Quang Xuan Truong Nguyen
email: nguyen-xuan.truong@usth.edu.vn

Dinh Hung Nguyen

Quang Nguyen

Truong

Nguyen Hung

Dinh Quang

Performance analysis of roof-mounted photovoltaic systems on the office building in Hanoi

Keywords: green building, photovoltaic source, simulation tool, rooftop, energy efficiency

I. INTRODUCTION

Energy consumption in buildings is studied in [START_REF] Pérez-Lombard | A review on buildings energy consumption information[END_REF] concluding that this sector represents more 40 % of total energy use in EU and USA. Most of this energy is for the provision of lighting, heating, cooling, and air conditioning. The green and sustainable development movement in the world has placed green building in high priority as it is able to meet the building demand while mitigating the negative impacts of construction sector. High energy performance level in building can be attained by taking advantage of passive design techniques and active solar technologies such as solar collectors for hot water and space heating, PV panels for generating electricity. PV array installed in the building contributes to the reduction of the primary, conventional energy supply, as well as the reduction of CO2 emissions in the surrounding environment. Distributed PV system can be installed as façade or rooftop applications in order to maximize the benefits of clean and quiet power generation. Façade installation can be optimized by observing the orientation and distance to length ratio. Rooftop application can observe the tilted angle and curved installation [START_REF] Deng | How to evaluate performance of net zero energy building e A literature research[END_REF][START_REF] Scognamiglioa | Photovoltaics in net zero energy buildings and clusters: Enabling the smart city operation[END_REF].

Energy consumption in Vietnam's building sector is growing quickly as a result of rapid industrialization and urbanization. In the report [START_REF] Chong | Viet Nam: Energy sector assessment, strategy, and road map[END_REF] have shown that, the residential sector is one of the main energy consumers in Vietnam, accounting for 33% share in 2012 and 35.6% in 2014, which will continue to increase in the future due to the pressure of increasing demand for buildings. The nZEB has become an important part of sustainable development and attracts the attention of scientists on the term of "green building" in Vietnam. However, the adoption of green building is still limited, the release of related regulations are slow and governmental support are lacking. It is recommended that the Vietnamese government should take stronger actions such as ratifying regulations or offering incentives to promote green building for sustainable development [START_REF] Nguyen | A critical review on Energy Efficiency and Conservation policies and programs in Vietnam[END_REF][START_REF] Nguyen | A review on Green Building in Vietnam[END_REF][START_REF] Millet | How to promote Green Building in Vietnam?[END_REF]. Besides, it has limited programs addressing renewable energy like PV generation deployment and energy efficiency use. Generally, the photovoltaic systems are yet to be used in the form of building-integrated systems nowadays. Such system is still at the research and demonstration stage. In this regard, it is required that the design and control strategies of green building are not straight forward since the buildings may involve complex integration of renewable sources, energy appliances, energy storages and may interact with the local smart grid.

Supported by the host university, our research project is a contribution for a better understanding of the problem. This paper mainly aims to: (1) present a platform for green building research and training; (2) estimate the building energy demand for lighting, HVAC, elevator, and other equipements; (3) use the PV system simulation software PVSyst to simulate performance of the 15 kWp PV system and investigate monthly energy production based on parameters including meteorological data of the installation site and PV array. Finally, simulation results will be used to advise the configurations of PV system for the building of the host university to contribute for the next phase of green building and nZEB research in our laboratory.

II. PLATFORM FOR GREEN BUILDING RESEARCH AND TRAINING AT USTH

A smart micro-grid platform at local level (building) with integrated Photovoltaic array and building energy management system is necessary to support green building research and eventually improve the current power grid by augmenting its performance and energy efficiency. This platform allows to study many aspects of smart building in Vietnam including building modeling with analysis of the differences between the model and the real system, understanding of distribution of energy in management issues under different scenarios. The power supply for this platform consists of a 15kWp PV generator to be installed on the rooftop, an electrical storage system and the power grid. The system extracts the maximum power obtainable from the PV array under different working conditions to provide a portion of the building power demand. The loads include the lighting system, HVAC system, laboratory equipment and the elevator.

Besides, the monitoring system allows data to be collected and utilized for the optimal operation. The energy controller reads all the data measured by the transducers to operate the equipment according to different selected modes. The energy manager controls the delivery of energy, and the charge or discharge of batteries.

The objective of the project is to benefit from PV power supply, to improve energy efficiency in the building and to reduce the cost of electrical consumption under different scenarios. To set up a reliable PV system, it is important to identify the important data which affect the performance of the system. The data collection for this study divided into two categories which are the meteorological database and solar radiation, and total energy used by requirement loads.

A. Meteorological database and solar radiation

The building is located at N°.18, Hoang Quoc Viet Street, Cau Giay District, Hanoi, in the northern of Vietnam at a latitude of 21°N. It has one 37.9 meter high block, with a 939 m 2 rooftop consisting of three available installation areas. Total surface area for two flat roof is 420 m 2 and a flat (free standing system) with an available area of 130m 2 , where PV panels can be installed (rooftop-mounted PV system).

The solar radiation climate database of a zone is extremely important to estimate the performance of a PV generation system. Vietnam is located in South East Asia, extending between latitudes 9°N and 23°N. The geographical location of the building and the local weather conditions influence the optimal tilt of the PV modules and is, therefore, of great importance. Several studies that addressed the solar resource maps show that global horizontal irradiation in annual daily average reaches around 3.4 kWh/m 2 /day in Hanoi. In the case of direct normal irradiation the annual daily average is around 2.5 kWh/m 2 /day, with approximately 1,500-1,700 hours of sunshine makes it a good site for promoting the use of PV systems [START_REF] Polo | Solar resources and power potential mapping in Vietnam using satellite-derived and GIS-based information[END_REF]. Besides, ambient temperature and wind speed are also the main factors in PV production system.

The meteorological data embedded in PVSyst for this project is a synthetic hourly or monthly meteorological file set from Meteonorm in 2005 (the data collected from 1991 to 2010). It is a database containing climatological data for solar engineering and could also be used to calculate solar radiation on arbitrarily oriented surfaces [START_REF] Mermoud | Use and Validation of PVSYST, a User-Friendly Software for PV-System Design[END_REF][START_REF] Muller | Handbook part 1: Software[END_REF]. Fig. 2a shows the monthly average global irradiation of Hanoi in which the minimum value is in January, about 67.6 kWh/m 2 , in contrast, the maximum value peak at 161.6 kWh/m 2 in July and August. And the average value is about 1,391 kWh/m 2 per year. Another data source, for instance, irradiance is Climate-SAF PVGIS which demonstrates the average daily global irradiance (Fig. 2b) with a relatively high amount of mean solar irradiation of 595 W/m 2 in July and August and 13 hours of sunshine. We obtained previous electric bills since 2015 as the case study and building energy audit to determine what can be done to reduce electricity usage. In this case, we did not take into account the energy demand of specialized laboratory equipment. The total amount of energy demand of the building for this purpose was estimated about 2193.8 kWh during a day, for example in June 2016 when energy consumption peaked, as shown in TABLE I. The energy demand consumption is estimated based on the daily power used and its operating time. To be certain, we studied same building's bill again by comparing its consumption with our estimated energy demand. As can be seen in TABLE I, II and III, building energy consumption in one month is acceptably lower than our estimated energy demand, thus this estimation is valid for our purposes.

