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Resource Graph Games: A Compact Representation for Games with Structured Strategy Spaces

Albert Jiang, Hau Chan, Kevin Leyton-Brown
2017 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
We propose Resource Graph Games (RGGs), the first general compact representation for games with structured strategy spaces, which is able to represent a wide range of games studied in literature.  ...  ., consisting of multiple sub-decisions - and hence can give rise to an exponential number of pure strategies. Examples include network congestion games, simultaneous auctions, and security games.  ...  We call a game with this property a game with polytopal strategy spaces. Multilinear Games When |S i | is exponential, representing a mixed strategy σ i explicitly would take exponential space.  ... 
doi:10.1609/aaai.v31i1.10618 fatcat:ldd5gppbrfg5pmsbroh5vi7mo4

Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games [chapter]

Tobias Harks, Veerle Timmermans
2016 Lecture Notes in Computer Science  
We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow  ...  On the other hand we show that matroidal set systems are in some sense necessary to guarantee uniqueness of equilibria: for every atomic splittable congestion game with at least three players and non-matroidal  ...  Polymatroid congestion games In polymatroid congestion games we assume that the strategy space for every player corresponds to a polymatroid base polytope.  ... 
doi:10.1007/978-3-319-45587-7_9 fatcat:qxub7ownjbhp7galio3px5m4mu

Uniqueness of equilibria in atomic splittable polymatroid congestion games

Tobias Harks, Veerle Timmermans
2017 Journal of combinatorial optimization  
We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow  ...  On the other hand we show that matroidal set systems are in some sense necessary to guarantee uniqueness of equilibria: for every atomic splittable congestion game with at least three players and non-matroidal  ...  Polymatroid congestion games In polymatroid congestion games we assume that the strategy space for every player corresponds to a polymatroid base polytope.  ... 
doi:10.1007/s10878-017-0166-5 fatcat:ntmjlvlj4jctzdkie6ynenvvq4

Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games [article]

Tobias Harks, Veerle Timmermans
2018 arXiv   pre-print
We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow  ...  On the other hand we show that matroidal set systems are in some sense necessary to guarantee uniqueness of equilibria: for every atomic splittable congestion game with at least three players and nonmatroidal  ...  Polymatroid Congestion Games In polymatroid congestion games we assume that the strategy space for every player corresponds to a polymatroid base polytope.  ... 
arXiv:1512.01375v3 fatcat:2ezdnhdulbgpjpphktskgagfoq

Multilinear Games [chapter]

Hau Chan, Albert Xin Jiang, Kevin Leyton-Brown, Ruta Mehta
2016 Lecture Notes in Computer Science  
(c) Finally, we show existence of an approximate NE with support-size logarithmic in the strategy polytope dimensions.  ...  However, this set of pure strategies is often structured, allowing it to be represented compactly, as in network congestion games, security games, and extensive form games.  ...  ) integers. 5 We call a game with this property a game with polytopal strategy spaces.  ... 
doi:10.1007/978-3-662-54110-4_4 fatcat:hytwfl3p3vf5hhbh5x3lgmm7ri

Matroids Are Immune to Braess' Paradox

Satoru Fujishige, Michel X. Goemans, Tobias Harks, Britta Peis, Rico Zenklusen
2017 Mathematics of Operations Research  
In this paper, we consider general nonatomic congestion games and give a characterization of the combinatorial property of strategy spaces for which the Braess paradox does not occur.  ...  The famous Braess paradox describes the counter-intuitive phenomenon in which, in certain settings, the increase of resources, like building a new road within a congested network, may in fact lead to larger  ...  [25] showed that integral-splittable congestion games with semi-convex cost functions always admit an equilibrium whenever each player's strategy space forms an integral polymatroid. 2 Nonatomic Congestion  ... 
doi:10.1287/moor.2016.0825 fatcat:qqmsftbd5revtjmtdvmut2u274

Computation and efficiency of potential function minimizers of combinatorial congestion games

Pieter Kleer, Guido Schäfer
2020 Mathematical programming  
We study the computation and efficiency of pure Nash equilibria in combinatorial congestion games, where the strategies of each player i are given by the binary vectors of a polytope P i .  ...  Our main goal is to understand which structural properties of such polytopal congestion games enable us to derive an efficient equilibrium selection procedure to compute pure Nash equilibria with attractive  ...  Our bounds for polytopal congestion games (significantly) improve upon the ones known for general congestion games: For example, the price of stability of congestion games with cost functions from the  ... 
doi:10.1007/s10107-020-01546-6 fatcat:rl4if7fm7fgcvhkcaaut4uobem

Matroids are Immune to Braess Paradox [article]

