A paradox of rationality
Kaushik Basu published the game in the American Economic Review under the subtitle "Paradoxes of Rationality in Game Theory". He later served as Chief Economist of the World Bank.
Each writes a whole number of dollars from 2 to 100 without talking to the other. Both are paid the lower of the two prices, but whoever wrote the lower one gets a $2 bonus and whoever wrote the higher one gets $2 less. Equal prices are paid as they are. Which number will you write? Lock in your answer, then check it with a game against bots, a simulation and the explanations.
If the other writes 100, you do better with 99: you get $101. But they reason the same way, and the "one dollar less" chain runs all the way down to $2. That is the game's only equilibrium. Yet real people don't play like that: in an experiment with game theorists only 3 of 45 wrote 2, and most wrote 95 to 100. Against such rivals 2 earns on average, while 98 earns .
The economist Kaushik Basu proposed this puzzle in 1994. The rules are simple; everything interesting lies in what you know about the other traveler.
Two travelers who don't know each other fly home from the same island with identical antique vases. The airline breaks both. The manager doesn't know the real price, so he asks each of them to write it down.
Each writes a whole number of dollars from 2 to 100. No talking, and nobody sees the other's number until both have written.
If the numbers are equal, each gets that number. If they differ, the manager takes the lower price as the true one: both get it, but whoever wrote the lower one gets a $2 bonus for honesty and whoever wrote the higher one gets $2 less as a penalty.
Which number should you write to get as much as possible?
"I'll write 100: the vase is worth it"
But if the other writes 100, then 99 gets you $101. If they notice this too and write 99, you are better off with 98. Strict logic doesn't stop until 2. Real people stop much earlier, and it pays them.
The second example comes from Basu's own article in Scientific American (2007).
A whole number from 2 to 100. It joins the site visitors' game: we will work out what it would earn against the numbers other people wrote.
One answer only: you can't change it after checking.
A breakdown of your number will appear here: what strict logic says and what it earns against real people.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
Suppose the other writes 100. Which is better for you to write, 100 or 99? Work out both payouts.
Now look through the other's eyes: they can take that step too. Where does the chain stop if both see it to the end and know the other sees it too?
You'll be playing a real person, not a perfect logician. How much do you lose by writing a large number if the other walks the chain to the end? And how much do you gain if they write a large one?
Each trip brings a new stranger and new broken vases. You write a number, the bot writes its own, and the manager pays by the rule. You don't know what is in the bot's head, but after each trip you see its number. At the end we compare your earnings with what a single fixed number would have made.
| # | You | Bot | Your $ | Its $ |
|---|
First let's walk the "one dollar less" chain to its end. Then we'll see what each number earns against a crowd of travelers, and how everything changes as the bonus grows.
The "people" shares come from the experiment by Becker, Carter and Naeve with game theorists: 45 entrants chose a single number for all games. We don't know exactly how the numbers spread within each group, so here they are uniform.
The "people" crowd is built from the shares in one real experiment. Who is in it and what they write opens after your answer in round 1.
"Noisy travelers" write the numbers that earn more against the rest of the crowd more often, but they make mistakes. A number that earns $5 less is chosen almost three times less often (2.7 times). For every bonus from $1 to $50 we find the crowd that responds to its own mistakes in exactly this way.
The first layer has no formulas, the second has a table and a strict proof, the third covers what Basu himself, the experiments and models of "imperfect" players say.
“In plain words” is hidden on all puzzles.
If the other writes 100 and you write 100, you get $100. If you write 99, yours is the lower price: you get $99 plus a $2 bonus, $101 in all. One step below the other beats standing next to them.
But the other thinks the same way. Expecting 99 from you, they write 98. You foresee that and write 97… If both see the chain to its end, it stops at 2, the lowest allowed. This card explains where the logic leads, but not why people don't go there. The next cards cover that.
Each step of the chain is sensible on its own, yet together they cost each player $98. This is Basu's point: strict logic that both know and both trust hurts them both here.
Compare 99 with 2. If the other writes 2, with 99 you get $0 instead of $2. If they write 3, you get $1 instead of $4. It never gets worse: the biggest loss is $3. But if the other writes 100, with 99 you get $101 instead of $4, $97 more.
For 2 to make sense you would have to be almost sure the other will also walk the chain to the end. A real person has no source for such certainty.
Against a crowd built from the experiment with game theorists, 98 earns per trip on average, 100 earns , and 2 only . No other number earns so little.
The best thing against people is to be a step or two below the crowd's typical number, not a hundred steps.
“Rigorous” is hidden on all puzzles.
Rows are your number, columns the other's number, each cell your payout. In each column the best response is underlined. It is always one below the other's number, and the best answer to 2 is 2 as well.
A pair of numbers is stable when each is the best response to the other. The underlined number meets itself only in column 2: only there is "answer with the same" best. There is exactly one stable pair, (2, 2).
Different numbers. Whoever wrote more gets the lower price minus $2. By writing 1 below the other they would get a price one dollar lower but plus $2, that is $3 more. And if the other wrote 2, writing 2 as well is enough: $2 instead of $0. So they can do better, and the pair is unstable.
Equal numbers above 2. Both get that number. Either one, by writing 1 less, would get $1 more: −1 on the price, +2 bonus. Unstable.
Both write 2. Whoever writes more gets 2 − 2 = $0 instead of $2. Lower is not allowed. Stable: this is the Nash equilibrium, and the only one.
