Two-Party Agreements as Circular Sets

In making an agreement with someone, I conditionally promise to perform a certain action, conditioning my obligation on their both making a corresponding promise and performing their action. What promise should I require? That they simply commit to perform is not enough. I should demand the kind of promise I am making myself, and they should demand the same of me. This makes our promises indirectly self-referential. Assuming the performance actions are specified, my promise can be characterized as a set of available promises, all those the other could make to activate my obligation. We have an agreement if each one’s promise is a member of the other’s promise. Assume that the set P of available promises satisfies (1) Aczel’s axiom for circular sets; (2) transitivity: if the obligation of $p \in P$ is activated by $p’$, then $p’ \in P$; and (3) superset closure: if $p \in P$ is activated by $p’$, $p$ is activated by any promise that implies (is a superset of) $p’$. The focus is on bargaining procedures that treat the parties symmetrically (e.g., no specified offerer or accepter.) Each party chooses an agreement promise $p*$ such that (4) if both make $p*$ and one performs, the other is obligated to perform; (5) if one makes $p*$ and the other does not, the former is not unilaterally obligated. It is shown that among available promise sets of a given size, exactly one contains an agreement promise and contains exactly one of them.