Link Biased Strategies in Network Formation Games

📝 Abstract

We show a simple method for constructing an infinite family of graph formation games with link bias so that the resulting games admits, as a \textit{pairwise stable} solution, a graph with an arbitrarily specified degree distribution. Pairwise stability is used as the equilibrium condition over the more commonly used Nash equilibrium to prevent the occurrence of ill-behaved equilibrium strategies that do not occur in ordinary play. We construct this family of games by solving an integer programming problem whose constraints enforce the terminal pairwise stability property we desire.

💡 Analysis

We show a simple method for constructing an infinite family of graph formation games with link bias so that the resulting games admits, as a \textit{pairwise stable} solution, a graph with an arbitrarily specified degree distribution. Pairwise stability is used as the equilibrium condition over the more commonly used Nash equilibrium to prevent the occurrence of ill-behaved equilibrium strategies that do not occur in ordinary play. We construct this family of games by solving an integer programming problem whose constraints enforce the terminal pairwise stability property we desire.

📄 Content

Link Biased Strategies in Network Formation Games Shaun Lichter and Terry Friesz Dept. Industrial and Manufacturing Engineering Penn State University E-mail: {tlf13, sml310}@psu.edu Christopher Griffin Applied Research Laboratory Penn State University E-mail: griffinch@ieee.org Abstract—We show a simple method for constructing an infinite family of graph formation games with link bias so that the resulting games admits, as a pairwise stable solution, a graph with an arbitrarily specified degree distribution. Pairwise stability is used as the equilibrium condition over the more commonly used Nash equilibrium to prevent the occurrence of ill-behaved equilibrium strate- gies that do not occur in ordinary play. We construct this family of games by solving an integer programming problem whose constraints enforce the terminal pairwise stability property we desire. I. INTRODUCTION The Network Science community has largely dedi- cated its efforts to the exposition and analysis of topolog- ical properties that occur in several real-world networks (e.g., scale-freeness [1]–[4]). Recently, there has been interest in showing that these topological properties may arise as a result of optimization, rather than some immutable physical law [5]. As Doyle et al. [5] point out, various networks such as communications networks and the Internet are designed by engineers with some objectives and constraints. While it is true that there is often not a single designer in control of the entire network, the network does not naturally evolve without the influence of designers. In each application, the net- work structure must be feasible with respect to some physical constraints corresponding to the tolerances and specifications of the equipment used in the network. For example, in a system such as the world wide web, a single web-page might have billions of connections, however it is not possible for a node to have such a degree in many other applications, such as collaboration or road networks. Certainly the structure of the network has a significant impact on its [the network’s] ability to function, its evolution, and its robustness. However network structures often arise as a result (locally) of optimized decision made by a single agent or multiple competitive or cooperative agents, who take network structure and function into account as a part of a col- lection of constraints and objectives. Recently, network formation has been modeled from a game theoretic perspective [6]–[9], and in [10], it was shown that there exists games that result in the formation of a stable graph with an arbitrary degree sequence k = (k1, . . . , kn). These models require that the players have similar objectives that are convex with minima near the desired ki values. These assumptions were necessary to show that a game can be constructed that admits a stable graph with an arbitrary degree sequence. However, this assumption is limiting in its modeling power because players (usually) do not have an exact number of links that they desire nor do they (usually) have an objective function precisely specifying this desire. Instead the ki arise endogenously as a result of other factors. In this paper, we present a model incorporating a player’s link bias - preference of one link over another. The incorporation of link bias allows the game to result in a stable graph of arbitrary degree without requiring the degree sequence to be precisely coded into the game. II. MODEL Let N = {1, 2, . . . n} be a set of nodes. We will assume n is fixed for the remainder of this paper. A link between two nodes is any subset of size 2 of set N. A graph g is any set of links and the complete graph gc is the set of all size two subsets of N. The set G is composed of all graphs over the node set N, that is, G = {g : g ⊆gc}. In a network formation game, each node is a player. Link bias is introduced by assuming player i has a cost function fi : G →R. For this paper, we assume a linear cost function: fi(g) = X j cijxij (1) where xij = ( 1 there is a link between i and j in g 0 else (2) arXiv:1106.3582v1 [math.OC] 17 Jun 2011 A player’s strategy is to determine to which nodes to link in order minimize cost (or maximize payoff). Following [7], a link between players i and j exists if and only if the two players decide to link. That is, a player may unilaterally reject a link. This is consistent with friending policies on Facebook or linking policies on LinkedIn. A. Stability The value of a graph g is the total value produced by agents in the graph; we denote the value of a graph as the function v : G →R and the set of of all such value functions as V . An allocation rule Y : V × G →Rn distributes the value v(g) among the agents in g. Denote the value allocated to agent i as Yi(v, g). Since, the allocation rule must distribute the value of the network to all players, it must be balanced; i.e., P i Yi(v, g) = v(g) for all (v, g) ∈V × G. The allocation rule governs how the value is distribute

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