Algorithmic game theory

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Algorithmic game theory (AGT) is an area in the intersection of game theory and computer science, with the objective of understanding and design of algorithms in strategic environments.

Contents

Typically, in Algorithmic Game Theory problems, the input to a given algorithm is distributed among many players who have a personal interest in the output. In those situations, the agents might not report the input truthfully because of their own personal interests. We can see Algorithmic Game Theory from two perspectives:

On top of the usual requirements in classical algorithm design (e.g., polynomial-time running time, good approximation ratio), the designer must also care about incentive constraints.

History

Nisan-Ronen: a new framework for studying algorithms

In 1999, the seminal paper of Noam Nisan and Amir Ronen [1] drew the attention of the Theoretical Computer Science community to designing algorithms for selfish (strategic) users. As they claim in the abstract:

We consider algorithmic problems in a distributed setting where the participants cannot be assumed to follow the algorithm but rather their own self-interest. As such participants, termed agents, are capable of manipulating the algorithm, the algorithm designer should ensure in advance that the agents’ interests are best served by behaving correctly. Following notions from the field of mechanism design, we suggest a framework for studying such algorithms. In this model the algorithmic solution is adorned with payments to the participants and is termed a mechanism. The payments should be carefully chosen as to motivate all participants to act as the algorithm designer wishes. We apply the standard tools of mechanism design to algorithmic problems and in particular to the shortest path problem.

This paper coined the term algorithmic mechanism design and was recognized by the 2012 Gödel Prize committee as one of "three papers laying foundation of growth in Algorithmic Game Theory". [2]

Price of Anarchy

The other two papers cited in the 2012 Gödel Prize for fundamental contributions to Algorithmic Game Theory introduced and developed the concept of "Price of Anarchy". In their 1999 paper "Worst-case Equilibria", [3] Koutsoupias and Papadimitriou proposed a new measure of the degradation of system efficiency due to the selfish behavior of its agents: the ratio of between system efficiency at an optimal configuration, and its efficiency at the worst Nash equilibrium. (The term "Price of Anarchy" only appeared a couple of years later. [4] )

The Internet as a catalyst

The Internet created a new economy—both as a foundation for exchange and commerce, and in its own right. The computational nature of the Internet allowed for the use of computational tools in this new emerging economy. On the other hand, the Internet itself is the outcome of actions of many. This was new to the classic, ‘top-down’ approach to computation that held till then. Thus, game theory is a natural way to view the Internet and interactions within it, both human and mechanical.

Game theory studies equilibria (such as the Nash equilibrium). An equilibrium is generally defined as a state in which no player has an incentive to change their strategy. Equilibria are found in several fields related to the Internet, for instance financial interactions and communication load-balancing[ citation needed ]. Game theory provides tools to analyze equilibria, and a common approach is then to ‘find the game’—that is, to formalize specific Internet interactions as a game, and to derive the associated equilibria.

Rephrasing problems in terms of games allows the analysis of Internet-based interactions and the construction of mechanisms to meet specified demands. If equilibria can be shown to exist, a further question must be answered: can an equilibrium be found, and in reasonable time? This leads to the analysis of algorithms for finding equilibria. Of special importance is the complexity class PPAD, which includes many problems in algorithmic game theory.

Areas of research

Algorithmic mechanism design

Mechanism design is the subarea of economics that deals with optimization under incentive constraints. Algorithmic mechanism design considers the optimization of economic systems under computational efficiency requirements. Typical objectives studied include revenue maximization and social welfare maximization.

Inefficiency of equilibria

The concepts of price of anarchy and price of stability were introduced to capture the loss in performance of a system due to the selfish behavior of its participants. The price of anarchy captures the worst-case performance of the system at equilibrium relative to the optimal performance possible. [5] The price of stability, on the other hand, captures the relative performance of the best equilibrium of the system. [6] These concepts are counterparts to the notion of approximation ratio in algorithm design.

