Three-player conflicting interest games and nonlocality

Abstract

We outline the general construction of three-player games with incomplete information which fulfil the following conditions: (i) symmetry with respect to the permutations of players; (ii) the existence of an upper bound for total payoff resulting from Bell inequalities; (iii) the existence of both fair and unfair Nash equilibria saturating this bound. Conditions (i)-(iii) imply that we are dealing with conflicting interest games. An explicit example of such a game is given. A quantum counterpart of this game is considered. It is obtained by keeping the same utilities but replacing classical advisor by a quantum one. It is shown that the quantum game possesses only fair equilibria with strictly higher payoffs than in the classical case. This implies that quantum nonlocality can be used to resolve the conflict between the players.

Güney-Hillery approach to Bell inequalities

“…Schur lemma one finds that it takes block-diagonal form in the basis in which the decomposition (13) is explicit. Moreover, if it happens that any irreducible representations D s appears in (13) with the multiplicity at most one, X(v A , v B ) is simply diagonal and reduces to the multiplicity of unit matrix in any irreducible subspace.…”

“…a permutation of such pairs. In what follows we will be interested in the G-orbits in the total space of bipartite system which are of the form [11]÷ [13] O…”

Bell inequalities for $$S_4$$ group: classical and quantum bounds for two-orbit case

“…Some particular examples of Bell's inequalities and their violation has been already discussed in [12] and [13]. Here we present a more detailed discussion based on the general construction of states described in [12] and sketched in previous section.…”

“…unitary (orthogonal) basis is described in Appendix. Some appropriate orbits have been constructed in [12], [13] by purely geometric means; essentially, using the fact that S 4 is the symmetry group of regular tetrahedron. However, in order to keep the discussion more general, we follow the algorithm outlined above.…”

“…They have been already studied in Refs. [12], [13], [14]. We restrict ourselves to the two-orbit case.…”