I don't have much more substantive to respond with respect to the mathematical logic involved. However, I object to Basu's repeated assertion that any selection other than 2 for a value chosen as "illogical", and stating as fact that people who choose values around 100 are "illogical" or "only thought it through partially" betrays what I see to be the root problem on relying solely on logic/mathematics as the only source of "truth".
When stating proofs you must be careful to discuss foundational assumptions. Basu gives us only what is stated in the game as foundational, and seems to deflect attempts to add in additional "rules". Nonetheless, there are axiomatic truths that are left unmentioned when talking about "money" that he seems to violate.
First, money doesn't grow on trees. When the airline manager is offering compensation, they do so because the airline wants your continued business, and expects that the amount paid will not only ameliorate your ill will, but also encourage you to fly the airline again. Paying you less than what it is actually worth will not make you feel better, so choosing the minimum amount cannot be satifying.
Second, compensation for injury is typically insured. The manager is not paying out of a bottomless pocket- he has either saved for such an occasion in an interest-bearing account, or has purchased some insurance against such a claim. In order to recoup the price of this insurance, I would suggest that he is communicating a means to allow the customer to share in the price. To do this, I will introduce the first player in my scenario.
Gallant is a basically ethical person. Gallant knows that they have paid $X for the antique. They wish to be compensated for this, and naively, will choose $X. However, they don't know what the other person has paid, and if that person has paid $X-1, they will receive $X-3 as compensation, a less than ideal amount. However, the other person may reason this too, and therefore they both choose $X-2, which is the minimum amount that would fully compensate them if the other person "guesses higher". Any lower, and they could never be fully compensated. In fact, by choosing $2 as the penalty/reward, the manager may be communicating the cost of the insurance- i.e. if you both paid the same and both act ethically, each of you will lose $2, which will compensate the manager for a $4 price of insurance, bought retroactively, a better deal than you will get with any insurance agency.
But, you might add, if both players agree to a higher price, they both could do better. This is the second player in my scenario.
Goofus is unethical, idiotic, and basically just wants to get the most out of everything. Furthermore, Goofus has a short attention span. Goofus chooses $100.
Goofus is the reason why well-run insurance companies sometimes falter- people begin to game the system, and try to get something for nothing. However, if we pit Goofus against Gallant, we see that the reward goes to Gallant, and Goofus winds up getting compensated minus the full cost of insurance for the manager: $X-4. But there is another character I want to introduce, called Mephisto.
Mephisto is dedicated and wants to win. But, Mephisto also wants to be compensated for his loss. Mephisto knows he might be playing against Goofus, in which $99 would be the correct choice. Mephisto also knows he might be plaing against Gallant, in which case $X-3 would balance winning with some compensation. Mephisto also knows he might be playing against another Mephisto, in which case he might even decide to go lower to win more than the other, but at some point, you're only hurting both of you by playing that game. Mephisto will in my opinion at least choose $X-3, because experimentally/experientially he knows that not everyone is that sophisticated, but also, there are some basically honest people too, and he may satisfice on that answer as resonable. He may also choose to gamble higher, trusting that there is a chance that the other sucker is Goofus.
Lastly, I want to come to the case of Basu himself, the Poindexter Game Theorist.
Poindexter decides that the Nash equilibrium, by his analysis, is $2, and chooses that. What does Poindexter have now? He has $4, in all likelyhood, because few people are Poindexters. He also has a net loss of $X-4, while the other player has a net loss of $X. Neither is properly compensated for their loss, and neither is happy or likely to fly the airline again. Poindexter also now has a subdural hematoma, because when he gets outside Mephisto and Goofus hold him down while Gallant beats the stuffing out of him. The other player is undercompensated and unhappy, and never flies that airline again. The insurance company makes a small temporary gain for not having to pay out on a claim, but eventually both it and the airline go bankrupt due to lack of business.
Lesson? It may be unethical to be Goofus, but it is very unwise to be a Poindexter.
note: I've had the stuffing kicked out of me for being a Poindexter, and it's no fun, and I don't recommend beating up Poindexters as an ethical strategy. But it did teach me one thing: don't be a Poindexter.
Logical analysis of games like Prisoner's Dilemma, and the like, miss the whole point of playing lose-lose type games; the ability to learn through experimenting with other people. Playing ethically, like Gallant, is a strategy to see if the other person is ethical as well and trustworthy. Playing rationally, like Mephisto, is a way to either go up or down in ante to gauge the ethics of the other person. In fact, Mephisto may ape Goofus, because there is no faith in the goodwill of others and you should grab it while you can. With the Prisoner's Dilemma, we already assume that both prisoners have behaved unethically to get where they are, so whether they keep the code or squeal like pigs is really going to determine how well choosing a life of crime will work for them in the long run.
All in all, it's hard to ferret out what the other player might be thinking, because your values of $X might not agree, which is a third basic ignorance on the part of Basu of the axioms of money: there is nothing "identical" when you're talking about money. One player may have coveted it more, and paid a higher sum for an identical object. One player may be shrewd, and paid less. Money is not directly exchangeable to mathematics (it's not even directly exchangeable for other money, sometimes!). And, for that matter neither is logic and reason directly exchangeable for choice. They're approximations, useful to be sure, but not truth.