IV. SYSTEM CONFIGURATION

A photovoltaic system mainly composes: solar panels; power electronic equipment such as charge-discharge controller, inverter, and battery for energy storage and auxiliary power generation equipment. The parameter selection of every component is very important to fulfil the system configuration. The performance of PV modules depends on the cell temperature, solar irradiance and module type. According our requirements, we selected crystalline silicon cells. A 15 kWp PV system was simulated in PVSyst using the modules of XL SW 335 rated at 335 Wp, whose specifications are shown in TABLE IV. Based on the space availability for the correct orientation of PV array, it was estimated that 48 modules with total peak power 16.09 kWp that can be installed on the building's rooftop.

A. PV Panels

To receive the maximum amount of solar radiation, all modules are placed on the same plane, the PV array needs to be placed at a certain angle. The placement of the PV array is described using the plane azimuth m A and the tilt angle m  .

The tilt angle, is the angle between the horizontal plane and the PV module. The azimuth has different definitions, but in this paper the plane azimuth, is defined as the angle between the orientation of the collector plane and south (in the northern hemisphere). For geographical locations installations with fixed modules in the northern hemisphere like Hanoi, typically the array should be oriented to face south at 0° azimuth. The optimal tilt angle will be presented in section V. For simulation, two main factors of concern are incident irradiance and cell temperature. In Fig. 3a, the maximum power of solar cell SW 335 mono-crystalline at its NOCT (45°C) is proportional to the incident irradiance. One panel can generate 306.7 W under incident irradiation 1,000 W/m 2 . The generated electric power drops quickly with decreasing incident irradiance, when the irradiance is 200 W/m 2 , the panel can only produce 59.6 W. In Fig. 3b, the power output depends strongly on temperature with negative coefficient. A PV cell generated power from 269 W to 356 W under 1,000 W/m 2 incident irradiance when the cell operating temperature varies from 10 o C to 70 o C.

B. Inverter and matching PV array/Inverter

The inverter takes the DC power from the PV array and converts it into standard AC power used by the building appliances. Some factors that we have to take into account in choosing inverter include:

-The generated power of PV array: The power of inverter is the output power of system. When the power of inverter is higher than generated power from the array, it increases cost unnecessarily. On the contrary, when the array generates more than the power that inverter can handle, the exceed power is waste and the efficiency of the system decreases.

-Voltage and current ranges: The highest output voltage and current of the array have to be under the peak voltage and current that inverter can handle. In operation mode, basing on the highest and lowest produced Vmpp of the array, we have to select inverter that having a suitable operating range. The DC input current range of inverter has to be considered similarly to be suitable with the DC output current of array.

-Energy appliances and installation: There are two kinds of inverter: Central inverter and string inverter. Central inverter is suitable for a homogenous system when all panels have the same orientation and the same DC output voltage. This kind of inverter can reduce the cost per unit of power due to the better performance and footprint size (in comparison with string inverter). However, when strings work in different conditions, the mismatch among them will reduce the efficiency of harvesting system. The complication of high DC voltage wiring could be another problem, which increases the risk of fire. In the other hands, string inverter is suitable for the system that contains various modules with different orientations.

The simulation only consider about choosing which type of PV module and inverters. PVSyst has a built in system that matches the number of inverters with the number of strings. It also proposes a number of modules in each series, and the number of strings based on the sizing parameter. This is to assure that all requirements regarding current, voltage and power levels are met. When an inverter have multiple MPPT inputs, in PVSyst, it divide the operating power between the inputs. Two inputs take half the nominal power at each input. It is not possible to share the power unequally between each MPPT input. The main reason for choosing two different modules of Sunmodule XL SW 335 and two kind of inverters is to compare the performance of different types of well known (TABLE V). These technique conditions are used together with the numbers proposed by PVSyst to set design the configurations in (TABLE VI).

The ratio between peak power of array (in standard testing condition 1,000 W/m2 irradiance and 25°C) and the power of inverter (15 kWp) is 1.07. This ratio is acceptable since the high efficiency of inverter choosing, and the high insolation in Hanoi [START_REF] Mondol | Optimizing the economic viability of grid-connected photovoltaic systems[END_REF][START_REF] Notton | Optimal sizing of a gridconnected PV system for various PV module technologies and inclinations, inverter efficiency characteristics and locations[END_REF].

C. Batteries and battery charge controller

Several studies have suggested that battery storage colocated with grid-integrated building-scale PV generation benefits electricity distributors in maintaining system voltages within acceptable limits [START_REF] Ratnam | An optimizationbased approach to scheduling residential battery storage with solar PV: Assessing customer benefit[END_REF]. Energy storage can supply more flexibility and balancing to the grid, providing a back-up to intermittent solar energy. Locally, it can improve the management of distribution network, reducing costs and improving efficiency. The design principle for sizing the system's battery is to compensate for daily variations in solar radiation and not to act as seasonal energy storage [START_REF] Galvin | A stand-alone photovoltaic supercapacitor battery hybrid energy storage system[END_REF]. PV modules are not an ideal source for battery charging. The PV output is heavily dependent on weather conditions, therefore an optimum charge/discharge cycle can be guaranteed, resulting in a low battery state of charge (SOC) [START_REF] Amstrong | Comparison battery charging algorithms for stand-alone photovoltaic system[END_REF]. Low battery SOC drives to shorten the life of the battery. The battery proposed is Lithium battery LiFePO4 with 100 Ah battery capacity, with three blocks of 5 kWh/each. Then the nominal voltage for each cell is 54 V. The battery has columbic efficiency by 97%.

V. PV SYSTEM PERFORMANCE ANALYSIS

The performance evaluation of this study is based on the PVSyst software simulation. This part discuss the output from the simulation tool such as the potential energy resources, the optimum tilted angle of PV array and energy production from the PV array as well as system losses.

A. Tilted angle optimization of PV array installation

To optimize the tilted angle m  for the rooftop PV array, an ideal case was built with azimuth equal to

0 0  m A
. The tilted angle of the free standing system was adjusted from 0° to 30°. The result of simulation is shown in Fig. 4 and5. The specific energy product and system energy product indicates that the best tilted angle is 15°, however the difference between the results is small for the range of tilt angles of 9, 12, 15, 18, and 21 degree. Therefore, the tilt angle of South faced PV system for this building can be flexible between 9 and 21 degree. In reality, optimum PV array output can also be achieved using tilt angle approximately equal to the site's latitude of 21°N for Hanoi. Mono crystalline cell is better choice than poly crystalline cell. Central inverter is better choice than string inverter. With the mono-crystalline and central inverter, simulation shows an annual system production of PV 15 kWp system of 19,348 kWh/year, and the specific yearly production is 1,205 kWh/kWp/year. The overall performance ratio (PR) is 0.849 and the normalized production is 3.3 kWh/kWp/day. The PR is a measure for the overall losses of a PV system and is defined as the ratio of final energy yield of the PV system in kWh/kWp to a reference yield, which takes only solar irradiation into account.

