Epistemic status: I think I'm over complicating the matter.
In the Traveller’s Dilemma (call it TD below for short), theoretically the only Nash Equilibrium is to have both players (I'll call them Alice and Bob) reasoning to give the lowest bid (Thanks Stuart_Armstrong for letting me notice this): starting with bidding 100 dollars, Alice would realise she can gain more by claiming 99, so Bob’s best choice is to claim 98, but Alice would also know this and claim 97 and so on…until the race to the bottom finishes at the lowest claim of $2.
Empirically this doesn’t seem to happen, both players would likely be cooperative and bid high. Which for me seems rather bizarre for a single round simultaneous game.
Notice that TD is very similar to the dollar auction/war of attrition: In the dollar auction, both players pay for their bids, with the higher bidding player receiving the auctioned dollar.
We can slightly modify the two-player dollar auction to make it even more similar to TD: the player with the lower bid would also have to pay for the same bid as the winner.
This modified dollar auction has the same payoff rules as a TD with the range of claims being (-infinity,0]
The only difference is the bidding process: in TD, both players choose their claims simultaneously, while in DA the two players engage in multiple rounds of competing bids.
Given the difference between TD and I believe there is something wrong with the assumed reasoning that we use to derive the Nash Equilibrium of TD. If Alice believes that Bob is fully rational, she would not believe that Bob would follow this line of reasoning in his own head and give a claim of $2.
Imagine that Alice and Bob are allowed to communicate before choosing their claims, but they only discussed their claims in approximate terms (eg: would it be a high claim close to $100? A low claim close to $2? Somewhere in between?)
Would a rational Alice want to make Bob convinced that she would give a low claim, or would she want to convince Bob that she would give a high claim?
If Alice convinced Bob that she would give a low claim, Bob’s best response is to give a low claim. Knowing this, Alice would give a low claim and both Alice and Bob will receive a low payoff.
While if Alice convinced Bob that she would give a high claim, Bob’s best response is to give a high claim. Knowing Bob will give a high claim, Alice would give a high claim and both Alice and Bob will receive a high payoff.
It appears that Alice has an incentive to convince Bob that she would give a high claim close to $100, instead of a low claim close to $2.
Also, even if Alice’s promise is not binding, her best response is to keep to it: she runs a greater risk of losing $2 if she claim high after promising a low claim, and will likely lose a lot if she claim low after promising a high claim. As a result, Bob would still trust Alice’s promises even when he knows that Alice is fully capable of lying.
If we imagine a round of “communication in approximate terms” for Alice and Bob, the line of reasoning for an equilibrium with both players bidding high becomes visible. A rational player would prefer to be believed that they will be cooperative in this game, and in this particular case they have the incentive to keep their promises of cooperation. Even if we disrupt the communication round and make each player’s promise invisible to the other (thereby we create a round with imperfect information, and the resulting game is functionally identical to the original TD), each player can still make a guess on what the other player would’ve communicated, and how they would plan their subsequent bid based on the unspoken communication.
I haven’t done anything to evaluate this process vigorously, as the “low”, “middle”, “high” bids are rather vague terms that would not allow me to draw clear boundaries for them. However, it appears to me that the strategy for both players on the communication round would be a mixed strategy that skews towards the cooperative (high bid) end.
As a result, the Bob who claims $2 in Alice’s imagination is probably not a rational player. A rational Bob will promise a high claim and keep with his promise.
I am aware that the vacuous terms of low, middle, and high claims are extremely slippery, but I believe the absence of precise information does stop TD from going continuously downhill: it is almost impossible to claim exactly $1 below the claim of the other player.
I think that's how it stays less disastrous than the dollar auction.
I think we can also apply this same logic to the centipede game and conclude why defecting in the first round is not empirically common: both players have the incentive to be believed that they will be cooperative until late in the game, and (depending on the parameters of the game) it is rational to keep the promise of long term cooperation if the other player trusts you.