review articles 1_CACM_V51.8.indb 74 7/21/08 10:13:35 AM review articles DOI:10.1145/1378704.1378721 University, under the leadership of John The most dramatic interaction between CS von Neumann, in the 1950s.a In this article I try to do two things: and GT may involve game-theory pragmatics. identify the main areas of interaction between computer science and game BY YOAV SHOHAM theory so far; and point to where the most interesting interaction yet may lie—in an area that is still relatively un- derexplored. The first part aims to be an unbiased Computer survey, but it is impossible to avoid bias altogether. Ten researchers survey- ing the interactions between CS and GT would probably write 10 different Science and types of reports. Indeed, several already have (as I will discuss). Moreover, in this brief discussion I cannot possibly do justice to all the work taking place Game Theory in the area. So I try to compensate for these limitations in two ways: I provide a balanced set of initial pointers into the different subareas, without regard to the amount or nature of work that has taken place in each; and I point the reader to other relevant surveys of the CS-GT interaction, each having its own take on things. GAME THEORY HAS influenced many fields, The second part is decidedly subjec- including economics (its initial focus), political tive, but it is still meant to be broadly science, biology, and many others. In recent years, relevant both to computer scientists and game theorists interested in the in- its presence in computer science has become teraction between the disciplines. impossible to ignore. GT is an integral part of Lessons from Kalai (1995) artificial intelligence (AI), theory, e-commerce, My departure point is a 13-year-old sur- networking, and other areas of computer science, vey paper by E. Kalai,16 a game theorist and it is routinely featured in the field’s leading with algorithmic sensibilities. Geared primarily toward computer scientists, journals and conferences. One reason is application the paper took stock of the interac- pull: the Internet calls for analysis and design of tions between game theory, operations systems that span multiple entities, each with its research, and computer science at the time. It points to the following areas: own information and interests. Game theory, for all 1. Graphs in games its limitations, is by far the most developed theory 2. The complexity of solving a game 3. Multiperson operations research of such interactions. Another reason is technology 4. The complexity of playing a game push: the mathematics and scientific mind-set of 5. Modeling bounded rationality. game theory are similar to those that characterize The reason I start with this paper, be- sides providing the interesting perspec- many computer scientists. Indeed, it is interesting tive of a non-computer scientist, is the to note that modern computer science and modern comparison with current CS-GT interac- game theory originated in large measure at the same a I thank Moshe Tennenholtz for this observa- ILLUSTRATION BY JEAN FRANCOIS PODEVIN BY ILLUSTRATION place and time—namely at Princeton tion, which is especially true of GT and AI. AUGUST 2008 | VOL. 51 | NO. 8 | COMMUNICATIONS OF THE ACM 75 1_CACM_V51.8.indb 75 7/21/08 10:13:36 AM review articles tion, as both the matches and mismatches is known to exist,27 the computation of maximally harmful. Recent work, how- are instructive. Looking at the interactions a sample Nash equilibrium was shown ever, has begun to bridge these gaps. between CS and GT taking place today, to be complete for this class,2 and the This third category blends into the one can identify the following foci: problem of computing Nash equilib- fourth one, which is research moti- a. Compact game representations; ria with specific properties was shown vated by specific applications that have b. Complexity of, and algorithms for, to be NP-hard.4, 10 At the same time, emerged in the past decade. For exam- computing solution concepts; algorithms—some quite sophisticated, ple, the domain of networking has given c. Algorithmic aspects of mecha- and all exponential in the worst case— rise to a literature on so-called “price nism design; have been proposed to compute Nash of anarchy” (which captures the inef- d. Game-theoretic analysis equilibria.11, 41 Somewhat surprisingly, ficiency of equilibria in that domain), inspired by specific applications; recent experiments have shown that a games of routing, networking-forma- e. Multiagent learning; relatively simple search algorithm sig- tion games, and peer-to-peer networks. f. Logics of knowledge and belief, nificantly outperforms more sophisti- Other domains include sponsored and other logical aspects of cated algorithms.31 This is an active area search auctions, information markets, games.b that promises