Dath Ilan vs. Sid Meier's Alpha Centauri: Pareto Improvements

Here's an example game tree:

(Kindly ignore the zeros below each game node; I'm using the dev version of GTE, which has a few quirks.)

Roughly speaking:

1. Bob either has something up his sleeve (an exploit of some sort), or does not.
2. Bob either:
   1. Offers a flat agreement (well, really surrender) to Alice.
   2. Offers a (binding once both sides agree to it) arbitrated (mediated in the above; I am not going to bother redoing the screenshot above) agreement by a third party (Simon) to Alice.
   3. Goes directly to war with Alice.
3. Assuming Bob didn't go directly to war, Alice either:
   1. Accepts (and abides by) said agreement.
   2. Goes to war with Bob.
   3. Does an internal audit^[1]/preparation^[2] and goes to war with Bob.

Wars cost the winner 2 and the loser 10, and also transfers 20 from the loser to the winner. (So normally war is normally +18 / -30 for the winner/loser.).

Alice doing an audit/preparing costs 3, on top of the usual, regardless of if there's actually an exploit.

Alice wins all the time unless Bob has something up his sleeve and Alice doesn't prepare. (+18/-30, or +15/-30 if Alice prepared.) Even in that case, Alice wins 50% of the time. (-6 / -6). Bob has something up his sleeve 50% of the time.

Flat offer here means 'do the transfer as though there was a war, but don't destroy anything'. A flat offer is then always +20 for Alice and -20 for Bob.

Arbitrated means 'do a transfer based on the third party's evaluation of the probability of Bob winning, but don't actually destroy anything'. So if Bob has something up his sleeve, Charlie comes back with a coin flip and the result is 0, otherwise it's +20/-20.

There are some 17 different Nash equilibria here, with an EP from +6 to +8 for Alice and -18 to -13 for Bob. As this is a lot, I'm not going to list them all. I'll summarize:

1. There are 4 different equilibria with Bob always going to war immediately. Payoff in all of these cases is +6 / -18.
2. There are 10 different equilibria with Bob always trying an arbitrated agreement if Bob has nothing up his sleeve, and going to war 2/3rds of the time if Bob has something up his sleeve (otherwise trying a mediated agreement in the other 1/3rd), with various mixed strategies in response. All of these cases are +8 / -14.
3. There are 3 different equilibria with Bob always going to war if Bob has something up his sleeve, and always trying a flat agreement otherwise, with Alice always accepting a flat agreement and doing various strategies otherwise. All of these cases are +7 / -13.

Notably, Alice and Bob always offering/accepting an arbitrated agreement is not an equilibrium of this game^[3]. None of these equilibria result in Alice and Bob always doing arbitration. (Also notably: all of these equilibria have the two sides going to war at least occasionally.)

There are likely other cases with different payoffs that have an equilibrium of arbitration/accepting arbitration; this example suffices to show that not all such games lead to said result as an equilibrium.

1. ^{^}

   I use 'audit' in most of this; I used 'prep' for the game tree because otherwise two options started with A.

2. ^{^}

   read: go 'uhoh' and spend a bunch of effort finding/fixing Bob's presumed exploit.

3. ^{^}

   This is because, roughly, a Nash equilibrium requires that both sides choose a strategy that is best for them given the other party's response, but if Bob chooses MediatedS / MediatedN, then Alice is better off with PrepM over AcceptM. Average payout of 15 instead of 10. Hence, this is not an equilibrium.