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12 pages, 281 KiB

Open AccessArticle

Bounding the Price of Anarchy of Weighted Shortest Processing Time Policy on Uniform Parallel Machines

by Felipe T. Muñoz and Rodrigo Linfati

Mathematics 2024, 12(14), 2223; https://doi.org/10.3390/math12142223 - 16 Jul 2024

Viewed by 641

This article investigates the performance of the Weighted Shortest Processing Time (WSPT) rule as a local sequencing policy in a scheduling game for uniformly related parallel machines, where the social objective is the total weighted completion time. Our research aims to establish improved [...] Read more.

This article investigates the performance of the Weighted Shortest Processing Time (WSPT) rule as a local sequencing policy in a scheduling game for uniformly related parallel machines, where the social objective is the total weighted completion time. Our research aims to establish improved upper bounds for the price of anarchy in this game. We determine two bounds, incorporating parameters that characterize the instance family, such as the speed of the fastest machine (

s_{m}

) and the number of machines (m). One bound establishes a fixed upper bound for the price of anarchy, while the other outperforms the parametric upper bound found in the existing literature. These newly derived bounds provide better insights into the performance of the scheduling game under study, proving that the price of anarchy is upper bounded by

min \{s_{m} (1 + 1 / 2 s_{m} - 1 / 2 m), m, 4\}

. Full article

10 pages, 368 KiB

Open AccessArticle

Nash Equilibria in Two-Resource Congestion Games with Player-Specific Payoff Functions

by Fatima Khanchouche, Samir Sbabou, Hatem Smaoui and Abderrahmane Ziad

Games 2024, 15(2), 7; https://doi.org/10.3390/g15020007 - 26 Feb 2024

Viewed by 1352

Abstract

In this paper, we examine the class of congestion games with player-specific payoff functions introduced by Milchtaich, I. (1996). Focusing on the special case of two resources, we give a short and simple method for identifying all Nash equilibria in pure strategies. We [...] Read more.

In this paper, we examine the class of congestion games with player-specific payoff functions introduced by Milchtaich, I. (1996). Focusing on the special case of two resources, we give a short and simple method for identifying all Nash equilibria in pure strategies. We also provide a computation algorithm based on our theoretical analysis. Full article

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19 pages, 545 KiB

Open AccessArticle

Fighting for Routes: Resource Allocation among Competing Planners in Transportation Networks

by Charlotte Roman and Paolo Turrini

Games 2023, 14(3), 37; https://doi.org/10.3390/g14030037 - 28 Apr 2023

Cited by 1 | Viewed by 1524

Abstract

In transportation networks, incomplete information is ubiquitous, and users often delegate their route choice to distributed route planners. To model and study these systems, we introduce network control games, consisting of multiple actors seeking to optimise the social welfare of their assigned subpopulations [...] Read more.

In transportation networks, incomplete information is ubiquitous, and users often delegate their route choice to distributed route planners. To model and study these systems, we introduce network control games, consisting of multiple actors seeking to optimise the social welfare of their assigned subpopulations through resource allocation in an underlying nonatomic congestion game. We first analyse the inefficiency of the routing equilibria by calculating the Price of Anarchy for polynomial cost functions, and then, using an Asynchronous Advantage Actor–Critic algorithm implementation, we show that reinforcement learning agents are vulnerable to choosing suboptimal routing as predicted by the theory. Finally, we extend the analysis to allow vehicles to choose their route planner and study the associated equilibria. Our results can be applied to mitigate inefficiency issues arising in large transport networks with route controlled autonomous vehicles. Full article

(This article belongs to the Special Issue Negotiations: The Good, the Bad, and the Ugly)

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14 pages, 302 KiB

Open AccessArticle

Additively Separable Hedonic Games with Social Context

by Gianpiero Monaco, Luca Moscardelli and Yllka Velaj

Games 2021, 12(3), 71; https://doi.org/10.3390/g12030071 - 18 Sep 2021

Viewed by 2395

Abstract

In hedonic games, coalitions are created as a result of the strategic interaction of independent players. In particular, in additively separable hedonic games, every player has valuations for all other ones, and the utility for belonging to a coalition is given by the [...] Read more.