Complexity of finding equilibria

The existence of an equilibrium in a game is typically established using non-constructive fixed point theorems. There are no efficient algorithms known for computing Nash equilibria. The problem is complete for the complexity class PPAD even in 2-player games. [7] In contrast, correlated equilibria can be computed efficiently using linear programming, [8] as well as learned via no-regret strategies. [9]

Computational social choice

Computational social choice studies computational aspects of social choice, the aggregation of individual agents' preferences. Examples include algorithms and computational complexity of voting rules and coalition formation. [10]

Other topics include:

And the area counts with diverse practical applications: [11] [12]

Journals and newsletters

Algorithmic Game Theory papers are often also published in Game Theory journals such as GEB, [15] Economics journals such as Econometrica, and Computer Science journals such as SICOMP. [16]

See also

Related Research Articles

<span class="mw-page-title-main">Gödel Prize</span> Computer science award

The Gödel Prize is an annual prize for outstanding papers in the area of theoretical computer science, given jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery Special Interest Group on Algorithms and Computational Theory. The award is named in honor of Kurt Gödel. Gödel's connection to theoretical computer science is that he was the first to mention the "P versus NP" question, in a 1956 letter to John von Neumann in which Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time.

In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their private observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from their strategy, the distribution from which the signals are drawn is called a correlated equilibrium.

<span class="mw-page-title-main">Christos Papadimitriou</span> Greek computer scientist (b. 1949)

Christos Charilaos Papadimitriou is a Greek theoretical computer scientist and the Donovan Family Professor of Computer Science at Columbia University.

The revelation principle is a fundamental result in mechanism design, social choice theory, and game theory which shows it is always possible to design a strategy-resistant implementation of a social decision-making mechanism. It can be seen as a kind of mirror image to Gibbard's theorem. The revelation principle says that if a social choice function can be implemented with some non-honest mechanism—one where players have an incentive to lie—the same function can be implemented by an incentive-compatible (honesty-promoting) mechanism with the same equilibrium outcome (payoffs).

In game theory, an epsilon-equilibrium, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.

Algorithmic mechanism design (AMD) lies at the intersection of economic game theory, optimization, and computer science. The prototypical problem in mechanism design is to design a system for multiple self-interested participants, such that the participants' self-interested actions at equilibrium lead to good system performance. Typical objectives studied include revenue maximization and social welfare maximization. Algorithmic mechanism design differs from classical economic mechanism design in several respects. It typically employs the analytic tools of theoretical computer science, such as worst case analysis and approximation ratios, in contrast to classical mechanism design in economics which often makes distributional assumptions about the agents. It also considers computational constraints to be of central importance: mechanisms that cannot be efficiently implemented in polynomial time are not considered to be viable solutions to a mechanism design problem. This often, for example, rules out the classic economic mechanism, the Vickrey–Clarke–Groves auction.

Distributed algorithmic mechanism design (DAMD) is an extension of algorithmic mechanism design.

The Price of Anarchy (PoA) is a concept in economics and game theory that measures how the efficiency of a system degrades due to selfish behavior of its agents. It is a general notion that can be extended to diverse systems and notions of efficiency. For example, consider the system of transportation of a city and many agents trying to go from some initial location to a destination. Here, efficiency means the average time for an agent to reach the destination. In the 'centralized' solution, a central authority can tell each agent which path to take in order to minimize the average travel time. In the 'decentralized' version, each agent chooses its own path. The Price of Anarchy measures the ratio between average travel time in the two cases.

In game theory, the price of stability (PoS) of a game is the ratio between the best objective function value of one of its equilibria and that of an optimal outcome. The PoS is relevant for games in which there is some objective authority that can influence the players a bit, and maybe help them converge to a good Nash equilibrium. When measuring how efficient a Nash equilibrium is in a specific game we often also talk about the price of anarchy (PoA), which is the ratio between the worst objective function value of one of its equilibria and that of an optimal outcome.

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Congestion games (CG) are a class of games in game theory. They represent situations which commonly occur in roads, communication networks, oligopoly markets and natural habitats. There is a set of resources ; there are several players who need resources ; each player chooses a subset of these resources ; the delay in each resource is determined by the number of players choosing a subset that contains this resource. The cost of each player is the sum of delays among all resources he chooses. Naturally, each player wants to minimize his own delay; however, each player's choices impose a negative externality on the other players, which may lead to inefficient outcomes.

In algorithmic game theory, a succinct game or a succinctly representable game is a game which may be represented in a size much smaller than its normal form representation. Without placing constraints on player utilities, describing a game of players, each facing strategies, requires listing utility values. Even trivial algorithms are capable of finding a Nash equilibrium in a time polynomial in the length of such a large input. A succinct game is of polynomial type if in a game represented by a string of length n the number of players, as well as the number of strategies of each player, is bounded by a polynomial in n.

