Social inequalities are ubiquitous, and here we show that the values of the Gini (g) and Kolkata (k) indices, two generic inequality indices, approach each other (starting from g=0 and k=0.5 for equality) as the competitions grow in various social institutions like markets, universities and elections. It is further shown that these two indices become equal and stabilize at a value (at g=kâ 0.87) under unrestricted competitions. We propose to view this coincidence of inequality indices as a generalized version of the (more than a) century old 80-20 law of Pareto. Furthermore, the coincidence of the inequality indices noted here is very similar to the ones seen before for self-organized critical (SOC) systems. The observations here, therefore, stand as a quantitative support toward viewing interacting socio-economic systems in the framework of SOC, an idea conjectured for years.