On effectivity functions of game forms

To each game form g an effectivity function (EFF) Eg is assigned. An EFF E is called formal (formal-minor) if E=Eg (respectively, E[less-than-or-equals, slant]Eg) for a game form g. (i) An EFF is formal iff it is superadditive and monotone. (ii) An EFF is formal-minor iff it is weakly superadditive. Theorem (ii) looks more sophisticated, yet, it is simpler than Theorem (i) and instrumental in its proof. In addition, (ii) has important applications in social choice, game, and even graph theories. Constructive proofs of (i) were given by Moulin, in 1983, and by Peleg, in 1998. Both constructions are elegant, yet, sets of strategies Xi of players i[set membership, variant]I might be doubly exponential in size of the input EFF E. In this paper, we suggest another construction such that Xi is only linear in the size of E. Also, we extend Theorems (i), (ii) to tight and totally tight game forms.