In hedonic games, coalitions are created as a result of the strategic interaction of independent players. In particular, in additively separable hedonic games, every player has valuations for all other ones, and the utility for belonging to a coalition is given by the sum of the valuations for all other players belonging to it. So far, non-cooperative hedonic games have been considered in the literature only with respect to totally selfish players. Starting from the fundamental class of additively separable hedonic games, we define and study a new model in which, given a social graph, players also care about the happiness of their friends: we call this class of games social context additively separable hedonic games (SCASHGs). We focus on the fundamental stability notion of Nash equilibrium, and study the existence, convergence and performance of stable outcomes (with respect to the classical notions of price of anarchy and price of stability) in SCASHGs. In particular, we show that SCASHGs are potential games, and therefore Nash equilibria always exist and can be reached after a sequence of Nash moves of the players. Finally, we provide tight or asymptotically tight bounds on the price of anarchy and the price of stability of SCASHGs. Full article

(This article belongs to the Section Algorithmic and Computational Game Theory)

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16 pages, 2055 KiB

Open AccessArticle

Route Selection Decision-Making in an Intermodal Transport Network Using Game Theory

by Lucija Bukvić, Jasmina Pašagić Škrinjar, Borna Abramović and Vladislav Zitrický

Sustainability 2021, 13(8), 4443; https://doi.org/10.3390/su13084443 - 15 Apr 2021

Cited by 12 | Viewed by 3684

Abstract

Traveling through a transport network, or ordering and delivering packets, involves fundamental decision-making processes which can be approached by game theory: Rather than simply choosing a route, individuals need to evaluate routes in the presence of the congestion resulting from the decisions made [...] Read more.

Traveling through a transport network, or ordering and delivering packets, involves fundamental decision-making processes which can be approached by game theory: Rather than simply choosing a route, individuals need to evaluate routes in the presence of the congestion resulting from the decisions made by themselves and everyone else. In this paper, a game theory model for resolving route choices in transport network graphs is used. In the process of doing this, discovering a rather unexpected result known as Braess’s paradox, which shows that adding capacity to a network can sometimes actually cause congestion and an increase in transport costs. The decisions are made by non-cooperative players in a game theory environment known as prisoner’s dilemma. These methods are used to analyze routing problems by competing logistics operators on the transport network consisting of three Eastern Adriatic ports and an intermodal terminal in Budapest. The congestion game can be used in route selection regarding a decrease in transport costs for the carriers who are considered as rational players choosing the most sustainable solution. Full article

(This article belongs to the Collection Sustainable Freight Transportation System)

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16 pages, 296 KiB

Open AccessArticle

Equilibrium Inefficiency and Computation in Cost-Sharing Games in Real-Time Scheduling Systems

by Eirini Georgoulaki, Kostas Kollias and Tami Tamir

Algorithms 2021, 14(4), 103; https://doi.org/10.3390/a14040103 - 25 Mar 2021

Cited by 4 | Viewed by 2241

Abstract

We study cost-sharing games in real-time scheduling systems where the server’s activation cost in every time slot is a function of its load. We focus on monomial cost functions and consider both the case when the degree is less than one (inducing positive [...] Read more.

