (Figure12). Then, for a given city, all intermediate values are summed. A weighting might be used: for example, first step on last step price/km metrics will be weighted by 25% in the final distance based score to avoid giving too much emphasis to tapering charges policy. Final score is calculated for the 10 cities and rescaled to run from 0 to 1 (Figure14). -"Ticket availability "is one of the factors proposed by Hodson to quantify each dimension of price discrimination: it is the number of differentiated tickets available. "Maximum discount" is the second one used by Hodson: it is the maximum price difference between a differentiated ticket and a ticket that is not differentiated, expressed as a proportion of the cost of the higher priced ticket. -The two additional factors added are the "maximum raise factor which is the opposite calculation of the maximum discount and the "marketing complexity factor" which is a more qualitative factor for innovation: peak /off peak fare, pay as you go card… Literature on urban planning gave many indications whether a city belongs to one category or another: monocentric, polycentric or composite morphologies. [START_REF] Murayama | Toward comparative study on spatial planning issues and approaches in diverse megacities: Tokyo and megacities around the world[END_REF] gave a complete approach of growth, city design, and sustainable development in several megacities around the world in particular London, Toronto, Seoul. For Amsterdam, (the Ramstad Polynet study, 2005) gave precious indications on trip flow between cities around Amsterdam. For Madrid region, (Caceres and Sánchez, 2009) made an analysis of the changing commuting patterns and gave several maps of trip flows that were compared to public transport network. Focus on flows between CBD and peripheral area determines the nature of trips needed. For Copenhagen, the synthesis on the Fingerplan made by [START_REF] Ostergard | Spatial planning in Denmark. 51 st IFHP world Congress[END_REF] produced a few maps of polycentric scenario in Copenhagen region.

III B-"DISTANCE-BASED" PRICE DISCRIMINATION SCORE: CALCULATION AND RESULTS

Selection of the cities

Main information on cities is summarized in the following table (Figure 7). 1. Monocentric cities have developed distance based-strategies with very accurate levels of price, reflecting the relative cost of mode or route. In Singapore, 39 levels of price are possible. In Tokyo as in Singapore, price per kilometre in the city centre is much higher than in suburbs: for example price /km in the CBD is 82% higher than price per km 40 km away.

2. Composite cities have a fare strategy that must fit needs of mobility in all areas: long distance commuters to historical CBD or short trips in suburbs. Composite cities have to face development of suburbs and new patterns from suburbs to suburbs without going through the city centre. Main challenge is that PT network is not always able to deliver good service: suburbs do not benefit from the same infrastructure that the one implemented in the city centre. People may go through the centre to reach another suburb. Composite cities have concentric zonal fare and very often, flat fare in the city centre.

3. Polycentric cities have another challenge: delivering services in all directions: trips can start anywhere in the area. Public transport use is very often multi modal. Fare strategy is cell zone fares; price is equal for a number of crossed cells wherever trip starts in the area. Price per km in the centre or 40 km away is nearly the same.

Calculation of fare/km from city center to 40 km away.

Levels of price per kilometre are always calculated to search if the pricing strategy is corresponding to an orientation of the use of the network in its geographical dimension:

1. Poly-centric cities deliver the same fare per km in all the area of the PTA. Price increases proportionally with distance covered through cells zones of equal distance.

The target is clearly facilitating trips in all the polycentric area. Price per km is the same everywhere which means that the level of service delivered is the same wherever the journey starts in the network. th WCTR, July 15-17, 2013 -Rio, Brazil

13 3. Composite cities use flat fares in the centre and concentric zonal fares: the level of degression is less important not to penalize suburban inhabitants that do not benefit from the same level of services. Flat fare policy in the centre is very different from one city to another. Flat fare is usually a 10 km area. It can be more, as in Toronto where the Toronto Transport Authority uses flat fare until 20 km on its network, which is not the case on the Go network (11 km). Go network is not supposed to be used in CBD and price /km in CBD is prohibitive compared to TTA network. The three studied sub scores give an idea of the global adequacy of fare policies to: city morphology, level of detail of the fare strategy and level of price from city center to 40 km away.

1. Tokyo or Singapore have a maximum score for adequacy of fare to city morphology with distance based strategies but do not differentiate level of fares the same way (5 steps for Tokyo, 20 steps for Singapore between 1 and 40 km). Singapore fully uses distance based pricing with a high level of sophistication: 39 levels of prices, 80 % difference between CBD and 40 km area. Singapore has adapted very accurately its offer of service through integrated card fare depending on route distance. Mono-centric cities use kilometric-based strategies. These cities are able to evaluate the cost of each route and the system of calculation reflects distances (mostly radial).

2. Amsterdam, Copenhagen and Berlin metro area have a real strategy to deliver the same price per km everywhere on the network (Figure .10). This strategy is linked to their polycentricism. Copenhagen and Amsterdam have a fare strategy that enhances peripheral patterns and full use of the network. These cities have low density of population. Their networks are not so dense in metropolitan areas and public transport modal shares in peripheral zones are not high. Strong fare integration is supposed to be a vector of development of public transport. Polycentric cities make low difference between price / km delivered in the CBD and 40 km away.

3. Madrid, London and Paris are composite cities, with radial networks that partially respond to the need of trips. Their fare strategy leads them to zonal fare with few levels but strong rebate between CBD and 40 km away. Seoul wants to develop PT between the city and new centres: Bundang, Ilsan, Pyeongchon, Sanbon, and Joongdong were planned around the city of Seoul. Seoul subway is interconnected with the other network and the railway, same for Madrid who added 2 zonal fares corresponding to areas of Castilla la Mancha.

The level of efficiency with regards to urban need is medium in composite cities where radial networks do not match needs of suburbs-to-suburbs trips. Zonal Fare strategies are very different from one city to another: level of fares, number of steps, reductions to take into account distance to city center and density of the network. Concentric zonal fares should be very accurate to maintain equity.

How to conceal radial networks and geographic equity? In Paris, London, Singapore, high income households are located very close to CBD: a high level of fare in the city centre reflects this strategy. Nevertheless, fare grid systems do not appear to cope well with the type of city. London still has a radial network, but city morphology is tending to a polycentric form. Is the zonal fare system still adapted compared to cell zones or distance based? Level of price is very high and is not a good signal for car users. Paris has a radial network but faces strong peripheral commuting patterns: public transport authority is reducing the number of zones and even mentions the possibility of a flat fare, which seems contradictory. Tokyo has long distance commuters but surprisingly low integration of routes (except on pay as you go cards) and no season ticket integrated.

Copenhagen has a polycentric fare structure but its network is still radial.

"Distance -based "price discrimination is the most common grid as it reflects mainly the urban form of the city with the limits enounced above.

III C-RESULTS OF THE STUDY OF "USE-BASED" PRICE DISCRIMINATION STRATEGIES

Synthesis of use-based score

As described in chapter III-A, all fares are compared using the percentage of discount or rise compared to the price of a single ticket paid cash. A score also takes into account the marketing complexity and the number of fares offered in each category. Score is calculated from 0 to 1: score 1 is the maximum attributed to the city that reached the best sub score. Use-based score performance is determined by the capability of the network to make price discrimination on several choices: time, mode, trip and means of payment: Singapore uses all possibilities, as London. Seoul focuses its strategy on mode (bus) and payment, no season ticket or special offers related to time use or trip use. Amsterdam or Tokyo do not make any difference of price for use of personal card (payment score) which is not the strategy of London, Seoul or Singapore that are making discount for users of smartcards. Cities like Paris do not use these tools for differentiating prices.

Time use

Existing peak / off peak prices makes the difference for the marketing complexity factor, for time differenciation score: Singapore, Tokyo, London. Some cities do not promote timebased pass. There is no season ticket in Seoul. Tokyo, with its "all line metro pass", has an expensive offer. Season tickets are very prohibitive in Singapore that has a very high level of raise factor for this offer. Amsterdam has its own strategy proposing a large offer of daytickets.

Trip use

Trip use offers are very common in Tokyo. This strategy is not used in Seoul and very limited in other cities (10 trips tickets only). In Singapore or Tokyo, access to unlimited trips products is prohibitive with a very high level of fare.

