One way for Rose to analyze the game is for her to sets up her stack of coins with a fraction x of the coins with H on top and a fraction (1-x) of the coins with T on top. She can then make her gain/loss independent of whether H or T shows up from Colin’s pile. If she does this, who will win and by how much per game?
The Game matrix is shown below:
Rose arranges her coins in a random way with the constraint that a fraction z of coins have H on top and a fraction 1-z have T on top.
If Colin shows H, Rose’s average return per game is
RH(z) = -9x+5(1-x).
If Colin shows T, then her return per game is
RT(z) = 5x - (1-x).
If she chooses x so that RH(x) = RT(x) then her win/loss is independent of Colin’s sequence.
-9x+5(1-x) = 5z-(1-x)
gives x = 0.3, RH(0.3) = RT(0.3) = 0.8.
Thus, if Rose arranges her stack with 30% coins showing H and 70% showing T, she wins 0.8 cents per game, regardless of what B does. 0.8 cents/game is called the “expected value” of the game and x = 0.3 is called Rose’s optimum strategy. Unfortunately, Colin has no winning strategy. Because this is a zero-sum game, he loses whatever Rose wins. If Colin knew some Game Theory or had read this post, he would refuse to play this game.
If you want to learn more about Game Theory, too excellent books are: “Game Theory and Strategy,” by Philip D Straffin, and “Evolution and the Theory of Games,” by John Maynard-Smith.
There is also a good Wiki page: https://en.wikipedia.org/wiki/Game_theory.