 The step from two to more players, often routine in epistemic logics, can be delicate in the logic of matrix games. Accessibility relations of type \([\approx_i]\), interpreted as identity of profiles except for the \(i^{}\)-coordinate, yield a product logic akin to the three-variable fragment of first-order logic which is known to be undecidable (Bezhanishvili 2006). However, with only relations of identity at the i-coordinate, i.e., \([\equiv_i]\), the logic remains decidable (Venema 1998; Van De Putte, Tamminga, &amp; Duijf 2017; Lomuscio, van der Meyden, &amp; Ryan 2000).