Payment use

Developing smartcards tools does not mean systematically offering a discount for using them. London, Singapore, Toronto have incentives to use pay-as-you-go cards, whereas Amsterdam and Tokyo do not give any incentive for using them. Seoul offers several products with discounts or raise. Paris ticket "T +" paid on bus is more expensive and makes part of this payment use fare policy even if it is not very innovative

Mode use

These products are very linked to the capability of differencing modes and combination of mode products. Marketing differentiation is strong when it exists a demand for express lines, specific combination as metro+bus, park & ride strategies, air-con lines … Seoul bus reengineering of the network is visible with a maximum raise factor on specific lines, the same for Singapore (air-con, express lines) … Amsterdam has a strong park & ride policy. Madrid metro fare are different from a line to another

III D-RESULTS OF THE COMBINATION OF DISTANCE BASED PRICE DISCRIMINATION SCORE AND USE BASED PRICE DISCRIMINATION SCORE

Use based and distance based strategies are complementary and their conjoint study gives a good glance of the PTA strategy to maximize revenue. th WCTR, July 15-17, 2013 -Rio, Brazil 18 To explore price discrimination mapping (Figure 14), it is necessary to question a few assumptions.

1-Use based pricing strategy is not linked to distance based strategy as it could be assumed: There are 2 high levels of use based discrimination scores in two different systems of network pricing (kilometric for Singapore and cell zones for Amsterdam). 2-Is there a link between distance-based price strategy and city morphology? : It seems to be the case except for Paris: Paris's low score, as far as distance based is concerned, is due to the inadequacy of its fare strategy to city morphology. Reducing number of zonal fare towards flat fares is the opposite of the strategies followed by cities facing expansion to multipolar activities. 3-It there a link between high level of fare and high level of discrimination? Singapore, London or Tokyo have a high level of fare and high scores for distance or use based fares, but Seoul, Berlin or Amsterdam are very well positioned for price discrimination whereas price in their core centre is not high. 4-Is use-based strategy a factor of strong differentiation between cities? Innovation on services leads to price discrimination in several cities and it seems obvious that use based system is a motor of price discrimination: Seoul's development of bus leads to innovative pricing on its network, London's pay-as-you-go card allows several practices as peak/off peak prices, Berlin's strategy to develop "short trip" with specific fares (40 % rebate) on a single fare, and on the opposite, Madrid raises the unit price if using public transport for more than 9 stops.

CONCLUSION

This scoring analysis of the fare structure of PT networks on a limited sample of large cities in the world shows a great diversity of situations. This is partly due to local historical reasons, but it appears price differentiation is developing, in relation with new pricing technologies. It is hoped the combination of distance-based and use-based principles could better cope with the morphology of cities and the mobility needs of the population. It should also help to increase patronage and fare box revenue, even if such assumptions are not easy to assess.

Fare grids can be a tool to respond more accurately to the changing needs of the different urban schemes and to customers' expectations : kilometric distance based, zonal fare, cells zonal fare correspond to different types of urban morphologies as mono-centric cities, composite cities or polycentric cities. But the study of fare grids enhances the complexity of combination of fare possibilities offered by fare management tools to implement "use-based" fare discrimination. In Asia, innovating aspect of fare grid is its capability to give a signal to customer expectations in matter of level of service and at the same time a tool to improve fare box ratio. We will go further by emitting the idea that fare grid is not only a lever to improve fare box but it has an effect on network usability and orientation of demand.

Use-based" grids are very complex and promote or restrict time, trip, mode, and payment uses of the network. These grids are used by some PTA as signals to orient demand toward a service or another: air-con bus, express line, pay-as-you-go card, short trips, park and ride…

Fare box can increase with fare grids that will combine the appropriate distance based policy (reflecting the urban structure) with the appropriate use based policy reflecting the structure of cost of the network and the willingness to pay of the travellers for specific services.

This first approach should be developed further, but this needs to have an access to more detailed data, such as the distribution of the fare box revenue according to the real consumption (trips per each fare). Price differentiation score should be then weighted with consumed quantities. Another improvement of the methodology would be to take into account the third type of differentiation, i.e. "Passenger-based" pricing, which is commonly used by PTAs for social reasons. Evolutions can be recently observed, with a progressive shift from individuals' status (i.e., children, retired people, students,) to households'.

Further developments of this study should also try to measure the elasticity of demand and the possible induction of patronage for a given price discrimination strategy chosen among "use-based" pricing possibilities.

Figure 1 -

 1 Figure 1 -The three major types of urban organisation and their influence on trip generation.

Figure 2 -

 2 Figure 2 -Correspondence between city morphology and public transport network structure

Figure 3 -

 3 Figure3-"Distance based "strategies, definitions[START_REF] Mezghani | Study on e-ticketing in public transport[END_REF]

  Figure 5 associates metrics to each definition. Distance-based score and use-based score, only have been calculated for this paper. These scores give an indication of how the cities use fare levers. Hodson's scoring methods are applied. Hodson calculated a composite of indices of what he called "journey-based price discrimination, ticket-based discrimination and discounts for students and young people". In this paper, metrics are slightly different from Hodson's ones, as some indicators and metrics are redefined and there is no calculation of a global score. The aim is to position each city one to another through their capability of using either use-based or distance-based strategies. Distance-based score and use-based score have been calculated as the result of intermediate calculations of the following metrics. Each intermediate score (Figure

Figure 4 -

 4 Figure 4 -Components of price discrimination

Figure 5 -

 5 Figure 5 -Methodology of "distance based" fare scoring

Figure 6 -

 6 Figure 6 -Methodology of "use-based" scoring

Figure 7 -

 7 Figure 7 -Selection and characteristics of the cities

Figure 9 -

 9 Figure 9 -Polycentric cities fare/km profile (1 to 40 km) -2012 season ticket price

Figu re 10 -

 10 Figu re 10 -Mono centric cities fare/km profile (1 to 40 km)

Figure 11 -

 11 Figure 11 -Composite cities fare/km profile (1 to 40 km)

Figure 12 -

 12 Figure 12 -Calculation of the "distance based" price discrimination score

  Use-based score can be split into 4 sub scores: 1. Time use: One-hour fare; daily, 2 days,3 days, 4 days, 5 days, 6 days, 7 days, weekly, monthly, 3 months, 6 months, off peak, peak, annual 2. Trip use: single, sold by 2 tickets, 10 tickets, 11 tickets, unlimited pack. 3. Payment use: cash, cash on board, pay as you go, travel card, payment 4. Mode use: metro, bus, bus and metro , train, cycle, ferries, express line, air con line, night bus line, combination of mode…

Figure 13 -

 13 Figure 13 -Use based score and representation

Figure 14 -

 14 Figure 14 -Price discrimination mapping; combination of distance based and use based score

  mode, multi operator) or payment use (value ticket, pay-as-you-go). With the notion of "use based strategy", four criteria of use can be introduced: time, trip, mode and mean of payment.

Mezghani calls it "ticket-based" strategy, due to strategy based on the way trips are sold. He considers all tickets in the same category no matter if these tickets are sold: considering time use (off peak ticket / night ticket), trip use (unlimited, season, multi journey tickets), mode use From "distance-based" price discrimination to "use based" price discrimination (BOUTEILLER, Catherine; FAIVRE D'ARCIER, Bruno) 13 th WCTR, July 15-17, 2013 -Rio, Brazil 7 (multi-

  The adequacy of fare to urban need is the result of two factors: the fare structure: flat, zonal, kilometric; and the urban structure characterized by commuting patterns (Figure8)

	From "distance-based" price discrimination to "use based" price discrimination
		(BOUTEILLER, Catherine; FAIVRE D'ARCIER, Bruno)
		Adequacy of	Steps	Last/first
		fare to urban		Price/km
		need
	Singapore	4	39	82%
	Tokyo	4	5	72%
	Paris	2	4	71%
	Madrid	3	4	71%
	Seoul	4	7	58%
	Berlin	4	3	54%
	London	3	9	53%
	Amsterdam	4	8	22%
	Copenhagen	3	9	20%
	Toronto	3	9	10%
	Figure 8 -Main metrics obtained for calculation of "distance based" price discrimination score
		13 th WCTR, July 15-17, 2013 -Rio, Brazil
				12

th WCTR, July 15-17, 2013 -Rio, Brazil


Flavio Pressacco

Giacomo Plazzotta

Laura Ziani

Bilateral symmetry and modified Pascal triangles in Parsimonious games *

Keywords: Homogeneous weighted majority games, bilateral symmetry, modified Pascal triangles, games representations

We discuss the prominent role played by bilateral symmetry and modified Pascal triangles in self twin games, a subset of constant sum homogeneous weighted majority games. We show that bilateral symmetry of the free representations unequivocally identifies and characterizes this class of games and that modified Pascal triangles describe their cardinality for combinations of m and k, respectively linked through linear transforms to the key parameters n, number of players and h, number of types in the game.

