Eric Remila

University of Lyon

We study the Maker-Maker version of the domination game introduced in 2018 by Duch\^ene et al. Given a graph, two players alternately claim vertices. The first player to claim a dominating set of the graph wins. As the Maker-Breaker version, this game is PSPACE-complete on split and bipartite graphs. Our main result is a linear time algorithm to so...

Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class...

This paper provides the first axiomatic characterization of a class of certification methods, dubbed proportional threshold methods, that builds on consistency properties across populations and profiles of binary opinions. We then compare proportional threshold methods with the approval voting method. We formally outline the similarities and differ...

A coalitional ranking describes a situation where a finite set of agents can form coalitions that are ranked according to a weak order. A social ranking solution on a domain of coalitional rankings assigns a social ranking, that is a weak order over the agent set, to each coalitional ranking of this domain. We introduce two lexicographic solutions...

We consider cooperatives games (TU-games) enriched by a system of a priori unions and a communication forest graph which are independent from each other. These two structures reflect the limitations of cooperation possibilities. In this framework, we introduce four Owen-type allocation rules, which are defined by a two-step application of an alloca...

In many real world situations, the design of social rankings over agents or items from a given raking over groups or coalitions, to which these agents or items belong to, is of big interest. With this aim, we revise the lexicographic excellence solution and introduce two novel solutions which, moreover, take into account the size of the groups. We...

In this study, we propose a new direction of research on the axiomatic analysis of approval voting, which is a common democratic decision method. Its novelty is to examine an infinite population setting, which includes an application to intergenerational problems. In particular, we assume that the set of the population is countably infinite. We pro...

We relax the assumption that the grand coalition must form by imposing the axiom of Cohesive efficiency: the total payoffs that the players can share is equal to the maximal total worth generated by a coalition structure. We determine how the three main axiomatic characterizations of the Shapley value are affected when the classical axiom of Effici...

Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACM-SIAM symposium on discrete algorithms (SODA), pp 1096–115, 2020. https://doi.org/10.1137/1.9781611975994.67) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar),...

We introduce the game influence, a scoring combinatorial game, played on a directed graph where each vertex is either colored black or white. The two players, Black and White, play alternately by taking a vertex of their color and all its successors (for Black) or all its predecessors (for White). The score of each player is the number of vertices...

Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class...

Decreasing sandpiles model the dynamics of configurations where each position i∈N contains a finite number of stacked grains hi, such that hi≥hi+1 (decrease property). Grains move according to a decreasing local rule R=(r1,r2,…,rp) such that rj≥rj+1, meaning that rj grains move from columns i to i+j for all 1≤j≤p, if it does not contradict the decr...

We introduce the game INFLUENCE, a scoring combinatorial game, played on a directed graph where each vertex is either colored black or white. The two players, Black and White play alternately by taking a vertex of their color and all its successors (for Black) or all its predecessors (for White). The score of each player is the number of vertices h...

Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a \emph{distributed interactive proof} for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a di...

A situation in which a finite set of agents can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. In the literature, various models of games with restricted cooperation can be found, in which only certain subsets of the agent set are allowed to form. In this article, we cons...

This paper deals with Harsanyi power solutions for cooperative games in which partial cooperation is based on specific union stable systems given by the winning coalitions derived from a voting game. This framework allows for analyzing new and real situations in which there exists a feedback between the economic influence of each coalition of agent...

We introduce three natural collective variants of the well-known axiom of desirability (Maschler and Peleg in Pac J Math 18:289–328, 1966), which require that if the (per capita) contributions of a first coalition are at least as large as the (per capita) contributions of a second coalition, then the (average) payoff in the first coalition should b...

Let ta and tb be a pair of relatively prime positive integers. We work on chains of n(ta+tb) agents which form together an upper and rightward directed path of the grid Z2 from O=(0,0) to M=(nta,ntb). We are interested on evolution rules such that, at each time step, an agent is randomly chosen on the chain and is allowed to jump to another site of...

We introduce the class of tree TU-games augmented by a linear order over the links, which reflects the formation process of the tree. We characterize a new allocation rule for this class of cooperative games by means of three axioms: Standardness, Top consistency and Link amalgamation. Then, we discuss both a bargaining foundation and two possible...

