Orders of rationality?

One of the fascinating problems of Game Theory is the Traveler’s Dilemma, a variant of the classical Prisoner’s Dilemma situation -where rational actions by individuals to maximize individual welfare leads to sub-optimal outcomes for each of them.

The game design is as follows: (Reference: Traveler’s Dilemma by Kaushik Basu, Scientific American:
link: http://www.sciam.com/print_version.cfm?articleID=7750A576-E7F2-99DF-3824E0B1C2540D47)

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?

The solution to the problem is that each would choose $2 as the value of the antique-hence getting only $2 each! Also, this is a dominant solution (i.e. it is the strategy to be adopted, not just one of the strategies)

Note: For the detailed proof visit the link given above

Now if $2 sounds ludicrously low….you’re right! When the Traveler’s Dilemma game was administered on actual players most players chose $100 or a number close to that rather than $2 though, by the tenets of Game Theory they ought to have chosen to play $2. And by deviating from the rational choice, they were better off!

Other fascinating aspects include how strategies changed with the change in reward/penalty, and on repeated play.

Why would people deviate from expected rational behaviour. Three possible explanations have been offered: irrationality, rational altruism and meta-rationality (Usage:mine).

The first explanation, that players were simply ignorant or unable to deduce the ‘correct’ strategy might be true in some cases but was dismissed as a general explanation since even Game theorists who arrived at the dominant (2,2) solution, chose to play a number close to 100.

The second explanation is based on the premise that in the real world people do not want to ‘cheat’ their partners to make an extra buck-, that some degree of “altruism” and a sense of “fairness” is hardwired in all of us.
As an insight, it is extremely useful, but the implications are simple to factor in, it wil involve just tinkering with the utility function ( a function that maps actual winnings to the ‘utility’ that players assign to them). Right now, the utility function is a simplistic f(x)=x, i.e. if I earn $50 it is ‘worth’ $50 to me, now we will have to design utlity functions which are a function of not just our earnings but possibly that of the other’s earnings ( “altruism”) and/or of the difference between earnings ( “fairness”). Constructing such utility functions by analyzing actual moves by players could be a very interesting empirical research study. However, is this explanation correct? Even if we control for this effect, is it possible that players will still opt to play an ‘unexpected’ strategy.

Which brings me to my favourite explanation- the second one, wherein lies the most fascinating insight (its fascinating for me- I came up with it! ),that of ‘orders of rationality’

‘Orders of’ something, is the level at which it loops back at itself. If that seems complex, think of the Friends episode where Rachel and Phoebe discover that Monica and Chandler are involved, and want to ‘play’ a prank on them. Set 1( Monica and Chandler) have first order knowledge- they know they are involved and have no idea what Phoebe and Rachel ( Set 2) know, whereas Set 2 has second-order knowledge since they know, AND they know that Set 1 doesn’t know that they know. ( Note: This brings out the distinction between not knowing that the other set knows and knowing that the other set definitely doesn’t know; absence of evidence is not evidence of absence).
As the episode goes along, successively the orders of knowledge possessed by each Set increases ( Chandler and Monica find out that Phoebe and Rachel know( which Phoebe and Rachel don’t know!) and decide to play a prank themselves ( in the words of Chandler : “The messers become the messees!!”) and this continues).

In a Game Theory situation, knowing the rules of the game would be first order knowledge, knowing that the other player knows the rules of the game is second order knowledge, knowing that the other person has second order knowledge is third order knowledge ( or more generally- knowing that the other person has nth order knowledge is (n+1)th order knowledge). If this process continues ad infinitum(if you are not Joey, you can conceive of this!), the players are said to have “common knowledge”. Most games ( even the simple ones like the Prisoner’s Dilemma) have a premise of common knowledge. Common knowledge, is hence, an infinite stack of the orders of knowledge.

Can we then conceive even of rational behavior- a basic premise of game theory, and indeed of economics in general, as a stack of decisions rather than as a one-shot one.

To make it more clear, consider this: we human beings have the ability to think about our thinking ( We do it all the time “ I was not thinking straight when I said that”; “ When I think about incidents, I find I index them by date”; “ I think about sequences more easily than I do of individual entities” etc). This thinking about thinking (or meta-thinking to put it more elegantly) is very interesting- we can ‘step-out’ of a mode to look at our own thought processes at a higher level of abstraction. This leads us to the next insight, that, similarly, we could also be deciding sequentially. (I can meta-think to about 4 levels, after that my head starts to hurt!).

So does the solution to the Traveler’s Dilemma puzzle lie in the insight that we can rationally decide to act rationally/irrationally! (To put it more vividly, to accurately reflect the sequential process- can we META-rationally decide to act rationally/irrationally). This is consistent with what players said that they knew that rationally they ought to have played 2 but they chose to ignore rationality in making the choice ( citation: Kaushik Basu).

This has far more profound implications for the way we study Game Theory. Unlike the previous explanation, which also challenges a basic premise of Game Theory ( i.e. of selfish rationality) but can easily be made endogenous to current literature merely by changing the functional specification of utility, this would radically alter the way we approach the problem solving in the abstract, and would require an entire supporting theory ( as opposed to mere empirical work) which discusses the orders of rationality and how at meta-rational levels, choices of rationality and irrationality are made.