Nonsensical Rationality...

The June 20007 Scientific American poses a challenge – a puzzle to introduce problems with the modern understanding of economics.  The puzzle is in the form of a game.  Two people must each pick a value from $2 to $100.  Both will win the minimum of the two dollar amounts, plus (if the values are different) a two dollar reward for the one who picked the lower of the two and a two dollar penalty for the one who picked the higher.  The creator of this “Traveler’s Dilemma” game outlines its noble goals:

... 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.

I relish paradox.  A paradox signals a failure to fully understand something, and resolving the paradox means that we either more fully understand the implications of a theory, or that the theory is wrong and must be modified to conform to reality.  Either way paradox is a road to knowledge.  So let’s play.  What number would you pick?

My first instinct is to go with the maximum: $100.  If I pick the maximum then the only way I can make significantly less is if the other person picks a smaller number.  It seems that I’ve given myself the highest headroom and not artificially limited the return I might get.  But there’s a catch to this – the reward or punishment schedule.  If I pick $100 and my opposite picks $99, then they get $101 and I only get $97.  If I really want to maximize my return perhaps I should pick a lower number in the hope of being rewarded for it.

As I see it there are two ways to resolve this dilemma.  One is to assume that my opposite (I hesitate to call them my “opponent” since we both stand to gain from the transaction) is unpredictable.  Essentially I’m playing against a random number generator.  In that case I just have to sum all the possible outcomes from each of my possible choices against all of the other player’s choices and weigh them equally to compute an average.  A few minutes in Excel and I have my answer.  The best outcome against a random player comes from either $96 or $97, both of which give an average yield of $49.08.  $95 and $98 are close with an average of $49.07, and my initial guess of $100 ties with $93 at $49.02.  So I could guess $97 and make on average about a nickel more.  Awesome.

The second approach gets more to the heart of the matter.  Let’s assume that I’m rational about my choice and that the other player is equally rational.  That would comport with the “rational agent” theory so popular in libertarian thinking which argues that people always pick the choice that maximizes their own return.  If I assume that my opposite is rational, as opposed to random, then I can assume that they will always pick the same value that I do.  After all they will use exactly the same kind of logic that I use to arrive at their choice.  Since we will naturally come to the same conclusion, we will both pick $100 since that is the maximum yield given that both players pick the same number.

So the right answer is either $96, $97 or $100.  Am I right?  Apparently not.

Traveler’s Dilemma 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.

I’m sorry; did you say two dollars?  There’s a guy there with a hundred bucks in his hand ready to give out and all I have to do is play the stupid game, and yet somehow the “logic” of the game dictates that I should ask for $2?  So the other guy picks $100 or close to it as most people do, and I get $4 and he gets zero.  Both of us could have had money for a nice date but instead I get a lousy latte and he gets nothing.  No wonder Mr. Spock seems so unhappy most of the time: logic is a harsh mistress.

But how does this so-called logic work?  Kaushik Basu explains:

To see why 2 is the logical choice, consider a plausible line of thought that Lucy [one of the hypothetical players] might persue: her first idea is that she should write the largest possible number, 100, which will earn her $100 if Pete is similarly greedy.  […]  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 the 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.

Implausible indeed.  It reminds me the scene in Princess Bride where the Sicilian (Wallace Shawn) goes through a litany of “I know this, but you know that I know it” to affirm and refute every possible move in his battle of wits against Wesley.  Or an equally applicable scene from Futurama where two robots sit down for a game of chess.  The robot playing white declares “mate in 106 moves.”  The other robot says “No, not again!”  This logic appears about as sound.  After all I have provided two quite plausible explanations of the common-sense solutions – why does the supposedly formal solution seem so absurd?

It turns out that $2 is the so-called “Nash equilibrium” for this game.  In formal terms, that means the solution that, given a choice, neither party would change to advantage themselves.  All other choices are unstable.  Both picking $100 is unstable because given information about the other player’s choice, a player will change their vote from $100 to $99 in order to get the extra dollar.  Likewise all other combinations.  But why is the Nash equilibrium considered to be the “logical” choice for this particular game?  It seems to me that a Nash solution would require a different game.

In particular the game we are presented with is a single choice with no input on what the other person is doing.  Nash assumes we’re looking for a stable solution under the option to change.  But even adding conditions for that is not enough.  Suppose we say you can pick a value and then change it after you hear what the other player said.  I’d still pick $100 and then modify it based on new information rather than trying to pick the stable solution right away.  Even if it was my opponent who got to change their value, they'd only get an extra dollar.  Who cares!

Suppose, then, that we modify the game so that it has to settle on the stable solution before we allow it to end.  That is, each player gets to change their choice and only when both are happy we end the game.  In the real world this would be a negotiation, where both parties can change their offers until both are satisfied or until the solution fails to stabilize.  Would we end up with $2 in that case?  No way.  If we start at $100/$100, party A would change their bid to $99.  Then party B would change their bid to $98.  Now party A would change their bid to $97.  Party B has a choice at this point.  They can go down a notch again to $96 so they get $98, but that’s less than what they got starting from $100.  Why would they do that?  Since they are rational they can look ahead and see how the game is going to go for the next several moves.  Instead of seeking immediate advantage they would pick $100 again.  Their return stays the same ($98), but on the next turn party A would be certain to pick $100 again so that their return would go from $99 to $100.  We’re now in a loop.  Any rational agent will see that to end the loop and get something rather than nothing they will have to say they are satisfied because no one is going to get the extra dollar.

There’s only one game for which $2 is the right answer.  That’s the game that says that the only wrong answer is one where the other player “wins”.  If I want to prevent the other player from getting the bonus at my expense, then it’s quite clear that picking the lowest possible value is the only way to do it.  But in real life that kind of compulsion is pathological, not rational.

I’ve often wondered why the “rational agent” model of economics seemed so bogus, and the libertarian ideology upon which it's based.  This article has made it very clear.  The paradox is resolved.  This interpretation of “rational” is completely bankrupt.

- jack*

UPDATE: more here.

Comments

[omgparticle]

Enjoyed your comments and your blog.

Looking at people in life as just an opponent is a bit silly.
what the "win" means or is at stake, and sometimes a value determination between the parties is where it seems to get boiled down.
If the stakes are life and death , most likely you must win at all costs provided you value your life more than theirs.
If the stakes are something else then it's best if both win to the maximum given the difference of what value each brings to the table.

[Ma'at's Feather]

Good analysis!

Whether or not real-live folks want to 'win' vs. maximize returns could be turned into some interesting experiments.

Also, whether or not they can compute the return on the fly, and whether behavior changes depending on how the return is explained!