Oh, did I leave out the part about the electric shocks? The book explains that points always go to Blue to make the analysis simpler. The reader is told to imagine that Red is compelled to play. Alternately, Red can demand to be paid to participate. Assuming Red uses optimal strategy, what is the minimum number of points per turn that Red should demand before agreeing to play? And how many points can Blue afford to offer Red on the side before each turn before Blue loses the advantage?
I'm either overthinking this or we're missing some vital piece of information. If Red is compelled to play under penalty of death, then Red doesn't care what move he makes as he gets no points either way. From Blue's perspective, Red then has a 50/50 chance between moves 1 and 2. As such, Blue has to choose between a 50/50 shot at 3 points or 6 points, or a 50/50 shot at 5 points or 4 points. This is a question of how risk averse Blue is, since he averages 4.5 points either way. If Blue has to pay Red to play, and assuming Red can refuse to play (thus denying Blue any points), then Red would demand 2 points to play at the minimum, as that's the only way Blue is guaranteed to win any points (move 1, 1 becomes 0 for Blue if Red demands 3 points). However, even if Red demands 3 points, It still comes down to how risk averse Blue is, does Blue choose move 2, winning either 5 points or 4 points and paying 3 points to Red, making his outcome either 2 points or 1 point, or does Blue choose move 1 and risk getting 0 points for the chance winning 3 points. This seems to be a question of how risk averse Blue is since Red's payoff isn't determined by which move he/she makes.
We suppose that Red wants Blue to win as few points as possible. If Red can negotiate a payment before each turn from Blue, both sides can strategize to end up with as many points as possible in the long run. It could be. In previous, simpler examples, the book says each player's priority is to avoid the worst possible loss. I am not sure if this is the only way it could work; perhaps a player could also prioritize having a chance to get the best possible outcome, without regard to the worst cases. These complications will probably become more important in more complex games.Red doesn't care what move he makes as he gets no points either way
This seems to be a question of how risk averse Blue is
I choose my numbers randomly, switching back and forth so Blue can't predict my choices, but favor the number 1 at least twice as often as I favor 2 to minimize Blue's winnings. I also call the officials a bunch of doody heads for not allowing me to win through belligerent uncoopertiveness.
The officials have noted your position, and wonder why you did not elect to play as Blue. Let's evaluate the strategy for Red to play 1 twice as often as 2. We assume Blue is a profit-maximizing mofo and will play to win as many points as possible. When blue plays 1, Red expects to pay (2 x 3) + (1 x 6) = 12 over three turns, an average of 4 per turn. When blue plays 2, Red expects to pay (2 x 5) + (1 x 4) = 14 over three turns, an average of 4 2/3 per turn. No matter what Blue does, this is an improvement over Red playing 1 all the time, which has a worst case of 5 per turn.