|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,
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
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
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
matrix--a square grid containing all the relevant information about the
potential choices and payoffs for each player . 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.
Game theory predicts that the Nash equilibrium will occur when Traveler's
Dilemma is played rationally
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
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 . 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
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
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
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.
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
KAUSHIK BASU is professor of economics, Carl Marks Professor of International
Studies and director of the Center for Analytic Economics at Cornell University.
He has written extensively in academic journals on development economics,
welfare economics, game theory and industrial organization. He also writes
for the popular media, including a monthly column in BBC News online. He
is a fellow of the Econometric Society.
23 May 2007 Scientific American
September 13, 2007
04:00:05 pm, Categories: Mathematics, 444 words
Traveler's Dilemma and a new kind of formal reasoning in game theory What
is rational and what is irrational?
That question lay at the core of a mountain of letters we received about
our June article "The Traveler's Dilemma" by Kaushik Basu. We ran a small
selection of the letters here in the blog along with responses from Basu.
A letter from Adam Brandenburger of New York University appears in the October
print edition of SciAm. Here is a somewhat longer version of his letter:
In "The Traveler's Dilemma" (June 2007), Kaushik Basu describes an intriguing
game he introduced into the game theory literature some years ago. (The
Traveler's Dilemma bears some similarity to the famous Prisoner's Dilemma.)
In the game, two players must each choose a number between 2 and 100. As
Basu explains, the game is constructed so that there is a unique Nash
equilibrium-at which each player chooses the number 2 (and each then receives
$2). Yet, when the game is actually played, much higher choices are often
seen. Often, both players choose numbers close to 100, in which case they
both receive much higher "payoffs" than in the Nash equilibrium.
Basu is right that experience with games such as the Traveler's Dilemma poses
a serious challenge to the use of the Nash-equilibrium concept in game theory.
However, game theory is not synonymous with Nash equilibrium. There are now
theorems in formal game theory giving conditions under which Nash equilibrium
emerges (see ). The conditions are very stringent: In particular, the
assumption that the players in a game are rational is far from sufficient
to yield Nash equilibrium. This is good news. There is no conflict between
game theory and what we observe in games like the Traveler's Dilemma-only
between Nash equilibrium and what we observe.
Basu asks that a "new kind of formal reasoning" be developed to deliver more
satisfactory analyses of many games. In fact, over the past two decades,
a subfield of game theory-called interactive epistemology-has emerged on
precisely this topic. It is now possible to analyze mathematically what it
means for the players in a game to be rational or irrational, to think that
other players are rational or irrational, and the like. (See  for a recent
survey.) This is different from the classical Nash-equilibrium analysis of
games, and often yields the more intuitive answers Basu wants.
J.P. Valles Professor
Stern School of Business
New York University
 "Epistemic Conditions for Nash Equilibrium," by Robert Aumann and Adam
Brandenburger, Econometrica, Vol. 63, pages 1161-1180 (1995). [Also available:
Unpublished 1991 version (pdf).]
 "The Power of Paradox: Some Recent Developments in Interactive Epistemology,"
(pdf) by Adam Brandenburger, International Journal of Game Theory, Vol. 35,
pages 465-492 (2007).
Posted by Graham P Collins · 2 comments
1 - 2 of 2
Joe Hilbig [Member] September 25, 2007 @ 5:56 pm writes:
There is required no new logic to resolve the apparent contradiction, and
it it not necessary to accept that participants are acting illogically. The
fact that participants end up with more money than a "logical" player would
have suggests that they have got something right. I know that if my partner
and I both choose 100, we will each get 98, and that if either of us chooses
less than 98, we will both get less. I know that my partner also knows that,
(being a logical person as well) and so, logically, we both choose 100. Joe
Flag as inappropriate
Ray Ladbury [Member] September 26, 2007 @ 9:05 pm writes:
Joe, A more realistic situation might be a labor negotiation between an
automobile manufacturer and unions in two different countries. Both Unions
are to come up with bids. If the two bits are equivalent, the countries split
the work. If one bid is better, that country gets twice the work of the other.
The result is most likely to be a race to the bottom--even though cooperation
would yield a better result for the