Example [efr-4CSL]

Consider the (Grothendieck) fibration \mathsf {Set}^\to \to \mathsf {Set}, which can be viewed as a Markov fibration. Let f: {\mathbb {R} \choose \Sigma } \otimes {\bar {X} \choose X} \leftrightarrows {\bar {Y} \choose Y} be a parameterized lens. Denote by \mathrm {argmax}_f the open game {\bar {X} \choose X} \to {\bar {Y} \choose Y} with parameters {\mathbb {R} \choose \Sigma }, underlying parameterized lens f, and equilibrium relation given by \mathrm {argmax}.

Then if g: {\mathbb {R} \choose \Sigma '} \otimes {\overline {X'} \choose X'} \leftrightarrows {\overline {Y'} \choose Y'} is another parameterized lens, we have \mathrm {argmax}_f \oplus \mathrm {argmax}_g \cong \mathrm {argmax}_{f \oplus g}, where by an abuse of notation f \oplus g denotes the paramterized lens {\mathbb {R} \choose \Sigma \times \Sigma '} \otimes ({\overline {X} \choose X} \oplus {\overline {X'} \choose X'}) \to {\overline {Y} \choose Y} \oplus {\overline {Y'} \choose Y'} given by distributing into the coproduct, then projecting into the relevant factor of the product \Sigma \times \Sigma ' and applying either f or g

A context for either of these games consists of an element of X + X' and a function k: Y + Y' \to \mathbb {R}. The external choice game can be seen as having two players, one who gets to play if the context chooses an element in X, who must output an element y \in Y and optimize k(y) (according to his private utility function \Sigma \times X \times \bar {Y} \to \mathbb {R}), the other playing when the input is in X' and who must choose an element in y'. The single argmax game can be seen as a single player, who is constrained to play inside the same "branch" of the game as the input (this constraint is encoded in the lens f \oplus g), and whose utility function is given by the first players' in the first branch, and the second players' in the second branch.x