This is one of those frustratingly silly thought experiments that drives me around the bend.
The idea that its rational to discount your bid only applies as long as you can possibly make more money using the discounting strategy. Once you have discounted yourself below that floor, it’s perfectly rational to stop and say “He may get more money than me, but I will get at least X and X is > 2”
I consider $100, and Paul considers any number at all. If he chooses $100, I get $100. If he chooses $99, I get $98.
So there’s my floor – I should not rationally choose a strategy that can possibly net me less than $98 dollars. So then I consider $99 – if Paul chooses $100, I’ll get $101, but if he chooses $98, I’ll get $97.
So that’s it. I might in theory do better if I dropped down a notch, but I might also do worse, since I can’t predict Paul’s behavior (assuming I can predict Paul’s behavior is the height of hubris, and the essence of why the Traveler’s Dilemma is so silly). Since I don’t have to choose any strategy that nets me less than $98, why bother? I bid $100, and whether I get $100 or $98, I have chosen a path that is maximal in the average case ($99). Choosing $99 also results in $99 in the average case, but the spread is higher, and why would I increase my volatility in such an uncertain environment?
The fact that the author indicts game theory and free market economics on the basis of the counter-intuitive “rationality” of the Traveler’s Dilemma is (to me) the most telling part of the article.