[tt] Scientific American: The Traveler's Dilemma

[Premise Checker]

This game theory stuff will be of interest to transhumanists, as it 
goes into the whole business of what rationality is. The good idea, of 
meta-rationality, comes at the end. If I were to play, I'd simply go 
for the maximum payoff for both of us (a payout of 100 for each of us 
in the original Traveler's Dilemma). My meta-reasoning is simply that 
the whole concept of rationality is not precise. True, one can draw up 
supposed rules of rationality, such that the Nash equilibrium (a 
payout of 2 for each of us) fits them. But these rules are highly 
dubious, for a whole bunch of reasons, which are so often enumerated 
by various pundits. Arrow's Impossibility Theorem and Sen's 
Impossibility of a Paretian Liberal ought to be enough to convince 
anyone that our intuitive notions of rationality conflict with one 
another.

I should meta-reason that my opponent will choose his own rules of 
rationality and that he would expect me to choose mine. Since neither 
of us know which of these putatively true rules the other will pick, 
it's the safest just to go for the maximum joint payoff (100 for each 
of us in the original game). I have to hope, to be sure, that my 
opponent will not be so conceited as to think he has come up with a 
set rules that will garner acceptance by everyone else, including me. 
No one to date has promulgated a set of rules of rationality that has, 
in fact, won consensus. I know this, and unless my opponent is wholly 
ignorant of this fact of failure to reach consensus, we'll jointly 
reason that the best thing to do is to be modest.

Surely, if my opponent wonders whether he has, at last, got *the* 
rules right, he will know that there is a whole history behind such 
attempts. His proposed set of *the* rules will have been arrived at, 
not by being the first to ever attempt to formulate such a set, but 
through examining earlier attempts and *hoping* to get a set that 
will, finally and at last, achieve the desired consensus. His deriving 
the *the* rules will have taken a long and arduous search in the 
literature!

Now both of us, if we have the least degree of modesty, will know 
this. Even if my opponent is not modest, being the greatest mind that 
ever tackled the problem, will know that I have a lesser mind--and 
will opt for the maximum payout (100 for each of us).

I suppose I could balloon my thoughts here and build a career around 
it. Ron Heiner, at George Mason, has authored some papers about a one 
shot prisoner's dilemma arguing something similar. I asked him where's 
the free lunch, meaning what assumptions he built into his model of a 
rational person that leads, deductively, to his conclusion. Premises 
in, conclusion out, in other words. I asked him repeatedly to send me 
drafts of his paper so I could look myself for the Premised (Checker 
or Unchecked) that let him to his conclusions. He never sent them.

The Traveler's  Dilemma
http://www.sciam.com/print_version.cfm?articleID=7750A576-E7F2-99DF-3824E0B1C2540D47
7.5.20
[Linked by Arts & Letters Daily.]

When playing this simple game, people consistently reject the
rational choice. In fact, by acting illogically, they end up reaping
a larger reward--an outcome that demands a new kind of formal
reasoning.

By Kaushik Basu

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
handicapped by being 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 the price of the antique as 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, in which one or more individuals have
choices to make and will be rewarded according to those choices, 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.

In the years since I devised the game, TD has taken on a life of its
own, with researchers extending it and reporting findings from
laboratory experiments. These studies have produced insights into
human decision making. 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. (If the antique actually cost her much
less than $100, she would now be happily thinking about the
foolishness of the airline manager's scheme.)

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. In that case, she could
deviate to 97 and earn $99. 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 in this fashion. That
does not matter (and is, in fact, the whole point)--this is where
the logic leads us.

Game theorists commonly use this style of analysis, called backward
induction. Backward induction predicts that each player will write 2
and that they will end up getting $2 each (a result that might
explain why the airline manager has done so well in his corporate
career). 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.

Traveler's Dilemma is related to the more popular Prisoner's
Dilemma, in which two suspects who have been arrested for a serious
crime are interrogated separately and each has the choice of
incriminating the other (in return for leniency by the authorities)
or maintaining silence (which will leave the police with inadequate
evidence for a case, if the other prisoner also stays silent). The
story sounds very different from our tale of two travelers with
damaged souvenirs, but the mathematics of the rewards for each
option in Prisoner's Dilemma is identical to that of a variant of TD
in which each player has the choice of only 2 or 3 instead of every
integer from 2 to 100.

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 on
opposite page]. 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.

Despite their names, Prisoner's Dilemma and the two-choice version
of Traveler's Dilemma present players with no real dilemma. Each
participant sees an unequivocal correct choice, to wit, 2 (or, in
the terms of the prisoner story line, incriminate the other person).
That choice is called the dominant choice because it is the best
thing to do no matter what the other player does. By choosing 2
instead of 3, Lucy will receive $4 instead of $3 if Pete chooses 3,
and she will receive $2 instead of nothing if Pete chooses 2.

In contrast, the 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, named after John F. Nash, Jr., of Princeton
University. (Russell Crowe portrayed Nash in the movie A Beautiful
Mind.) A Nash equilibrium is an outcome from which no player can do
better by deviating unilaterally. Consider the outcome (100, 100) in
TD (the first number is Lucy's choice, and the second is Pete's). If
Lucy alters her selection to 99, the outcome will be (99, 100), and
she will earn $101. Because Lucy is better off by this change, the
outcome (100, 100) is not a Nash equilibrium.

TD has only one Nash equilibrium--the outcome (2, 2), whereby Lucy
and Pete both choose 2. The pervasive use of the Nash equilibrium is
the main reason why so many formal analyses predict this outcome for
TD.

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.
Of course, TD is not the only game to challenge the belief that
people always make selfish rational choices [see "The Economics of
Fair Play, by Karl Sigmund, Ernst Fehr and Martin A. Nowak;
Scientific American, January 2002]. But it makes the more puzzling
point that even if players have no concern other than their own
profit, it is not rational for them to play the way formal analysis
predicts.