Besides, we derive the whole set of self twin games in the form of a genealogical tree obtained through a simple constructive procedure in which each game of a generation, corresponding to a given value of m, is able to give birth to one child or two children (depending on the parity of m), self twin games of the next generation. The breeding rules are, given the parity of m, invariant through generations and quite simple.

Introduction

Bilateral Symmetry (BS) and Modified Pascal Triangles (M P T ) are issues that play a significant role in various fields of sciences. Just to recall some of the most important papers on these subjects let us quote, for BS, the milestone book of Weyl ([19], 1952) and the papers by Gardner ([6], 1971), Møller and Thornill ( [START_REF] Møller | Bilateral Symmetry and Sexual Selection: A Meta-Analysis[END_REF], 1998), Finnerty ([5], 2003), Song et al ( [START_REF] Song | Planar cell polarity breaks bilateral symmetry by controlling ciliary positioning[END_REF], 2010), Palmer ( [START_REF] Palmer | Symmetry Breaking and the Evolution of Development[END_REF], 2004), and for M P T , Ando ( [START_REF] Ando | A Triangular Array with Hexagon Property, Dual to Pascal's Triangle, Application of Fibonacci numbers[END_REF], 1988), Granville ( [START_REF]Zaphod Beeblebrox's Brian and the Fifty-ninth Row of Pascal's Triangle Andrew Granville[END_REF], 1992), Barry ([2], 2006), Trzaska ([17], 1991) and Bollinger ([3], 1993).

In this paper we will see that both issues jointly emerge in a game theory framework: precisely, as a key to self twin1 "parsimonious" games, a class of games introduced by Isbell ( [START_REF] Isbell | A class of majority games[END_REF], 1956) in the early stage of game theory, which at the best of our knowledge did not receive then any attention.

Parsimonious games (hereafter P games) are the subset of constant sum homogeneous weighted majority games characterized by the "parsimony property" to have, for any number n of non dummy players in the game, the smallest number (i.e. exactly n) of minimal winning coalitions.

For such games the incidence matrix M is the binary square n × n matrix whose entries are 1 (or 0) if column player i belongs (does not belong) to the row minimal winning coalition j. A twin (dual) relationship on P games has been introduced by Isbell ( [START_REF] Isbell | A class of majority games[END_REF], p. 185) through the following property of their incidence matrices: the transposed M T of the incidence matrix M of any P game G is the incidence matrix M of a P game G called the twin of G.

G is self twin (self dual) if G is identical to G. We denote by ST P the set of self twin P games. Our starting point is that an alternative and more friendly description of the twin relationship may be obtained making recourse to a specific property of the vectorial representations of P games.

Usually constant sum homogeneous weighted majority games, including P games, are described by their classical minimal homogeneous representation (q; w) where w = {w 1 , w 2 , . . . , w n } is the ordered (w 1 = 1, w i ≤ w i+1 for any i) vector of individual minimal homogeneous integer weights and q the minimal winning quota given by q = 1+ i∈N w i 2

and satisfying q = i∈S w i , for any coalition S of the so called minimal winning set 2 .

As shown in sect. 2.2, alternative unequivocal representations of a P game may be given either through its binary representation or through its type representation. In particular, the binary representation is the ordered n dimensional vector b with components b i = 1 if w i > w i-1 , b i = 0 otherwise (w i = w i-1 ). Note that the condition w 0 = 0, which gives zero weight to the fictitious player (i = 0) representing the set of dummies, implies b 1 = 1. The type representation is the ordered h (2 ≤ h ≤ n-2) dimensional vector x = (x 1 , x 2 , . . . , x h ) whose component x t is the number of players of type t (sharing the same common type weight w t > w t-1 ) in the game.

Yet, some of the information embedded in such representations is redundant; indeed, the first two and the last two components of b, like the last component of x, are the same in any P game. Then, what really matters in a P game is the, n -4 dimensional, free binary representation f b = (b 3 , . . . , b n-3 ), or respectively the, h -1 dimensional, free type representation f x = (x 1 , x 2 , . . . , x h-1 ).

In terms of these free representations, ST P games are (as shown in Pressacco-Plazzotta, 2013, [START_REF] Pressacco | Symmetry and twin relationships in Parsimoniuos games[END_REF], sect. 4.2, p. 9) the subset of P games characterized by bilateral symmetry. Exploiting bilateral symmetry it is straightforward to obtain closed form formulae for the cardinality of ST P games for feasible combinations of (n, h) and to show that a properly M P T , Γ, with entries Γ(m, k), describes the cardinality of ST P games with n = m + 4 players and h = k + 2 types. Indeed we will show that the rule which governs the evolution with m of the M P T mimics the one holding in the classical Pascal triangle (i.e. the "Pascal equation"), except for internal entries corresponding to combinations of even m and odd k. In detail, for such combinations is it Γ(m, k) = Γ(m -1, k) -Γ(m -1, k -1), so as the difference rather than the sum of the adjacent entries of the previous row is to be computed (obviously this modification has a feedback on the whole triangle).

On the way we describe also the regularities of other related M P T : ∆, which describes the cardinality (still as a function of the couple m, k) of the P non self twin games, and Θ = ∆/2 the cardinality of couples of (non identical) twins.

Besides these counting results, we found also that the whole set of ST P may be described in the form of a genealogical tree, derived through a constructive procedure. In this approach successive generations of ST P are associated to the sequence of m values, and each game may be seen as a parent which gives birth to one child or two children (depending on the even or odd character of the parent generation), ST P games of the next generation. The breeding rules, defined in terms of the free type representation, are simple and generations invariant, albeit parity specific.

A key role in the evolution of the genealogical tree is played by the pivot components of the subset of ST P games, whose f x representation has an odd number of components (shortly OSTP games). We show that the structure of the pivot's set is described by other "regular" triangles.

The plan of the paper is as follows. Section 2 recalls basic concepts of majority games, resumes relevant results on P games3 and introduces the alternative binary and type representations, which play a key role in the subsequent sections. Section 3 is devoted to define bilateral symmetry seen as a particular aspect of the symmetry (or twin) relationship on P games. In Section 4 closed form expressions of the number of ST P games for different feasible combinations of the parameters n, h or m, k are obtained. Section 5, the core of the paper, suggests and discusses an interpretation in terms of M P T of such formulas. In Section 6 we build a genealogical tree of ST P games, through the application of simple evolutionary rules that may be translated in high speed algorithms. Section 7 introduces the pivot triangles and give results on their main evolutionary properties. Section 8 gives some examples and conclusions follow in Section 9.

2 Definitions and main results on P games

Basics of weighted majority games

As usual N = (1, . . . , n) denotes the set of all (non dummy) players. Definition 2.1. A coalition S is a subset of the set N . A game G in coalitional function form is defined by the coalitional function of the game, that is the real function v :

P(N ) → R. Definition 2.2. A game in coalitional function form is simple if its v function has values in {0, 1}. A coalition S is winning if v(S) = 1, losing if v(S) = 0. Definition 2.3. A simple game is constant sum if, for any S ∈ P(N ), v(S) + v( S) = 1.
Definition 2.4. A coalition S is said to be minimal winning if v(S) = 1 and, for any T S, v(T ) = 0. The set of minimal winning coalitions is denoted by W M . A player i who does not belong to any minimal winning coalition is said dummy; at the other extreme, it is a dictator if it is v(i) = 1. We will consider games free of dictator and dummies. Definition 2.5. A simple, weighted majority game is described by a representation (q; w) where w is a vector (w 1 , w 2 , . . . , w n ) of positive weights, q > 1 2 • w(N ) = 1 2 • i∈N w i is the winning quota and v(S) = 1 ⇔ w(S) = i∈S w i ≥ q. Definition 2.6. A representation (q; w) of a weighted majority game is homogeneous if w(S) = q for any S of W M . Definition 2.7. Homogeneous weighted majority games are games for which (at least) a homogeneous representation exists.