We study cost-sharing games in real-time scheduling systems where the server’s activation cost in every time slot is a function of its load. We focus on monomial cost functions and consider both the case when the degree is less than one (inducing positive congestion effect for the jobs) and when it is greater than one (inducing negative congestion effect for the jobs). For the former case, we provide tight bounds on the price of anarchy, and show that the price of anarchy grows to infinity as a polynomial of the number of jobs in the game. For the latter, we observe that existing results provide constant and tight (asymptotically in the degree of the monomial) bounds on the price of anarchy. We then turn to analyze payment mechanism with arbitrary cost-sharing, that is, when the strategy of a player includes also its payment. We show that our mechanism reduces the price of anarchy of games with n jobs and unit server costs from

Θ (\sqrt{n})

to 2. We also show that, for a restricted class of instances, a similar improvement is achieved for monomial server costs. This is not the case, however, for unrestricted instances of monomial costs, for which we prove that the price of anarchy remains super-constant for our mechanism. For systems with load-independent activation costs, we show that our mechanism can produce an optimal solution which is stable against coordinated deviations. Full article

(This article belongs to the Section Analysis of Algorithms and Complexity Theory)

14 pages, 343 KiB

Open AccessArticle

Network Creation Games with Traceroute-Based Strategies

by Davide Bilò, Luciano Gualà, Stefano Leucci and Guido Proietti

Algorithms 2021, 14(2), 35; https://doi.org/10.3390/a14020035 - 26 Jan 2021

Cited by 3 | Viewed by 2042

Abstract

Network creation games have been extensively used as mathematical models to capture the key aspects of the decentralized process that leads to the formation of interconnected communication networks by selfish agents. In these games, each user of the network is identified by a [...] Read more.

Network creation games have been extensively used as mathematical models to capture the key aspects of the decentralized process that leads to the formation of interconnected communication networks by selfish agents. In these games, each user of the network is identified by a node and selects which link to activate by strategically balancing his/her building cost with his/her usage cost (which is a function of the distances towards the other player in the network to be built). In these games, a widespread assumption is that players have a common and complete information about the evolving network topology. This is only realistic for small-scale networks as, when the network size grows, it quickly becomes impractical for the single users to gather such a global and fine-grained knowledge of the network in which they are embedded. In this work, we weaken this assumption, by only allowing players to have a partial view of the network. To this aim, we borrow three popular traceroute-based knowledge models used in network discovery: (i) distance vector, (ii) shortest-path tree view, and (iii) layered view. We settle many of the classical game theoretic questions in all of the above models. More precisely, we introduce a suitable (and unifying) equilibrium concept which we then use to study the convergence of improving and best response dynamics, the computational complexity of computing a best response, and to provide matching upper and lower bounds to the price of anarchy. Full article

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19 pages, 391 KiB

Open AccessArticle

On Nash Equilibria in Non-Cooperative All-Optical Networks

by Vittorio Bilò, Michele Flammini and Luca Moscardelli

Algorithms 2021, 14(1), 15; https://doi.org/10.3390/a14010015 - 9 Jan 2021

Cited by 1 | Viewed by 2014

Abstract

We consider the problem of determining a routing in all-optical networks, in which some couples of nodes want to communicate. In particular, we study this problem from the point of view of a network provider that has to design suitable payment functions for [...] Read more.

We consider the problem of determining a routing in all-optical networks, in which some couples of nodes want to communicate. In particular, we study this problem from the point of view of a network provider that has to design suitable payment functions for non-cooperative agents, corresponding to the couples of nodes wishing to communicate. The network provider aims at inducing stable routings (i.e., routings corresponding to Nash equilibria) using a low number of wavelengths. We consider three different kinds of local knowledge that agents may exploit to compute their payments, leading to three corresponding information levels. Under complete information, the network provider can design a payment function, inducing the agents to reach a Nash equilibrium mirroring any desired routing. If the price to an agent is computed only as a function of the wavelengths used along connecting paths (minimal level) or edges (intermediate level), the most reasonable functions either do not admit Nash equilibria or admit very inefficient ones, i.e., with the largest possible price of anarchy. However, by suitably restricting the network topology, a constant price of anarchy for chains and rings and a logarithmic one for trees can be obtained under the minimal and intermediate levels, respectively. Full article

(This article belongs to the Section Analysis of Algorithms and Complexity Theory)

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23 pages, 414 KiB

Open AccessArticle

On Multidimensional Congestion Games

by Vittorio Bilò, Michele Flammini, Vasco Gallotti and Cosimo Vinci

Algorithms 2020, 13(10), 261; https://doi.org/10.3390/a13100261 - 15 Oct 2020

Cited by 3 | Viewed by 2911

Abstract

We introduce multidimensional congestion games, that is, congestion games whose set of players is partitioned into

d + 1

clusters

C_{0}, C_{1}, \dots, C_{d}

. Players in

C_{0}

have full information about all the other participants [...] Read more.

