Multi-Player Diffusion Games on Graph Classes

We study competitive diffusion games on graphs introduced by Alon et al. [1] to model the spread of influence in social networks. Extending results of Roshanbin [8] for two players, we investigate the existence of pure Nash equilibria for at least three players on different classes of graphs including paths, cycles, grid graphs and hypercubes; as a main contribution, we answer an open question proving that there is no Nash equilibrium for three players on (m x n) grids with min(m, n) >= 5. Further, extending results of Etesami and Basar [3] for two players, we prove the existence of pure Nash equilibria for four players on every d-dimensional hypercube.Comment: Extended version of the TAMC 2015 conference version now discussing hypercube results (added details for the proof of Proposition 1

Voronoi Choice Games

We study novel variations of Voronoi games and associated random processes that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios, and they further lead to many interesting, non-trivial random processes that appear worthy of study. As an example of the type of problem we study, suppose a group of $n$ miners are staking land claims through the following process: each miner has $m$ associated points independently and uniformly distributed on an underlying space, so the $k$th miner will have associated points $p_{k1},p_{k2},\ldots,p_{km}$. Each miner chooses one of these points as the base point for their claim. Each miner obtains mining rights for the area of the square that is closest to their chosen base, that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the $n$ chosen points. Each player's goal is simply to maximize the amount of land under their control. What can we say about the players' strategy and the equilibria of such games? In our main result, we derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game on the circle where a player owns the arc starting from their point and moving clockwise to the next point. This result uses interesting properties of random arc lengths on circles, and demonstrates the challenges in analyzing these kinds of problems. We also provide several other related results. In particular, for the 1-dimensional game on the circle, we show that a pure Nash equilibrium always exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists in the variant when each player owns the arc starting from their point clockwise to the next point

The Discrete Voronoi game in ℝ\u3csup\u3e2\u3c/sup\u3e

In this paper we study the last round of the discrete Voronoi game in ℝ2, a problem which is also of independent interest in competitive facility location. The game consists of two players P1 and P2, and a finite set U of users in the plane. The players have already placed two disjoint sets of facilities F and S, respectively, in the plane. The game begins with P1 placing a new facility followed by P2 placing another facility, and the objective of both the players is to maximize their own total payoffs. In this paper we propose polynomial time algorithms for determining the optimal strategies of both the players for arbitrarily located existing facilities F and S. We show that in the L1 and the L∞ metrics, the optimal strategy of P2, given any placement of P1, can be found in O(n log n) time, and the optimal strategy of P1 can be found in O(n5 log n) time. In the L2 metric, the optimal strategies of P2 and P1 can be obtained in O(n2) and O(n2) and O(n8) times, respectively

Discrete hotelling pure location games: potentials and equilibria

We study two-player one-dimensional discrete Hotelling pure location games assuming that demand f(d) as a function of distance d is constant or strictly decreasing. We show that this game admits a best-response potential. This result holds in particular for f(d) = wd with 0 < w ≤ 1. For this case special attention will be given to the structure of the equilibrium set and a conjecture about the increasingness of best-response correspondences will be made

Balancing Graph Voronoi Diagrams

Abstract—Many facility location problems are concerned with minimizing operation and transportation costs by par-titioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice. We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s+1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s logn. Keywords-graph Voronoi diagram; balancing; facility loca-tion; territorial design I

New Variations of the Maximum Coverage Facility Location Problem

Consider a competitive facility location scenario where, given a set U of n users and a set F of m facilities in the plane, the objective is to place a new facility in an appropriate place such that the number of users served by the new facility is maximized. Here users and facilities are considered as points in the plane, and each user takes service from its nearest facility, where the distance between a pair of points is measured in either L1 or L2 or L∞ metric. This problem is also known as the maximum coverage (MaxCov) problem. In this paper, we will consider the k-MaxCov problem, where the objective is to place k (⩾1) new facilities such that the total number of users served by these k new facilities is maximized. We begin by proposing an O(nlogn) time algorithm for the k-MaxCov problem, when the existing facilities are all located on a single straight line and the new facilities are also restricted to lie on the same line. We then study the 2-MaxCov problem in the plane, and propose an O(n2) time and space algorithm in the L1 and L∞ metrics. In the L2 metric, we solve the 2-MaxCov problem in the plane in O(n3logn) time and O(n2logn) space. Finally, we consider the 2-Farthest-MaxCov problem, where a user is served by its farthest facility, and propose an algorithm that runs in O(nlogn) time, in all the three metrics