We study a non linear weighted Shapley value () for cooperative games with transferable utility, in which the weights are endogenously given by the players' stand-alone worths. We call it the proportional Shapley value since it distributes the Harsanyi dividend () of all coalitions in proportion to the stand-alone worths of its members. We show tha...

We provide a strategic implementation of the sequential equal surplus division rule (Béal et al. in Theory Decis 79:251–283, 2015). Precisely, we design a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the sequential equal surplus division outcome of a superadditive rooted tree TU-game. This mechanis...

A new class of values combining marginalistic and egalitarian principles is introduced for cooperative TU-games. It includes some modes of solidarity among the players by taking the collective contribution of some coalitions to the grand coalition into account. Relationships with other class of values such as the Egalitarian Shapley values and the...

Emergence is easy to exhibit, but very hard to formally explain. This paper deals with square sand grains moving around on nicely stacked columns in one dimension (the physical sandpile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff sandpile model is a discrete dynamical system describing the evolution of fini...

This article introduces a discount parameter and a weight function in the classical model of cooperative games with restrictions on cooperation, represented by a tree. We provide axiomatic characterizations of solutions that are inspired by the hierarchical outcomes.

The congested clique model is a message-passing model of distributed computation where k players communicate with each other over a complete network. Here we consider synchronous protocols in which communication happens in rounds (we allow them to be randomized with public coins). In the unicast communication mode, each player i has her own n-bit i...

New and recent axioms for cooperative games with transferable utilities are introduced. The non-negative player axiom requires to assign a non-negative payoff to a player that belongs to coalitions with non-negative worth only. The axiom of addition invariance on bi-partitions requires that the payoff vector recommended by a value should not be aff...

Many axiomatic characterizations of values for cooperative games invoke axioms which evaluate the consequences of removing an arbitrary player. Balanced contributions (Myerson, 1980) and balanced cycle contributions (Kamijo and Kongo, 2010) are two well-known examples of such axioms. We revisit these characterizations by nullifying a player instead...

In this paper we prove that the general avalanche problem AP is in NC for all decreasing sandpile models in one dimension. It extends the developments of [5], and requires a careful attention on the general rule set considered, stressing the importance of the decreasing property. This work continues the study of dimension sensitive problems since i...

Sand is a proper instance for the study of natural algorithmic phenomena. Idealized square/cubic sand grains moving according to “simple” local toppling rules may exhibit surprisingly “complex” global behaviors. In this paper we explore the language made by words corresponding to fixed points reached by iterating a toppling rule starting from a fin...

The CONGEST model is a synchronous, message-passing model of distributed computation in which each node can send (possibly different) messages of O(log n) bits along each of its incident communication links in each round, where n is the number of computing nodes in the system. In the particular case where the communication network is a complete gra...

This paper studies values for cooperative games with transferable utility. Numerous such values can be characterized by axioms of (Formula presented.)-associated consistency, which require that a value is invariant under some parametrized linear transformation (Formula presented.) on the vector space of cooperative games with transferable utility....

In this paper we prove that the general avalanche problem AP is in NC for the Kadanoff sandpile model in one dimension, answering an open problem of [2]. Thus adding one more item to the (slowly) growing list of dimension sensitive problems since in higher dimensions the problem is P-complete (for monotone sandpiles).

A new class of allocation rules combining marginalistic and egalitarian principles is introduced for cooperative TU-games. It includes some modes of solidarity among the players by taking the collective contribution of some coalitions to the grand coalition into account. Relationships with other class of allocation rules such as the Egalitarian Sha...

Reciprocal preferences have been introduced in the literature of social choice theory in order to deal with preference intensities. They allow individuals to show preference intensities in the unit interval among each pair of options. In this framework, majority based on difference in support can be used as a method of aggregation of individual pre...

In Béal et al. (forthcoming) two new axioms of invariance, called Addition invariance and Transfer invariance respectively, are introduced to design allocation rules for TU-games. Here, we derive direct-sum decompositions of the linear space of TU-games by using the TU-games selected to construct the operations of Addition and Transfer. These decom...

The Banzhaf value is characterized on the (linear) space of all TU-games on a fixed player set by means of the Dummy player axiom and Strong transfer invariance. The latter axiom indicates that a player’s payoff is invariant to a transfer of worth between two coalitions he or she belongs to. To prove this result we derive direct-sum decompositions...

