Aiming　for　efficient　coordination　mechanisms　for　the　distributed　weighted　set　cover　problem,　we　study　from　ordinal　potential　game　theoretic　learning　and　propose　a　Nash　equilibrium　selection　algorithm　(NESA).　An　ordinal　potential　game　model　is　established,　where　the　local　utility　function　is　designed　by　incorporating　a　greedy　heuristic.　To　distinguish　Nash　equilibria　of　different　global　fitness,　we　further　classify　them　into　the　inferior　Nash　equilibrium　(INE)　and　the　superior　Nash　equilibrium　(SNE),　and　show　that　the　optimal　solution　must　be　an　SNE.　High-quality　SNE　solutions　are　obtained　by　assigning　each　player　a　local　stochastic　rule　based　on　its　category　and　a　finite　memory.　By　demonstrating　the　existence　of　a　finite　improvement　path　from　each　INE　to　an　SNE,　we　prove　finite-time　convergence　of　the　NESA.　Numerical　experiments　are　carried　out　and　comparisons　against　representative　methods　are　presented,　which　demonstrate　the　effectiveness　as　well　as　the　superiority　of　our　methodology　to　the　state-of-the-art.