The building requires single phase voltage for lighting and office equipment like computers, laptops and three phase voltage to supply the elevators and HVAC system, therefore three separable single phase inverters configuration is chosen. The following section presents the analysis of simulation results to evaluate the system performance for the monocrystalline and string inverter variant. , PV mono-crystalline modules and string inverter are mentioned in this simulation, when the yearly data is averaged over 24 h period for each month by estimated in PVSyst, we get the average power production of PV array profile for different months as shown in Fig. 6. It can be observed that:

-The average monthly peak power ranges between 8.7 kW in August and 5.2 kW in January. This indicates that the PV array on an average only produces around 60 % of its rated power even in the sunniest month of the year. The rated power of the PV array using 48 modules of XL SW 335 is estimated under standard test conditions, i.e. 1,000 W/m 2 and 25°C. However, according to Fig. 2, the average global irradiance in August is 595 W/m 2 and the ambient temperature is about 37.5°C.

-PV array generates energy for 13 hours during summer months while it is only 11 hours in winter.

C. System Losses

PV array failed to generate 100% the energy delivered from the sun because of some losses. Fig. 9 explored the overall system loss diagram for site installation. A collector plane received about 1,391 kWh/m2 of global incident irradiation. But the effectiveness plane receives the irradiance only at 1,366 kWh/m2. The biggest losses happen in PV array production energy. Energy produced from the PV array affected by several factors such as ambient temperature which takes the main part of loss (9.4%), solar incidence, manufacture mismatch and ohmic wiring. The loss in inverter is 3.1% since the efficiency of inverter in specification of producer is 96.9%. Other losses in inverter operation is zero, shows that PV array and inverter perfectly fit each other. The actual energy supplied to the load can identify after through the losses of regulator and battery storage. These annual system production of PV 15 kWp system of 19,138 kWh/year. Fig. 9. Overall system losses in case: azimuth at 0° and tilted angle at 15°, simulation result from PVSyst [START_REF] Mermoud | Use and Validation of PVSYST, a User-Friendly Software for PV-System Design[END_REF][START_REF] Muller | Handbook part 1: Software[END_REF] VI. CONCLUSION This paper presents the use of PVSyst as a simulation tool to evaluate the PV array installed on building's rooftop. This software enables the PV project designer to select system configurations as well as predict its energy production. System sizing depends strongly on the geographical location of the installation site. This work studied a PV system which is applied for low-voltage grid connection at building-scale, with electrical storage system in the scope of our research on energy efficiency dedicated to the green building. The simulations shown that the average monthly peak power ranges from 8.7 kW in August to 5.2 kW in January. This indicates the PV array on an average only produces of 60 % of its rated power even in the sunniest month of the year. It is also helpful as a pre-decision of installation. The results indicated that the optimal tilt for PV modules installed on building's rooftop in Cau Giay-Hanoi to get maximum yield is 15°.

For future works, the sizing of grid-connected PV system should be extended to all component and an economic analysis should also be performed. In addition, measurement equipment such as pyrometer can be used to obtain on-site irradiance data in real time. Such measuring equipment could be used monthly in order to correct the simulation results provided by PVsyst with the output and real production from the PV installation.

Fig. 1 .

 1 Fig. 1. Building energy management platform for University building

Fig. 2 .

 2 Fig. 2. Value of average global irradiation in kWh/m 2 from Meteonorm (a) and in W/m 2 from Climate-SAF PVGIS (b)

Fig. 4 .

 4 Fig. 4. Specific product of PV system for

Fig. 6 .

 6 Fig. 6. Average power output of 15 kWp PV array as a function of time of the day for different months.

Fig. 7 .

 7 Fig. 7. Average daily yield of 15 kWp PV array for different months.

Fig. 7

 7 Fig.7 and 8show the average daily yield of PV system for different months. The different of PV yield of two time can be observed between January and August. In August, PV yield always reaches a high level with almost no fluctuation around the average value approximately 64.290 kWh/day. There are also cloudy days with low daily yield about of 30 kWh which is showed in these months from May to November. By contrast, in January, PV yield fluctuates dramatically among days in month. There are cloudy days with low daily yield of < 5 kWh and sunny days with yield > 70 kWh. This trend also occurs in the winter: in December, January, February and March.

Fig. 8 .

 8 Fig. 8. Average daily yield of 15 kWp PV array for January and August.

TABLE I .

 I

	Load		Power (kW)	Use (hour)	Energy (kWh)
	HVAC		369.8	4			1479.2
	Elevator	38.4	2			76.8
	Lighting	51.6	4			206.4
	Office equipment	71.9	6			431.4
	Total		531.7				2193.8
	TABLE II.	BUILDING ENERGY CONSUMPTION IN MONTHS (KWH) 2015
	Jan	Feb	Mar	Apr		May	Jun
	21,600	14,480	21,120	28,880	34,080	42,400
	Jul	Aug	Sep	Oct		Nov	Dec
	36,160	37,680	37,040	24,400	31,840	31,840

BUILDING ENERGY DEMAND FOR A DAY IN JUNE 2016

TABLE III .

 III BUILDING ENERGY CONSUMPTION IN MONTHS (KWH) 2016

	Jan	Feb	Mar	Apr	May	Jun
	27,200	18,080	21,680	26,240	38,080	48,800
	Jul	Aug	Sep	Oct	Nov	Dec
	44,000	40,400	---	---	---	---

TABLE V

 V

		.	PARAMETERS OF INVERTERS
	Inverter	Number	Nominal power	Operation voltage	DC input peak
	Ingecon Sun 15 TL M	1	15 kW	300-820 V	50A (2 inputs 20/30 A)
	Conext RL 5000E	3	5 kW	100-540 V	36A
	TABLE VI.	SIMULATED ARRAY/INVERTER COMBINATIONS
	Inverter	Num.	Mod. in series	String	MPPT input/String input
	Ingecon Sun 15 TL M	1 (three phase)	12	4		2/2
	Conext RL 5000E	3 (mono-phase)	8	6		1/2


Stefano Moretti

Meltem Öztürk

Ordinal power relations and social rankings 1

Several real-life complex systems, like human societies or economic networks, are formed by interacting units characterized by patterns of relationships that may generate a group-based social hierarchy. In this paper, we address the problem of how to rank the individuals with respect to their ability to "influence" the relative strength of groups in a society. We also analyse the effect of basic properties in the computation of a social ranking within specific classes of (ordinal) coalitional situations. We show that the pairwise combination of these natural properties yields either to impossibility (i.e., no social ranking exists), or to flattening (i.e., all the individuals are equally ranked), or to dictatorship (i.e., the social ranking is imposed by the relative comparison of coalitions of a given size).

Introduction

Ranking is a fundamental ingredient of many real-life situations, like the ranking of candidates applying to a job, the rating of universities around the world, the distribution of power in political institutions, the centrality of different actors in social networks, the accessibility of information on the web, etc. Often, the criterion used to rank the items (e.g., agents, institutions, products, services, etc.) of a set N also depends on the interaction among the items within the subsets of N (for instance, with respect to the users' preferences over bundles of products or services). In this paper we address the following question: given a finite set N of items and a ranking over its subsets, can we derive a "social" ranking over N according to the "overall importance" of its single elements?

