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.
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.
In 1999, the seminal paper of Noam Nisan and Amir Ronen drew the attention of the Theoretical Computer Science community to designing algorithms for selfish (strategic) users. As they claim in the abstract:
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".[1]
See main article: 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",[2] 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.)
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.
See main article: 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.
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.[3] The price of stability, on the other hand, captures the relative performance of the best equilibrium of the system.[4] These concepts are counterparts to the notion of approximation ratio in algorithm design.
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.[5] In contrast, correlated equilibria can be computed efficiently using linear programming,[6] as well as learned via no-regret strategies.[7]
See main article: 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.
Other topics include:
And the area counts with diverse practical applications:[8] [9]
Algorithmic Game Theory papers are often also published in Game Theory journals such as GEB, Economics journals such as Econometrica, and Computer Science journals such as SICOMP.[12]