In 2003, scientists from HP Labs in Palo Alto, California, posted a paper on the Internet showing how a quantum public goods game provides strategies that reduce the temptation to free-load. When people make economic or social decisions, they don't always choose based on self-interest alone, but may be influenced by social norms and expectations—sort of the way properties of a photon are influenced by distant measurements. So if you send your pledge via a quantum information channel, its message can depend on the messages from the other contributors. Therefore, the HP scientists suggested, entangled photons transmitted by laser beams through optical fibers could in theory be used for pledging donations in real-life community projects. Using quantum-entangled photons to communicate their intentions could allow a coordination of commitments that otherwise couldn't be guaranteed.
"Quantum mechanics offers the ability to solve the free-rider problem in the absence of a third-party enforcer," wrote Kay-Yut Chen, Tad Hogg, and Raymond Beausoleil in their paper.8
QUANTUM VOTING
The same principle could be applied to other sorts of community communication issues, including voting, especially in elections with multiple candidates. You wouldn't need runoffs, since the multiple possible outcomes could be encoded in quantum information.
Here, I think, is a real potential for coping with some of the mathematical problems inherent in today's democratic system of voting. For instance, when three candidates are running for office, the ultimate winner may not reflect the desire of the majority of the voters. Here's how it can work:
In a primary election, Candidate A gets 37 percent of the votes, Candidate B gets 33 percent, and Candidate C gets 30 percent. So candidate A and B get into a runoff. But for most of candidate B's voters, C was the second choice. And for most of Candidate A's voters, C was also the second choice. So if C were running against A alone, C would win. If C were running against B alone, C would win. But in the primary, C finished third so the ultimate winner will be A or B. Since a majority of the voters prefer C to either A or B, the winner is clearly not the electorate's optimal choice. A quantum voting scheme could, by incorporating multiple possibilities in the voting, reach a more "democratic" election result.
It sounds far-fetched, but its mere possibility affirms the potentially dramatic value of invoking quantum weirdness to cope with the complexities of the ordinary world. And it may even be possible that quantum game theory underlies much deeper aspects of nature and of life. In the mushrooming literature on quantum games are papers suggesting that quantum strategies at the molecular level may mimic aspects of evolutionary game-theoretic descriptions of the competition between organisms. In particular, Azhar Iqbal of the University of Hull in England argues that quantum entanglement could influence the interactions of molecules leading to a more stable mix of ingredients than would otherwise occur (in analogy to an evolutionary stable strategy for organisms in an ecosystem). A quantum entanglement "strategy," he suggests, could determine whether a population of molecules can "with-stand invasion" from a small number of new molecules (corresponding to mutants in evolutionary biology).9 If there's anything to this—and it would seem to be far too early to say—then you could imagine something like quantum game theory playing a role in the origin of stable sets of self-replicating molecules—in other words, life itself. (In which case the Code of Nature might turn out to be solvable only with quantum cryptography.)
In any case, quantum game theory offers a new perspective on both games and physics, with implications awaiting further exploration. At the very least, quantum physics and games share one obvious similarity—probability distributions, as with the mixed strategies of games and the mixed realities of quantum mechanics. Life and physics, it seems, are all mixed up. Sorting it all out will require a closer look at the power of probability.
* * *
11
Pascal's Wager Games, probability, information, and ignorance
Although this may seem a paradox, all exact science is dominated by the idea of approximation.
—Bertrand Russell
As a teenager in 17th-century France, Blaise Pascal seemed destined for mathematical greatness. He wrote a genius-caliber treatise on geometry at age 16 and invented a rudimentary computer to assist the calculations of his tax-collector father. But as an adult, Pascal was seduced by religion, forgoing math to produce a series of philosophical musings assembled (after his death) into a book called Pensées. He died at 39, leaving a legacy, in the words of the mathematician E. T. Bell, as "perhaps the greatest might-have-been in history."1
Still, Pascal remains a familiar name in today's mathematics textbooks, thanks to a favor he did for a French aristocrat who desired assistance with his gambling habit. What Pascal offered was not religious counseling on the evils of gambling, but mathematical advice on how to win. In his correspondence on this question with Pierre Fermat, Pascal essentially invented probability theory. What's more, out of Pascal's religious ruminations came an idea about probability that was to emerge centuries later as a critical concept in mathematics, with particular implications for game theory.
When it comes to making bets, Pascal observed, it is not enough to know the odds of winning or losing. You need to know what's at stake. You might want to take unfavorable odds if the payoff for winning would be really huge, for example. Or you might consider playing it safe by betting on a sure thing even if the payoff was small. But it wouldn't seem wise to bet on a long shot if the payoff was going to be meager.
