"Paradoxes of Rationality"

Are people irrational? The claim is that this research "undermines both the libertarian idea that unrestrained selfishness is good for the economy and the game-theoretic tenet that people will be selfish and rational," but I'm not so sure about that:

The Traveler's Dilemma, by Kaushik Basu, Scientific American, June 2007: Lucy and Pete, returning from a remote Pacific island, find that the airline has damaged the identical antiques that each had purchased. An airline manager says that he is happy to compensate them but is ... clueless about the value of these strange objects. Simply asking the travelers for the price is hopeless, he figures, for they will inflate it.

Instead he devises a more complicated scheme. He asks each of them to write down ... any dollar integer between 2 and 100 without conferring together. If both write the same number, he will take that to be the true price, and he will pay each of them that amount. But if they write different numbers, he will assume that the lower one is the actual price and that the person writing the higher number is cheating. In that case, he will pay both of them the lower number along with a bonus and a penalty—the person who wrote the lower number will get $2 more as a reward for honesty and the one who wrote the higher number will get $2 less as a punishment. For instance, if Lucy writes 46 and Pete writes 100, Lucy will get $48 and Pete will get $44.

What numbers will Lucy and Pete write? What number would you write?

Scenarios of this kind ... are known as games by the people who study them (game theorists). I crafted this game, "Traveler’s Dilemma," in 1994 with several objectives in mind: to contest the narrow view of rational behavior and cognitive processes taken by economists and many political scientists, to challenge the libertarian presumptions of traditional economics and to highlight a logical paradox of rationality.

Traveler’s Dilemma (TD) achieves those goals because the game’s logic dictates that 2 is the best option, yet most people pick 100 or a number close to 100—both those who have not thought through the logic and those who fully understand that they are deviating markedly from the "rational" choice. Furthermore, players reap a greater reward by not adhering to reason in this way. Thus, there is something rational about choosing not to be rational when playing Traveler’s Dilemma. ... Nevertheless, open questions remain about how logic and reasoning can be applied to TD.

Common Sense and Nash To see why 2 is the logical choice, consider a plausible line of thought that Lucy might pursue: her first idea is that she should write the largest possible number, 100, which will earn her $100 if Pete is similarly greedy. ...

Soon, however, it strikes her that if she wrote 99 instead, she would make a little more money, because in that case she would get $101. But surely this insight will also occur to Pete, and if both wrote 99, Lucy would get $99. If Pete wrote 99, then she could do better by writing 98, in which case she would get $100. Yet the same logic would lead Pete to choose 98 as well. ... And so on. Continuing with this line of reasoning would take the travelers spiraling down to the smallest permissible number, namely, 2. It may seem highly implausible that Lucy would really go all the way down to 2... That does not matter...—this is where the logic leads us. ...

Virtually all models used by game theorists predict this outcome for TD—the two players earn $98 less than they would if they each naively chose 100 without thinking through the advantages of picking a smaller number. ...

Game theorists analyze games without all the trappings of the colorful narratives by studying each one’s so-called payoff matrix—a square grid containing all the relevant information about the potential choices and payoffs for each player [see box...]. Lucy’s choice corresponds to a row of the grid and Pete’s choice to a column; the two numbers in the selected square specify their rewards. ...

[T]he full version of TD has no dominant choice. If Pete chooses 2 or 3, Lucy does best by choosing 2. But if Pete chooses any number from 4 to 100, Lucy would be better off choosing a number larger than 2.

When studying a payoff matrix, game theorists rely most often on the Nash equilibrium... A Nash equilibrium is an outcome from which no player can do better by deviating unilaterally. ...

TD has only one Nash equilibrium—the outcome (2, 2), whereby Lucy and Pete both choose 2. ... Game theorists do have other equilibrium concepts—strict equilibrium, the rationalizable solution, perfect equilibrium, the strong equilibrium and more. Each of these concepts leads to the prediction (2, 2) for TD. And therein lies the trouble. Most of us, on introspection, feel that we would play a much larger number and would, on average, make much more than $2. Our intuition seems to contradict all of game theory.

Implications for Economics The game and our intuitive prediction of its outcome also contradict economists’ ideas. Early economics was firmly tethered to the libertarian presumption that individuals should be left to their own devices because their selfish choices will result in the economy running efficiently. The rise of game-theoretic methods has already done much to cut economics free from this assumption. Yet those methods have long been based on the axiom that people will make selfish rational choices that game theory can predict. TD undermines both the libertarian idea that unrestrained selfishness is good for the economy and the game-theoretic tenet that people will be selfish and rational.

In TD, the "efficient" outcome is for both travelers to choose 100 because that results in the maximum total earnings by the two players. Libertarian selfishness would cause people to move away from 100 to lower numbers with less efficiency in the hope of gaining more individually.

And if people do not play the Nash equilibrium strategy (2), economists’ assumptions about rational behavior should be revised. ... All these considerations lead to two questions: How do people actually play this game? And if most people choose a number much larger than 2, can we explain why game theory fails to predict that? On the former question, we now know a lot; on the latter, little.

How People Actually Behave Over the past decade researchers have conducted many experiments with TD, yielding several insights. A celebrated lab experiment using real money with economics students as the players was carried out... The experiment confirmed the intuitive expectation that the average player would not play the Nash equilibrium strategy...

[The research] also studied how the players’ behavior might alter as a result of playing TD repeatedly. Would they learn to play the Nash equilibrium, even if that was not their first instinct? Sure enough, when the reward was large the play converged, over time, down toward the Nash outcome... Intriguingly, however, for small rewards the play increased toward the opposite extreme...

