The standard Penney's game is played by two players. Player $A$ chooses a sequence of $k>2$ bits and $B$ (seeing $A$'s selection) chooses a different sequence of $k$ bits. A fair coin is flipped repeatedly; whichever player's sequence appears *first* in the (long) sequence of random coin flips is the winner.

Paradoxically, no matter what sequence $A$ chooses, $B$ can choose a different sequence that is more likely to appear before $A$'s. For instance, for $k=3$, if $A$ chooses $HHH$, then $B$ should choose $THH$. In $1/8$ games, the $HHH$ will appear as the first three coin tosses and $A$ will win; in *all other games* $B$ will win because before $HHH$ can appear, $B$'s sequence $THH$ *must* appear. If instead $A$ chose $THH$, then $B$ should choose $TTH$, which is more likely to appear first. Here are some simulations illustrating the effect:

- $HHH\underline{TTH}TTTTHHTTT$: B wins
- $H\underline{TTH}TTHTHHTHHTT$: B wins
- $H\underline{TTH}THHHHHHTHTT$: B wins
- $H\underline{TTH}THHTTHTHHHT$: B wins
- $HHHTT\underline{TTH}THHTHTH$: B wins
- $TTTTTT\underline{THH}THHTTH$: A wins
- $\underline{TTH}TTTHTHTHHHTH$: B wins
- $HH\underline{TTH}TTTTTHHTTT$: B wins
- $HHHHHHH\underline{THH}TTHHT$: A wins
- $\underline{TTH}TTTHHHHTHTHT$: B wins
- $\underline{THH}HTHTTTTHTHHT$: A wins
- $TH\underline{THH}TTTHTTHTTH$: A wins

And so on. Thus the available sequences exhibit non-transitive dominance: $\alpha_1 > \alpha_2 > \ldots > \alpha_1$, where the $\alpha_i$ are distinct $k$-bit sequences and "$>$" means "is more likely to appear first in a sequence randomly generated by a fair coin."

In short, if $B$ plays optimally, he is guaranteed (in probability) to win most the games.

Let's call that traditional Penney game a *one-level* game, because only one sequence match is needed for termination.

Consider a *two-level* generalization, in which $A$ first chooses *two* different $k$-bit sequences and $B$ (seeing $A$'s choices) chooses two different sequences (not already chosen by $A$). The game proceeds as before, with random coin flips, but now the winner is the first to have *both* his chosen sequences appear, in either order and at any separation. (Allow overlaps of sequences.)

*Question*

Is there an optimal strategy for $B$ such that he is guaranteed (in probability) to win most two-level Penney games?