An efficiency ordering of k-price auctions under complete information

We study $k$-price auctions in a complete information environment and characterize all pure-strategy Nash equilibrium outcomes. In a setting with $n$ agents having ordered valuations, we show that any agent, except those with the lowest $k-2$ valuations, can win in equilibrium. As a consequence, worst-case welfare increases monotonically as we go from second-price auction ($k=2$) to lowest-price auction ($k=n$), with the first-price auction achieving the highest worst-case welfare.