Consensus Strategies for Signed Profiles on Graphs

The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+,-}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a - sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes.median;consensus function;median graph;majority rule;plurality strategy;Graph theory;Hamming graph;Johnson graph;halfcube;scarcity strategy;Discrete location and assignment;Distance in graphs

Consensus Strategies for Signed Profiles on Graphs

The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+,-}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a - sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes

Graphical potential games

We study the class of potential games that are also graphical games with respect to a given graph $G$ of connections between the players. We show that, up to strategic equivalence, this class of games can be identified with the set of Markov random fields on $G$. From this characterization, and from the Hammersley-Clifford theorem, it follows that the potentials of such games can be decomposed to local potentials. We use this decomposition to strongly bound the number of strategy changes of a single player along a better response path. This result extends to generalized graphical potential games, which are played on infinite graphs.Comment: Accepted to the Journal of Economic Theor

"Graph Entropy, Network Coding and Guessing games"

We introduce the (private) entropy of a directed graph (in a new network coding sense) as well as a number of related concepts. We show that the entropy of a directed graph is identical to its guessing number and can be bounded from below with the number of vertices minus the size of the graph’s shortest index code. We show that the Network Coding solvability of each specific multiple unicast network is completely determined by the entropy (as well as by the shortest index code) of the directed graph that occur by identifying each source node with each corresponding target node. Shannon’s information inequalities can be used to calculate up- per bounds on a graph’s entropy as well as calculating the size of the minimal index code. Recently, a number of new families of so-called non-shannon-type information inequalities have been discovered. It has been shown that there exist communication networks with a ca- pacity strictly ess than required for solvability, but where this fact cannot be derived using Shannon’s classical information inequalities. Based on this result we show that there exist graphs with an entropy that cannot be calculated using only Shannon’s classical information inequalities, and show that better estimate can be obtained by use of certain non-shannon-type information inequalities