Does Game Theory Work?

AI AND GAME THEORY Editor: michael Wooldridge, University of Oxford, [email protected]

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 theory 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 rational 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

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

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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. If it comes up follow social norms, but we often dilemma. heads, you get $1,000; if it comes up do, because often such norms are tails, you get nothing. self-policing. In the case of our two Models of Utility secretaries, the punishment for Sub- There are other reasons why deci- B: You get $450. ject 1 transgressing would be the sions people make in life and in the shame of being viewed as greedy at a laboratories of game theorists don’t Would you choose A or B? Ex- colleague’s expense, or being forced necessarily correspond to the solu- pected utility theory says that you to fetch your own coffee every day. tion concepts we find in textbooks. can expect to earn $500 from option But as the secretaries themselves in- One important one is that the utility A or $450 from option B, so option A dicated, for some sufficiently large model used in game-theoretic analy- is the rational choice. But I wouldn’t value of M + G, Subject 1 would de- sis doesn’t necessarily correspond to choose A, nor, I think, would most mand a larger piece of the pie: the people; I know how grumpy I would benefits offered by $M + (G/2) start be if the coin came up tails, and it to outweigh the social stigma of isn’t worth the chance of that extra greed or the tedium of always fetch- People tend to be loss- $50. (If we were to play the game 100 ing your own coffee. Nevertheless, times, I would choose differently.) an enforcement mechanism appears averse: they will make Even with such a simple experiment, to be behind the social norm here. expected utility theory fails. We generally view social norms— choices to avoid potential Kahneman and Tversky’s prospect and, in particular, fairness norms theory suggests that when making de- such as splitting the difference—as perceived losses. cisions, people will broadly behave playing havoc with game-theoretic as follows. First, they don’t dispas- models of rational choice. They seem sionately view all outcomes on some to be responsible for the behavior in how people assess utility. Perhaps the simple linear scale, with higher being Flood’s prisoner’s dilemma experi- most famous work in this area comes better and lower being worse. Rather, ments, and people seem to adhere to from Daniel Kahneman and Amos they evaluate outcomes with respect them quite commonly, both in real Tversky—work for which Kahneman to some fixed reference point, which life and in experimental settings. So, received a Nobel prize in 2002 (Tver- captures a status quo. People view where do they come from, and how sky passed away in 1996). They put outcomes below this reference point and why do they persist? forward a theory of how people make as losses, and outcomes above this Ken Binmore is a prominent game decisions under uncertainty called point as gains. In the aforementioned theorist with an interest in game the- prospect theory.4 Their starting point example, you might fix on $450 as a ory’s role in the social sciences. He’s was the observation that expected reference point, because you’re guar- written extensively on game theory utility theory doesn’t predict how anteed this amount by choosing op- and social norms, and while I can’t people make decisions in practice. tion B. Alternatively, consider some- do justice to the breadth or depth of Expected utility theory is the best- body who receives a bonus at work. his viewpoints here, I can crudely known and most widely studied the- Suppose they believed in advance that summarize his views as follows:3 ory of decision making under uncer- they would receive $50,000; this be- tainty. The mathematics of expected comes the reference point. If they A social norm (convention, social con- utility theory is elegant and compel- then in fact receive $100,000, they tract, and so on) is a self-policing agree- ling, but a wealth of evidence shows perceived this as a gain—it’s above ment between members of a society that that it doesn’t fare well at predicting the reference point. In contrast, if

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IS-27-06-Game.indd 78 11/6/12 2:08 PM they receive $40,000, then they per- • Players will adapt their behavior just one example, a standard game- ceive this as a loss, despite the fact that over time toward more rational theoretic assumption is that players they’re considerably better off than outcomes, if they receive suffi- have common knowledge of the game they were originally. Prospect theory cient opportunity for trial-and-error at hand: everybody knows who the suggests that people tend to be loss- learning. players are, their available choices, averse: they will make choices to avoid and their preferences, and everybody potential perceived losses. Much work remains on reconcil- knows that everybody knows this, Prospect theory is by no means ing the theoretical models and solu- and so on. It isn’t hard to see that the only alternative to expected util- tions of game theory with observed such assumptions simply don’t hold ity theory that’s been proposed, al- human behavior. Such issues are cur- in many settings, and where they though it’s probably the most suc- rently studied in the field of behav- don’t hold, we can’t hope to rely on cessful such approach in recent times. ioral game theory. game-theoretic advice. So, if we want Irrespective of prospect theory’s ad- to use a game-theoretic model to ob- vantages and disadvantages as a tool Normative Interpretations tain advice about how to act, we had for predicting human decisions, how- Let’s now turn to the normative in- better first be sure that the model and ever, we can see that conventional terpretation of game theory, which its assumptions really fit the circum- game theory assumes a model of de- describes what players ought to do in stances at hand. cision making that doesn’t necessar- a game. Under a normative interpre- Despite this word of caution, some ily reflect how people actually make tation, we think of game theory as prominent examples exist of game- choices. theoretic advice being successfully used in important real-world settings. Lessons Let’s briefly look at two. So what can we learn from the previ- Often, game-theoretic ous discussion?1 Auction Design models are applied in In the 1990s, cell phones became • In real-life settings, social norms widely used, and governments world- (and, in particular, norms of co- entirely inappropriate wide acted to regulate access to the operation) often play a part in how electromagnetic frequencies that these people make decisions. However, circumstances. phones use to communicate. At the if the incentives at hand are suffi- end of the 1990s, mobile phone tech- ciently large, then these incentives nology was moving toward a “third- can start to override social norms. providing advice to players about how generation” (3G) set of standards, • More generally, for incentives (such best to play games, or to game de- enabling phones to access rich data as payments) to be sufficient to in- signers about how best to design them services. These 3G services used a new fluence behavior, they must be ad- (this latter issue is called mechanism range of frequencies, and many gov- equate. Many game theory labora- design). The normative interpretation ernments decided to use auctions to tory experiments fail because the works if the advice is good—that is, allocate these frequencies to 3G ser- rewards involved are insufficient to if it helps people to make better de- vice providers. They recruited game influence behavior. cisions or helps governments or other theorists to design these auctions. • The model of choice that people designers of economic mechanisms to In the UK, a team of game theorists use will not always be based on design better ones. So, what evidence was charged with designing auctions expected utility-type models. We is there that game-theoretic advice that would allocate frequency ranges must understand their baseline ex- is good? with the following goals:5 pectations and whether they see The first point to make here is that outcomes as gains or losses; and we often, game-theoretic models are ap- • assign the electromagnetic spectrum must understand the tendency to plied in entirely inappropriate circum- efficiently—that is, allocate spec- be loss-averse. stances. Such models are predicated trum to those who would best use it; • For players to make rational choices, on a host of assumptions—some that • promote competition; and the game they’re playing must be are easily justifiable, others that are • realize the spectrum’s full economic sufficiently simple. perhaps harder to justify. To pick value—that is, maximize revenue.