Pascal framed this issue in his religious writings, specifically in the context of making a wager about the existence of God. Choosing to believe in God was like making a bet, he said. If you believe in God, and that belief turns out to be wrong, you haven't lost much. But if God does exist, believing wins you an eternity of heavenly happiness. Even if God is a low-probability deity, the payoff is so great (basically, infinite) that He's a good bet anyway. "Let us weigh the gain and the loss in wagering that God is," Pascal wrote. "Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is."2
Pascal's reasoning may have been theologically simplistic, but it certainly was mathematically intriguing.3 It illustrated the kind of reasoning that goes into calculating the "mathematical expectation" of an economic decision—you multiply the probability of an outcome by the value of that outcome. The rational choice is the decision that computes to give the highest expected value. Pascal's wager is often cited as the earliest example of a math-based approach to decision theory.
In real life, of course, people don't always make their decisions simply by performing such calculations. And when your best decision depends on what other people are deciding, simple decision theory no longer applies—making the best bets becomes a problem in game theory. (Some experts would say decision theory is just a special case of game theory, in which one player plays the game against nature.) Still, probabilities and expected payoffs remain intertwined with game theory in a profound and complicated way.
For that matter, all of science is intertwined with probability theory in a profound way—it's essential for the entire process of observation, experiment, and measurement, and then comparing those numbers with theory. And probability arises not only in making measurements and testing hypotheses, but also in the very description of physical phenomena, particularly in the realm of statistical physics. In the social sciences, of course, probability theory is also indispensable, as Adolphe Quetelet argued almost two centuries ago. So game theory's intimate relationship with probabilities, I'd wager, is one of the reasons why it finds such widespread applicability in so many different scientific contexts. And no doubt it's this aspect of game theory that has positioned it so strategically as an agent for merging social and physical statistics into a physics of society—something like Asimov's psychohistory or a Code of Nature.
So far, attempts to devise a sociophysics for describing society have mostly been based not on ga
me theory, but on statistical mechanics (as was Asimov's fictional psychohistory). But game theory's mixed strategy/probabilistic formulas exhibit striking similarities to the probability distributions of statistical physics. In fact, the mixed strategies used by game players to achieve a Nash equilibrium are probability distributions, precisely like the distributions of molecules in a gas that statistical physics quantifies.
This realization prompts a remarkable conclusion—that, in a certain sense, game theory and statistical mechanics are alter egos. That is to say, they can be expressed using the same mathematical language. To be more precise, you'd have to say that certain versions of game theory share math identical to particular formulations of statistical mechanics, but the deep underlying connection remains. It's just that few people have noticed it.
STATISTICS AND GAMES
If you search the research literature thoroughly, though, you will find several papers from the handful of scientists who have begun to exploit the game theory–statistical physics connection. Among them is David Wolpert, a physicist-mathematician at NASA's Ames Research Center in California.
Wolpert is one of those creative thinkers who refuse to be straitjacketed by normal scientific stereotypes. He pursues his own intuitions and interests along the amorphous edges separating (or connecting) physics, math, computer science, and complexity theory. I first encountered him in the early 1990s while he was exploring the frontiers of interdisciplinary science at the Santa Fe Institute, discussing such issues as the limits of computability and the nature of memory.
In early 2004, Wolpert's name caught my eye when I noticed a paper he posted on the World Wide Web's physics preprint page.4 His paper showed how to build a bridge between game theory and statistical physics using information theory (providing, incidentally, one of the key inspirations for writing this book). In fact, as Wolpert showed in the paper that attracted my attention to this issue in the first place, a particular approach to statistical mechanics turns out to use math that is equivalent to the math for noncooperative games.
Wolpert's paper noted that the particles described by statistical physics are trying to minimize their collective energy, like the way people in a game try to reach the Nash equilibrium that maximizes their utility. The mixed strategies used by players to achieve a Nash equilibrium are probability distributions, just like the distribution of energy among particles described by statistical physics.
After reading Wolpert's paper, I wrote him about it and then a few months later discussed it with him at a complexity conference outside Boston where he was presenting some related work. I asked what had motivated him to forge a link between game theory and statistical physics. His answer: rejection.
Wolpert had been working on collective machine learning systems, situations in which individual computers, or robots, or other autonomous devices with their own individual goals could be coordinated to achieve an objective for the entire system. The idea is to find a way to establish relationships between the individual "agents" so that their collective behavior would serve the global goal. He noticed similarities in his work to a paper published in Physical Review Letters about nanosized computers. So Wolpert sent off one of his papers to that journal.
