If the fairness constraints are all pairwise (ie each player has fairness curves for each opponent), then the scheme generalizes directly. Slightly more generally, if each player’s fairness set is weakly convex and closed under componentwise max, the scheme still generalizes directly (in effect the componentwise max creates a fairness curve which can be intersected with the xyz=Ai surfaces to get the (xi,yi,zi) points.
In order to generalize fully, the agents should each precommunicate their fairness sets. In fact, after doing this, the algorithm is very simple: player X can compute what it believes is the optimal-xyz feasible-and-fair-according-to-everyone point (x,y,z) (which is unique because these are all convex sets), and if PA proves the outcome will be fair-according-to-X, then output x; otherwise output 0.
I’d like for this to work with as little prior coordination as possible, so I’m not keen on assuming the agents precommunicate their fairness sets. But the generalization with only the Ai pre-coordinated is neat.
What’s the harm in requiring prior coordination, considering there’s already a prior agreement to follow a particular protocol involving Ais? (And something earlier on in the context about a shared source of randomness to ensure convexity of the feasible set.)
The actual problem we want to work toward is one where all the prior coordination info is in the environment independent of the particular agents (e.g. the existence of Schelling points), and the agents are just deducing things about each other. For instance, two FairBots work in a source code swap Prisoner’s Dilemma against one another even if written in different programming languages.
I’m willing to accept “accepting a natural ordering on the payoff set” and “accepting a natural set of outcome products” as things that could conceivably be Schelling points in a simple environment, but “know the shape of each others’ fairness sets” looks like an infinite pre-exchange of information that cannot be gleaned from the environment.
(And “generate mutual random bits” is a cooperative thing that can be viewed as an atomic action in the environment.)
If the fairness constraints are all pairwise (ie each player has fairness curves for each opponent), then the scheme generalizes directly. Slightly more generally, if each player’s fairness set is weakly convex and closed under componentwise max, the scheme still generalizes directly (in effect the componentwise max creates a fairness curve which can be intersected with the xyz=Ai surfaces to get the (xi,yi,zi) points.
In order to generalize fully, the agents should each precommunicate their fairness sets. In fact, after doing this, the algorithm is very simple: player X can compute what it believes is the optimal-xyz feasible-and-fair-according-to-everyone point (x,y,z) (which is unique because these are all convex sets), and if PA proves the outcome will be fair-according-to-X, then output x; otherwise output 0.
I’d like for this to work with as little prior coordination as possible, so I’m not keen on assuming the agents precommunicate their fairness sets. But the generalization with only the Ai pre-coordinated is neat.
What’s the harm in requiring prior coordination, considering there’s already a prior agreement to follow a particular protocol involving Ais? (And something earlier on in the context about a shared source of randomness to ensure convexity of the feasible set.)
The actual problem we want to work toward is one where all the prior coordination info is in the environment independent of the particular agents (e.g. the existence of Schelling points), and the agents are just deducing things about each other. For instance, two FairBots work in a source code swap Prisoner’s Dilemma against one another even if written in different programming languages.
I’m willing to accept “accepting a natural ordering on the payoff set” and “accepting a natural set of outcome products” as things that could conceivably be Schelling points in a simple environment, but “know the shape of each others’ fairness sets” looks like an infinite pre-exchange of information that cannot be gleaned from the environment.
(And “generate mutual random bits” is a cooperative thing that can be viewed as an atomic action in the environment.)