For instance, consider a company with three employees 1, 2 and 3 working in the same department. According to the opinion of the manager of the company, the job performance of the different teams S ⊆ N = {1, 2, 3} is as follows: {1, 2, 3} {3} {1, 3} {2, 3} {2} {1, 2} {1} ∅ (S T , for each S, T ⊆ N , means that the performance of S is at least as good as the performance of T ). Based on this information, the manager asks us to make a ranking over his three employees showing their attitude to work with others as a team or autonomously. Intuitively, 3 seems to be more influential than 1 and 2, as employee 3 belongs to the most successful teams in the above ranking. Can we state more precisely the reasons driving us to this conclusion? And what can we say if we have to decide who between 1 and 2 is more productive and deserves a promotion? In this paper we analyse different properties of ordinal social rankings in order to get some answers to such questions.

The problem studied in this paper can be seen as an ordinal counterpart of the one about how to measure the power of players in simple games, which are coalitional games where coalitions may be winning or not [START_REF] Banzhaf | Weighted voting doesn't work: A mathematical analysis[END_REF][START_REF] Dubey | Value theory without efficiency[END_REF]. However, our framework is different for at least two reasons: first, we face coalitional situations where only a qualitative (ordinal) comparison of the strength of coalitions is given; second, we look for a ranking over the single objects in N , and we do not require a quantitative assessment of the "power" of the players. As far as we know, the only attempt in the literature to generalize the notions of coalitional game and power index within an ordinal framework has been provided in [START_REF] Moretti | An axiomatic approach to social ranking under coalitional power relations[END_REF], where, given a total preorder representing the relative strength of coalitions, a social ranking over the player set is provided according to a notion of ordinal influence and using the Banzhaf index [START_REF] Banzhaf | Weighted voting doesn't work: A mathematical analysis[END_REF] of a "canonical" coalitional game.

In the literature of simple games, related questions deal with the ordinal equivalence of power indices (see, for instance, [START_REF] Lambo | Ordinal equivalence of power notions in voting games[END_REF][START_REF] Freixas | On ordinal equivalence of the Shapley and Banzhaf values[END_REF][START_REF] Lucchetti | Ranking sets of interacting objects via semivalues[END_REF]) and the analysis of the differences between rankings generated by alternative power indices on special classes of simple games (e.g, the papers [START_REF] Saari | Some surprising properties of power indices[END_REF]8]). Similarly to our work, in [START_REF] Taylor | Simple games: desirability relations, trading, pseudoweightings[END_REF] the authors investigated alternative notions of ordinal power on different classes of simple games. All the aforementioned papers focus on the notion of simple game, that is a numerical representation of a dichotomous power relation (i.e., winning or loosing coalitions), a much more restricted domain than the one considered in this work, where a power relation can be whatever total preorder over the coalitions.

In a still different context, a model of coalition formation has been introduced in [START_REF] Piccione | Coalition formation under power relations[END_REF], where the relative strength of disjoint coalitions is represented by an exogenous binary relation and the players try to maximize their position in a social ranking. We also notice a connection with some kind of "inverse problems", precisely, how to derive a ranking over the set of all subsets of N in a way that is "compatible" with a primitive ranking over the single elements of N (see, for instance, [START_REF] Barberà | Ranking sets of objects[END_REF]; see [START_REF] Moretti | Ranking sets of possibly interacting objects using shapley extensions[END_REF] for an approach using coalitional games).

In this paper, a social ranking is defined as a map associating to each power relation (i.e., a total preorder over the set of all subsets of N ) a total preorder over the elements of N . The properties for social rankings that we analyse in this paper have classical interpretations, such as anonymity and symmetry, saying that the ranking should not depend on the identity of the elements of N , or the dominance, saying that an element i ∈ N should be ranked higher than an element j ∈ N whenever i dominates j, i.e. a coalition S ∪ {i} is stronger than S ∪ {j} for each S ⊂ N not containing neither i nor j. Another property we study in this paper is the independence of irrelevant coalitions, saying that the social ranking between two elements i and j should only depend on their respective contributions when added to coalitions not containing neither i nor j (in other words, the information needed to rank i and j is provided by the relative comparison of coalitions U, W ⊂ N such that U \ {i} = W \ {j}).

We use these properties to axiomatically analyse social rankings on particular classes of power relations. We first notice that two natural properties, precisely, dominance and symmetry, are not compatible over the class of all power relations (see Theorem 1 in Section 4), despite the fact that, in some related axiomatic frameworks (see, for instance, [START_REF] Barberà | Ranking sets of objects[END_REF]), similar axioms have been successfully used in combination. On the other hand, the properties of independence of irrelevant coalitions and symmetry in combination determine a flattening of the social ranking, where all the agents are equivalent (see Proposition 2 in Section 4). Finally, we proved that the property of independence of irrelevant coalitions and dominance property determine a kind of 'dictatorship of the cardinality' when a relation of strong dominance among coalitions of the same size holds: in this case, the only social ranking satisfying those two properties is the one imposed by the relation of dominance of a given cardinality s ∈ {1, . . . , |N |} (see Theorem 2 in Section 5).

The structure of the paper is the following. In the next section, we present some related approaches from the literature and our main contributions. Basic notions and definitions are presented in Section 2. In Section 3 we introduce and discuss some properties for social rankings. In Section 4 we study the compatibility of certain axioms and their effect on some elementary notions of social ranking. In Section 5 we focus on the analysis of social rankings that satisfy both the dominance property and the property of independence of irrelevant coalitions, and that, on particular power relations, are specified by the ordering of coalitions of the same size. Section 6 concludes with some future research directions.

A binary relation R on a finite set N = {1, . . . , n} is a collection of ordered pairs of elements of N , i.e. R ⊆ N × N . ∀x, y ∈ N , the more familiar notation xRy will be often used instead of the more formal one (x, y) ∈ R. We provide some standard properties for R. Reflexivity: for each x ∈ N , xRx; transitivity: for each x, y, z ∈ N , xRy and yRz ⇒ xRz; totality: for each x, y ∈ N , x = y ⇒ xRy or yRx; antisymmetry: for each x, y ∈ N , xRy and yRx ⇒ x = y. A reflexive and transitive binary relation is called preorder. A preorder that is also total is called total preorder. A total preorder that also satisfies antisymmetry is called linear order. The notation ¬(xRy) means that xRy is not true. We denote by 2 N the power set of N and we use the notations T N and T 2 N to denote the set of all total preorders on N and on 2 N , respectively. Moreover, the cardinality of a set S ∈ 2 N is denoted by |S|. In the remaining of the paper, we will also refer to an element S ∈ 2 N as a coalition S.

Consider a total preorder ⊆ 2 N × 2 N over the subsets of N . Often we will use the notation S T to denote the fact that S T and ¬(T S) (in this case, we also say that the relation between S and T is 'strict'), and the notation S ∼ T to denote the fact that S T and T S. For each i, j ∈ N , i = j, and all k = 1, . . . , n-2, we denote by Σ k ij = {S ∈ 2 N \{ij} : |S| = k} the set of all subsets of N not containing neither i nor j with k elements. Moreover, for each i, j ∈ N , we define the set D k ij ( ) = {S ∈ Σ k ij : S ∪ {i} S ∪ {j}} as the set of coalitions S ∈ 2 N \{ij} of cardinality k such that S ∪ {i} is in relation with S ∪ {j} (and, changing the ordering of i and j, the set

D k ji ( ) = {S ∈ Σ k ij : S ∪ {j} S ∪ {i}}).
3 Axioms for social rankings

In the following of these notes, we interpret a total preorder on 2 N as a power relation, that is, for each S, T ∈ 2 N , S T stands for 'S is considered at least as strong as T according to the power relation '.

