15.6 Duality, games, and approachability

Scalar regret is too narrow for games and vector-valued targets. Let a decision profile produce a value in an ordered or enriched object \(V\), and let \(S\hookrightarrow V\) be a declared acceptable region. An approachability observer should return not only a distance to \(S\), but the comparison data that identify which face, player, or compatibility condition fails.