Consider now the class of all simple, constant sum n person homogeneous weighted majority games 4 .

Result 2.1. All games of such a class admit a minimal homogeneous representation, that is a (homogeneous) representation (q; w) such that all weights are integers, there are players with (minimum) weight 1 and q = w(N )+1 2 , which implies for any S ∈ W M , w(S)w( S) = 1.

Remark 2.1. Hereafter we will suppose that in such a representation the vector w is ordered according to the convention w 1 = 1, w i ≤ w i+1 for any i.

Result 2.2. The cardinality of the W M set of our class may be either greater or equal (but not lower) than n. See [START_REF] Isbell | A class of majority games[END_REF], p. 185.

Parsimoniuos games and their alternative representations

Definition 2.8. We call Parsimoniuos games (hereafter P games) the subset of constant sum homogeneous weighted majority games characterized by the parsimony property to have for any given number n of non dummy players in the game the smallest number, i.e. exactly n, of minimal winning coalitions.

The nice properties of this class of games have been studied in a ground breaking paper (Isbell, 1956, [9]) going back to the early stage of game theory development. In particular, there it has been found that a P game is univocally described also by an alternative binary representation. Definition 2.9. Given a P game G with minimal homogeneous representation (q; w), its binary representation is the ordered (according to the non decreasing weight labelling) n dimensional vector b with components 5 : b 1 = 1 and for

i = 2, . . . , n, b i = 1 if w i > w i-1 , b i = 0 otherwise, i.e if w i = w i-1 .
Result 2.3. The following constraints ( [START_REF] Isbell | A class of majority games[END_REF], p. 185) hold, in any P game, on the first two and on the last two components of b:

b 1 = 1, b 2 = 0, b n-1 = 0, b n = 1.
Apart from these constraints, the choice of the other components of the binary representation is fully free.

Definition 2.10. The free binary representation of a P game with binary representation b is the ordered n -4 dimensional vector of the internal components of b. We denote such a vector by

f b = (b 3 , . . . , b n-2 ).
4 At the origins of game theory, homogeneous weighted majority games (h.w.m.g.) have been introduced in [START_REF] Neumann | Theory of games and economic behaviour[END_REF] by Von Neumann-Morgenstern and have been studied mainly under the constant sum condition. Subsequent treatments in the absence of the constant sum condition (with deadlocks) may be found e.g. in [START_REF] Ostmann | On the minimal representation of homogeneous games[END_REF] by Ostmann, who gave the proof that any h.w.m.g. (including non constant sum ones) has a unique minimal homogeneous representation, and in [START_REF] Rosenmüller | Homogeneous games: recursive structure and Mathematics of Operations Research[END_REF]. Generally speaking, the homogeneous minimal representation is to be thought in a broader sense but hereafter the restrictive application concerning the constant sum case is used. 5 The initial condition w0 = 0 which gives weight zero to the fictitious player labelled 0, representing the set of all dummy players, would give b1 = 1, coherently with the general rule.

Result 2.4.

There is a one-to-one correspondence between the set of all feasible choices of f b and the set of P games; hence, putting (for any n > 3)6 m = n -4, there are exactly 2 m P games with n = m + 4 (non dummy) players.

Another alternative representation of a P game may be obtained from the following considerations. Given the minimal homogeneous representation of the game, let us divide the players in h non overlapping subsets, according to their weight. For any t = 1, . . . , h let w t denote the type weight (i.e. the common individual weight of all players of the group labelled t). Types too are labelled according to the (now) strictly increasing weight convention, i.e. w 1 = 1, w t-1 < w t for any t.

Definition 2.11. The (complete) type representation of a P game is the ordered h dimensional vector x = (x 1 , x 2 , . . . , x h ) whose component x t is the number of players of group t in the game.

Note that

Result 2.5. (n -2) ≥ h ≥ 2
Indeed there are as many types in a game as the number of 1's components in its binary representation and we know that there are at least two zeros as well as at least two 1's. Moreover it turns out that in any P game the following constraints hold on x : Result 2.6.

x h = 1

x 1 ≥ 2 x h-1 ≥ 2
Proof. Immediate consequence of the constraints given by Result 2.3.

Remark 2.2. For h = 2, h -1 = 1 and x 1 = n -1.
Keeping account of the surely binding constraint (x h = 1), the really relevant (indeed full) information on a P game is embedded in the first (h -1) components of x. Definition 2.12. The truncated or free7 type vector f x = (x 1 , x 2 , . . . , x h-1 ) is the h -1 dimensional vector obtained by deletion of the last (obliged) component of x.

To summarize: besides by their minimal homogeneous representation (q; w) P games are unequivocally described either by f b or by f x.

Indeed the connection between f x and (q; w) is given by the following recursive relation on type weights:

w 0 = 0 w 1 = 1 w t = x t-1 • w t-1 + w t-2 , t = 2, . . . , h -1 w h = (x h-1 -1) • w h-1 + w h-2 (2.1)
Example. Just to give an example consider the nine person P game with five types and free type structure f x = (2, 2, 1, 3). There are two players of the first type with individual (and type) weight w 1 = w 2 = 1; the next two players (labelled 3 and 4) of type 2 with type weight w 2 = x 1 • w 1 + w 0 = 2, so that w 3 = w 4 = 2; one player of type 3 with type weight (1,1,2,2,5,7,7,7,19) and the corresponding minimal winning quota is q = 26.

w 3 = x 2 • w 2 + w 1 =
After that, even if it has not been explicitly stated in the Isbell paper, the same argument used there to derive the number of n person P games may be applied to obtain immediately that:

Result 2.7. The number of P games with n = m + 4 players and h = k + 2 types is given, for any k = 0, . . . , m, by:

C m,k = m! k!(m -k)! (2.2)
Hence putting m = 0, 1, . . . , n -4 on rows and k = 0, . . . , m on columns, the classic Pascal triangle gives the number of different P games with n = m + 4 players and h = k + 2 types. Keeping account of this coincidence, we will denote by

C (with components C(m, k) = C m,k = m! k!(m-k)! ) this triangle.
In the next sections we wish to extend these old results to obtain a M P T , Γ, which gives for any combination of (m, k), the number Γ(m, k) of ST P games with n = m + 4 players and h = k + 2 types.

Bilateral symmetry in P games

In this section we introduce bilateral symmetry in P games Definition 3.2. The free type representation f x of a P game G is bilaterally symmetric or self symmetric if, for any t = 1, . . . , h -1:

x t = x h-t
The following result links bilateral symmetry of the f b and f x representations of a game: Proof is given in Appendix A.

Definition 3.3. A P game is bilaterally symmetric (shortly is BSP ) if its free representations (binary or type) are bilaterally symmetric.

In the next sections we will often use the terminology "self twin" games in place of "bilaterally symmetric" games. More general symmetry relationships in P games come along the following lines.

b i = b n+1-i Remark 3.1.
This property too may be given equivalently in terms of symmetry of the free type representations.

Definition 3.5. A couple G, G of P games are each other symmetric if their free type representations f x and f x are each other symmetric, i.e if for any t = 1, . . . , h -1:

x t = x h-t
Definition 3.6. A couple G, G of each other symmetric P games are also called twins. Definition 3.7. A game G, whose symmetric is just itself, is said self symmetric or self twin. Remark 3.2. Obviously G is self symmetric if and only if G is bilaterally symmetric. Each P game G has just one symmetric G; a given couple of symmetric games surely have free type representations with the same number (h -1) of components.

In conclusion self twin games are a particular subset of P games characterized by bilateral symmetry of their free representations; in the next section we will derive the cardinality of such subsets as a function of n or respectively of the couple (n, h).