Satoru Fujishige, Michel X. Goemans, Tobias Harks, Britta Peis and Rico Zenklusen
2017 arXiv   pre-print
In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur.  ...  The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs  ...  [19] showed that integral-splittable congestion games with semi-convex cost functions always admit an equilibrium whenever each player's strategy space forms an integral polymatroid.  ... 
arXiv:1504.07545v2 fatcat:urckk3k5lnfzzfqchdx74m7t5m

An FPTAS for Computing Nash Equilibrium in Resource Graph Games

Hau Chan, Albert Xin Jiang
2018 Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence  
We consider the problem of computing a mixed-strategy Nash equilibrium (MSNE) in resource graph games (RGGs), a compact representation for games with an exponential number of strategies.  ...  polytope) has bounded treewidth.  ...  Consider a security game as defined above, but with a more complex attacker strategy space.  ... 
doi:10.24963/ijcai.2018/21 dblp:conf/ijcai/ChanJ18 fatcat:5hs7oh44n5aolnztwrbe6mz4ea

Computing Equilibria in Atomic Splittable Polymatroid Congestion Games with Convex Costs [article]

Tobias Harks, Veerle Timmermans
2018 arXiv   pre-print
In this paper, we compute ϵ-approximate Nash equilibria in atomic splittable polymatroid congestion games with convex Lipschitz continuous cost functions.  ...  to atomic splittable polymatroid congestion games implying that we can compute ϵ-approximate Cournot-Nash equilibria within pseudo-polynomial time.  ...  Similar to atomic splittable congestion games, the complete strategy space of the game is defined by P k :" ś iPN P k i pd i q.  ... 
arXiv:1808.04712v1 fatcat:dv246fkbdbdqngq5e5ubdqesbe

Pricing in Resource Allocation Games Based on Lagrangean Duality and Convexification [article]

Tobias Harks
2020 arXiv   pre-print
We show that for finite strategy spaces or certain concave games, the equilibrium existence problem reduces to solving a well-structured LP.  ...  We consider a basic resource allocation game, where the players' strategy spaces are subsets of R^m and cost/utility functions are parameterized by some common vector u∈ R^m and, otherwise, only depend  ...  Let (N , X, (π i ) i∈N ) be a congestion game with homogeneous nondecreasing cost functions and polytopal strategy spaces with aggregation polyhedron P N . Let u be minimal for X.  ... 
arXiv:1907.01976v3 fatcat:uboddyscdbb2na7ids7cxukn44

Sensitivity Analysis for Convex Separable Optimization over Integral Polymatroids [article]

Tobias Harks and Max Klimm and Britta Peis
2016 arXiv   pre-print
In addition, there is a game where the players strategies are isomorphic to the non-polymatroid region and that does not admit a pure Nash equilibrium.  ...  the polytope.  ...  (i) congestion games with matroidal strategies and player-specific costs (studied by Ackermann et al.  ... 
arXiv:1611.05372v2 fatcat:vb5ogwmmlfdkvml6uoyxl3bx4m

Totally Unimodular Congestion Games [article]

Alberto Del Pia, Michael Ferris, Carla Michini
2016 arXiv   pre-print
We investigate a new class of congestion games, called Totally Unimodular (TU) Congestion Games, where the players' strategies are binary vectors inside polyhedra defined by totally unimodular constraint  ...  Network congestion games belong to this class.  ...  Another class of symmetric congestion games for which a pure Nash equilibrium can be computed in polynomial time are matroid congestion games, i.e. congestion games where the strategy space of each player  ... 
arXiv:1511.02784v2 fatcat:huq3ovo52vhvtjsq36cmi33xoi

Average Case Performance of Replicator Dynamics in Potential Games via Computing Regions of Attraction [article]

Ioannis Panageas, Georgios Piliouras
2016 arXiv   pre-print
in networked coordination and congestion games with large gaps between price of stability and price of anarchy.  ...  Based on these geometric characterizations, we propose a novel quantitative framework for analyzing the efficiency of potential games with many equilibria.  ...  Replicator Dynamics on Congestion/Network Coordination Games Congestion Games A congestion game is defined by the tuple (N ; E; (S i ) i∈N ; (c e ) e∈E ) where N is the set of agents (with N = |N |),  ... 
arXiv:1403.3885v6 fatcat:7su2bhez7zevlaebpnwr7lqisq

Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries [article]

Paul W. Goldberg, Francisco J. Marmolejo-Cossío
2019 arXiv   pre-print
We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.  ...  the players has a constant number of pure strategies.  ...  [1] ; upper and lower bounds for equilibrium computation in bimatrix games, congestion games [11] and anonymous games [16] ; upper and lower bounds for randomised algorithms computing approximate  ... 
arXiv:1807.06170v2 fatcat:bw2mcquqqnfilaefxigxkxjm6i
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