Given a class C 2 N ⊆ T 2 N of power relations, we call a map ρ : C 2 N -→ T N , assigning to each power relation in C 2 N a total preorder on N , a social ranking solution or, simply, a social ranking. Then, given a power relation , we will interpret the total binary relation ρ( ) associated to by the social ranking ρ, as the relative power of players in a society under relation . Precisely, for each i, j ∈ N , iρ( )j stands for 'i is considered at least as influential as j according to the social ranking ρ( )', where the influence of an agent is intended as her/his ability to join coalitions in the strongest positions of a power relation. Note that we require that ρ( ) is a total preorder over the elements of N , that is we always want to express the relative comparison of two agents, and such a relation must be transitive.

A social ranking ρ : C 2 N -→ T N such that iρ( )j ⇔ {i} {j} for each ∈ C 2 N and each i, j ∈ N is said to be primitive (i.e., it neglects any information contained in about the comparison of coalitions of cardinality different from 1). A social ranking ρ : C 2 N -→ T N such that iρ( )j and jρ( )i for all i, j ∈ N is said to be unanimous (N is an indifference class with respect to ρ( )). Now we introduce some properties for social rankings. The first axiom is the dominance one: if each coalition S containing agent i but not j is stronger than coalition S with j in the place of i, then agent i should be ranked higher than agent j in the society, for any i, j ∈ N . Precisely, given a power relation ∈ T 2 N and i, j ∈ N we say that i dominates j in if S ∪ {i} S ∪ {j} for each S ∈ 2 N \{i,j} (we also say that i strictly dominates j in if i dominates j and in addition there exists S ∈ 2 N \{i,j} such that S ∪ {i} S ∪ {j}).

Definition 1 (DOM). A social ranking

ρ : C 2 N -→ T N satisfies the dominance (DOM) property on C 2 N ⊆ T 2 N if and only if for all ∈ C 2 N and i, j ∈ N , if i dominates j in then iρ( )j [and ¬(jρ( )i) if i strictly dominates j in ].
The following axiom states that the relative strength of two agents i, j ∈ N in the social ranking should only depend on their effect when they are added to each possible coalition S not containing neither i nor j, and the relative ranking of the other coalitions is irrelevant. Formally:

Definition 2 (IIC). A social ranking ρ : C 2 N -→ T N satisfies the Independence of Irrele- vant Coalitions (IIC) property on C 2 N ⊆ T 2 N iff iρ( )j ⇔ iρ( )j
for all i, j ∈ N and all power relations , ∈ C 2 N such that for each S ∈ 2 N \{i,j}

S ∪ {i} S ∪ {j} ⇔ S ∪ {i} S ∪ {j}. Let ∈ T 2 N and i, j, p, q ∈ N be such that |D k ij | = |D k pq | and |D k ji | = |D k qp |
for each k = 0, . . . , n -2. Differently stated, for coalitions S of fixed cardinality, we have that the number of times that S ∪{i} is stronger than S ∪{j} equals the number of times that S ∪{p} is stronger than S ∪{q} (and the number of times that S ∪{j} is stronger than S ∪{i} equals the number of times that S ∪ {q} is stronger than S ∪ {p}). In this symmetric situation, the following axiom states a principle of equivalence between the pairs i, j and p, q.

Definition 3 (SYM). A social ranking ρ : C 2 N -→ T N satisfies the symmetry (SYM) property on C 2 N ⊆ T 2 N iff iρ( )j ⇔ pρ( )q for all i, j, p, q ∈ N and ∈ C 2 N such that |D k ij | = |D k pq | and |D k ji | = |D k qp | for each k = 0, . . . , n -2.
Remark 1. Note that if a social ranking ρ satisfies the SYM axiom on C 2 N ⊆ T 2 N , then for every ∈ C 2 N and i, j

∈ N , if |D k ij | = |D k ji |
for each k = 0, . . . , n -2, then iρ( )j and jρ( )i, that is i and j are indifferent in ρ( ) (to see this, simply take p = i and q = j in Definition 3).

Remark 2. If we want to check if a given social ranking rule satisfies DOM, IIC, or SYM only partial information on is needed. In fact, conditions on the ranking ρ( ) between two elements {i, j} only depend on the comparisons of subsets having the same cardinality and sharing the same subset S ∈ 2 N \{i,j} not containing neither i nor j.

We conclude this section with an example showing that an apparently natural procedure (namely, the majority rule) to rank the agents of N may fail to provide a transitive social ranking. We first formally introduce such a procedure.

Definition 4 (Majority rule). A majority rule (denoted by M ) is a map assigning to each power relation

∈ T 2 N a total binary relation M ( ) on N such that iM ( )j ⇔ d ij ( ) ≥ d ji ( ).
where

d ij ( ) = |{S ∈ 2 N \{i,j} : S ∪ {i} S ∪ {j}}| for each i, j ∈ N .
Example 1. One can easily check that the majority rule M satisfies the property of DOM, IIC and SYM on the class T 2 N . On the other hand, it is also easy to find an example of power relation such that M ( ) is not transitive. Consider for instance the power relation ∈ T

Primitive and unanimous social rankings

In this section we study the relations between the axioms introduced in the previous section and the social ranking solutions. In the following, we show that DOM and SYM are not compatible in a general case, for N > 3 (see Theorem 1), whereas SYM and IIC determine a unanimous social ranking on particular power relations.

We start with showing some consequences of using the axioms introduced in the previous section when the cardinality of the set N is 3 or 4. The analysis for cardinality |N | = 3 is easy since we can enumerate all the cases. As we will present in the following, the notion of complementarity plays an important role in this case. We denote by S * the complement of the subset S (S * = N \S), and we say that a social ranking ρ such that iρ( )j ⇔ {j} * {i} * for each ∈ T 2 N and each i, j ∈ N is complement primitive (i.e., it neglects any information contained in about the comparison of coalitions of cardinality different from n -1). A relation which provides coherent comparisons with respect to the complement of objects is said "self-reflecting" . The notion of "self-reflecting" is introduced in [START_REF] Fishburn | Signed orders and power set extensions[END_REF]. More formally, if we denote by S * the complement of the subset S (S * = N \ S), we say that the power relation is self-reflecting if and only if for all S, Q ∈ N , S Q implies Q * S * .

Corollary 1. If |N | = 3 and the power relation is self-reflecting, then the DOM condition is sufficient in order to determine the social ranking and it corresponds to a primitive social rule.

Proof. Let N = {i, j, k}. Self-reflecting implies that ∀i, j ∈ N i j ⇔ j * i * ⇔ ik jk. By DOM we get ∀i, j, k ∈ N iρ( )j ⇔ i j ⇔ j * i * ⇔ ik jk.