The cardinality of self twin P games

In this section we resume the results on the cardinality of self twin or bilaterally symmetric games. First of all, using the notation Γ(m) = Γ(n -4) with m = 0, 1, . . . , n -4 for the number of ST P games with n = m + 4 players, it is:

Result 4.1. Γ(m) = 2 m/2 = 2 (n-4)/2 , m (or n) even (4.1a) Γ(m) = 2 (m+1)/2 = 2 (n-3)/2 , m (or n) odd (4.1b) Remark 4.1.
For n even, this is just the squared root of the overall number of n person P games.

Then for n = 4 (m = 0), the only P game is self twin; for n = 6 (m = 2) there are four P games, two of which self twin; for n = 8 (m = 4), four self twin out of sixteen P games and so on; the number of self twin games multiplies by two as we pass from an even n to the next odd n + 1.

Proof. For n even, we have a free binary choice for each component of the first half of the free binary representation vector, that is m/2 = (n -4)/2 components (while of course the second half is fully constrained by symmetry); for n odd, we can freely choose the first (m -1)/2 components and the pivot, that is (m + 1)/2 = (n -3)/2.

As for the number Γ(m, k) = Γ(n -4, h -2) of ST P games with n players and h types 9 , it is: 9 Obviously, n and m and, respectively, h and k have the same parity. No bilateral symmetry at all may be obtained. c) If m (and then n -4) is odd and k even, bilateral symmetry may be obtained only if the pivot (median) component of the free binary representation is 0. Then, we are left with m -1 = n -5 (free components) and k = h -2 (free elements equal to 1) both even, and we are brought back, keeping account of the symmetry constraint, to the result sub a). d) If both m and k are odd, bilateral symmetry may be obtained only if the pivot component of the free binary vector is 1. Then both m -1 = n -5 (free components) and k -1 = h -3 (free elements equal to 1) are even and we are brought back again to the result sub a).

MP T in self twin games

The results of the previous section give rise to a modified version Γ of the classic Pascal triangle, as may be perceived by Table 1 (on rows m = n -4, starting from m = 0 or n = 4; on columns k = h -2, going from k = 0 or h = 2 to k = m or h = n -2), where you find the number of ST P games for combinations of m and k. We wish to underline that the definition of (modified) Pascal triangle for Γ is not unfounded. Indeed the entries of Γ are far from being merely a collection of non negative integers summing up for any row to a power of 2; on the contrary, they are all binomial coefficients (or zero) whose evolution obeys almost wholly (with only a slight modification) the basic recurrent combinatorial rule (Pascal equation) of the classic Pascal triangle

k 0 1 2 3 4 5 6 7 8 ... m 0 1 1 1 1 2 1 0 1 3 1 1 1 1 4 1 0 2 0 1 5 1 1 2 2 1 1 6 1 0 3 0 3 0 1 7 1 1 3 3 3 3 1
C(m, k) = C(m -1, k) + C(m -1, k -1).
Hereafter the evolutionary rules of the Γ triangle.

Γ(m, k) = Γ(m -1, k) + Γ(m -1, k -1) (5.2)
For k = 1, . . . , m -1 and (m + k) odd:

Γ(m, k) = Γ(m -1, k) -Γ(m -1, k -1) (5.3)
Remark 5.1. In the case m odd, k even, it is (m -1) even and (k -1) odd, and by Formula (4.2b) Γ(m -1, k -1) = 0. Hence, in this case, the Pascal equation would not be really challenged. Then the rows with odd m go on following the classic Pascal recurrence equation, which needs to be adapted only for the case m even, k odd.

The straightforward but tedious proof of Formulas (5.2) and (5.3) is given in Appendix B. Let us shortly comment here the result (5.1).

For any given n (or equivalently m = n -4), there is just one P game with only two types (h = k + 2 = 2). Obviously it corresponds to the case in which the free binary representation of the game is the null vector. This implies that the free type representation has only one element x 1 = n -1. Hence, this P game is surely characterized by bilateral symmetry (i.e. it is self twin). Then Γ(m, 0) = 1. We recall that these games, well known in literature, are called Apex games (see [START_REF] Horowitz | The Competitive Bargaining Set for Cooperative n-Person Games[END_REF], 1973).

Symmetrically, for any given n, there is just one P game with h = n -2 types. It corresponds to the case in which the free binary representation of the game is the unit vector. This implies that the free type representation has n -3 elements (the largest possible, given n), with x 1 = x n-3 = 2 and all the other elements x t = 1. Hence, also this P game is characterized by bilateral symmetry (i.e. it is self twin). Then Γ(m, m) = 1. These games have been introduced by Isbell in ( [START_REF] Isbell | A class of majority games[END_REF], p. 185) who underlined that the sequence of the individual weights is the Fibonacci one, with the only slight modification that the weight of the vicetop player is repeated twice or, more formally, denoting by f i the sequence of Fibonacci numbers:

w i = f i for i = 1, . . . , n -2, w n-1 = f n-2 , w n = f n-1 .
Hence such games may be called Fibonacci games. For example, the sequence of individual weights of the Fibonacci game with n = 10 players is {1, 1, 2, 3, 5, 8, 13, 21, 21, 34}.

If we are interested in counting the number of non self twin, we find another M P T , ∆, whose entries (see Table 2) are obviously defined for any m, k combination by: It turns out that also ∆ is a M P T , whose evolutionary rules are as follows:

∆(m, k) = C(m, k) -Γ(m, k) (5.4)
Result 5.2. For any m and for k = 0 or k = m:

∆(m, 0) = ∆(m, m) = 0 (5.5)
Proof obvious. For k = 1, . . . , m -1 and (m + k) even:

∆(m, k) = ∆(m -1, k) + ∆(m -1, k -1) (5.6)
Proof.

∆(m, k) = C(m, k) -Γ(m, k) = = C(m -1, k) + C(m -1, k -1) -(Γ(m -1, k) + Γ(m -1, k -1)) = C(m -1, k) -Γ(m -1, k) + C(m -1, k -1) -Γ(m -1, k -1) = ∆(m -1, k) + ∆(m -1, k -1)
(5.7)

For k = 1, . . . , m -1 and (m + k) odd:

∆(m, k) = ∆(m -1, k -1) + ∆(m -1, k) + 2Γ(m -1, k -1) (5.8) Proof. ∆(m, k) = C(m, k) -Γ(m, k) = = C(m -1, k) + C(m -1, k -1) -(Γ(m -1, k) -Γ(m -1, k -1)) = C(m -1, k) + C(m -1, k -1) -(Γ(m -1, k) + Γ(m -1, k -1) -2Γ(m -1, k -1)) = C(m -1, k) -Γ(m -1, k) + C(m -1, k -1) -Γ(m -1, k -1) + 2Γ(m -1, k -1) = ∆(m -1, k) + ∆(m -1, k -1) + 2Γ(m -1, k -1)
(5.9)

Remark 5.2. Recall that for odd m, even k it is Γ(m -1, k -1) = 0. Once more this implies that in this case the classic Pascal equation is not challenged, so that it needs to be really updated only for the combinations even m, odd k.

Among other things it turns out that all non zero components of ∆ are even; this allows to introduce another, associated to ∆, modified Pascal triangle Θ, whose entries are defined for any m, k by:

Θ(m, k) = ∆(m, k)/2 (5.10)
which gives the number of pairs of non identical twins to be added to the identical twins to obtain the overall number of P games for any combination of m, k. See Table 3. See section 8 for a presentation of some examples.

k

A constructive approach to self twin games

In this section we propound a constructive evolutionary approach to ST P games in which successive generations of such games (each generation being associated to a value of m) are generated by ST P games of the previous generation through the application of simple rules. The base of this approach is the free type representation of a P game f x = (x 1 , x 2 , . . . , x h-1 ) introduced in the final part of section 2.2, with ST P games characterized by bilateral symmetry of their free type representation. We discuss this approach starting from the following picture 1, which gives the initial evolution of ST P games (described by the above representation), starting from the seed, the "Adamo" game with representation (3), that is the only P game with 4 players (which is ST P ). Let us give here rather informally the evolutionary rules that characterize the tree.