100 is never better than 99: against 100 it gets $100 instead of $101, against 99 it gets $97 instead of $99, and against everything else the same. A sensible player doesn't write 100. But then 99 is never better than 98 among the numbers left. Cross out 99, then 98…
After 98 such rounds only 2 is left. Each round requires one more level of "I know that they know that I know…". This is what theorists call common knowledge of rationality.
The payout for your number x when the other writes y. R is the bonus and penalty ($2 in Basu's version).
One step below an equal number pays as soon as the bonus exceeds the step. So for R > 1 no pair y = y > 2 is stable, while for R = 1 every equal pair is.
Together the two get twice the lower price: the bonus only moves money from one to the other. So the average payout in a crowd is the average lower of two prices.
Writing the top H instead of the bottom L: the largest possible loss and the largest possible gain (for R ≥ 1).
“Deeper” is hidden on all puzzles.
Kaushik Basu described the game in the American Economic Review in 1994 as a paradox of rationality, and wrote about it in Scientific American in 2007. He doesn't think people who write a large number are simply wrong. In his view a person may reason: "I'll write something large, say 95, and I expect the other to write something similar".
Basu admits that formally describing such behaviour, where the rational answer is rationally rejected, is very hard. That is why the game is still an argument in the debate about what rationality is.
In 1999 Monica Capra, Jacob Goeree, Rosario Gomez and Charles Holt played the dilemma with students for real money: prices from 80 to 200 cents, a bonus from 5 to 80 cents. With a 5-cent bonus the average number was 180, with an 80-cent bonus 120. In repeated games a large bonus pushed players towards the equilibrium of 80, and a small one towards the opposite end, 200.
So the equilibrium isn't "wrong": it becomes plausible when the bonus is large compared with the range of prices. Round 5 has a simple rule for when exactly.
The "noisy travelers" in round 3 follow a model proposed in 1995 by Richard McKelvey and Thomas Palfrey: players pick better options more often, but not always, and expect others to behave the same way. When the bonus is small a mistake on the high side is cheap, so the crowd stays near the top. When the bonus is large every high-side mistake gets costly, and the crowd slides down to 2. The model needs no "kindness" or "trust": it is enough that people are imperfect and know this about each other.
Another view is levels of thinking: some take one "one dollar less" step, some two, hardly anyone a hundred. The same "and they'll think that I'll think…" chain appears in "Guess 2/3 of the average": there too the winner is not the deepest thinker but the one a step deeper than the others.
First the maths (when the equilibrium becomes plausible), then the logic (when shared rationality hurts everyone), then practice (a price war between two shops).
Prices from $2 to H, bonus and penalty R. We compare "write H" with "write 2".
Whoever writes the top instead of the bottom risks at most 2R − 1 dollars and may gain up to H − L − R. While the bonus is small the risk is tiny and the gain large, so writing high is sensible. Once 2R − 1 catches up with H − L − R, that is when R reaches about a third of the range, trust becomes expensive. In Basu's game that is R ≈ 33, and at that bonus most of our noisy travelers already write 2.
| Experiment | Prices | R | Threshold | What people wrote |
|---|
The threshold is a third of the range, (H − L + 1)/3. It is a rough rule from our calculation, not a law, but the experiments agree with it.
Step zero is writing 100. Each further step is "the other will take the previous one, so I go a dollar lower".
Whoever thinks exactly one step more than the other gets the most. Whoever thinks many steps more just gives money away: their number is needlessly low. And if both think without end, and each knows the other does too — common knowledge of rationality — both end up at 2.
Common knowledge doesn't always hurt. In "The blue-eyed islanders" one sentence everyone already knew makes the knowledge common and sets off a long chain of conclusions. In the traveler's dilemma the same kind of chain costs both players $98.
Two shops sell the same item at a unit cost of 20. A hundred buyers each buy one where it is cheaper; at equal prices they split in half.
At prices of 100 each shop earns 50 × 80 = 4,000. Whoever cuts to 99 takes every buyer and earns 100 × 79 = 7,900. The rival answers with 98… Check it with the sliders: the best response to any price above 22 is one less, and the war ends near cost, at 20–22. This model of competition was described in 1883 by the French mathematician Joseph Bertrand.
Real shops rarely go all the way to cost, for the same reasons people write high in the traveler's dilemma: products differ a little, buyers don't always go for the cheapest, and rivals meet every day and fear retaliation. The model fits best where the product is identical and prices are visible at once. Tenders won by the lowest price follow a similar logic, but they have rules of their own, so the conclusion shouldn't be carried over directly.
Kaushik Basu published the game in the American Economic Review under the subtitle "Paradoxes of Rationality in Game Theory". He later served as Chief Economist of the World Bank.
Tilman Becker, Michael Carter and Jörg Naeve invited members of the Game Theory Society to play for real money. Of the 45 who chose one number for all games, only three wrote 2, ten wrote 100, and 23 chose 95 to 99.
In that experiment one player was picked at random and paid $20 times their average payout in the games. The winner averaged $85 and received $1,700. The number 2 could never average more than $4.
In the experiment by Capra, Goeree, Gomez and Holt prices ran from 80 to 200 cents. With a 5-cent bonus students wrote 180 on average, with an 80-cent bonus 120. By the logic the number should have been 80 both times.
Ariel Rubinstein put a version with prices from 180 to 300 to about 2,500 people from seven countries online, with no money involved. 55% wrote the maximum, 300, and fewer than one in seven chose the equilibrium, 180.
100 never earns more than 99 and sometimes earns less. Theorists call such a strategy dominated. Even so, ten of the 45 game theorists in the experiment chose it.
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