Next theorem shows that on the class T 2 N (all possible total preorders) the properties of DOM and SYM are not compatible. We rewrite the relevant informations about and the elements 1, 2 and 3 by means of the following Table 2. By Remark 1, a social ranking rule ρ : T 2 N -→ T N which satisfies SYM should be such that 2ρ( )3, 3ρ( )2, 1ρ( )3, 3ρ( )1. By the DOM property, we should have 1ρ( )2, and ¬(2ρ( )1), which yields a contradiction with the transitivity of the ranking ρ( ). Now, consider power relations in T 2 N , with N ⊇ {1, 2, 3, 4}, that are obtained from the power relation defined above and assigning all the additional subsets of N not contained in {1, 2, 3, 4} in the same indifference class. More precisely, let N ⊇ {1, 2, 3, 4}, and take ∈ T 2 N such that U W :⇔ U W for all the subsets U, W ⊆ {1, 2, 3, 4}, and U W , W U for all the other subsets of N not included in {1, 2, 3, 4}.

The following proposition shows that the adoption of properties IIC and SYM yields a unanimous social ranking over all those power relations ∈ T N such that, for some k ∈ {0, . . . , |N | -2}, D t ji ( ) = D t ij ( ) for all cardinalities t = k and all i, j ∈ N , and |. Suppose now that iρ( )j. By IIC, we have iρ( )j. By SYM, jρ( )l and lρ( )i. By transitivity of ρ( ), jρ( )i. By IIC we conclude that jρ( )i too. In a similar way, if we suppose jρ( )i, then we end up with the conclusion that iρ( )j too, and the proof follows.

|D k ji ( )| is not necessarily equal to |D k ij ( )| (provided that D k ij ( ) \ D k ji ( ) = ∅ and D k ji ( ) \ D k ij ( ) = ∅).
Note that |D k ji ( )| = |D k li ( )| = |D k jl ( )| and |D k ij ( )| = |D k il ( )| = |D k lj ( )
An interesting consequence of Proposition 2 is that if the only information making a difference between two objects is given by comparisons of a fixed cardinality, then it is sufficient to have one discordance in order to declare an indifference (with IIC and SYM). Proposition 2 suggests how to deal with situations where coalitions are of a fixed size (such situations are not so eccentric in real life). For instance, let us imagine that we have committees with a given number (k) of persons and that we have a ranking on them (for instance N = {1, 2, 3, 4} and k = 2, with 12 13 14 34 24 23). Since committees are always formed by two persons, no information is available on subsets of N with l = k elements (or such information is irrelevant). How to define a social ranking in this case? One solution could be to consider all the other comparisons indifferent. Then, by Proposition 2, we know that SYM and IIC properties can be used in order to support a unanimous social ranking.

Table 3: An illustrative example of the six possible cases for a power relation as the one considered in Proposition 2 with N = {1, 2, 3, i, j, l}, k = 2 and D = {{1, 2}, {2, l}}. i vs j i vs. l j vs. l case 4): S = {1, 2} {1, 2, i} {1, 2, j} {1, 2, i} {1, 2, l} {1, 2, j} {1, 2, l} case 6): S = {1, 3} {1, 3, i} {1, 3, j} {1, 3, i} {1, 3, l} {1, 3, j} {1, 3, l} case 2): S = {2, 3} {2, 3, i} {2, 3, j} {2, 3, i} {2, 3, l} {2, 3, j} {2, 3, l} case 5): S = {1, l} {1, l, i} {1, i, j} {1, i, j} {1, j, l} {1, i, j} {1, i, l} case 3): S = {2, l} {2, l, i} {2, i, j} {2, i, j} {2, j, l} {2, i, j} {2, i, l} case 1):

S = {3, l} {3, l, i} {3, i, j} {3, i, j} {3, j, l} {3, i, j} {3, i, l} |D ij ( )| = 4 |D il ( )| = 2 |D jl ( )| = 4 |D ji ( )| = 2 D li ( )| = 4 |D lj ( )| = 2
5 Dictatorship of the coalition size

In this section, we define a special class of power relations (namely, the per size-strong dominant relations) characterized by the fact that a relation of dominance always exists with respect to coalitions of the same size, but the dominance may change with the cardinality (for instance, an element i could dominate another element j when coalitions of size s are considered, but j could dominate i over coalitions of size t = s). We first need to introduce the notion of s-strong dominance.

Definition 5. Let ∈ T 2 N , i, j ∈ N and s ∈ {0, . . . , n -2}. We say that i s-strong dominates j in , iff

S ∪ {i} S ∪ {j} for each S ∈ 2 N \{i,j} with |S| = s. (1)
Definition 6. We say that ∈ T 2 N is per size-strong dominant (shortly, ps-sdom) iff for each s ∈ {0, . . . , n -2} and all i, j ∈ N , we have either

[i s-strong dominates j in ] or [j s-strong dominates i in ].
The set of all ps-sdom power relations is denoted by

S 2 N ⊆ T 2 N .
We first study the effect of the combination of the properties of DOM and IIC on a specific instance of ps-sdom power relations where there exist elements that are always placed at the top or at the bottom in the rankings of coalitions of equal cardinality. We rewrite the relevant informations about by means of Table 4.

Note that for each s ∈ {0, 2}, it holds that either S ∪ {1} S ∪ {l} for each S ⊆ N \ {1} with |S| = s and all l ∈ N \ S (i.e., coalitions S ∪ {1} are ranked above all coalitions S ∪ {l}, with l = i and S containing 0 or 2 elements), and S ∪ {1} S ∪ {l} for each S ⊆ N \ {1} with |S| = 1 and all l ∈ N \ S (i.e., coalitions S ∪ {1} are ranked below all coalitions S ∪ {l}, with l = i and S containing precisely 1 element). Similar considerations can be done for element 4. So, elements 1 and 4 are two "extreme" ones. Let us remark that there can be at most two "extreme" elements of a power relation in S 2 N . In Proposition 3 we argue that on this kind of power relations, a social ranking satisfying both DOM and IIC cannot rank "extreme" elements (in this case 1 and 4) in between two others. The following proposition shows the effect of DOM and IIC on the social position of the "extreme" elements.

Then, [iρ( )j for all j ∈ N ] OR [jρ( )i for all j ∈ N ].

Proof. Suppose on the contrary that there exist j, k ∈ N \ {i}, such that jρ( )i and iρ( )k.

Define ∈ T 2 N such that

S ∪ {i} = S ∪ {j} ⇔ S ∪ {i} S ∪ {j} for all S ⊆ N \ {i, j}, (5)
S ∪ {i} = S ∪ {k} ⇔ S ∪ {i} S ∪ {k} for all S ⊆ N \ {i, k},

and S ∪ {k} = S ∪ {j} for all S ⊆ N \ {j, k}.

[note that each coalition S ∪ {i}, with S ⊆ N \ {i}, by condition (2) and (3), is ranked strictly higher or lower than each other coalition S ∪ {j}, j = i, so the rearrangement of coalitions in to obtain is feasible.] By IIC, we have that iρ( )j ⇔ iρ( )j and iρ( )k ⇔ iρ( )k.

So, by relation (4), jρ( )i and iρ( )k. On the other hand, by DOM we have kρ( )j and ¬(jρ( )k), which yields a contradiction with the transitivity of ρ( ).

Proposition 3 shows that if there is an element i ∈ N having "contradictory" and "radical" behavior depending on the size of coalitions (very well for size k and very bad for size l), then the social ranking satisfying IIC and DOM can not give him an intermediate position: the element i will be the "best" one or the "worst" one in the social ranking.