Proposition 6.1. The seed of the genealogical tree (the generation zero game) is the ST P "Adamo" four person game, whose free type representation is 3.

Definition 6.1. A ST P game with an odd (respectively an even) number of components in its free type representation f x is called odd (even) self twin parsimonious, or shortly OST P (EST P ) game. Proposition 6.4. The pivot of an OSTP of any (zero or) even generation is an odd number.

Proposition 6.5. The pivot of an OSTP of any odd generation is an even number.

Proposition 6.6. An OSTP G of any even generation, with pivot x h/2 gives birth to a couple of children STP games of the next (odd) generation, let us denote them by G 1 and G 2 . G 1 is OSTP and G 2 is ESTP.

Proposition 6.7. The child G 1 replicates the representation of the parent G except for the pivot which is now (coherently with prop. 6.5) surely an even number x 1 h/2 = x h/2 + 1.

Proposition 6.8. The child G 2 is obtained by an almost full replication of the brother G 1 but (coherently with prop. 6.3) splitting the pivot x h/2 + 1 into a couple of equal central components (x h/2 + 1)/2. Proposition 6.9. An OSTP G 1 of any odd generation, with even pivot x 1 h/2 = x h/2 + 1 gives birth to a unique child OSTP game G 11 of the next (even) generation. The child replicates the structure of the father (and of the grandfather also) except for the pivot which is now (coherently with prop. 6.5) surely an odd number

x 11 h/2 = x 1 h/2 + 1 = x h/2 + 2.
Proposition 6.10. An ESTP G 2 of any odd generation gives birth to a unique child OSTP game G 21 of the next (even) generation. The child replicates the structure of the father but with the insertion of the pivot 1 in the previous even representation of G 2 . Proposition 6.11. All the games built according to these rules are by construction bilaterally symmetric (hence STP).

Proposition 6.12. The number of STP (half OSTP, half ESTP) of any odd generation is twice the number of the STP (all OSTP) of the previous generation.

Proposition 6.13. The number of STP (all OSTP) of any even generation m is the same of the STP of the previous odd generation m -1 and twice the number of STP of the previous even m -2 generation.

Proposition 6.14. Prop. 6.1, 6.12 and 6.13 jointly grant that the STP of our genealogical tree exhaust the cardinality of STP games for any value of n (or m). Check it through a simple comparison with the values of Γ(m) given by Result 4.1.

We may conclude that: Proposition 6.15. The genealogical tree introduced in this section fully and unequivocally describes all ST (bilaterally symmetric) Parsimonious games.

In this section we found that the pivot components of OST P played a key role in the evolutionary rules of the genealogical tree. In the next section we will see that other triangles, no more of the M P T type but nevertheless characterized by significant regularities, are able to describe the evolutionary rule behind the pivot behaviour.

The pivot triangles

A pivot triangle is a triangle of numbers whose rows correspond to generations according to their m value; in each row we find the sequences, in increasing order, of all feasible values of the pivots of the OST P games for the related generation. Columns are labelled by natural numbers c = 1, 2, . . .. The value of the pivot in row m and column c is x m,c , while r m,c is the number of OST P games of m generation whose pivot is x m,c (shortly repetitions of x m,c ).

We will distinguish hereafter even triangles, associated to even generations (m zero or even), and odd triangles associated to odd values of m.

The pivot triangles of even generations

Let us consider in this section only the even generations of the tree. For any of the initial even generations, we list here (Table 4) the values x m,c of the pivots involved, as well as (in brackets) the number r m,c of repetitions of each pivot's value.

Table 4: Even generation pivots triangle

To understand the table note that the evolutionary rules of the genealogical tree imply that going from one even generation, m, to the next even, m + 2, each pivot x m,c gives rise to a couple of grandchildren pivots, whose values are, respectively, x m+2,c+1 = x m,c + 2 and x m+2,1 = 1. Hence, if in the m generation the odd pivot x m,c is repeated r m,c times, the odd pivot x m,c + 2 is repeated r m,c = r m+2,c+1 times in the following m + 2 generation.

On the other side, Formula (4.1a) implies that the odd pivot x m,1 = 1 is repeated r m+2,1 = 2 m/2 times in the m + 2 generation 10 .

These rules, jointly with the initial condition that in generation 0 there is the unique pivot x 0,1 = 3, give a straightforward explanation of the whole table. Let us enter now in some detail.

Proposition 7.1. For any row m the number of columns, that is of different pivots of m generation, is m 2 + 1. Indeed, it is the cardinality of the set of all positive odd integers not greater than m + 3, except for m + 1.

The pivots' values are as follows:

Result 7.1.

x 0,1 = x 0,(m/2)+1 = 3 there is just one pivot (7.1a)

x m,c = 2c -1 for m even and c = 1, . . . , (m/2) (7.1b)

x m,c = 2c + 1 for m even and c = (m/2) + 1 (

Put Z(m) = c x m,c , the sum of all pivots' values (without repetitions) of the m generation. Then, it is:

Result 7.2. Z(m) = 2 + m 2 + 1 2 or (7.2a) Y (m) = Z(m) -2 = m 2 + 1 2 (7.2b)
Proof. Subtracting 2 to the last entry of each row (the other entries unchanged), we find exactly the well known triangle of squared numbers, i.e. the triangle having on the rows the sequences of all odd integers up to m + 1, whose sum of entries (on each row) is m 2 + 1

2

, that is the sequence of squared natural numbers (1, 4, 9, 16, . . .). See Table 5. Table 5: Squared numbers triangle.

The sum of the row m + 2 is obtained adding m + 3 to the sum of the (non bracketed) entries of the previous row m (see Table 4):

Result 7.3. Z(m + 2) = Z(m) + (m + 3) (7.3)
Proof. Immediate by Formula (7.2a). An enlightening pivotal explanation is the following: we know that each row (label m) has exactly 1 + (m/2) columns. Going from the m generation to the following, each of the (m + 2)/2 pivot with value x generates a pivot of value x + 2; moreover, the pivot 1 is to be added (here we do not care the repetitions). The global increment is then 1 + (2

• (m + 2)/2) = 1 + (m + 2) = m + 3.
As for the repetitions, we have: It turns out that the highest rightward diagonal (corresponding to the first row m = 0) has constant entries equal to 1, while those starting from the other rows (m even), have constant entries 2 (m/2)-1 . See Table 6.

(1) (1) m 6 (4) (2) (1) (1) 2 (1) (1) 4 (2)
(1) (1)

10 ( 16
) (8) (4) (2) (1) (1) 12
(32) (16) (8) (4) (2) (1) (1)
Table 6: Pivot repetitions triangle Finally, as for the sum Φ(m) of all pivots value on each row (keeping account of the repetitions), it is for any m (zero or even):

Result 7.6. Φ(m) = c x m,c • r m,c = 2 m/2 • 3 (7.6)
Proof. Immediate by induction. It is true for m = 0; let us check that if it is true for any even m, it is true for the next even (m + 2). It is:

Φ(m + 2) = c x m+2,c • r m+2,c = c (x m,c + 3) • r m,c = = c x m,c • r m,c + 3 • c r m,c = Φ(m) + 3 • 2 m/2 = = 2 m/2 • 3 + 3 • 2 m/2 = 2 • 2 m/2 • 3 = 2 (m+2)/2 • 3 (7.7)

The pivot triangles of odd generations

Let us consider now the behaviour of the odd generations (m = 1, 3, . . .) and in particular the pivot elements of the subset with odd representation. It turns out that it mimics (mutatis mutandis) the rule driving the even generation one; this fact may be perceived by the following table:

c 1 2 3 4 5 1 4 (1) 3 2(1) 6(1) m 5 2(2) 4(1) 8(1) 7 2(4) 4(2) 6(1) 10(1) 9 2(8) 4(4) 6(2) 8(1) 12(1)
m ′ ,c = x m,c + 1 (7.8) r ′ m ′ ,c = r m,c (7.9)
Exploiting Formulas (7.8) and (7.9) it is easy to obtain the counterpart of all results given in the previous subsection. Here, we provide only the analogous of Formula (7.6). Putting

Ψ(m ′ ) = c x ′ m ′ ,c • r m ′ ,c
it is:

Result 7.7. Ψ(m ′ ) = 2 (m ′ -1)/2 • 4 (7.10) Proof. It is: Ψ(m ′ ) = c x ′ m ′ ,c • r m ′ ,c = c (x m,c + 1) • r m,c = = c x m,c • r m,c + c r m,c = Φ(m) + 2 m/2 = = 2 m/2 • 3 + 2 m/2 = 4 • 2 m/2 = 4 • 2 (m ′ -

Conclusions

In this paper we discuss the role of modified Pascal triangles in describing the cardinality of self twin (bilaterally symmetric) Parsimonious games for any combination of the relevant parameters m, k associated respectively to the number n = m+4 of players and h = k+2 of types in the game. In detail, we show that the entries of the modified triangles follow almost wholly (with a slight modification) the evolutionary rule embedded in the basic combinatorial relation (Pascal equation) which gives any binomial coefficient as the sum of two adjacent coefficient of the previous row of the classic Pascal triangle. In addition we also provide a genealogical tree of self twin games, in which each game of a given generation (corresponding to a value of m) is able to give birth to one or two (depending on the parity of m) children self twin games of the next generation. The breeding rules, defined in terms of the free type representation, are, given the parity, invariant across generations. They are quite simple and may be translated in an high speed pen and pencil constructive procedure to obtain all self twin games for small enough values of n; obviously a simple computational routine produces the set of all self twin games also for large values of n with the only constraint of the computational power (recall that the number of self twin games explodes at the rhythm of the square root of 2 n-4 ). The analysis of the genealogical tree revealed that a key role in its evolution is played by the pivot components of the subset of self twin Parsimonious games whose free type representation has an odd number of components. We found that other triangles (pivot triangles) describe the structure of the pivot's set and give a synthesis of their evolutionary pattern.

We are aware that our paper is wholly theoretical so we leave to subsequent research to look for practical applications to hard or social sciences. Indeed, we hope that such results could follow given the prominent role played by bilateral symmetry in many fields of the life of the universe 11 .

A Appendix: proof of the proposition 3.1

Define I(b) = (I 1 , I 2 , . . . , I h ) as the vector of ordinal labellings of 1's of b.

For example, if b = (1, 0, 1, 0, 1, 1, 0, 1, 0, 1), I(b) = [START_REF] Ando | A Triangular Array with Hexagon Property, Dual to Pascal's Triangle, Application of Fibonacci numbers[END_REF][START_REF] Bollinger | Extended Pascal triangles[END_REF][START_REF] Finnerty | The Origins of axial patterning in metazoa: how old is bilateral symmetry[END_REF][START_REF] Gardner | Bilateral Symmetry and Interneuronal Organization in the Buccal Ganglia of Aplysia[END_REF][START_REF] Horowitz | The Competitive Bargaining Set for Cooperative n-Person Games[END_REF][START_REF] Møller | Bilateral Symmetry and Sexual Selection: A Meta-Analysis[END_REF].

It follows that, for t = 1, . . . , h -1, it is:

x t = I t+1 -I t and x h = 1 Conversely, for any t > 2, it is:

I t = 1 + t-1 j=1
x j and I 1 = 1

In the example: We show that this implies that also f x is bilaterally symmetric. Property A.1. For any couple of symmetric subscripts t and h + 1t (sum of subscripts h + 1) it is:

x 1 = 3 -1 = 2,
I t + I h+1-t = n + 1

In the example, it is: Considering now the difference (x tx h-t ), it is for any t = 1, . . . , h -1:

I
x tx h-t = (I t+1 -I t ) -(I h+1-t -I h-t ) = (I t+1 + I h-t ) -(I t + I h+1-t ) = (n + 1) -(n + 1) = 0

The last but one equality is a consequence of Property A.1, keeping account that the sum of indices of the terms in both brackets is h + 1. Hence x tx h-t = 0, which means that f x is bilaterally symmetric.

Conversely, hypothesis: f x bilaterally symmetric. We show that this implies that also f b is bilaterally symmetric.

By hypothesis, for any t = 1, . . . , h -1, it is:

x t = x h-t
Substitution gives: I t+1 -I t = I h+1-t -I h-t and create order, beauty and perfection" ( [START_REF] Weyl | Symmetry[END_REF], p. 5); "Symmetry considerations dominate modern fundamental physics both in quantum theory and in relativity" ( [START_REF] Brading | Symmetries in physics: philosophical reflections[END_REF], p. ix preface); "Symmetry plays an essential role in science" ([14], editor foreword); "Bilateral symmetry is a hallmark of Bilateralia" ([5], abstract); "Understanding the origin and evolution of bilateral organism requires an understanding of how bilateral symmetry develops, starting from a single cell" ( [START_REF] Wiener | The origin evolution and development of bilateral symmetry in multicellular organism, Tissues and Organs[END_REF], abstract).

  5 hence individual weight w 5 = 5; three players of type 4 and type weight w 4 = x 3 • w 3 + w 2 = 7 and hence individual weights w 6 = w 7 = w 8 = 7 and finally one top player (type 5) whose type weight is given by w 5 = (x 4 -1) • w 4 + w 3 = 19. The sequence w of individual weights in the minimal homogeneous representation is then

8 . 3 . 1 .

 831 Definition The free binary representation f b of a P game G is bilaterally symmetric or self symmetric if, for any i = 3, . . . , n -2: b i = b n+1-i Proposition 3.1. Bilateral symmetry of f b is equivalent to bilateral symmetry of b. Proof. Immediate keeping account that, in any P game, b 1 = b n = 1 and b 2 = b n-1 = 0.

Result 3 . 1 .

 31 Let f b and f x free binary and free type representations of a game G; f b is bilaterally symmetric if and only if f x is bilaterally symmetric.

Definition 3 . 4 .

 34 A couple G, G of P games are each other symmetric if their free binary representations f b and f b are each other symmetric, i.e if, for any i = 3, . . . , n -2:

Result 4. 2 .

 2 Γ(m, k) = C m/2,k/2 m even, k even (4.2a) Γ(m, k) = 0 m even, k odd (4.2b) Γ(m, k) = C (m-1)/2,k/2 m odd, k even (4.2c) Γ(m, k) = C (m-1)/2,(k-1)/2 m odd, k odd (4.2d) Proof. a) Both m and k even. There is a free choice of the k/2 = (h -2)/2 components equal to 1 among the m/2 = (n -4)/2 places in the first half of the free binary representation; by bilateral symmetry there is no free choice in the second half. b) If m even and k odd.

Result 5 . 1 . 1 ( 5 . 1 )

 51151 For any m and for k = 0 or k = m: Γ(m, 0) = Γ(m, m) = For k = 1, . . . , m -1 and (m + k) even:

Figure 1 :

 1 Figure 1: Genealogical tree of self twin games (rows m from 0 to 5 or n from 4 to 9) expressed by their free type representation.

Proposition 6 . 2 .Proposition 6 . 3 .Definition 6 . 2 .

 626362 All STP games of zero or even generation (m zero or even), are OSTP i.e. they have h -1 odd. Half of the STP games of any odd generation (m odd) are OSTP (and obviously the other half ESTP). The central component x h/2 of the free type representation of an OSTP is called pivot of the representation (or shortly of the game).

Result 7. 4 .

 4 r m,c = 2 (m/2)-c for m even and c = 1, . . . , (m/2) (7.4a) r m,(m/2)+1 = 1 for m (zero or even) and c = (m/2) c = 2 • r m-1,c for m even and c = 1, . . . , (m/2) -1 (7.5b) r m,c = 1 for m even and c = (m/2), (m/2) + 1 (7.5c)

  x 2 = 5 -3 = 2, x 3 = 6 -5 = 1, x 4 = 8 -6 = 2, x 5 = 10 -8 = 2 and x 6 = 1, or f x = (2, 2, 1, 2, 2). Hypothesis: f b bilaterally symmetric, hence b bilaterally symmetric. See Proposition 3.1.

Proposition A. 1 .

