  We present a new compositional approach to compositional game theory (CGT)
based upon Arrows, a concept originally from functional programming, closely
related to Tambara modules, and operators to build new Arrows from old. We
model equilibria as a bimodule over an Arrow and define an operator to build a
new Arrow from such a bimodule over an existing Arrow. We also model strategies
as graded Arrows and define an operator which builds a new Arrow by taking the
colimit of a graded Arrow. A final operator builds a graded Arrow from a graded
bimodule. We use this compositional approach to CGT to show how known and
previously unknown variants of open games can be proven to form symmetric
monoidal categories.