<span class="mw-page-title-main">Noam Nisan</span> Israeli computer scientist

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<span class="mw-page-title-main">Tim Roughgarden</span> American computer scientist

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Fisher market is an economic model attributed to Irving Fisher. It has the following ingredients:

The Prize in Game Theory and Computer Science in Honour of Ehud Kalai is an award given by the Game Theory Society. The prize is awarded for outstanding articles at the interface of game theory and computer science. Following the eligibility rules of the Gödel Prize, preference is given to authors who are 45 years old or younger at the time of the award. It was established in 2008 by a donation from Yoav Shoham in honor of the Ehud Kalai's contributions in bridging these two fields.

<span class="mw-page-title-main">Price of anarchy in auctions</span>

The Price of Anarchy (PoA) is a concept in game theory and mechanism design that measures how the social welfare of a system degrades due to selfish behavior of its agents. It has been studied extensively in various contexts, particularly in auctions.

The Price of Anarchy (PoA) is a concept in game theory and mechanism design that measures how the social welfare of a system degrades due to selfish behavior of its agents. It has been studied extensively in various contexts, particularly in congestion games (CG).

Amir Ronen is an Israeli computer scientist.

References

  1. ↑ Nisan, Noam; Ronen, Amir (1999), "Algorithmic mechanism design", Proceedings of the 31st ACM Symposium on Theory of Computing (STOC '99), pp. 129–140, doi: 10.1145/301250.301287 , ISBN   978-1581130676, S2CID   8316937
  2. ↑ "ACM SIGACT Presents Gödel Prize for Research that Illuminated Effects of Selfish Internet Use" (Press release). New York. Association for Computing Machinery. 2012-05-16. Archived from the original on 2013-07-18. Retrieved 2018-01-08.
  3. ↑ Koutsoupias, Elias; Papadimitriou, Christos (May 2009). "Worst-case Equilibria". Computer Science Review. 3 (2): 65–69. doi:10.1016/j.cosrev.2009.04.003. Archived from the original on 2016-03-13. Retrieved 2018-01-08.
  4. ↑ Papadimitriou, Christos (2001), "Algorithms, games, and the Internet", Proceedings of the 33rd ACM Symposium on Theory of Computing (STOC '01), pp. 749–753, CiteSeerX   10.1.1.70.8836 , doi:10.1145/380752.380883, ISBN   978-1581133493, S2CID   207594967
  5. ↑ Tim Roughgarden (2005). Selfish routing and the price of anarchy. MIT Press. ISBN   0-262-18243-2.
  6. ↑
    • Anshelevich, Elliot; Dasgupta, Anirban; Kleinberg, Jon; Tardos, Éva; Wexler, Tom; Roughgarden, Tim (2008). "The Price of Stability for Network Design with Fair Cost Allocation". SIAM J. Comput. 38 (4): 1602–1623. doi:10.1137/070680096. S2CID   2839399.
  7. ↑
    • Chen, Xi; Deng, Xiaotie (2006). Settling the complexity of two-player Nash equilibrium. Proc. 47th Symp. Foundations of Computer Science. pp. 261–271. doi:10.1109/FOCS.2006.69. ECCC   TR05-140..
  8. ↑ Papadimitriou, Christos H.; Roughgarden, Tim (2008). "Computing correlated equilibria in multi-player games". J. ACM. 55 (3): 14:1–14:29. CiteSeerX   10.1.1.335.2634 . doi:10.1145/1379759.1379762. S2CID   53224027.
  9. ↑ Foster, Dean P.; Vohra, Rakesh V. (1996). "Calibrated Learning and Correlated Equilibrium". Games and Economic Behavior.
  10. ↑ Felix Brandt; Vincent Conitzer; Ulle Endriss; Jérôme Lang; Ariel D. Procaccia, eds. (2016), Handbook of Computational Social Choice (PDF)
  11. ↑ Tim Roughgarden (2016). Twenty lectures on algorithmic game theory. Cambridge University Press. ISBN   9781316624791.
  12. ↑ "EC'19 || 20th ACM Conference on Economics and Computation".
  13. ↑ TEAC
  14. ↑ SIGEcom Exchanges
  15. ↑ Chawla, Shuchi; Fleischer, Lisa; Hartline, Jason; Tim Roughgarden (2015), "Introduction to the Special Issue – Algorithmic Game Theory – STOC/FOCS/SODA 2011", Games and Economic Behavior , 92: 228–231, doi:10.1016/j.geb.2015.02.011
  16. ↑ SICOMP
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