We introduce multidimensional congestion games, that is, congestion games whose set of players is partitioned into

d + 1

clusters

C_{0}, C_{1}, \dots, C_{d}

. Players in

C_{0}

have full information about all the other participants in the game, while players in

C_{i}

, for any

1 \leq i \leq d

, have full information only about the members of

C_{0} \cup C_{i}

and are unaware of all the others. This model has at least two interesting applications:

(i)

it is a special case of graphical congestion games induced by an undirected social knowledge graph with independence number equal to d, and

(i i)

it represents scenarios in which players have a type and the level of competition they experience on a resource depends on their type and on the types of the other players using it. We focus on the case in which the cost function associated with each resource is affine and bound the price of anarchy and stability as a function of d with respect to two meaningful social cost functions and for both weighted and unweighted players. We also provide refined bounds for the special case of

d = 2

in presence of unweighted players. Full article

(This article belongs to the Special Issue Graph Algorithms and Applications)

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17 pages, 649 KiB

Open AccessArticle

Formation of Stable and Efficient Social Storage Cloud

by Pramod C. Mane, Nagarajan Krishnamurthy and Kapil Ahuja

Games 2019, 10(4), 44; https://doi.org/10.3390/g10040044 - 1 Nov 2019

Cited by 2 | Viewed by 4720

Abstract

In this paper, we study the formation of endogenous social storage cloud in a dynamic setting, where rational agents build their data backup connections strategically. We propose a degree-distance-based utility model, which is a combination of benefit and cost functions. The benefit function [...] Read more.

In this paper, we study the formation of endogenous social storage cloud in a dynamic setting, where rational agents build their data backup connections strategically. We propose a degree-distance-based utility model, which is a combination of benefit and cost functions. The benefit function of an agent captures the expected benefit that the agent obtains by placing its data on others’ storage devices, given the prevailing data loss rate in the network. The cost function of an agent captures the cost that the agent incurs to maintain links in the network. With this utility function, we analyze what network is likely to evolve when agents themselves decide with whom they want to form links and with whom they do not. Further, we analyze which networks are pairwise stable and efficient. We show that for the proposed utility function, there always exists a pairwise stable network, which is also efficient. We show that all pairwise stable networks are efficient, and hence, the price of anarchy is the best that is possible. We also study the effect of link addition and deletion between a pair of agents on their, and others’, closeness and storage availability. Full article

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26 pages, 618 KiB

Open AccessArticle

Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations

by Alexander Aurell

Games 2018, 9(4), 88; https://doi.org/10.3390/g9040088 - 1 Nov 2018

Cited by 5 | Viewed by 7069

Abstract

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum, non-cooperating, differential games where the players’ state dynamics solve backward stochastic differential equations (BSDE) that depend on the marginal [...] Read more.

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum, non-cooperating, differential games where the players’ state dynamics solve backward stochastic differential equations (BSDE) that depend on the marginal distributions of player states. Players try to minimize their individual cost functionals, also depending on the marginal state distributions. Under some regularity conditions, we derive necessary and sufficient conditions for existence of Nash equilibria. Player behavior is illustrated by numerical examples, and is compared to a centrally planned solution where the social cost, the sum of player costs, is minimized. The inefficiency of a Nash equilibrium, compared to socially optimal behavior, is quantified by the so-called price of anarchy. Numerical simulations of the price of anarchy indicate how the improvement in social cost achievable by a central planner depends on problem parameters. Full article

