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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 of all players involve a ground set of boolean variables, and every player controls some of these variables. The payoff of a player is the weighted sum of the values of his boolean formulas. We consider pure Nash equilibria [18] and their well-studied refinement of payoff-dominant equilibria [12], where every player is no-worse-off than in any other pure Nash equilibrium. We study both structural and complexity properties for both decision and search problems. - We consider a subclass of weighted boolean formula games, called mutual weighted boolean formula games, which make a natural mutuality assumption. We present a very simple exact potential for mutual weighted boolean formula games. We also prove that each weighted, linear-affine (network) congestion game with player-specific constants is polynomial, sound monomorphic to a mutual weighted boolean formula game. In a general way,we prove that each weighted, linear-affine (network) congestion game with player-specific coefficients and constants is polynomial, sound monomorphic to a weighted boolean formula game. - We present a comprehensive collection of high intractability results. These results show that the computational complexity of decision (and search) problems for both payoff-dominant and pure Nash equilibria in weighted boolean formula games depends in a crucial way on five parameters: (i) the number of players; (ii) the number of variables per player; (iii) the number of boolean formulas per player; (iv) the weights in the payoff functions (whether identical or nonidentical), and (v) the syntax of the boolean formulas. These results show that decision problems for payoff-dominant equilibria are considerably harder than for pure Nash equilibria (unless the polynomial hierarchy collapses).