Weighted Boolean Formula Games

We introduce a new class of succinct games, called weighted boolean formula games. Here,each player has a set of boolean formulas he wants to get satisfied. The boolean formulas ofall players involve a ground set of boolean variables, and every player controls some of thesevariables. The payoff of a player is the weighted sum of the values of his boolean formulas. For thesegames, we consider pure Nash equilibria [42] and their well-studied refinement of payoff-dominantequilibria [30], where every player is no worse-off than in any other pure Nash equilibrium. Westudy both structural and complexity properties for both decision and search problems with respectto the two concepts:• We consider a subclass of weighted boolean formula games, called mutual weighted booleanformula games, which make a natural mutuality assumption on the payoffs of distinct players.We present a very simple exact potential for mutual weighted boolean formula games. Wealso prove that each weighted, linear-affine (network) congestion game with player-specificconstants is polynomial, sound Nash-Harsanyi-Selten homomorphic to a mutual weightedboolean formula game. In a general way, we prove that each weighted, linear-affine (net-work) congestion game with player-specific coefficients and constants is polynomial, soundNash-Harasanyi-Selten homomorphic to a weighted boolean formula game. These homomor-phisms indicate some of the richness of the new class.• We present a comprehensive collection of high intractability results. These results show thatthe computational complexity of decision (and search) problems for both payoff-dominantand pure Nash equilibria in weighted boolean formula games depends in a crucial way on fiveparameters: (i) the number of players; (ii) the number of variables per player; (iii) the numberof boolean formulas per player; (iv) the weights in the payoff functions (whether identical ornon-identical), and (v) the syntax of the boolean formulas. (For example, we prove thatdeciding the existence of a payoff-dominant equilibrium isΘ3 -complete even if weights areidentical and there are only four players.) Our completeness results show that decision (andsearch) problems for payoff-dominant equilibria are considerably harder than for pure Nashequilibria (unless the polynomial hierarchy collapses). Due to the space constraints, some technical proofs are shifted to the Appendix. This work has been partially supported by the IST Program of the European Union under contract numbers IST-2004-001907 (DELIS) and 15964 (AEOLUS).Department of Computer Science, University of Cyprus, Nicosia CY-1678, Cyprus. Currently visiting Faculty ofComputer Science, Electrical Engineering and Mathematics, University of Paderborn, 33102 Paderborn, Germany. Email:[email protected] of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, 33102 Paderborn,Germany. Email: [email protected] für Theoretische Informatik, Institut für Informatik, Julius-Maximilians-Universität Würzburg, 97074Würzburg, Germany. Email: [email protected]