Kaushik Basu has posed a seemingly irresolvable problem in Game Theory, but his exercise plays into many of the weaknesses of "rational" thought. It is in the very nature of Academia to base its efforts in language, and every expression in language robs its subject of context, the rich background against which any linguistic distillation is only a foreground.
A possible solution from within economics itself concerns opportunity costs. To take the scenario given in Scientific American as an example, we can ask "Why should Lucy and Pete strain themselves for hours to arrive at an optimal solution, all for the presumed advantage of gaining a mere two dollars more than the figure they name?". If you spend even one hour solving the problem, and the best that can be achieved is x + 2, then in essence, you are paying yourself two dollars an hour, a rather paltry sum for an hour's worth of agonizing over the problem. Why not jump straight at $100 and hope for the best? If the other player, say Pete, turns in any smaller figure, over which you have *no* control in any case, you lose only the two dollars you might have otherwise received had you guessed a figure *perfectly*. This two dollars is simply not worth the time and hassle involved, and the time could be put to better use making more money in vastly more productive endeavors.
A second entirely economic approach to resolving this "dilemma" also involves considering the real uses of Game Theory as used in its applications to Economics. If we believe that Game Theory is being applied because of the large number of players, and the inability to account for all their possible tactics, or that the situation allows for too many tactics by even a handful of players, then choosing the winning tactic must take these effects into account. The "winning" move becomes easy to identify, in that it wins much more value over a great variety of "opponent" strategies: go big, or go home!
This is the area under the curve for each of Lucy's possible choices of value. It's clear that she wins much more value much more often if she chooses the highest values.
These solutions by themselves are enough to solve the problem as given, but some of the larger implications of TD remain. It's in the nature of the TD scenario that it drags Game Theory and Economics into successively deeper problems. If one objects that the Prisoners' Dilemma and TD are simply the starting points of analyzing economic behavior, that doesn't change the fact that they miss the entire nature of economic behavior in the modern world.
A second problem of the use of Game Theory in Economics is the necessity of eliminating any possibility of communication or negotiation between the players under study, while all other parties are capable of any communication they like. (The Prisoners' Dilemma is the scenario that first comes to mind, but many other scenarios, including TD, also restrict communication in a lopsided way. The "airline manager" can communicate however and with whomever he wishes, and sets the rules of the game with no effective opposition in terms of how the rules might be different.)
This may have some use when there are a large number of players, in say, a stock or futures market, but that still leaves plenty of room for either having a sense of the fundamental values of the negotiable instruments or commodities, or simply not playing the market and putting those funds to better use. Brokers are much like psychics, always pointing out the huge gains, but never the huge losses. Considering they make money on transaction fees even in the worst of markets, that's no surprise -- they make money even when there are deep and broad losses in market value, but trumpeting a 15% return (while it lasts) is the driving force behind yet more commissions and fees. Either pay the exponentially growing information costs to keep your investments safe, resign yourself to gambling at a craps table, or get out of the market. The market in which these "guessing games" work best is called, surprisingly, a speculative market. You are forced to speculate what the other guy will do. As soon as he figures out that you have figured him out, he will start doing something else entirely! Market speculation, when far removed from concerns of the changing relative worth of goods, are of little productivity in the sense that most people use the term.
Third, Game Theory, as presented, ignores all but the most immediate consequences to the players, the nature of the players' relationships to each other and their environment (except as specially formulated for the problem), and describes "optimum" behavior as "rational", "objective", and "efficient". This leaves out almost any possibility of players being moral, visionary, and effective. They can play not just to "win" a battle, but to win a war. This process begins with players deciding who are their allies, and who are their opponents. The players assign economic values to other players, and can engage in a wide variety of transactions which support their grand strategies.
By assigning other economic players each a variety of values, a player may begin to formulate a set of strategies which preserve the features they value while eliminating the features they find threatening to their own interests. Each player exists in an economic environment or habitat which can be altered to benefit those players and their allies.
The Real World Intrudes
Given the litigious society we find ourselves in, it's unimaginable that it wouldn't occur to either Lucy or Pete that the airline manager (let's give him a name here, say, Jeff) is being entirely unreasonable, unprofessional, and simply, a real ass. This little scheme of his doesn't include any discussion of insurance, appraisal, or any of the more technical approaches to resolving this "dilemma". Scads of sources of information are being willfully ignored, all to present Lucy and Pete with a set of choices which seem borderline sado-masochistic.
Further, even if Lucy and Pete cannot prevail in their attempts to have a different method of solution altogether, Jeff has stipulated that Lucy and Pete may not communicate with each other, as he fears they may act in "collusion" to rob him blind! He should indeed be afraid, for at almost every turn he has demonstrated his ignorance and folly! He cannot establish a fair price for the antiquities in question, and yet somehow he believes that he can enforce the ground rules he has laid down. How can he assure himself that Lucy and Pete aren't at the Nefarious Antiquities Club, plotting to rob him repeatedly, when he obviously hasn't a clue how to use a telephone or an internet search engine? Lucy and Pete have ample ways of communicating with each other, and Jeff will have to work *awfully hard* to assure himself that they are not. Now Jeff is facing opportunity costs of his own, vastly higher than simply giving Lucy and Pete $100 each.
The costs of information, communication, and negotiation figure well beyond the scenario itself. Will Lucy or Pete think twice about flying on a different airline if they find Jeff's solution inequitable? Will they Tweet and Facebook Jeff into oblivion if they feel they have been treated poorly? Will Lucy or Pete turn to their professional colleagues for support? Do either have the authority or political power to make Jeff submit to their will, turning the tables on him? Jeff is a seemingly clueless idiot, as he simply attempts to dictate terms to people he doesn't know in the least, and with no thought of future consequences.