Example [efr-BNT1]

Let G: {* \choose X} \to {R \choose Y} be a game, representing an agent who is optimizing the return value r \in R in some sense. Then G \otimes (1_I \oplus 1_I = 1_{I + I}) represents the same agent, whose payoff may now depend on an additional bit (a value in I + I), but whose decisions may not depend on that bit (his selection function may still depend on its distribution).

On the other hand, G \oplus G (\cong G \otimes 1_I \oplus G \otimes 1_I) represents the same situation, but where the player's strategy may depend on the bit---he provides two strategies \sigma _1,\sigma _2, one for each possibility.

This proves that \otimes does not distribute over \oplus