The Price of Selfishness in Network Coding

Marden, Jason R.; Effros, Michelle

doi:10.1109/tit.2011.2177576

“…This results from combining (22) and (26). (iii) BR and BL are also not equilibria, because in these action profiles, players j and i respectively have incentives to deviate -the arguments are identical to cases (a) and (b) above, respectively.…”

Section: (D) Basis Distribution Rules Computed By Recursion (27)mentioning

confidence: 90%

“…Then, one possibility is to set r i = 10 i so that i∈S r i can be decoded to obtain the set of players S. Therefore, c r can be defined such that for all S ⊆ N , c r i∈S r i = Cr(S) |S| . Other notable specializations of our model that focus on the design of distribution rules are network coding (Marden and Effros [26]), graph coloring (Panagopoulou and Spirakis [37]), and coverage problems (Marden and Wierman [28], Marden and Wierman [29]). Designing distribution rules in our cost sharing model also has applications in distributed control (Gopalakrishnan et al [16]).…”

Section: Examples Of Distribution Rulesmentioning

confidence: 99%

See 1 more Smart Citation

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

Gopalakrishnan

¹

,

Marden

²

,

Wierman

³

2013

Proceedings of the Fourteenth ACM Conference on Electronic Commerce

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We consider the problem of designing distribution rules to share 'welfare' (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., the Shapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantee the existence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class of scalable and separable games, which includes a variety of applications such as facility location, routing, network formation, and coverage games. Given arbitrary local welfare functions W, we prove that a distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weighted Shapley value on some 'ground' welfare functions W ′ , which can be distinct from W. However, if budget-balance is required in addition to the existence of a Nash equilibrium, then W ′ must be the same as W. We also provide an alternate characterization of this space in terms of 'generalized' marginal contributions, which is more appealing from the point of view of computational tractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games with any fixed local welfare functions, it is necessary to work within the class of potential games.

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“…Other problems that have been formulated in terms of potential games are found in routing [9], [97], [156], [157], [186], [203], BS/AP selection [98], [99], [112], [161], [172], cooperative transmissions [4], [139], secrecy rate maximization [11], code design for radar [146], broadcasting [35], spectrum market [87], network coding [115], [147], [148], data cashing [94], social networks [40], computation offloading [42], localization [79], and demand-side management in smart grids [77], [183]. …”

Section: Discussionmentioning

confidence: 99%

A Comprehensive Survey of Potential Game Approaches to Wireless Networks

Yamamoto

¹

2015

IEICE Trans. Commun.

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SUMMARY Potential games form a class of non-cooperative games where the convergent of unilateral improvement dynamics is guaranteed in many practical cases. The potential game approach has been applied to a wide range of wireless network problems, particularly to a variety of channel assignment problems. In this paper, the properties of potential games are introduced, and games in wireless networks that have been proven to be potential games are comprehensively discussed. key words: potential game, game theory, radio resource management, channel assignment, transmission power control IntroductionThe broadcast nature of wireless transmissions causes cochannel interference and channel contention, which can be viewed as interactions among transceivers. Interactions among multiple decision makers can be formulated and analyzed using a branch of applied mathematics called game theory [61], [131]. Game-theoretic approaches have been applied to a wide range of wireless communication technologies, including transmission power control for code division multiple access (CDMA) cellular systems [153] and cognitive radios [132]. For a summary of game-theoretic approaches to wireless networks, we refer the interested reader to [68] [189].In this paper, we focus on potential games [126], which form a class of strategic form games with the following desirable properties:• The existence of a Nash equilibrium in potential games is guaranteed in many practical situations [126] (Theorems 1 and 2 in this paper), but is not guaranteed for general strategic form games. Example 2 in Sect. 2. We provide an overview of problems in wireless networks that can be formulated in terms of potential games. We also clarify the relations among games, and provide simpler proofs of some known results. Problem-specific learning algorithms [92], [168] are beyond the scope of this paper.The remainder of this paper is organized as follows: In Sect. 2, 3, and 4, we introduce strategic form games, potential games, and learning algorithms, respectively. We then discuss various potential games in Sects. 5 to 18, as shown in Table 1. Finally, we provide a few concluding remarks in Sect. 19.The notation used here is shown in Table 2. Unless the context indicates otherwise, sets of strategies are denoted by calligraphic uppercase letters, e.g., A i , strategies are denoted by lowercase letters, e.g., a i ∈ A i , and tuples of strategies are denoted by boldface lowercase letters, e.g., a. Note that a i is a scalar variable when A i is a set of scalars or indices, a i is a vector variable when A i is a set of vectors, and a i is a set variable when A i is a collection of sets.We use R to denote the set of real numbers, R + to denote the set of nonnegative real numbers, R ++ to denote the set of positive real numbers, and C to denote the set of complex numbers. The cardinality of set A is denoted by |A|. The power set of A is denoted by 2 A . Finally, ½condition is the indicator function, which is one when condition is true and is zero otherwise.We treat many sy...

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“…Then, one possibility is to set ρ i = 10 i so that i∈S ρ i can be decoded to obtain the set of players S. Therefore, c r can be defined such that for all S ⊆ N , c r i∈S ρ i = Cr(S) |S| . Other notable specializations of our model that focus on the design of distribution rules are network coding (Marden and Effros [26]), graph coloring (Panagopoulou and Spirakis [37]), and coverage problems (Marden and Wierman [28], Marden and Wierman [29]). Designing distribution rules in our cost sharing model also has applications in distributed control (Gopalakrishnan et al [16]).…”

Section: 2mentioning

confidence: 99%

Potential Games are Necessary to Ensure Pure Nash Equilibria in Cost Sharing Games

Gopalakrishnan¹,

Marden²,

Wierman³

2014

Preprint

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0

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We consider the problem of designing distribution rules to share 'welfare' (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., the Shapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantee the existence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class of scalable and separable games, which includes a variety of applications such as facility location, routing, network formation, and coverage games. Given arbitrary local welfare functions W, we prove that a distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weighted Shapley value on some 'ground' welfare functions W ′ , which can be distinct from W. However, if budget-balance is required in addition to the existence of a Nash equilibrium, then W ′ must be the same as W. We also provide an alternate characterization of this space in terms of 'generalized' marginal contributions, which is more appealing from the point of view of computational tractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games with any fixed local welfare functions, it is necessary to work within the class of potential games.

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The Price of Selfishness in Network Coding

Cited by 34 publications

References 33 publications

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

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

A Comprehensive Survey of Potential Game Approaches to Wireless Networks

Potential Games are Necessary to Ensure Pure Nash Equilibria in Cost Sharing Games

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