<p>In this paper, we investigate the concentration behaviors of ground states to stationary Mean-field Games systems (MFGs) with the nonlocal coupling in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\ge 2.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> With the mass critical exponent imposed on Riesz potentials, we first discuss the existence of ground states to potential-free MFGs, which corresponds to the establishment of Gagliardo-Nirenberg type’s inequality. Next, with the aid of the optimal inequality, we classify the existence of ground states to stationary MFGs with Hartree-type coupling in terms of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm of population density defined by <i>M</i>. In addition, under certain types of coercive potentials, the asymptotics of ground states to ergodic MFGs with the nonlocal coupling are given. Moreover, if the local polynomial expansions are imposed on potentials, we study the refined asymptotic behaviors of ground states and show that they concentrate on the flattest minimum points of potentials.</p>