In this article, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We first derive direct-sum decompositions of the space of TU-games on a fixed tree, and two new basis for these spaces of TU-games. We then focus our atte...

We study the consequences of a solidarity property that specifies how a value for cooperative games should respond if some player forfeits his productivity, i.e., becomes a null player. Nullified solidarity states that in this case either all players weakly gain together or all players weakly lose together. Combined with efficiency, the null game p...

If a player is removed from a game, what keeps the payoff of the remaining players unchanged? Is it the removal of a special player or its presence among the remaining players? This article answers this question in a complement study to Kamijo and Kongo (2012). We introduce axioms of invariance from player deletion in presence of a special player....

Let \(t_a\) and \(t_b\) a pair of relatively prime positive integers. We work on chains of \(n(t_a + t_b)\) agents, each of them forming an upper and rightward directed path of the grid \(\mathbb {Z}^2\), from \( O = (0, 0)\) to \( M = (n t_a, n t_b)\). We are interested on evolution rules such that, at each time step, an agent is randomly chosen o...

We introduce a new allocation rule, called the sequential equal surplus division for rooted forest TU-games. We provide two axiomatic characterizations for this allocation rule. The first one uses the classical property of component efficiency plus an edge deletion property. The second characterization uses standardness, an edge deletion property a...

Reciprocal preferences have been introduced in the literature of social choice theory in order to deal with preference intensities. They allow individuals to show preference intensities in the unit interval among each pair of options. In this framework, majority based on difference in support can be used as a method of aggregation of individual pre...

Emergence is a concept that is easy to exhibit, but very hard to formally handle. This paper is about cubic sand grains moving around on nicely packed columns in one dimension (the physical sandpile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff Sandpile Model is a discrete dynamical system describing the evol...

This article shows that, for any transferable utility game in coalitional form with a nonempty coalition structure core, the number of steps required to switch from a payoff configuration out of the coalition structure core to a payoff configuration in the coalition structure core is less than or equal to $(n^2+4n)/4$ , where $n$ is the cardina...

The river sharing problem deals with the fair distribution of welfare resulting from the optimal allocation of water among a set of riparian agents. Ambec and Sprumont [Sharing a river, J. Econ. Theor. 107, 453–462] address this problem by modeling it as a cooperative TU-game on the set of riparian agents. Solutions to that problem are reviewed in...

Sand pile models are dynamical systems describing the evolution from $N$ stacked grains to a stable configuration. It uses local rules to depict grain moves and iterate it until reaching a fixed configuration from which no rule can be applied. Physicists L. Kadanoff {\em et al} inspire KSPM, extending the well known {\em Sand Pile Model} (SPM). In...

This paper is about cubic sand grains moving around on nicely packed columns in one dimension (the physical sand pile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff Sand Pile Model is a discrete dynamical system describing the evolution of a finite number of stacked grains --as they would fall from an hourglas...

For any transferable utility game in coalitional form with a nonempty core, we show that that the number of blocks required to switch from an imputation out of the core to an imputation in the core is at most n-1, where n is the number of players. This bound exploits the geometry of the core and is optimal. It considerably improves the upper bounds...

We replace the axiom of fairness used in the characterization of the Myerson value (Myerson, 1977) by fairness for neighbors in order to characterize the component-wise egalitarian solution. When a link is broken, fairness states the two players incident to the link should be affected similarly while fairness for neighbors states that a player inci...

We introduce new axioms for the class of all TU-games with a fixed but arbitrary player set. These axioms require either invariance of an allocation rule or invariance of the payoff assigned by an allocation rule to a specified player in two related TU-games. Combinations of these new axioms are used to characterize the Shapley value, the Equal Div...

We present the axiom of weighted component fairness for the class of forest games, a generalization of component fairness introduced by Herings, Talman and van der Laan (2008) in order to characterize the average tree solution. Given a system of weights, component eciency and weighted component fairness yield a unique allocation rule. We provide an...

In this article we study cooperative multi-choice games with limited cooperation possibilities, represented by an undirected forest on the player set. Players in the game can cooperate if they are connected in the forest. We introduce a new (single-valued) solution concept which is a generalization of the average tree solution defined and character...

For any transferable utility game in coalitional form with nonempty core, we show that that the number of blocks required to switch from an imputation out of the core to an imputation in the core is less than or equal to n(n-1)/2, where n is the cardinality of the player set. This number considerably improves the upper bounds found so far by Koczy...

