The traveler’s dilemma and subway seats

With the preparations for Europe going on at full steam, I find myself drawn toward psychology articles about traveling. Take, for example, this article in Scientific American. Kaushik Basu explains the “traveler’s dilemma,” a scenario in which identical items purchased by two travelers are both damaged in transit. The airline agent is worried that they’ll claim the price of the item is higher than what they actually paid, so he devises the following 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.

Your first hunch is probably just like mine: Lucy and Peter should both write \$100, and then they’d each get a hundred bucks. But there’s a problem with this logic: if Lucy “cheats,” and writes \$99, then she’ll get \$101, so it’d be better for her to write \$99. Then the logic collapses in a chain reaction:

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.

This, of course, is why economists aren’t psychologists. Any normal person would just write \$100. Psychologists have taken a different approach to the problem and conducted a study of how people really behave in the travelers’ dilemma. A team led by C. Monica Capra found that as the reward for guessing a lower number than the other traveler increased, the guessed value decreased. In their study, volunteers played with real money, with an allowed range of guesses from 80 to 200 cents.

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.

The remainder of the article is quite interesting. You might also want to check out Mind Hacks’ take on the article, which connects it to nuclear proliferation.

I saw a rather more benign connection, to this post, where a London commuter rates the preferred seating positions in a Tube train car, complete with a handy diagram.

Position 1 is supreme, because you have only one neighbour, and (usually) a wider seat, with windows behind and beside you, plus a door which might give some hope of a breeze.

Position 2 has only one neighbour, and the breeze factor, though usually someone’s arse squished up against the glass partition beside you.

Meg says the trains fill up in roughly the order she indicates, which strikes me as a bit odd. Why would someone take position 2 next to someone if 3 and 4 were available? Don’t people want to maximize the area around them? But perhaps commuters realize that eventually all the seats will be taken, so it’s preferable to sit next to someone in a temporarily inferior seat with the knowledge that eventually it will be the preferred position.

I suspect the reality is much more like the results of Capra’s study on the traveler’s dilemma. Even though passengers know that in a full car seat 2 is better than 3 or 4, they’d rather take the optimistic view that the car won’t fill up completely and they’ll have more personal space sitting there compared to sitting next to someone in seat 1. Even when it’s certain the train will eventually fill, several people will sit in seats 3 and 4 before sitting next to a stranger by occupying seat 2.

1. #1 Allen Varney
May 29, 2007

Seating strategies are culturally conditioned. In Indonesia, for example, it is more polite to sit next to a stranger on an empty bus or in an empty theater than it is to sit further away from him.

2. #2 Kevin
May 29, 2007

I seem to remember seeing something a while back along the same lines as the ‘seat on train’ map a couple years ago. However, this was a map/quiz thing about ‘which urinal should you choose’ given certain circumstances in a men’s bathroom.

So Allen, is it more polite stand next to the stranger in the empty men’s bathroom in Indonesia?

3. #3 Dave Munger
May 29, 2007

Allen:

Interesting. But then I imagine the subway car map in Indonesia would look entirely different.

Also, Meg’s map doesn’t take into account the politeness factor where pregnant women and the elderly are given seats. Though perhaps it does — by sitting away from the exits, you decrease the social pressure to give up your seat when a more deserving individual boards the train.

4. #4 bigTom
May 29, 2007

When given a problem such as travelers dilemma I tend to think of my patners response as not being a definite number but rather a probability distribution. Of course I won’t both with a formal solution. But if one takes the simplest distribution uniform random, the optimal choice wouldn’t be far from the actually observed values. It seems a little far fetched to think the average Joe is using this algorithm, but its an intriging result.

5. #5 Tony Jeremiah
May 29, 2007

In relation to Allen’s comment, the logic underlying the Nash equilibrium appears to have a cultural/contextual bias–choosing the lowest value would only make sense if one assumes a competitive/winner-take-all strategy, which seems typical of the thought pattern of a society based on a capitalist economic system.

It seems to me that the only logical reason why anyone would follow the Nash equilibrium, is by having an implicit understanding that an underlying rule of the game is such that the person writing the highest value would get nothing, while the person choosing the lowest value would get whatever value they wrote (i.e., winner-take-all strategy). For example, if Lucy writes 99 and Peter writes 100, Lucy gets \$99 while Peter gets \$0. A winner-take-all strategy predicts the Nash equilibrium, because assuming the underlying motive is that you want to be the winner, you are going to chose the lowest value since that is the value that ultimately gets the reward.

However, the rules underlying the traveller’s dilemma is more connected to a cooperative strategy (which is likely more closely connected to cultures having a collectivistic (rather than individualistic) mentality. Specifically, the rules of the TD game are such that both players get a reward. Their combined score (which really is essentially a measure of cooperative/collectivistic thinking), should really dictate their reasoning in this instance.

Logically, the maximum profit that can be achieved by both is \$200, if both write \$100. Any other decision results in a lower score. For example, if Lucy chooses \$99 and Peter chooses \$100, the maximum dollar they can get based on the actual rules of the game is \$197– Lucy will get \$101 [(99+2)] and Peter will get \$97 [99-2].

Assuming that most people playing these games aren’t actually thinking like mathematicians (and are more likely following implicit social rules), it is most likely that the underlying explanation for the results of these games are a combination of the extent to which an individual playing the game is engaged in competitive or cooperative thinking, and, the extent to which the rules of the game induce competitive/cooperative decision making.

