Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On Characterizations of Potential and Ordinal Potential Games (2405.06253v1)

Published 10 May 2024 in cs.GT and math.OC

Abstract: This paper investigates some necessary and sufficient conditions for a game to be a potential game. At first, we extend the classical results of Slade and Monderer and Shapley from games with one-dimensional action spaces to games with multi-dimensional action spaces, which require differentiable cost functions. Then, we provide a necessary and sufficient conditions for a game to have a potential function by investigating the structure of a potential function in terms of the players' cost differences, as opposed to differentials. This condition provides a systematic way for construction of a potential function, which is applied to network congestion games, as an example. Finally, we provide some sufficient conditions for a game to be ordinal potential and generalized ordinal potential.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (31)
  1. Aggregate comparative statics. Games and Economic Behavior, 81:27–49, 2013.
  2. A review of transport market modeling using game-theoretic principles. European Journal of Operational Research, 291(3):808–829, 2021.
  3. Robustness of dynamics in games: A contraction mapping decomposition approach. Automatica, 155:111142, 2023.
  4. Characterization of potential games: Application in aggregative games. In 2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1–6. IEEE, 2023.
  5. Dynamic incentives for congestion control. IEEE Transactions on Automatic Control, 60(2):299–310, 2014.
  6. Amir Beck. Introduction to nonlinear optimization: Theory, algorithms, and applications with MATLAB. SIAM, 2014.
  7. Convex analysis and optimization, volume 1. Athena Scientific, 2003.
  8. Convex optimization. Cambridge university press, 2004.
  9. Flows and decompositions of games: Harmonic and potential games. Mathematics of Operations Research, 36(3):474–503, 2011.
  10. Price of stability in polynomial congestion games. ACM Transactions on Economics and Computation (TEAC), 4(2):1–17, 2015.
  11. Rahul Deb. Interdependent preferences, potential games and household consumption. Potential Games and Household Consumption (January 15, 2008), 2008.
  12. Christian Ewerhart. Ordinal potentials in smooth games. Economic Theory, 70(4):1069–1100, 2020.
  13. Congestion models and weighted bayesian potential games. Theory and Decision, 42:193–206, 1997.
  14. Distributed nash equilibrium seeking for aggregative games with directed communication graphs. IEEE Transactions on Circuits and Systems I: Regular Papers, 69(8):3339–3352, 2022.
  15. A mean field control approach for demand side management of large populations of thermostatically controlled loads. In 2015 European Control Conference (ECC), pages 3548–3553. IEEE, 2015.
  16. Strategic decompositions of normal form games: Zero-sum games and potential games. arXiv preprint arXiv:1602.06648, 2016.
  17. Martin Kaae Jensen. Aggregative games and best-reply potentials. Economic theory, 43(1):45–66, 2010.
  18. Distributed algorithms for aggregative games on graphs. Operations Research, 64(3):680–704, 2016.
  19. Demand response using linear supply function bidding. IEEE Transactions on Smart Grid, 6(4):1827–1838, 2015.
  20. Market-based coordination of thermostatically controlled loads—part i: A mechanism design formulation. IEEE Transactions on Power Systems, 31(2):1170–1178, 2015.
  21. Game theory. Cambridge University Press, 2020.
  22. The complexity of welfare maximization in congestion games. Networks, 59(2):252–260, 2012.
  23. Potential games. Games and economic behavior, 14(1):124–143, 1996.
  24. Yurii Nesterov. Lectures on convex optimization, volume 137. Springer, 2018.
  25. Robert W. Rosenthal. A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory, 2:65–67, 1973.
  26. Walter Rudin. Principles of mathematical analysis, volume 3. McGraw-hill New York, 1976.
  27. Margaret E. Slade. What does an oligopoly maximize? The Journal of Industrial Economics, pages 45–61, 1994.
  28. Takashi Ui. A shapley value representation of potential games. Games and Economic Behavior, 31(1):121–135, 2000.
  29. Mark Voorneveld. Best-response potential games. Economics letters, 66(3):289–295, 2000.
  30. Congestion games and potentials reconsidered. International Game Theory Review, 1(03n04):283–299, 1999.
  31. Distributed nash equilibrium seeking in an aggregative game on a directed graph. IEEE Transactions on Automatic Control, 66(6):2746–2753, 2020.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now