Scharpf's distinction between legitimacy from participation and responsiveness (government by the people) and legitimacy from effective problem-solving (government for the people); Schmidt added throughput legitimacy for the quality of the process in between.
In more detail
Fritz Scharpf developed the distinction in Governing in Europe (1999) to analyse the European Union, which he argued could draw on output legitimacy where national democracies drew on input. Vivien Schmidt (2013) added throughput legitimacy: efficacy, accountability, transparency, inclusiveness and openness of governance processes. She argued that poor throughput can undermine both input and output, while good throughput cannot by itself compensate for weak input or output. The triad is now used for international organisations, regulatory agencies, citizens' assemblies and digital platforms. Critics note that output claims can be used to justify unaccountable rule.