We introduce the sequential equal surplus division for sharing the total welfare resulting form the cooperation of agents along a river with a delta. This allocation rule can be seen as a generalization of the contribution vectors introduced by Ju, Borm and Ruys (2007) in the context of TU-games. We provide two axiomatic characterizations of the se...

Consider a system composed of n sensors operating in synchronous rounds. In each round an input vector of sensor readings x is produced, where the r-th entry of x is a value, selected in a finite set of potential values, produced by the r-th sensor. The sequence of input vectors is assumed to be smooth: exactly one entry of the vector changes from...

Sand pile models are dynamical systems describing the evolution from $N$ stacked grains to a stable configuration. It uses local rules to depict grain moves and iterate it until reaching a fixed configuration from which no rule can be applied. The main interest of sand piles relies in their {\em Self Organized Criticality} (SOC), the property that...

Sand pile models are dynamical systems emphasizing the phenomenon of Self Organized Criticality (SOC). From N stacked grains, iterating evolution rules leads to some critical configuration where a small disturbance has deep consequences on the system, involving numerous steps of grain fall. Physicists L. Kadanoff et al. inspire KSPM, a model presen...

We introduce a new notion in self-assembly, that of trans- forming the dynamics of assembly. This notion allows us to have trans- formation of the plane computed within the assembly process. More specically, we study a zooming transformation. First we show that the possibility of doing that transformation depends on the regularity of the assembly p...

We study strategies that minimize the instability of a fault-tolerant consensus system. More precisely, we find the strategy than minimizes the number of output changes over a random walk sequence of input vectors (where each component of the vector corresponds to a particular sensor reading). We analyze the case where each sensor can read three po...

In this paper, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We introduce natural extensions of the average (rooted)-tree solution (see [Herings, P., van der Laan, G., Talman, D., 2008. The average tree solution for c...

In this paper, we study domino tilings of polygons. We are especially interested in what happens when the domino prototiles become smaller and smaller. This study is done using tiling height functions, which are a numerical way to encode tilings. The main result of this paper is an analytic characterization of functions which are limits of height f...

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give a representation in terms of formation of the grand coalition according...

The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by the study of a quasicrystal growth model, we are here interested in better understanding how "tight" rhombus til...

We consider communication situations games being the combina-tion of a TU-game and a communication graph. We study the average tree (AT) solutions introduced by Herings et al. [9] and [10]. The AT solutions are defined with respect to a set, say T , of rooted spanning trees of the commu-nication graph. We characterize these solutions by efficiency,...

It is known that any two rhombus tilings of a polygon are flip-accessible, that is, linked by a finite sequence of local transformations called flips. This paper considers flip-accessibility for rhombus tilings of the whole plane, asking whether any two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on...

In this paper we consider cooperative graph games being TU-games in which players cooperate if they are connected in the communication graph. We focus our attention to the average tree solutions introduced by Herings, van der Laan and Talman [6] and Herings, van der Laan, Talman and Yang [7]. Each average tree solution is defined with re- spect to...

Consider a system composed of n sensors operating in synchronous rounds. In each round an input vector of sensor readings x is produced, where the i-th entry of x is a binary value produced by the i-th sensor. The sequence of input vectors is assumed to be smooth: exactly one entry of the vector changes from one round to the next one. The system im...

Self-assembling tile systems are a model for assembling DNA-based nano artefacts. In the cur- rently known constructions, most of the effort is put on guaranteeing the size of the output object, whereas the geometrical efficiency of the assembling of the shape itself is left aside. We propose in this paper a framework to obtain provably time effici...

It is known that any two domino tilings of a polygon are flip-accessible, i.e., linked by a finite sequence of local transformations, called flips. This paper considers flip-accessibility for domino tilings of the whole plane, asking whether two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on tilings...

We study spaces of tilings, formed by tilings which are on a geodesic between two fixed tilings of the same domain (the distance is defined using local flips). We prove that each space of tilings is homeomorphic to an interval of tilings of a domain when flips are classically directed by height functions.

In this paper we construct fixed finite tile systems that as- semble into particular classes of shapes. Moreover, given an arbitrary n, we show how to calculate the tile concentrations in order to ensure that the expected size of the produced shape is n. For rectangles and squares our constructions are optimal (with respect to the size of the syste...

