We define a notion of morphisms between open games, exploiting a surprising
connection between lenses in computer science and compositional game theory.
This extends the more intuitively obvious definition of globular morphisms as
mappings between strategy profiles that preserve best responses, and hence in
particular preserve Nash equilibria. We construct a symmetric monoidal double
category in which the horizontal 1-cells are open games, vertical 1-morphisms
are lenses, and 2-cells are morphisms of open games. States (morphisms out of
the monoidal unit) in the vertical category give a flexible solution concept
that includes both Nash and subgame perfect equilibria. Products in the
vertical category give an external choice operator that is reminiscent of
products in game semantics, and is useful in practical examples. We illustrate
the above two features with a simple worked example from microeconomics, the
market entry game