many additional results. and reputation systems. This combina- The crude mapping between this list The third match is somewhat less tight tion of the third and fourth categories and Kalai’s is as follows: than the first two. There are at least two is arguably the most active area today kinds of optimization one could speak at the interface of CS and GT, and many 1995 2008 about in a game-theoretic setting. The aspects of it are covered in Nisan et al.,25 1 •˲ a first is computing a best response to a which is an extensive edited collection 2 •˲ b fixed decision by the other agents; this is of surveys. The popularity of this area is 3 •˲ c, d of course the quintessential single-agent perhaps not surprising. The relevancies 4 optimization problem of operations re- of specific applications speak for them- 5 search and AI, among other fields. The selves (although arguments remain e, second is the optimization by the design- about whether the traditional game-the- f er of a mechanism aimed at inducing oretic analysis is an appropriate one). games with desirable equilibria. More generally, it is not surprising that Here, I discuss the areas that match This so-called “mechanism design” mechanism design struck a chord in up (1•˲a, 2•˲b, 3•˲c, d), then turn to has been the focus of much work in CS, given that much of CS’s focus is on the currently active areas that were not computer science. One reason is the the design of algorithms and protocols. discussed by Kalai (e, f), and finish with interesting interaction between tradi- Mechanism design is the one area with- the orphans on the other side (4, 5) that tional CS problems (such as optimi- in GT that adopts such a design stance. were discussed by Kalai but not yet vig- zation and approximation) and tradi- The fifth category active today is mul- orously pursued. tional mechanism-design issues (such tiagent learning, also called “interactive There has been substantial work as incentive compatibility, individual learning” in the game-theory literature.c on compact and otherwise specialized rationality, and social-welfare maximi- Multiagent learning, long a major focus game representations. Some of them zation). A good example is the interac- within game theory, has been rediscov- are indeed graph-based—graphical tion between the Vickrey-Clarke-Groves ered with something of a vengeance in games,18 local-effect games,21 MAIDS,19 mechanism and shortest-path computa- computer science and in particular AI; and Game networks,20 for example. The tion;26 another is the literature on com- witness special issues devoted to it in graph-based representations extend binatorial auctions,6 which combine the Journal of Artificial Intelligence39 and also to coalition game theory.7 But spe- a weighted-set-packing-like NP-hard the Machine Learning Journal.12 For com- cialized representations exist that are optimization problem with incentive puter science, the move from single- not graph based, such as those that are issues. The interplay between mecha- agent learning to multiagent learning multi-attribute based5 and logic based.15 nism design and cryptography is worth is interesting not only because it calls I believe this area is ripe for additional particular mention. Though both are in for new solutions but also because the work—regarding, for example, the strat- the business of controlled dissemina- very questions change. When multiple egy space of agents described using con- tion of information, they are different in agents learn concurrently, one can- structs of programming languages. significant ways. For one thing, they are not distinguish between learning and The complexity of computing a sam- dual in the following sense: mechanism teaching, and the question of “optimal” ple Nash equilibrium (as well as other design attempts to force the revelation learning is no longer well defined (just solution concepts) has been the focus of information, while cryptography at- as the more general notion of an “op- of much interest in CS, especially with- tempts to allow its hiding. For another, timal policy” ceases to be meaningful in the theory community. A new com- they traditionally embody quite differ- when one moves to the multiagent set- plexity class—PPAD—was proposed to ent models of paranoia. Game theory ting). For a discussion of this phenom- handle problems for which a solution assumes an even-keeled expected utility enon, see the Journal of Artificial Intelli- maximization on the part of all agents, gence special issue cited earlier.39 b This current survey originated in a presenta- while cryptography is more simple- tion made at a December 2007 festschrift in minded: it assumes that “good” agents c Kalai’s omission of this area is ironic, as he co- honor of E. Kalai. act as instructed, while “bad” agents are authored one of its seminal papers.