HHI is a pretty interesting metric. It’s calculated by taking each firm’s market share, squaring it, and summing across all firms.
This gives the probability that two randomly chosen customers belong to the same firm.
In one micro models of oligopoly, Cournot competition, it lines up directly with the markup firms can sustain.
Outside of theory, it’s an intuitive way to average together the market power of all firms, with increases in market share for bigger players being weighted more heavily.