It would be interesting to see what results are obtained from games when you have persons from individualistic vs. collectivistic cultures, or even strangers vs. friends.

6. #6 codeSuidae
May 29, 2007

I don’t know much about game theory, but I think perhaps the strategy analysis in this case depends heavily on what constitutes ‘winning’. As another poster suggests, since you are walking away from the game with cash you have incentive to stick with the high numbers. That way, even if your ‘opponent’ gets a couple more bucks, you still get the cash.

If you describe the game differently and call the dollars ‘points’, players would probably shift strategy, since points are worthless for anything other than determining the winner of the game. They would probably also quickly figure out that the game is even less entertaining than tic-tac-toe. In a tournament situation you might find a tit-for-tat strategy most successful, as with P.D., depending on the exact rules of play.

7. #7 Alan Kellogg
May 29, 2007

Dave,

Your example scenario has a huge problem. It assumes that two people will pay the same price for the same item. Not so. At one store one may pay 100 dollars for an item another person paid 80 dollars for at a different store. Thus assuming the person who claims the higher cost is lying makes no sense.

8. #8 Alex
May 30, 2007

The cultural effect on these sort of games was wonderfully proven by Reinhard Selten (and awarded with a Nobel prize).

The typical ultimo game:
Two players were given 100 monetary units, the first one needs to make an offer how to split the money between them – if the second agrees both will get it, otherwise they will lose it. The equilibrium is of course at 1 for the one who makes the offer 99 for the second person.

Selten played this game throughout the world. Thus assessing what is considered to be fair. In the Western cultural hemisphere more than half of the people offered a 50-50 split. And he reports also of a certain tribe in the southern pacific where player are giving away the money completely.

On a side note – Ockenfels one of Selten’s followers found the following:
In the Western Hemisphere, 41% of the player will turn a 80-20 split down if offered by a human player. But only 6% turn it down if offered by a computer.

9. #9 Jenny
May 30, 2007

As a frequent subway-sitter, I’m going to have to pull the ever-popular psychology cop out on this one: “It’s not that simple!”

When I take the Metro here in DC, the following factors are taken into consideration:

In rush hour:
1. I need to be in the car that stops closest to the elevator at the stop where I get off the train.
2. Depending on the stop, the Metro doors will open on one side of the train or the other, but not both. I try to be on the side of the car with the door that opens at the stop where I get off.

These apply at all times:
1. If it’s a long ride, I sit in the middle so other people can get on and off the train.
2. If it’s a short ride, I stand, or at least sit by the door.
3. If I’m riding with my friend who gets motion sickness on the Metro, we have to sit facing front (half the seats face backwards).
4. If I’m riding with family or friends who are playing tourist for the weekend, we sit on the left side of the yellow line so they can see the monuments between the Pentagon and L’Enfant Plaza.
5. Otherwise, if I’m by myself, I always sit as far away as possible from the sketchiest person in the car. Or the smelliest.

10. #10 Dave S.
May 30, 2007

My long experience in riding city busses suggests to me that the filling of the seats is more in line with the Aufbau Principle, as its described for the filling of molecular orbitals in atoms. Basically, the lowest energy states (seats near the front or rear) are filled by a single electron (rider) before a second electron (another rider) occupies that same orbital.

11. #11 roseindigo
June 1, 2007

. . . customs official scenario was exactly the same as Alan Kellog. The whole premise is ridiculous because people may actually have paid different prices in different countries or different cities or different stores, and no one needs to be lying except the customs agent who is making false assumptions.

12. #12 Abby
June 4, 2007

Missing from the analysis of the real-live reactions is that the two players may be influenced by the third entity in the scenario: the ticket agent.

Participants may be unconsciously judging that an authority figure can (and in real life might) arbitrarily change the rules if both choose the highest possible number. It’s suspicious and impolite to take all that’s offered. This may explain some of the high but less than maximum choices.

13. #13 rAgAv
June 5, 2007

I think that the key in the subway seating case is “flexibility”. Suppose, I board a train with only one person, I will choose to sit at the most favourite seat (let’s assume that its the one that gives the most personal space). As I see that the car is filling up at the subsequent stations, I’ll move to the seat that gives me more personal space. To execute this strategy perfectly, I’ll need more information such as how many people are likely to board on the next station. And ofcourse, any strategy that’ll require most information(relevant) is the one that is most accurate.

If you choose to be non-flexible( i.e. commited) in a scenario that has a varying configuration, chances are that you wouldn’t get to your objective.

Let’s take a relevant example – Suppose that you’re living in a country that you do not like and that you’re given the freedom to choose any country that you can live in. Top in the priority list for the deciding criteria is, say, how tech savvy the people in the country are. So, you choose to go to Germany(say). Then you realize that after a few years, people in Germany fall behind and people in India take over. Now that the configuration of the scenario has changed, you’ll have to move to India to get to your favourite place. If you’re commited to stay in Germany, it will be a blow to your objective i.e. to stay in the nation with the most tech savvy people. So, the commitment hardly helps in a varying scenario.

Another instance that can elucidate this better – suppose there’s a wide road which is divided into two lanes by painting a line on the road. Also suppose that each lane has ‘x’ no. of potholes that are distributed at different frequencies. If you’re commited to travel in one lane, then you’ll fail your objective – to take the minimum possible potholes possible. On the other hand, if you choose to switch between the lanes, then you’re flexible;thus handling the varying scenario successfully.

Pardon me if you don’t find this relevant. But, this is the fact that I rely on most of the times when I make decisions and I thought I’d better be informative.

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