(This article belongs to the Special Issue Mean-Field-Type Game Theory)

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266 KiB

Open AccessArticle

Bargaining Mechanisms for One-Way Games

by Andrés Abeliuk, Gerardo Berbeglia and Pascal Van Hentenryck

Games 2015, 6(3), 347-367; https://doi.org/10.3390/g6030347 - 8 Sep 2015

Cited by 2 | Viewed by 6263

Abstract

We introduce one-way games, a two-player framework whose distinguishable feature is that the private payoff of one (independent) player is determined only by her own strategy and does not depend on the actions taken by the other (dependent) player. We show that the [...] Read more.

We introduce one-way games, a two-player framework whose distinguishable feature is that the private payoff of one (independent) player is determined only by her own strategy and does not depend on the actions taken by the other (dependent) player. We show that the equilibrium outcome in one-way games without side payments and the social cost of any ex post efficient mechanism can be far from the optimum. We also show that it is impossible to design a Bayes–Nash incentive-compatible mechanism for one-way games that is budget-balanced, individually rational and efficient. To address this negative result, we propose a privacy-preserving mechanism based on a single-offer bargaining made by the dependent player that leverages the intrinsic advantage of the independent player. In this setting the outside option of the dependent player is not known a priori; however, we show that the mechanism satisfies individual rationality conditions, is incentive-compatible, budget-balanced and produces an outcome that is more efficient than the equilibrium without payments. Finally, we show that a randomized multi-offer extension brings no additional benefit in terms of efficiency. Full article

(This article belongs to the Special Issue Bargaining Games)

408 KiB

Open AccessArticle

The Price of Anarchy for Network Formation in an Adversary Model

by Lasse Kliemann

Games 2011, 2(3), 302-332; https://doi.org/10.3390/g2030302 - 23 Aug 2011

Cited by 12 | Viewed by 6107

Abstract

We study network formation with n players and link cost α > 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player ν incorporates the expected number of players to which ν [...] Read more.

We study network formation with n players and link cost α > 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player ν incorporates the expected number of players to which ν will become disconnected. We focus on unilateral link formation and Nash equilibrium. We show existence of Nash equilibria and a price of stability of 1 + ο(1) under moderate assumptions on the adversary and n ≥ 9. We prove bounds on the price of anarchy for two special adversaries: one removes a link chosen uniformly at random, while the other removes a link that causes a maximum number of player pairs to be separated. We show an Ο(1) bound on the price of anarchy for both adversaries, the constant being bounded by 15 + ο(1) and 9 + ο(1), respectively. Full article

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619 KiB

Open AccessArticle

Incentive Compatible and Globally Efficient Position Based Routing for Selfish Reverse Multicast in Wireless Sensor Networks

by Stephan Eidenbenz, Gunes Ercal-Ozkaya, Adam Meyerson, Allon Percus and Sarvesh Varatharajan

Algorithms 2009, 2(4), 1303-1326; https://doi.org/10.3390/a2041303 - 14 Oct 2009

Viewed by 9146

Abstract

We consider the problem of all-to-one selfish routing in the absence of a payment scheme in wireless sensor networks, where a natural model for cost is the power required to forward, referring to the resulting game as a Locally Minimum Cost Forwarding (LMCF). [...] Read more.

We consider the problem of all-to-one selfish routing in the absence of a payment scheme in wireless sensor networks, where a natural model for cost is the power required to forward, referring to the resulting game as a Locally Minimum Cost Forwarding (LMCF). Our objective is to characterize equilibria and their global costs in terms of stretch and diameter, in particular finding incentive compatible algorithms that are also close to globally optimal. We find that although social costs for equilibria of LMCF exhibit arbitrarily bad worst-case bounds and computational infeasibility of reaching optimal equilibria, there exist greedy and local incentive compatible heuristics achieving near-optimal global costs. Full article

(This article belongs to the Special Issue Sensor Algorithms)

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