In the following we argue that if a power relation is in S 2 N and a social ranking satisfies both DOM and IIC on the set of ps-sdom power realtions S 2 N , then it must exist a cardinality t * ∈ {0, . . . , n -2} whose relation of t * -strong dominance (dictatorially) determines the social ranking. We first need to introduce the next lemma, where a given element i plays an important role.

Lemma 1. Let i ∈ N and ρ : S 2 N -→ T N be a social ranking satisfying IIC and DOM on S 2 N . There exists t * ∈ {0, . . . , n -2} such that jρ( )k ⇔ j t * -strong dominates k in , for all j, k ∈ N \ {i} and ∈ S 2 N . Proof. Given a power relation ∈ S 2 N , define another power relation 0 ∈ S 2 N such that for each S ⊆ N \ {i} we have

S ∪ {l} 0 S ∪ {i} for all l ∈ N \ (S ∪ {i}), (8)
and

U 0 W :⇔ U W
for all the other possible pairs of coalitions U, W whose comparison is not already considered in (8). Roughly speaking, the only difference between 0 and is that coalitions of size s containing i are placed at the bottom of the ranking induced by over the coalitions of the same size. By DOM, it follows that lρ( 0 )i for every l ∈ N . Now, for each t ∈ {0, . . . , n -2}, define a power relation t ∈ T 2 N such that S ∪ {i} t S ∪ {l} for each l ∈ N and S ∈ 2 N \{i,l} with |S| = s,

where s ∈ {0, . . . , t}, and

U t W :⇔ U t-1 W
for all the other possible pairs of coalitions U, W whose comparison is not already considered in [START_REF] Lucchetti | Ranking sets of interacting objects via semivalues[END_REF]. So, the only difference between t and t-1 , for each t ∈ {1, . . . , n -2}, is that in t coalitions of size t containing i are placed at the top of the ranking induced by t-1 over coalitions of the same size t, and all the remaining comparisons remain the same as in t-1 . Note that by Proposition 3, we have that either lρ( t )i for every l ∈ N , or iρ( t )l for every l ∈ N . Moreover, By DOM, it follows that iρ( n-2 )l for every j ∈ N .

Let t * be the smallest number in {0, . . . , n -2} such that lρ( t * -1 )i for every l ∈ N and iρ( t * )l for every l ∈ N (for the considerations above such a t * must exist, being, at most, t * = n -2).

Next, we argue that for every j, k ∈ N \ {i}, the social ranking between j and k in is imposed by the relation of t * -strong dominance in .

W.l.o.g., suppose that S ∪ {j} S ∪ {k} (and, as a consequence, S ∪ {j} t * S ∪ {k}) for each S ∈ 2 N \{j,k} , and |S| = t * . Consider another power relation ∈ T 2 N obtained by t * and such that:

S ∪ {j} = S ∪ {i} for each S ∈ 2 N \{i,j} with |S| = t * , ( 10
) S ∪ {i} = S ∪ {k} for each S ∈ 2 N \{i,k} with |S| = t * , (11)
S ∪ {j} = S ∪ {k} for each S ∈ 2 N \{j,k} \ 2 N \{i,j} ∪ 2 N \{i,k} , and

|S| = t * , (12)
and, finally,

U V :⇔ U t * V (13)
for all the other relevant pairs of coalitions U, W of size s = t * + 1. By IIC jρ( )i (since in the comparisons between coalitions containing i and j are precisely as in t * -1 and, as previously stated, jρ( t * -1 )i) and iρ( )k (since in the comparisons between coalitions containing i and k are precisely as in t * and, as previously stated, iρ( t * )k). Then, by transitivity of ρ( ) we have jρ( )k. Note that by IIC, jρ( )k ⇔ jρ( t * )k ⇔ jρ( )k. We have then proved that whenever j t * -dominates k, then jρ( )k.

We can now formulate the following theorem stating the "dictatorship of the coalition's size". Theorem 2. Let ρ : S 2 N -→ T N be a social ranking satisfying IIC and DOM on S 2 N . There exists t * ∈ {0, . . . , n -2} such that iρ( )j ⇔ i t * -strong dominates j in , for all i, j ∈ N and ∈ S 2 N . Proof. Given a power relation ∈ S 2 N , let i ∈ N and define t * starting from and i precisely as in the proof of Lemma 1. Now take k ∈ N \ {i} and apply Lemma 1 with k in the role of i. Consequently, we have that there exists t ∈ {0, . . . , n -2} such that hρ( )l ⇔ h t-strong dominates l in , for each h, l ∈ N \ {k}, and in particular iρ( )l ⇔ i t-strong dominates l in , for whatever complete power relation ∈ S 2 N .

But in the proof of Lemma 1 we have shown that iρ( )l ⇔ i t * -strong dominates l in t * (remember that t * in the proof of Lemma 1 is the smallest number in {0, . . . , n -2} such that lρ( t * -1 )i for every l ∈ N and iρ( t * )l for every l ∈ N ). Then it must be t = t * , and the proof follows.

Conclusions

In this paper we introduced and studied the problem of how to rank the objects of a set N according to their ability to influence the ranking over the subsets of N . As far as we know, this is the first time that a social ranking is proposed using an axiomatic approach (and without the quantitative notion of power index from cooperative game theory, like in [START_REF] Moretti | An axiomatic approach to social ranking under coalitional power relations[END_REF]). A possible direction for future research is the open question about which axioms could be used to characterize a social ranking over the domain of all possible power relations. In view of our results, each combination of the axioms we propose in this paper is not satisfactory. In this respect, it is worth noting that all the properties that we analysed are based on the comparison of subsets having the same number of elements. Therefore, it would be interesting to study properties based on the comparison among subsets with different cardinalities. For instance, if N = {1, 2, 3}, the information of the type {1} {2, 3} {1, 3} {2} could be used to establish that 1 is socially stronger than 2 (note that 1 strictly dominates 2 on coalitions of size 1, and 2 strictly dominates 1 of coalitions of cardinality 2, but the "interval" between {2, 3} and {1, 3} is smaller than the one between {1} and {2}). Of course, social ranking solutions taking into account the relative comparison of coalitions with different size, would necessarily face the important problem of dealing with a much larger amount of information.

Another interesting direction is to focus on alternative classes of power relations. Similar to the classical approaches adopted in the analysis of power indices for simple games, one could be interested in studying power relations that satisfy a monotonicity property, that is to only consider those power relations ∈ T 2 N such that T S for each coalitions S, T ∈ 2 N with S ⊆ T . However, as already noticed, it is worth remarking that the constraints imposed by our axioms apply to coalitions of the same size, and therefore a restriction based on the monotonicity property has no relevant impact on the results provided in this paper. Moreover, the property of monotonicity, as many others studied in the literature on ranking sets of objects (see [START_REF] Barberà | Ranking sets of objects[END_REF]), prevents a kind of interaction among the elements of N that is often observed in practice, i.e., the possibility that the power of a coalition could be deteriorated by the addition of a new individual which is incompatible or redundant with those already contained in it (see [START_REF] Moretti | Ranking sets of possibly interacting objects using shapley extensions[END_REF] for a detailed discussion on this issue).

