Conditions ensuring BOB’s winning strategy in the exponential game MultiG1(G2)

Determine non-trivial conditions on an undetermined infinite game G2 and/or on an infinite game G1 that guarantee the existence of a winning strategy for BOB in the exponential game MultiG1(G2), defined as follows: players alternately construct a chronological mapping f from the game tree T1 of G1 to the game tree T2 of G2 (ALICE specifies f on odd-length moments and BOB on even-length moments), yielding a total chronological map f = lim fn at the end of the run; ALICE wins if and only if f is an A-morphism from (T1,A1) to (T2,A2), and BOB wins otherwise.

Background

The paper proves that the category GameA is cartesian closed and introduces the exponential game MultiG1(G2), which serves as the internal hom. In MultiG1(G2), ALICE and BOB alternately define a chronological mapping from the game tree of G1 to that of G2; the payoff criterion is that ALICE wins exactly when the resulting mapping is an A-morphism.

Proposition 9.24 establishes that ALICE has a winning strategy in MultiG1(G2) if and only if ALICE has a winning strategy in G2, and that if BOB has a winning strategy in G2 then BOB also has a winning strategy in MultiG1(G2). Motivated by the case where G2 is undetermined (neither player has a winning strategy), the authors raise a problem asking for structural conditions on G1 and/or G2 that ensure BOB’s victory in MultiG1(G2).