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IS-27-06-Game.indd 79 11/6/12 2:08 PM Much effort went into designing the scarce resources to best effect. This can work under both its descriptive auctions that were ultimately used, problem has provided game theory and normative interpretations, al- and the design process included labo- with one of its most innovative and though it might often appear that it ratory experiments to determine how compelling recent application areas. doesn’t. Applied in the right way— individuals would actually behave for The basic idea is to view the prob- by understanding the motivations various sets of possible auction sce- lem as a game played between the of those who participate and the so- narios. The auctions themselves oc- security organization and its adver- cial norms that guide them—game curred in the first half of 2000 (which saries. The moves available to the ad- theory can successfully predict how was, coincidentally, the peak of the versary typically correspond to at- people will behave and, similarly, its dot-com boom, when wild specula- tacking one of the defender’s assets, techniques can help in economic sys- tive investment was made in IT and while the moves available to the de- tem design. But of course, much re- telecommunications technology). The fender typically correspond to allo- search remains before we can fully results were staggering. In total, the cating security teams to these assets. understand the scope, applicabil- auctions raised $34 billion in revenue The game-theoretic analysis provides ity, and usefulness of game-theoretic for the UK government. This is an a randomized strategy, which indi- techniques. Expect to see much more astonishing amount: to recoup their cates how to assign security resources work on this subject in the years investment, those that obtained the to assets. These techniques have been to come. licenses would have to receive pay- ment for 3G services of at least $550 for every man, woman, and child in References the UK! Although the UK govern- Game theory can . 1 K. Binmore, Does Game Theory Work? ment was naturally jubilant at this The Bargaining Challenge, MIT Press, windfall, telecom companies quickly work under both its 2007. regretted the scale of their ambi- 2. M.M. Flood, “Some Experimental tions and began to complain about descriptive and normative Games,” research memorandum how much they had paid for licenses. RM-789, RAND Corp., 1952; www. Whatever side you take in this debate interpretations, although rand.org/pubs/. (and I have to say, complaints about 3. K. Binmore, Playing Fair, MIT Press, the auctions weren’t received with it might often appear that 1994. much sympathy by the game theo- 4. D. Kahneman and A. Tversky, “Pros- rists who designed them), the game- it doesn’t. pect Theory: An Analysis of Decision theoretic auction design process cer- under Risk,” Econometrica, vol. 47, tainly paid off spectacularly from the no. 2, 1979, pp. 263–292. government’s viewpoint. deployed in numerous real-world set- 5. P. Klemperer and K. Binmore, “The tings. Most famously, they currently Biggest Auction Ever: The Sale of the Security Games inform security strategy at Los Ange- British 3G Telecom Licenses,” working Since 9/11, the world has become les World Airports (LAX). The secu- paper, University College London, security obsessed. Time-consuming rity games paradigm was largely the 2001. security checks and screening proce- work of Milind Tambe and his group, 6. M. Tambe, Security Games, Cambridge dures are now standard in interna- and Tambe’s book6 represents the Univ. Press, 2012. tional travel, and those charged with state of the art in this most challeng- ensuring our personal safety invest ing domain. huge amounts of time and money into M ichael Wooldridge is a professor of com- these checks and procedures. But de- puter science at the University of Oxford. spite their best efforts, security is an Contact him at [email protected]. inexact science. One key problem As is often the case in science, the is that the resources available to se- answer to an apparently simple ques- curity organizations are inherently tion has a frustratingly complex an- Selected CS articles and columns limited, in which case a fundamen- swer, wrapped almost to obscurity in are also available for free at tal problem is how to allocate these caveats and disclaimers. Game theory http://ComputingNow.computer.org.

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