Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure
We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by potential function arguments. Unfortunately, this proof of existence is ineff...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Conference Paper |
| Language: | English |
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2013
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| Online Access: | https://hdl.handle.net/10356/98777 http://hdl.handle.net/10220/12552 |
| _version_ | 1811686184637693952 |
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| author | Caragiannis, Ioannis Fanelli, Angelo Gravin, Nick Skopalik, Alexander |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Caragiannis, Ioannis Fanelli, Angelo Gravin, Nick Skopalik, Alexander |
| author_sort | Caragiannis, Ioannis |
| collection | NTU |
| description | We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by potential function arguments. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. Do such equilibria exist? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an "approximation" of such games by a new class of potential games that we call ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5/2+O(γ) for arbitrarily small γ > 0; for latency functions with maximum degree d ≥ 2, it is d 2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1/γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a d O(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of ψ-games. The concept of approximating non-potential games by potential ones is interesting in itself and might have further applications. |
| first_indexed | 2024-10-01T04:56:23Z |
| format | Conference Paper |
| id | ntu-10356/98777 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T04:56:23Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | ntu-10356/987772020-03-07T12:31:20Z Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure Caragiannis, Ioannis Fanelli, Angelo Gravin, Nick Skopalik, Alexander School of Physical and Mathematical Sciences Conference on Electronic Commerce (13th : 2012 : Valencia, Spain) We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by potential function arguments. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. Do such equilibria exist? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an "approximation" of such games by a new class of potential games that we call ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5/2+O(γ) for arbitrarily small γ > 0; for latency functions with maximum degree d ≥ 2, it is d 2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1/γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a d O(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of ψ-games. The concept of approximating non-potential games by potential ones is interesting in itself and might have further applications. 2013-07-31T03:19:52Z 2019-12-06T19:59:34Z 2013-07-31T03:19:52Z 2019-12-06T19:59:34Z 2012 2012 Conference Paper Caragiannis, I., Fanelli, A., Gravin, N., & Skopalik, A. (2012). Approximate pure nash equilibria in weighted congestion games. Proceedings of the 13th ACM Conference on Electronic Commerce - EC '12, 284-301. https://hdl.handle.net/10356/98777 http://hdl.handle.net/10220/12552 10.1145/2229012.2229036 en |
| spellingShingle | Caragiannis, Ioannis Fanelli, Angelo Gravin, Nick Skopalik, Alexander Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure |
| title | Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure |
| title_full | Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure |
| title_fullStr | Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure |
| title_full_unstemmed | Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure |
| title_short | Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure |
| title_sort | approximate pure nash equilibria in weighted congestion games existence efficient computation and structure |
| url | https://hdl.handle.net/10356/98777 http://hdl.handle.net/10220/12552 |
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