We consider mutual weighted boolean formula games (MWBFG), a subclass of WBFG making a natural mutuality assumption on the formulas of players. We present a very simple exact potential for MWBFG. We establish a polynomial monomorphism from certain classes of weighted congestion games to subclasses of WBFG and MWBFG, respectively, indicating their rich structure.
We present a collection of complexity results about decision (and search) problems for both pure and payoff-dominant equilibria in WBFG. The precise complexities depend crucially on five parameters: (i) the number of players; (ii) the number of variables per player; (iii) the number of formulas per player; (iv) the weights in the payoff functions (whether identical or not), and (v) the syntax of the formulas. These results imply that, unless the polynomial hierarchy collapses, decision (and search) problems for payoff-dominant equilibria are harder than for pure equilibria.