 1 Bilateral symmetry of b is equivalent to the following property of I(b):

1 + I 6 = 1 + 10 =

 16110 I 2 + I 5 = 3 + 8 = I 3 + I 4 = 5 + 6 = 10 + 1

Table 1 :

 1 Triangle Γ with entries Γ(m, k), the cardinality of ST P games with n = m + 4 players and h = k + 2 types.

	1

Table 2 :

 2

						k
			0	1	2	3	4	5	6	7	8	...
		0	0
		1	0	0
		2	0	2	0
		3	0	2	2	0
	m	4	0	4	4	4	0
		5	0	4	8	8	4	0
		6	0	6	12	20	12	6	0
		7	0	6	18	32	32	18	6	0
		8	0	8	24	56	64	56	24	8	0
		...	... ...	...	...	...	...	... ... ... ...

Cardinality ∆(m, k) = C(m, k) -Γ(m, k) of non ST P games for any combination of m, k.

Table 3 :

 3

Cardinality Θ(m, k) = ∆(m, k)/2 of pairs of non identical twins for any combination of m, k.

Table 7 :

 7 Odd generation pivot triangleTo understand the behaviour of the new triangle requires nothing but to adapt all properties.

	Let x	′ m ′ ,c and r m ′ ′ ,c be the entries of the odd triangle, with m	′ = m + 1 (m even). It is immediate
	to check that, for any m	′ , c, we have:
	Proposition 7.2.
			x	′

  The first example keeps m = 4, that is n = 8. There are 2 4 = 16 P games, 4 of which are self twin, while the other 12 may be seen as 6 pairs of non identical twins. Table3reveals that those pairs may be divided in 3 groups with respectively three, four or five types. Each group contains 2 pairs of non identical twins. Here the complete list of the 16 P games, grouped by their h value and expressed through their free type representation (in parenthesis) with h -1 components.

	h = 2:	(8;1,1,1,1,1,1,1,1,7)
	h = 3:	(17;1,1,1,1,4,4,4,4,13)
		(16;1,1,1,3,3,3,3,3,13) (16;1,1,1,1,1,5,5,5,11)
		(13;1,1,2,2,2,2,2,2,11) (13;1,1,1,1,1,1,6,6,7)
	h = 4:	(20;1,1,2,2,2,2,9,9,11)	(7.11)
		(24;1,1,1,3,3,7,7,7,17)
		1)/2 (17;1,1,2,3,3,3,3,3,14) (17;1,1,1,1,1,5,6,6,11)
		(22;1,1,2,2,5,5,5,5,17) (22;1,1,1,1,4,4,9,9,13)
		(23;1,1,2,2,2,7,7,7,16) (23;1,1,1,3,3,3,10,10,13)
		(19;1,1,1,3,4,4,4,4,15) (19;1,1,1,1,4,5,5,5,14)
	h = 5: (29;1,1,2,2,5,5,12,12,17)
	8 Examples	(25;1,1,1,3,4,7,7,7,18) (23;1,1,2,3,5,5,5,5,18) (23;1,1,1,1,4,5,9,9,14)
	(7) (27,1,1,2,3,3,8,8,8,19) (27;11,1,1,3,3,7,10,10,17) self twin (25;1,1,2,3,3,3,11,11,14) (25;1,1,2,2,2,7,9,9,16) (26;1,1,1,3,4,4,11,11,15) (26;1,1,2,2,5,7,7,7,19) h = 6: (30;1,1,2,3,3,8,11,11,19) (31;1,1,2,3,5,5,13,13,18) (31;1,1,2,2,5,7,12,12,19) (29;1,1,2,3,5,8,8,8,21) (29;1,1,1,3,4,7,11,11,18) Example 8.1. h = 2: h = 7: (34;1,1,2,3,5,8,13,13,21)
	h = 3:	(3,4) and (4,3)	first pair of twins
		(2,5) and (5,2)	second pair of twins
	h = 4:	(2,3,2)	first self twin
		(3,1,3)	second self twin
		(2,1,4) and (4,1,2)	first pair of twins
		(2,2,3) and (3,2,2)	second pair of twins
	h = 5:	(2,1,2,2) and (2,2,1,2)	first pair of twins
		(2,1,1,3) and (3,1,1,2)	second pair of twins
	h = 6:	(2,1,1,1,2)	self twin
	The corresponding rows of our M P T are (m = 4, k from 0 to 4):
		C : 1, 4, 6, 4, 1
		Γ : 1, 0, 2, 0, 1
		∆ : 0, 4, 4, 4, 0
		Θ : 0, 2, 2, 2, 0
	The minimal homogeneous representations are:
	h = 2:	(7;1,1,1,1,1,1,1,6)
	h = 3: (13;1,1,1,3,3,3,3,10)	(13;1,1,1,1,4,4,4,9)
		(11;1,1,2,2,2,2,2,9)	(11;1,1,1,1,1,5,5,6)
	h = 4:	(16;1,1,2,2,2,7,7,9)
		(15;1,1,1,3,4,4,4,11)
		(14;1,1,2,3,3,3,3,11)	(14;1,1,1,1,4,5,5,9)
		(17;1,1,2,2,5,5,5,12)	(17;1,1,1,3,3,7,7,10)
	h = 5: (19;1,1,2,3,3,8,8,11)	(19;1,1,2,2,5,7,7,12)
		(18;1,1,2,3,5,5,5,13)	(18;1,1,1,3,4,7,7,11)
	h = 6: (21;1,1,2,3,5,8,8,13)

In Isbell terminology, self dual games.

Note that N is the "grand" coalition of all players.

Some of these results are directly given by Isbell, while others are derived as straightforward consequences of his approach, including the role of the classic Pascal triangle in describing the cardinality of P games with n players and h types.

We will consider here P games with n > 3. Indeed, for n = 3 there is a unique P game in which all players share the same weight, while all P games with n > 3 have at least two types of players.

Note that the other constraints on x1 and x h-1 may leave some freedom in the choice of the related components or be fully binding depending on the values of the couple (n, h).

Elsewhere(Pressacco-Plazzotta, 2013, [13], sect. 4, Theorem 4.2) we show that bilateral symmetry is the exact counterpart of self duality in P games a property suggested, as said in the introduction, byIsbell ([9], pp. 185-186) on the basis of special character of the incidence matrices of such games.

Note that this must be equal to the sum of numbers within brackets in the m generation

There is a huge literature concerning symmetry in hard sciences; besides the references given in the introduction, let us recall here some prominent sentences: "Symmetry is one idea by which man through the ages has tried to comprehend

Acknowledgements

We wish to thank Michele Giacomini, former student of the 2013 Game Theory short course at the Scuola Superiore Università di Udine, for a lot of valuable suggestions and comments on an earlier draft of the paper.

It is interesting to observe that (as proved in [START_REF] Pressacco | Symmetry and twin relationships in Parsimoniuos games[END_REF],Theorem T5.1, sect. 5.1), all pairs of twins have the same minimal winning quota q (of course pair dependent).

Example 8. [START_REF] Barry | On integer sequence based constructions of generalized Pascal triangles[END_REF] The second example keeps n = 9 (m = 5). There are 32 P games, 8 of which self twin, while the other 24 may be divided in 12 pairs of twins.

Here the complete list of the 32 P games, grouped by their h value:

forth pair of twins The minimal homogeneous representations are: equivalent to:

The sum of subscripts in both sides of the equation is h + 1, thus Property A.1 is satisfied and, by proposition A.1 this implies bilateral symmetry of b.

B Appendix: proofs of evolutionary rules of the MP T Γ

Proof. Case m even, k even. Recalling Formula (4.2a), it is:

Now, denoting by m ′ = m -1 and k ′ = k -1 both odd and keeping account of Formula (4.2c):

while, applying Formula (4.2d):

we found that Formula (5.2) is satisfied.

Proof. Case m odd, k odd. Recalling Formula (4.2d), it is:

In turn, for Formula (4.2b):

while, applying Formula (4.2a):

so that we found again that Formula (5.2) is satisfied.

Proof. Case m odd, k even. Recalling Formula (4.2c), it is:

Now, keeping account of Formula (4.2a):

on the other side, applying Formula (4.2b) (m -1 even, k -1 odd):

So that Formula (5.3) is satisfied.

Proof. Case m even, k odd. Recalling Formula (4.2b), it is: Again Formula (5.3) is satisfied.