We consider linear steady Stokes flow under the action of a finite number of particles located inside the flow domain. The particles exert point-wise forces on the fluid, and are unknown in number, location and magnitude. We are interested in the determination of these point-wise forces, using only a single pair of partially available Cauchy boundary measurements. The inverse problem then couples two harsh problems : identification of point-wise sources and recovery of missing boundary data. We reformulate it as a threeplayer Nash game. The first two players aim at recovering the Dirichlet and Neumann missing data, while the third one aims at the point-forces reconstruction of the number, location and magnitude of the point-forces. To illustrate the efficiency and robustness of the proposed algorithm, we finally present several numerical experiments for different geometries and source distribution, including the case of noisy measurements.