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.

11 Comments

1. 1. adamnash 10:01 PM 11/13/07

   I wrote a blog post explaining my theory to explain the results of these experiments:
   
   http://psychohistory.wordpress.com/2007/11/13/the-travelers-dilemma-irrational-choices-altriuism-or-implicit-collusion/
   
   Let me know what you think.

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2. 2. hacksoncode 11:03 PM 1/24/08

   One simple answer is that people know, either consciously or unconsciously, that they can't accurately predict others' behaviors, and therefore should assess anything from total to partial randomness to their partner's choice.
   
   If we assume our partner will select 100% randomly, the maximum expected value of the outcome is to play 100.
   
   Only if you assume the total and ruthless rationality of your opponent (and one identical to your own, at that) does 2,2 become the most rational choice.
   
   There is, however, increased downward pressure as the penalty becomes higher (which is consistent with experiment).
   
   Another explanation consistent with the experimental results is minimizing regrets. You might be angry with your partner if you picked 100 and they picked 2, but you'd be angry with yourself if the roles were reversed.
   
   Additionally, in the presence of low penalties, a certain "gambling" aspect creeps in. If you choose 2, the *best* you can do is 4.

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3. 3. aik212 09:45 PM 2/25/08

   I suspect that the absolute values involved are a major factor determining how participants play. Participants make a risk/reward calculation--by bidding a high number, the participant is risking $2 (he receives $0 if the opponent bids $2) in order to potentially gain almost $100. $2 is not particularly valuable, so there is virtually no downside risk compared to a fairly large upside potential ($100). Only someone with extreme competitive instinct or obsessed with proving his deductive abilities would not risk $2 to have a pretty good shot at gaining $100. However, say the participants had to bid $10,000 to $50,000, with the reward/punishment set at $10,000. I would imagine that more participants (assuming they are not so wealthy that $10,000 is an academic sum) would play the Nash Equilibrium. 2 reasons: A) The downside risk is much greater B) The incentive for the both parties to play non-cooperatively is much higher, increasing the chance that the other party will act selfishly.
   
   --
   Edited by aik212 at 02/25/2008 3:16 PM

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4. 4. imag94 04:21 PM 3/2/08

   If we assume the vase costs 40 initially, then bounded rationality stipulates both players to choose 40 as the price to be given to the authority. If they decides to co-operate as in 'co-operative game theory' then they will choose 100, each keeping in mind the self interest of the counterpart player.
   
   If the players are competitive then there can be un desireble outcomes of each trying to out do the other.

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5. 5. gnhillenbrand 08:21 PM 1/29/09

   Dear Kaushik Basu,
   
   I found your article on the travelers dilemma in the May 2007 article to be very stimulating. I think about it often, even now a year later. This is not a credentialed comment; my exposure to game theory is minimal. Please forebear naivety on my part.
   
   The assertion that the reasoning leading to 2 (shorted here) is logical is troubling: Lucy& first idea 100, which will earn her $100 & 99 would make $101. But this will occur to Pete& all the way down to 2& this is where logic leads us.
   
   However, a truly logical Lucy will not reason that 99 will earn her more than a bid of 100 because symmetric logic does not permit an outcome of (100, 99). Only (same, same) outcomes are possible when identical logic is used; A bid of 100 is logical because it is the best outcome for Lucy given the logical symmetry. she can recognize without cooperation. If you cant beat them, join them.
   
   If, in contrast, Lucy expects Pete to be illogical she still bids 100 unless she knows Pete had studied game theory and will to make a bid of 2. The study of game theory seems to be a vector of hidden cooperation.
   
   For multi-round bids, it is logical to respond with a bid of 100 to an opponent's bid of either 2 or 100, and to respond with a bid of 2 to any other opponents bid. The downward spiral is unilaterally blocked by avoiding bids other than 2 or 100. Outcomes other than (100,100), are rendered unattainable without any cooperation beyond an expectation that the opponent will always act in self-interest and will observe that stable outcomes other than (100,100) are unattainable. Note that an opponent who treats (100, 2) as a stable outcome by not re-bidding 99 or 100 is playing beat-the-opponent and not travelers dilemma. It is not in their logical self-interest to avoid a re-bid that would improves their own outcome.
   
