Potential games are necessary to ensure pure nash equilibria in cost sharing games

We study multi-hop broadcast in wireless networks with one source node and multiple receiving nodes. The message flow from the source to the receivers can be modeled as a tree-graph, called broadcast-tree. The problem of finding the minimum-power broadcast-tree (MPBT) is NP-complete.Unlike most of the existing centralized approaches, we propose a decentralized algorithm, based on a non-cooperative cost-sharing game. In this game, every receiving node, as a player, chooses another node of the network as its respective transmitting node for receiving the message. Consequently, a cost is assigned to the receiving node based on the power imposed on its chosen transmitting node.In our model, the total required power at a transmitting node consists of (i) the transmit power and (ii) the circuitry power needed for communication hardware modules. We develop our algorithm using the marginal contribution (MC) cost-sharing scheme and show that it is the only scheme by which the optimum broadcast-tree is always a Nash equilibrium (NE) of the game. Simulation results demonstrate that our proposed algorithm outperforms conventional algorithms for the MPBT problem. Besides, we show that the circuitry power, which is usually ignored by existing algorithms, significantly impacts the energy-efficiency of the network. Index TermsEnergy-efficiency; minimum-power multi-hop broadcast; potential game; optimization. is disseminated through the network with the help of some intermediate nodes which re-transmit the message. This problem is known as the minimum-power broadcast-tree (MPBT) problem since the connections between the source and the receiving nodes form a tree-graph rooted at the source, called the broadcast-tree [1]. MPBT construction has been studied by researchers extensively during the past two decades [1][2][3][4][5][6][7][8][9][10][11][12]. NP-completeness of the MPBT can be shown by reducing the Steiner tree problem to it [13]. This means that a polynomial-time algorithm to find the optimum broadcast-tree unlikely exists. Although many algorithms have been proposed for the MPBT problem, most of them are centralized heuristics [1-6]. Since multi-hop device-todevice communication is seen as a promising technique for improving the capacity of 5G cellular networks [14] and due to the variety of its applications, e.g., video streaming [15], vehicular communications [16], etc., it is vital to revisit the MPBT problem to find a decentralized yet efficient algorithm for it.Seeing the MPBT problem from a decentralized optimization point of view, every individual node has to play its own role in forming the broadcast-tree by establishing a communication link to another node. This can be suitably modeled by game theory, in which the players of the game, here the nodes, are typically modeled as selfish agents seeking to minimize (maximize) their own cost (revenue). In our work, the action of a node is to choose another node in the network as its respective transmitting node to receive the source's message from. As a consequence of i...

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“…Lemma 3. Applying the SV to a non-cooperative CSG makes the game potential for which an NE always exists [39] [24].…”

Section: Lemmamentioning

confidence: 99%

“…Hence, we just use the CBBTC algorithm representing both. The results for the network power for all the algorithms are normalized to the average 24 Total no. of nodes in the network, |Q| The normalized network power is then denoted byP net (a) = P net (a)/P max in which P net (a)…”

Section: A Simulation Setupmentioning

confidence: 99%

Cost Sharing Games for Energy-Efficient Multi-Hop Broadcast in Wireless Networks

Mousavi

Al-Shatri

Klein

2020

IEEE Trans. Wireless Commun.

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“…We proceed to show a lower bound on the price of stability of Θ(nH n ) for any uniform cost sharing protocol. To prove this, we use a result of Gopalakrishnan et al [7,Theorem 1] who showed that the set of generalized weighted Shapley cost sharing protocols is the unique maximal set of uniform cost sharing protocols that is stable. To state their result, we first recall the definition of the generalized weighted Shapley cost sharing protocols [9].…”

Section: Price Of Stabilitymentioning

confidence: 99%

“…Proposition 12 (Gopalakrishnan et al [7]). For every budget-balanced, uniform and stable cost sharing protocol ξ there is a weight system w such that ξ is equivalent to the generalized weighted Shapley cost sharing protocol with weight system w.…”

Section: Price Of Stabilitymentioning

confidence: 99%

Sharing Non-anonymous Costs of Multiple Resources Optimally

Klimm

Schmand

2015

Lecture Notes in Computer Science

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In cost sharing games, the existence and efficiency of pure Nash equilibria fundamentally depends on the method that is used to share the resources' costs. We consider a general class of resource allocation problems in which a set of resources is used by a heterogeneous set of selfish users. The cost of a resource is a (non-decreasing) function of the set of its users. Under the assumption that the costs of the resources are shared by uniform cost sharing protocols, i.e., protocols that use only local information of the resource's cost structure and its users to determine the cost shares, we exactly quantify the inefficiency of the resulting pure Nash equilibria. Specifically, we show tight bounds on prices of stability and anarchy for games with only submodular and only supermodular cost functions, respectively, and an asymptotically tight bound for games with arbitrary set-functions. While all our upper bounds are attained for the well-known Shapley cost sharing protocol, our lower bounds hold for arbitrary uniform cost sharing protocols and are even valid for games with anonymous costs, i.e., games in which the cost of each resource only depends on the cardinality of the set of its users.

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“…Precisely speaking, by designing proper utility functions to each agent, the overall welfare (or overall cost) is considered as the potential function. Then, the techniques developed for game theoretic control (GTC) are applicable to find pure Nash equilibriums, which provide candidates of optimal solutions [7], [8].…”

Section: Introductionmentioning

confidence: 99%

Cooperative Control via Congestion Game Approach

Hao

Pan

Qiao

et al. 2018

IEEE Trans. Automat. Contr.

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The optimization of facility-based systems is considered. First, the congestion game is converted into a matrix form, so that the matrix approach is applicable. Then, a facilitybased system with a system performance criterion is considered. A necessary and sufficient condition is given to assure that the system is convertible into a congestion game with the given system performance criterion as its potential function by designing proper facility-cost functions. Using this technology, for a dynamic facility-based system the global optimization may be reached when each agent optimizes its payoff functions. Finally, the approach is extended to those systems which are partly or nearly convertible.

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Potential games are necessary to ensure pure nash equilibria in cost sharing games

Cited by 23 publications

References 48 publications

Cost Sharing Games for Energy-Efficient Multi-Hop Broadcast in Wireless Networks

Cost Sharing Games for Energy-Efficient Multi-Hop Broadcast in Wireless Networks

Sharing Non-anonymous Costs of Multiple Resources Optimally

Cooperative Control via Congestion Game Approach

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