Algorithmic game theory
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Algorithmic game theory is an area in the intersection of game theory and algorithm design, whose objective is to design algorithms in strategic environments. 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. On top of the usual requirements in classical algorithm design, say polynomial-time running time, good approximation ratio, ... the designer must also care about incentive constraints. We can see Algorithmic Game Theory from two perspectives:
- Analysis: look at the current implemented algorithms and analyze them using Game Theory tools: calculate and prove properties on their Nash equilibria, price of anarchy, best-response dynamics ...
- Design: design games that have both good game-theoretical and algorithmic properties. This area is called algorithmic mechanism design
The field was started when Nisan and Ronen in STOC'99  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.
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. 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
The main areas of research in algorithmic game theory include:
- Algorithmic mechanism design
- Inefficiency of equilibria (price of anarchy)
- Complexity of finding equilibria
- Market equilibrium
- Multi-agent systems
- Computational social choice
And the area counts with diverse practical applications:
- 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.
- John von Neumann, Oskar Morgenstern (1944) Theory of Games and Economic Behavior. Princeton Univ. Press. 2007 edition: ISBN 978-0-691-13061-3
- Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007), Algorithmic Game Theory, Cambridge, UK: Cambridge University Press, ISBN 0-521-87282-0.