TD has other implications for our understanding of real-world
situations. The game sheds light on how the arms race acts as a
gradual process, taking us in small steps to ever worsening
outcomes. Theorists have also tried to extend TD to understand how
two competing firms may undercut each other's price to their own
detriment (though in this case to the advantage of the consumers who
buy goods from them).

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 at the University of Virginia by C. Monica Capra, Jacob K.
Goeree, Rosario Gomez and Charles A. Holt. The students were paid $6
for participating and kept whatever additional money they earned in
the game. To keep the budget manageable, the choices were valued in
cents instead of dollars. The range of choices was made 80 to 200,
and the value of the penalty and reward was varied for different
runs of the game, going as low as 5 cents and as high as 80 cents.
The experimenters wanted to see if varying the magnitude of the
penalty and reward would make a difference in how the game was
played. Altering the size of the reward and penalty does not change
any of the formal analysis: backward induction always leads to the
outcome (80, 80), which is the Nash equilibrium in every case.

The experiment confirmed the intuitive expectation that the average
player would not play the Nash equilibrium strategy of 80. With a
reward of 5 cents, the players' average choice was 180, falling to
120 when the reward rose to 80 cents.

Capra and her colleagues 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 of 80. Intriguingly,
however, for small rewards the play increased toward the opposite
extreme, 200.

The fact that people mostly do not play the Nash equilibrium
received further confirmation from a Web-based experiment with no
actual payments that was carried out by Ariel Rubinstein of Tel Aviv
University and New York University from 2002 to 2004. The game asked
players, who were going to attend one of Rubinstein's lectures on
game theory and Nash, to choose an integer between 180 and 300,
which they were to think of as dollar amounts. The reward/penalty
was set at $5.

Around 2,500 people from seven countries responded, giving a
cross-sectional view and sample size infeasible in a laboratory.
Fewer than one in seven players chose the scenario's Nash
equilibrium, 180. Most (55 percent) chose the maximum number, 300
[see box on next page]. Surprisingly, the data were very similar for
different subgroups, such as people from different countries.

The thought processes that produce this pattern of choices remain
mysterious, however. In particular, the most popular response (300)
is the only strategy in the game that is "dominated--which means
there is another strategy (299) that never does worse and sometimes
does better.

Rubinstein divided the possible choices into four sets of numbers
and hypothesized that a different cognitive process lies behind each
one: 300 is a spontaneous emotional response. Picking a number
between 295 and 299 involves strategic reasoning (some amount of
backward induction, for instance). Anything from 181 to 294 is
pretty much a random choice. And finally, standard game theory
accounts for the choice of 180, but players might have worked that
out for themselves or may have had prior knowledge about the game.

A test of Rubinstein's conjecture for the first three groups would
be to see how long each player took to make a decision. Indeed,
those who chose 295 to 299 took the longest time on average (96
seconds), whereas both 181 to 294 and 300 took about 70 seconds--a
pattern that is consistent with his hypothesis that people who chose
295 to 299 thought more than those who made other choices.

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 and therefore make irrational choices
unwittingly. This explanation must be true in some cases, but it
does not account for all the results, such as those obtained in 2002
by Tilman Becker, Michael Carter and Jörg Naeve, all then at the
University of Hohenheim in Germany. In their experiment, 51 members
of the Game Theory Society, virtually all of whom are professional
game theorists, played the original 2-to-100 version of TD. They
played against each of their 50 opponents by selecting a strategy
and sending it to the researchers. The strategy could be a single
number to use in every game or a selection of numbers and how often
to use each of them. The game had a real-money reward system: the
experimenters would select one player at random to win $20
multiplied by that player's average payoff in the game. As it turned
out, the winner, who had an average payoff of $85, earned $1,700.

Of the 51 players, 45 chose a single number to use in every game
(the other six specified more than one number). Among those 45, only
three chose the Nash equilibrium (2), 10 chose the dominated
strategy (100) and 23 chose numbers ranging from 95 to 99.
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

Tthe 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.

Suppose you and I are two of these smart, ruthless players. What
might go through our minds? I expect you to play a large
number--say, one in the range from 90 to 99. Then I should not play
99, because whichever of those numbers you play, my choosing 98
would be as good or better for me. But if you are working from the
same knowledge of ruthless human behavior as I am and following the
same logic, you will also scratch 99 as a choice--and by the kind of
reasoning that would have made Lucy and Pete choose 2, we quickly
eliminate every number from 90 to 99. So it is not possible to make
the set of "large numbers that ruthless people might logically
choose a well-defined one, and we have entered the philosophically
hard terrain of trying to apply reason to inherently ill-defined
premises.

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.

MORE TO EXPLORE

On the Nonexistence of a Rationality Definition for Extensive Games.
Kaushik Basu in International Journal of Game Theory, Vol. 19, pages
33-44; 1990.

The Traveler's Dilemma: Paradoxes of Rationality in Game Theory.
Kaushik Basu in American Economic Review, Vol. 84, No. 2, pages
391-395; May 1994.

Anomalous Behavior in a Traveler's Dilemma? C. Monica Capra et al.
in American Economic Review, Vol. 89, No. 3, pages 678-690; June
1999.

The Logic of Backwards Inductions. G. Priest in Economics and
Philosophy, Vol. 16, No. 2, pages 267-285; 2000.

Experts Playing the Traveler's Dilemma. Tilman Becker et al. Working
Paper 252, Institute for Economics, Hohenheim University, 2005.

Instinctive and Cognitive Reasoning. Ariel Rubinstein. Available at
arielrubinstein.tau.ac.il/papers/Response.pdf