We produce an algorithm that is optimal with respect to both space and execution time to generate all the lozenge (or domino) tilings of a hole-free, general-shape domain given as input.We first recall some useful results, namely the distributive lattice structure of the space of tilings and Thurston's algorithm for constructing a particular tiling...

We study the spaces of rhombus tilings, i.e. the graphs whose vertices are tilings of a fixed zonotope. Two tilings are linked if one can pass from one to the other by a local transformation, called a flip. We first use a decomposition method to encode rhombus tilings and give a useful characterization for a sequence of bits to encode a tiling. We...

Let F be a figure formed from a finite set of cells of the planar square lattice. We prove that the problem of tiling such a figure with bars formed from 2 or 3 cells can be reduced to the the logic problem 2-SAT and we deduce a linear-time algorithm of tiling with these bars. Afterwards, we prove that the similar problem im the triangular lattice...

A discrete rotation algorithm can be apprehended as a parametric application $f_\alpha$ from $\ZZ[i]$ to\ $\ZZ[i]$, whose resulting permutation ``looks like'' the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotate d copies of an image for angles in-between 0...

A discrete rotation algorithm can be apprehended as a parametric application $f\_\alpha$ from $\ZZ[i]$ to $\ZZ[i]$, whose resulting permutation ``looks like'' the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotate d copies of an image for angles in-between 0...

2D-gon tilings with parallelograms are a model used in physics to study quasicrys-tals, and they are also important in combinatorics for the study of aperiodic struc-tures. In this paper, we study the graph induced by the adjacency relation between tiles. This relation can been used to encode simply and efficiently 2D-gon tilings for algorithmic ma...

We fix two rectangles with integer dimensions. We give a quadratic time algorithm which, given a polygon F as input, produces a tiling of F with translated copies of our rectangles (or indicates that there is no tiling). Moreover, we prove that any pair of tilings can be linked by a sequence of local transformations of tilings, called flips. Thi...

A discretized rotation acts on a pixel grid: the edges of the neighborhood relation are affected in particular way. Two types of configurations (i.e. applications from Z2 to a finite set of states) are introduced to code locally the transformations of the neighborhood. All the characteristics of discretized rotations are encoded within the configur...

International audience Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positi...

We study the spaces of rhombus tilings, i.e. the graphs whose vertices are tilings of a fixed zonotope, and two tilings are linked if one can pass from one to the other one by a local transformation, called flip. We first use a decomposition method to encode rhombus tilings and give a useful characterization for a sequence of bits to encode a tilin...

A discretized rotation is the composition of an Euclidean rotation with the rounding operation. For 0 < α < π/4, we prove that the discretized rotation [ r α ] is bijective if and only if there exists a positive integer k such as \(\{cos\alpha, sin\alpha\}=\{\frac{2k+1}{2k^{2}+2k+1},\frac{2k^{2}+2k}{2k^{2}+2k+1}\}\) The proof uses a particular subg...

We fix two rectangles with integer dimensions. We give a quadratic time algorithm which, given a polygon F as input, produces a tiling of F with translated copies of our rectangles (or indicates that there is no tiling). Moreover, we prove that any pair of tilings can be linked by a sequence of local transformations of tilings, called flips. This s...

Fix a polygon P with vertical and horizontal sides. We first recall how each tiling of P with dominoes (i.e. rectangles 2×1) can be encoded by a height function. Such an encoding induces a lattice structure on the set TP of the tilings of P. We give some applications of this structure, and we especially describe the order of meet irreducible elemen...

We give a linear time algorithm which, given a simply connected figure of the plane divided into cells, whose boundary is crossed by some colored inputs and outputs, produces non-intersecting directed flow lines which match inputs and outputs according to the colors, in such a way that each edge of any cell is crossed by at most one line. Th...

We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called flips.This study allows us to formulate an exhaustive generation algorithm an...

We give a linear time algorithm to elect a leader. This problem originated in networking and distributed computing research. Given a graph, its vertices represent processors (here finite state machines), and its edges communication lines (here synchronous). The leader election problem consists in finding a protocol for a family of graphs which, upo...

This paper studies the tilings with colored-edges triangles constructed on a triangulation of a simply connected orientable surface such that the degree of each interior vertex is even (such as, for (fundamental) example, a part of the triangular lattice of the plane). The constraints are that we only use three colors, all the colors appear in each...