"The editor actually came back and said ‘Well, … what you're doing just plain isn't physics,'" Wolpert said. "And I was annoyed." So he started thinking about physics and game theory. After all, a bunch of agents with their own agendas, yet pursuing a common goal, is entirely analogous to players in a game seeking a Nash equilibrium. "And then I said, OK, I'm going to try to take that and completely translate it into a physics system," he recalled.5
Games deal with players; physics deals with molecules. So Wolpert worked on the math that would represent a player's strategy like a molecule's state of motion. The mix of all the players' strategies would then be like the combined set of motion states of all the molecules, as ordinarily described by statistical physics. The formulas he came up with would allow you to calculate a good approximation to the actual set of any individual player's strategies in a game, given some limited knowledge about them. You could then do exactly the same sort of calculation for the combined strategies of all the players in a game. Basically, Wolpert showed how the math of statistical physics turns out to be the same as the math for games where players have limited rationality.
"Those topics are fundamentally one and the same," he wrote in his paper. "This identification raises the potential of transferring some of the powerful mathematical techniques that have been developed in the statistical physics community to the analysis of noncooperative game theory."6
Wolpert's mathematical machinations were rooted in the idea of "maximum entropy," a principle relating standard statistical physics to information theory, the math designed to quantify the sending and receiving of messages. The maximum entropy (or "maxent") idea was promoted by the maverick physicist Edwin Jaynes in a 1957 paper that was embraced by a number of physicists but ignored by many others. Wolpert, for one, calls Jaynes's work "gloriously beautiful" and thinks that it's just what scientists need in order "to bring game theory into the 21st century."
Jaynes's principle is simultaneously intriguing and frustrating. It seems essentially simple but nevertheless poses tricky complications. It is intimately related to the physical concept of entropy, but is still subtly different. In any event, its explanation requires a brief excursion into the nature of probability theory and information theory, the essential threads tying game theory and statistical physics together.
PROBABILITY AND INFORMATION
For centuries, scientists and mathematicians have argued about the meaning of probability. Even today there exist separate schools of probabilistic thought, generally referred to by the shorthand labels of "objective" and "subjective." But those labels conceal a tangle of subarguments and technical subtleties that make probability theory one of the most contentious and confusing realms of math and science.
In a way, that's a bit surprising, since probability theory really lies at the very foundation of science, playing the central role in the process of analyzing experimental data and testing theories. It's what doing science is all about. You'd think they'd have it all worked out by now. But establishing rules for science is a little like framing a constitution for Iraq. There are different philosophies and approaches to science. The truth is that science (unlike mathematics) is not built on a rock-solid foundation of irreducible rules. Science is like grammar. Grammar arises from regularities that evolve in the way native speakers of a language form their words and string them together. A true grammarian does not tell people how they should speak, but codifies the way that people actually do speak. Science does not emanate from a cookbook that provides recipes for revealing nature's secrets, but from a mix of methods that somehow succeed in rendering nature comprehensible. That's why science is not all experiment, and not all theory, but a complex interplay of both.
Ultimately, though, theory and experiment must mesh if the scientist's picture of nature is to be meaningful and useful. And in most realms of science, you need math to test the mesh. Probability theory is the tool for performing that test. (Different ideas about how to perform the test, then, lead to different conceptions of probability.)
Before Maxwell, probability theory in science was mostly limited to quantifying things like measurement errors. Laplace and others showed the way to estimate how far off your measurement was likely to be from the true value for a particular degree of confidence. Laplace himself applied this approach to measuring the mass of Saturn. He concluded that there was only one chance in 11,000 that the true mass of Saturn would deviate from the then-current measurement by more than 1 percent. (As it turned out, today's best measurement indeed differs from the one in Laplace's day by only 0.6 percent.) Probability theory has developed into an amazingly precise way of making such estimates.
But what does probability itself really mean? If you ask people who ought to know, you'll get different answers.
The "objective" school of thought insists that the probability of an event is a property of the event. You observe in what fraction of all cases that event happens and thereby measure its objective probability. The subjective view, on the other hand, argues that probability is a belief about how likely something is to happen. Measuring how often something happens gives you a frequency, not a probability, the subjectivists maintain.
There is no point here delving into the debates about the relative merits of these two views. Dozens of books have been devoted to that controversy, which is largely irrelevant to game theory. The fact is that the prevailing view today, among physicists at least, is that the subjectivist approach contains elements that are essential for a sound assessment of scientific data.
Subjective statistics often goes under the label of Bayesian, after Thomas Bayes, the English clergyman who discussed an approach of that nature in a paper published in 1763 (two years after his death). Today a formula known as Bayes' theorem is at the heart of practicing the subjective statistics approach (although that precise theorem was actually worked out by Laplace). In any case, the Bayesian viewpoint today comes in a variety of flavors, and there is much disagreement about how it should be interpreted and applied (perhaps because it is, after all, subjective).
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