  The category of open games, which provides a strongly compositional
foundation of economic game theory, is intermediate between symmetric monoidal
and compact closed. More precisely it has counits with no corresponding units,
and a partially defined duality. There exist open games with the same types as
unit maps, given by agents with the strategic goal of predicting a future
value. Such agents appear in earlier work on selection functions. We explore
the algebraic properties of these agents via the symmetric monoidal bicategory
whose 2-cells are morphisms between open games, and show how the resulting
structure approximates a compact closed category with a family of lax
commutative bialgebras.