   Respectfully,
   
   George Hillenbrand
   

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6. 6. matts2 10:49 AM 5/6/09

   I don't see the game theory logic at all. Let us suppose I pick 99. The best I can get is 101, the worst is 0. Let us suppose I pick 2. The best I can get is 4, the worst is 2. Picking 99 has a much better chance of giving me a good payoff. As long as my opponent picks anything higher than 6 I am better off than picking 2.

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7. 7. mgreenaw 03:19 PM 6/15/09

   This only makes any sense if you assume there is a competitive nature to the process. It's not stated in the original problem that the idea is to get more money than the other participant. I would have thought that there are really only three goals behind this, one of which precludes the others, namely:
   
   1. Make the maximum profit out of the airline,
   2. Recoup the cost of the artifact, or
   3. If the above are not possible because it was worth more than $100 then try to minimise your loss (automatically requires either a $99 or $100 bid)
   
   To me, this set of drivers will always be more powerful than an unstated aim of doing better out of the airline than the other participant. I don't know if I'm alone, but screwing myself over just to cause someone else a $4 greater loss isn't something I make a habit of.
   
   To me, only the item's real value (which is known only to the participants, but presumably the same value for both) and $100 are worth bidding. If you are bidding the real value, that's the worst you can reasonably expect to get, because if the other person bids lower than the real value then they are prepared to lose money on the transaction. You will only lose what they are prepared to lose, and if their bid tends towards $2 then they wouldn't have bothered making a fuss about it in the first place. If you've ever claimed lost baggage, you know your time is worth well over $2!
   
   All this does, for me, is to reinforce my belief that human decision making is never going to be based on a simple set of rules; it's based on a lifetime of experiences and values, coupled with our individual attitude to risk.

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8. 8. CyberGarp 11:59 AM 6/18/09

   I doubt that there is a Nash Equilibrium at all. If the logical number of 2 is derived by both players. There is little value in 2, and there is the possibility that the other player realizes this as well, and both of you should bid 100, and maximize both your sums. However, if this is realized, then you should bid 99 and get a little extra, and the cycle of logic down to 2 begins again. One could shorten the logic and only consider values which maximize both players take, i.e. 100 and 99, 98. If I bid 99 and the other player bids 100, I get 102 he gets 98. If either of us bids below 98, then we've shorted ourselves and it's not a good idea to fall below this threshold--and the other player could realize this as well. This logic still fails to achieve an equilibrium, and goes into a cycle, as one considers 100 -> 99 -> 98 -> 100 -> 99 -> 98 -> 100 -> .... This is the omega of lambda calculus, or the Turning incomplete problem in the guise of game theory.

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9. 9. CyberGarp in reply to CyberGarp 02:03 PM 6/18/09

   Further thoughts: of course, human behavior will cut the cycle and make a choice. Logic will not suffice when a cycle exists. This cycle is also at the heart of Gödel's theorem in that a system (in this case game theory) is either incomplete or inconsistent, this being an example of an incomplete problem. By Gödel's theorem, if game theory is consistent such a case has to exist.

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10. 10. Dimitris1988 10:00 PM 9/19/09

    If both are clever enough they will write "100$" xD

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11. 11. goddarddc 09:01 PM 5/17/10

    OK, this one has been bothering me ever since the article appeared. The answer is simple--the "experts" are analyzing the wrong game here! This is actually a 3-way game, between the 2 travelers and the manager. If there were no bonus/malus, the travelers are not competing against each other, just the manager, and should write 100. With 100% penalty (only the low bid gets anything, high bid gets nothing) it is the 2 travelers against each other and $2 might be the right answer. But in between it's a 3-way contest. Each traveler is competing partly against the manager, and partly against the other traveler; the mix depends on the size of the penalty. I'll bet a 3-player game theory would give the whole spectrum of answers from $100 to $2, depending on the penalty.

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