Game theorists have made a number of attempts to explain why a lot of players do not choose the Nash equilibrium in TD experiments. Some analysts have argued that many people are unable to do the necessary deductive reasoning... This explanation must be true in some cases, but it does not account for all the results, such as those obtained ...[when] 51 members of the Game Theory Society, virtually all of whom are professional game theorists, played the original 2-to-100 version of TD. ... Presumably game theorists know how to reason deductively, but even they by and large did not follow the rational choice dictated by formal theory.

Superficially, their choices might seem simple to explain: most of the participants accurately judged that their peers would choose numbers mainly in the high 90s, and so choosing a similarly high number would earn the maximum average return. But why did everyone expect everyone else to choose a high number?

Perhaps altruism is hardwired into our psyches alongside selfishness, and our behavior results from a tussle between the two. We know that the airline manager will pay out the largest amount of money if we both choose 100. Many of us do not feel like "letting down" our fellow traveler to try to earn only an additional dollar, and so we choose 100 even though we fully understand that, rationally, 99 is a better choice for us as individuals.

To go further and explain more of the behaviors seen in experiments such as these, some economists have made strong and not too realistic assumptions and then churned out the observed behavior from complicated models. I do not believe that we learn much from this approach. As these models and assumptions become more convoluted to fit the data, they provide less and less insight.

Unsolved Problem The challenge that remains, however, is not explaining the real behavior of typical people presented with TD. Thanks in part to the experiments, it seems likely that altruism, socialization and faulty reasoning guide most individuals’ choices. Yet I do not expect that many would select 2 if those three factors were all eliminated from the picture. How can we explain it if indeed most people continue to choose large numbers, perhaps in the 90s, even when they have no dearth of deductive ability, and they suppress their normal altruism and social behavior to play ruthlessly to try to make as much money as possible? Unlike the bulk of modern game theory, which may involve a lot of mathematics but is straightforward once one knows the techniques, this question is a hard one that requires creative thinking.  ...

If I were to play this game, I would say to myself: "Forget game-theoretic logic. I will play a large number (perhaps 95), and I know my opponent will play something similar and both of us will ignore the rational argument that the next smaller number would be better than whatever number we choose." What is interesting is that this rejection of formal rationality and logic has a kind of meta-rationality attached to it. If both players follow this meta-rational course, both will do well. The idea of behavior generated by rationally rejecting rational behavior is a hard one to formalize. But in it lies the step that will have to be taken in the future to solve the paradoxes of rationality that plague game theory and are codified in Traveler’s Dilemma.

Game Theory vs. Ordinary Decision Theory: I Know that You Know that I Know ... I heard this tale in India. A hat seller, on waking from a nap under a tree, found that a group of monkeys had taken all his hats to the top of the tree. In exasperation he took off his own hat and flung it to the ground. The monkeys, known for their imitative urge, hurled down the hats, which the hat seller promptly collected. Half a century later his grandson, also a hat seller, set down his wares under the same tree for a nap. On waking, he was dismayed to discover that monkeys had taken all his hats to the treetop. Then he remembered his grandfather’s story, so he threw his own hat to the ground. But, mysteriously, none of the monkeys threw any hats, and only one monkey came down. It took the hat on the ground firmly in hand, walked up to the hat seller, gave him a slap and said, "You think only you have a grandfather?" This story illustrates an important distinction between ordinary decision theory and game theory. In the latter, what is rational for one player may depend on what is rational for the other player. For Lucy to get her decision right, she must put herself in Pete’s shoes and think about what he must be thinking. But he will be thinking about what she is thinking, leading to an infinite regression. Game theorists describe this situation by saying that "rationality is common knowledge among the players." In other words, Lucy and Pete are rational, they each know that the other is rational, they each know that the other knows, and so on. The assumption that rationality is common knowledge is so pervasive in game theory that it is rarely stated explicitly. Yet it can run us into problems. In some games that are played over time, such as repeated rounds of Prisoner’s Dilemma, players can make moves that are incompatible with this assumption. I believe that the assumption that rationality is common knowledge is the source of the conflict between logic and intuition and that, in the case of Traveler’s Dilemma, the intuition is right and awaiting validation by a better logic. The problem is akin to what happened in early set theory. At that time, mathematicians took for granted the existence of a universal set—a set that contained everything. The universal set seemed extremely natural and obvious, yet ultimately several paradoxes of set theory were traced to the assumption that it existed, which mathematicians now know is flawed. In my opinion, the common knowledge of rationality assumed by game theorists faces a similar demise. —K.B.

If game theory is your thing, please jump in here and set the record straight, but he seems to reject the possibility that people are rational maximizers after all, the behavior only looks irrational because the wrong model is being used to evaluate behavior in these experiments. That is, he says "some economists have made strong and not too realistic assumptions and then churned out the observed behavior from complicated models," and then dismisses this whole approach to explaining the experimental results.

Update: Andrew Gelman of the Statistical Modeling, Causal Inference, and Social Science blog emails that:

I just wanted to comment that selfishness and rationality are not the same thing. For example, an unselfish philianthropist might want to use rational principles in giving away his money. (Conversely, we all know lots of people who pursue selfish goals irrationally.) We distinguish selfishness from rationality in the particular case of voting in this paper: http://www.stat.columbia.edu/~gelman/research/published/rational_final6.pdf (coauthored with an economist!).