A generalized quantifier Q_đŠ is called a CSP-quantifier if its defining class đŠ consists of all structures that can be homomorphically mapped to a fixed finite template structure. For all positive integers n â„ 2 and k, we define a pebble game that characterizes equivalence of structures with respect to the logic L^k_{âÏ}(CSP^+_n), where CSP^+_n is the union of the class Qâ of all unary quantifiers and the class CSP_n of all CSP-quantifiers with template structures that have at most n elements. Using these games we prove that for every n â„ 2 there exists a CSP-quantifier with template of size n+1 which is not definable in L^Ï_{âÏ}(CSP^+_n). The proof of this result is based on a new variation of the well-known Cai-FĂŒrer-Immerman construction.