Yeah. Usually the centipede game is used to teach this lesson, but your game is also very nice :-)

Nit: I think this game is more standardly referred to in the literature as the "traveler's dilemma" (Google seems to return no relevant hits for "almost free lunches" apart from this post).

So the situation is as bad as it could possibly be.

You mean, it is as bad as it could possibly be for the Nash equilibrium to be a good strategy and a good predictor in this setup? Yep, absolutely. All models tend to have their domain of validity, and this game shows the limits of the Nash equilibrium model of decision making.

Unilateral precommitment lets people win at "Almost Free Lunches".

One way to model precommitment is as a sequential game: first player 1 chooses a number, then player 1 has the option of either showing that number to player 2 or keeping it hidden, then player 2 chooses a number. Optimal play is for player 1 to pick £1,000,000 and show that number, and then for player 2 to choose £999,999.99.

An interesting feature of this is that player 1's precommitment helped player 2 even more than it helped player 1. Player 1 is "taking one for the team", but still winning big. This distinguishes it from games like chicken, where precommitment is a threat that allows the precommitter to win the larger share. Though this means that if either player can precommit (rather than one being pre-assigned to go first as player 1) then they'd both prefer to have the other one be the precommitter.

This benefit of precommitment does not extend to the two option version (n2 vs. n1). In that version, player 2 is incentivized to say "n1" regardless of what player 1 commits to, so unilateral precommitment doesn't help them avoid the Nash Equilibrium. As in the prisoner's dilemma.

Doesn't the existence of the rule that says that no money changes hands if there's a tie alter the incentives? If we both state that we want 1,000,000 pounds, then we both get it and we both walk away happy. What incentive is there for either of the two agents to name a value that is lower than 1,000,000?

It checks out empirically. War is a Nash semi-equilibrium - both sides would be better off if they could coordinate an end to it, but they usually can't (in the near term). If Stanislav Petrov had been a bit less chill, the Cold War would have ended in 1983 in the "everybody launches all the nukes; cockroaches win" Nash equilibrium. If that's not arbitrarily bad, it's plenty bad enough.

It is interesting that experimental results of traveller's dilemma seems to give results which deviate strongly from the Nash Equilibrium, and in fact quite close to the Pareto Optimal Solution.

This is pretty strange for a game that has only one round and no collusion (you'd expect it to end as Prisoner's Dilemma, no?)

It is rather different from what we would see from the dollar auction, which has no Nash Equilibrium but always deviate far away from the Pareto optimal solution.

I suspect that the this game being one round-only actually improves the Pareto efficiency of its outcomes:

Maybe if both participants are allowed to change their bid after seeing each other's bid they WILL go into a downward spiral one cent by one cent until they reach zero or one player gives up at some point with a truce, just like how dollar auctions always stop at some point.

When there is only one round, however, there is no way for a player to make their bid exactly 1 or 2 cents less than the other player, and bidding any less than that is suboptimal compared to bidding more than the other player, so perhaps there is an incentive against lowering one's bidding indefinitely to 0 before the game even starts, just like no one would bid $1000 in the dollar auction's first round.