Acknowledgement

This work benefited from the support of the projects AMANDE ANR-13-BS02-0004 and CoCoRICo-CoDec ANR-14-CE24-0007 of the French National Research Agency (ANR).

Proposition 1 .

 1 If |N | = 3, then there are only two social ranking solutions satisfying the DOM and SYM conditions: the primitive solution and the complement primitive one. Proof. Let N = {1, 2, 3} with 1 2 3. Then six cases may occur in : case 1) 13 23 12, case 2) 13 12 23, case 3) 23 13 12, case 4) 12 13 23, case 5) 23 12 13 and case 6) 12 23 13. DOM and SYM impose that: case 1) by DOM :1ρ( )2, by SYM (1ρ( )3 and 2ρ( )3) or (3ρ( )1 and 3ρ( )2). Hence we have 1ρ( )2ρ( )3 (primitive) or 3ρ( )1ρ( )2 (complement primitive) case 2) by DOM :1ρ( )2 and 1ρ( )3. We can have 2ρ( )3 or 3ρ( )2. Hence we have 1ρ( )2ρ( )3 (primitive) or 1ρ( )3ρ( )2 (complement primitive) case 3) by SYM : (1ρ( )2, 1ρ( )3 and 2ρ( )3) or (2ρ( )1 , 3ρ( )1 and 3ρ( )2). case 4) by DOM 1ρ( )2ρ( )3 case 5) by DOM :2ρ( )3, by SYM (1ρ( )2 and 1ρ( )3) or (2ρ( )1 and 3ρ( )1). Hence we have 1ρ( )2ρ( )3 (primitive) or 2ρ( )3ρ( )1 (complement primitive) case 6) by DOM :1ρ( )3 and 2ρ( )3. We can have 1ρ( )2 or 2ρ( )1. Hence we have 1ρ( )2ρ( )3 (primitive) or 2ρ( )1ρ( )3 (complement primitive)

Theorem 1 .

 1 Let |N | > 3. There is no social ranking rule ρ : T 2 N -→ T N which satisfies DOM and SYM on T 2 N . Proof. We first show a particular situation where DOM and SYM are not compatible. Consider a power relation ∈ T 2 N with N = {1, 2, 3, 4} and such that 1 ∼ 2 ∼ 3 13 23 12 24 ∼ 14 34 1234 ∼ 123 ∼ 124 ∼ 134 ∼ 234

Proposition 2 . 3 )

 23 Let ρ : T 2 N -→ T N be a social ranking satisfying IIC and SYM. Let ∈ T 2 N and k ∈ {0, . . . , |N | -2} be such that S ∪ {i} S ∪ {j} and S ∪ {j} S ∪ {i}, for all S ∈ 2 N \{i,j} with|S| = k, D k ij ( ) \ D k ji ( ) = ∅ and D k ji ( ) \ D k ij ( ) =∅ for all i, j ∈ N . Then iρ( )j and jρ( )i for each i, j ∈ N . Proof. Take i, j ∈ N such that |D k ij ( )| ≥ |D k ji ( )|. Define another power relation ∈ T 2 N such that S ∪ {i} S ∪ {j} ⇔ S ∪ {i} S ∪ {j} for each S ∈ 2 N \{i,j} with |S| = k, and S T and T S for all the other coalitions S, T ∈ 2 N with |S| = |T | = k + 1. We still need to define relation on the remaining coalitions of size k. Take l ∈ N \ {i, j}. Let D ⊆ D k ij ( ) be such that |D| = |D k ji ( )|. By Remark 3 (see Section 8 Appendix), define the remaining comparisons in as follows (an illustrative example of these cases are given in Table 3): case 1) for each S ∈ D k ji ( ) with l ∈ S, let S ∪ {i, j} \ {l} S ∪ {i} and S ∪ {i, j} \ {l} S ∪ {j}; case 2) for each S ∈ D k ji ( ) with l / ∈ S, let S ∪ {i} S ∪ {l} and S ∪ {j} S ∪ {l}; case For each S ∈ D with l ∈ S, let S ∪ {i, j} \ {l} S ∪ {i} and S ∪ {i, j} \ {l} S ∪ {j}; case 4) for each S ∈ D with l / ∈ S, let S ∪ {i} S ∪ {l} and S ∪ {j} S ∪ {l}; case 5) for each S ∈ D k ij \ D with l ∈ S, let S ∪ {i, j} \ {l} S ∪ {i} and S ∪ {i, j} \ {l} S ∪ {j}; case 6) for each S ∈ D k ij \ D with l / ∈ S, let S ∪ {i} S ∪ {l} and S ∪ {j} S ∪ {l}.

Example 2 .

 2 Consider a power relation ∈ S 2 N with N = {1, 2, 3, 4} and such that

Proposition 3 .

 3 Let ρ : S 2 N -→ T N be a social ranking satisfying IIC and DOM on S 2 N . Let ∈ S 2 N and i ∈ N be such that for each s ∈ {0, . . . , n -2} either[S ∪ {i} S ∪ {j} for all j ∈ N \ {i} and S ∈ 2 N \{i,j} with |S| = s](2)or [S ∪ {j} S ∪ {i} for all j ∈ N \ {i} and S ∈ 2 N \{i,j} with |S| = s].

Table 1 (

 1 From now, we will sometimes omit braces and commas to separate elements, for instance, ij denotes the set {i, j}). Note that d 12 ( ) = 2, d 21 ( ) = 3, d 23 ( ) = 2, d 32 ( ) = 3, d 13 ( ) = 3 and

	2 1 3
	23 13 12 14 34 24
	134 ∼ 124 ∼ 234
	We rewrite the relevant information about	by means of

2 N with N = {1, 2, 3, 4} such that

Table 1 :

 1 The relevant information about of Example 1. So, we have that 2M ( )1, 3M ( )2 and 1M ( )3, but ¬(3M ( )1)): M ( ) is not a transitive relation.

	1 vs. 2	2 vs. 3	1 vs. 3
	1 ≺ 2	2 3	1 3
	13 ≺ 23	12 ≺ 13	12 ≺ 23
	14 24	24 ≺ 34	14 34
	134 ∼ 234 124 ∼ 134 124 ∼ 234
	d 31 ( ) = 2.

Table 2 :

 2 The relevant informations about and the elements 1, 2 and 3.

	1 vs. 2	2 vs. 3	1 vs. 3
	1 ∼ 2	2 ∼ 3	1 ∼ 3
	13 23	12 ≺ 13	12 ≺ 23
	14 ∼ 24	24 34	14 34
	134 ∼ 234 124 ∼ 134 124 ∼ 234

Table 4 :

 4 The relevant informations about of Example 2.

	1 vs. 2	2 vs. 3	1 vs. 3	1 vs. 4	2 vs. 4	3 vs. 4
	1 2	2 3	1 3	1 4	2 4	3 4
	13 ≺ 23	12 ≺ 13	12 ≺ 23	12 ≺ 24	12 ≺ 14	13 ≺ 14
	14 ≺ 24	24 ≺ 34	14 ≺ 34	13 ≺ 34	23 ≺ 34	23 ≺ 24
	134 234 124 ≺ 134 124 234 123 234 123 134 123 124

A longer version of this paper has been submitted to an international journal.