https://hal.inria.fr/hal-01408751Lenisa, MarinaMarinaLenisaUniversità degli Studi di Udine - University of Udine [Italie]Coalgebraic MultigamesHAL CCSD2014[INFO] Computer Science [cs]Ifip, HalMarcello M. Bonsangue2016-12-05 13:23:492016-12-05 15:34:562016-12-05 15:34:56enConference papershttps://hal.inria.fr/hal-01408751/document10.1007/978-3-662-44124-4_3application/pdf1<i>Coalgebraic games</i> have been recently introduced as a generalization of Conway games and other notions of games arising in different contexts. Using coalgebraic methods, games can be viewed as elements of a <i>final coalgebra</i> for a suitable functor, and operations on games can be analyzed in terms of (generalized) <i>coiteration schemata</i>. Coalgebraic games are <i>sequential</i> in nature, <i>i.e</i>. at each step either the Left (L) or the Right (R) player moves (<i>global polarization</i>), moreover only a <i>single</i> move can be performed at each step. Recently, in the context of Game Semantics, <i>concurrent</i> games have been introduced, where global polarization is abandoned, and multiple moves are allowed. In this paper, we introduce <i>coalgebraic multigames</i>, which are situated half-way between traditional sequential games and concurrent games: global polarization is still present, however <i>multiple</i> moves are possible at each step, <i>i.e</i>. a team of L/R players moves in parallel. Coalgebraic operations, such as sum and negation, can be naturally defined on multigames. Interestingly, sum on coalgebraic multigames turns out to be related to Conway’s <i>selective sum</i> on games, rather than the usual (sequential) <i>disjoint sum</i>. Selective sum has a parallel nature, in that at each step the current player performs a move in <i>at least</i> one component of the sum game, while on disjoint sum the current player performs a move in <i>exactly</i> one component at each step. A monoidal closed category of coalgebraic multigames in the vein of a Joyal category of Conway games is then built. The relationship between coalgebraic multigames and games is then formalized via an equivalence of the multigame category and a monoidal closed category of coalgebraic games where tensor is selective sum.