AI AND GAME THEORY Editor: Michael Wooldridge, University of Oxford, [email protected] Does Game Theory Work? Michael Wooldridge, University of Oxford he past decade has witnessed a huge explo- of rational choice. Nash equilibrium is the most sion of interest in issues that intersect com- famous example of a solution concept. A Nash T equilibrium is a selection of choices for players puter science and game theory. We see game theory such that no player would prefer to unilaterally tracks at all major AI conferences, and algorithmic deviate from this selection. game theory has been one of the most high-profi le So, given this discussion, how exactly are we to growth areas in theoretical computer science in re- interpret game theory? Two obvious interpreta- cent times. Given this level of interest, it might be tions exist: worth stepping back and asking, “Does game the- ory actually work?” (with apologies to Ken Bin- • Under a descriptive interpretation, we can view more, whose eponymous 2007 book inspired this game theory as attempting to predict how (hu- article1). Here, I discuss some ways of viewing this man) players will behave in strategic settings. question and review research on the topic. • Under a normative interpretation, we can view game theory as prescribing courses of action for It Depends What You players—that is, game theory tells players how Mean By “Work” they ought to act. “Does game theory work?” seems at fi rst sight to be a fairly straightforward question, but we need These two interpretations present very different to be a little more precise before we can begin to criteria for the question of whether game theory answer it. Before we tighten up the question at works. The descriptive interpretation suggests that hand, let’s fi rst recall what game theory is (and we should look for whether game theory success- isn’t). fully predicts how people will make choices in set- Game theory is the mathematical theory of in- tings that we can model as games. The normative teractions between self-interested agents. In par- interpretation suggests that we should examine ticular, it focuses on decision making in settings whether, by following game theory’s prescriptions, where each player’s decision can infl uence the out- we can obtain outcomes that are better than what comes (and hence the well-being) of other players. we might otherwise have obtained. In the remain- In such settings, each player must consider how der of this article, I discuss these two interpretations each other player will act in order to make an op- and ask whether game theory “works” for them. timal choice. In game theory, the term “game” means an abstract mathematical model of a multi- Descriptive Interpretations agent decision-making setting; the aim is typically Does game theory work as a tool for predicting to include in such a model all and only those as- how people will actually behave in game-like set- pects of the domain that are germane to the deci- tings? Conventional wisdom says “no.” However, sions that players must make. Game theory puts the truth is more complex, subtle, and interesting forward a number of solution concepts that are than this simple answer suggests. typically intended to formulate some notion of ra- tional choice in a game-theoretic setting. Social Norms Solution concepts are thus at the heart of game From as far back as the 1950s, when game theory theory. They’re so called because they formulate so- was in its infancy, researchers wanted to investi- lutions to games: game outcomes that could result gate the extent to which the exciting new theorems if the players employed the corresponding notion and models they were developing could predict 76 1541-1672/12/$31.00 © 2012 IEEE IEEE INTELLIGENT SYSTEMS Published by the IEEE Computer Society IS-27-06-Game.indd 76 11/6/12 2:08 PM actual behavior. Merrill Flood con- Of course, we must treat Flood’s received $G/2—that is, the sub- ducted one of the most famous early experiments—and his analysis—with jects would share the “surplus” G, experiments to investigate this ques- a good degree of caution. We can while Subject 1 also received the en- tion.2 This work introduced the pris- easily pick holes in the experimental tire amount M. What actually hap- oner’s dilemma, now regarded as one setup and question the conclusions pened is that the secretaries divided of the most important formal games drawn. But this was among the first the total amount G + M equally be- in the game theory canon. The point attempts to experimentally investi- tween themselves, in fact resulting about the prisoner’s dilemma is that gate whether game theory’s solution in Subject 1 receiving less than he or the standard game-theoretic analysis concepts had any relation to actual she would have simply by refusing to leads to players selecting an outcome observed human behavior, and one strike a deal with Subject 2. When (“mutual defection”) that’s worse for conclusion Flood drew—that the questioned, the secretaries reported both of them than another outcome players were converging to “splitting that they had agreed on this split in (“mutual cooperation”). This fact has the difference”—seems to reflect be- advance, but that if the values M and led many commentators to suggest havioral patterns that we see often G in question were much larger, then that the game-theoretic analysis must when people interact. they wouldn’t have felt compelled to be wrong. We can better understand this be- abide by the agreement. Flood’s con- Flood believed that “the axiomatic havior if we look at an even simpler clusion was that the social relation- structures [of game theory] should be experiment that Flood carried out, ships between the secretaries acted tested for applicability and useful- to bring new factors into the game ness in controlled experimental situa- that weren’t reflected in simply try- tions.”2 Jointly with Melvin Dresher, ing to maximize the portion of M and he organized a series of experiments Social norms play G that the subjects received. In other (admittedly rather ad hoc experi- words, social factors change the utili- ments by today’s rigorous standards) an important role in ties of the secretaries. Once again, with the goal of doing just this. In we can easily criticize the experimen- one experiment, they had two people determining how we tal setup, but this misses the point, play the iterated prisoner’s dilemma which is that even this crude experi- for 100 rounds; that is, they played a behave when interacting ment demonstrates that other factors game with 100 rounds in which each are at work when players determine round was the prisoner’s dilemma. with others. how to act. But what are those fac- The players knew how long they were tors, exactly? going to play for, and were both edu- Subsequent research on this cated and mathematically proficient which involved two secretaries play- topic—and the literature is now individuals with some knowledge of ing the following game: enormous—suggests that social game theory. So, what did they do? norms play an important role in de- Well, they didn’t choose to follow the The experimenter offers to give Subject 1 termining how we behave when in- prescriptions of game theory, which an amount $M but to give Subjects 1 and teracting with others. Social norms in this case points to mutual defec- 2 together a greater amount $M + G if are nothing more than standards of tion in every round of the game. In they can agree how to share the larger behavior: conventions, or customs. fact, mutual cooperation occurred amount. They’re rules of conduct that we nearly two-thirds of the time, adopt and follow as part of our up- although the players’ comments along This is, of course, a rather cruel bringing and culture. We learn social the way (“dope … I’ll be damned! ... game. Subject 1 is clearly in a stron- norms from our parents (“Flush the the stinker! ... he’s crazy … I’ll teach ger position—he or she can always toilet!”), from our friends at school him”) suggest that the road to this gain at least $M, irrespective of (“Everyone should have a turn with outcome wasn’t without some obsta- Subject 2, and Subject 2 needs Sub- the cool toy”), and when we start a cles. Flood’s interpretation was that ject 1’s agreement to gain anything. job (“It isn’t the done thing to hog the players were “rapidly learning … Flood assumed that the secretaries the printer all the time”). Going back and converging to a split-the-difference would strike a deal in which Subject 1 to our two secretaries, it seems likely [outcome].”2 received $M + (G/2), while Subject 2 that the social norm would simply NOVEMBER/DECEMBER 2012 www.computer.org/intelligent 77 IS-27-06-Game.indd 77 11/6/12 2:08 PM say that a windfall of this kind allows them to coordinate on a particu- how people make decisions. This is should be shared equally. As the lar outcome. easy to demonstrate. Consider the stakes become higher, though, the following game:4 temptation to transgress becomes The outcome in question will often greater. So what happens if one of be one that’s efficient; such norms You are given a choice between options the subjects transgresses—if Subject 1 can potentially explain why people A and B. insists on receiving $M + (G/2), for actually manage to cooperate (as op- example? Nothing compels us to posed to defecting) in the prisoner’s A: We flip a fair coin.