You and an opponent play a game by rolling a fair 6 sided dice- the one who gets the higher number wins 1\$. If you lose you get nothing. If you tie you both just roll again.
(a) What are the expected winnings.
(b) A modified version of the game- if you tie, you just end the game instead (with neither getting anything). Now what are the expected winnings.
(c) We continue with the version of the game in (b). Now suppose you have the option to pay $0.25$\$ for a 'boost' of 2 points- if your roll yields $3$ you can boost to get $5$, if your roll yields $6$ you can boost to get $8$ and so on. When should you pay for the boost? You can only decide to boost after seeing your die roll but not your opponent's.
Answers: It's only (c) I'm unsure about and would like some verification.
(a) $\mathbb{E}[Winning] = 0.5(1) + 0.5(0) = 0.5$ by symmetry.
(b) $\mathbb{E}[Winning] = 0.5(\mathbb{P}(\text{no tie}))(1) + 0 = 0.5 \times (30/36) =0.4333$
(c) This is the one I'm not sure about because of the answer I got. A quick guess tells me that you are safe to do it if you get $\geq 2$ since the other guy's roll will be $3.5$ on average.
But consider this:
if you rolled $1$ your expected winning is $0$ (since ties just end the game).
If you rolled $3$ your expected winnings is $1/3 > 0.25$.
So it seems even if you got a $1$ in aggregate it would be worth it to boost.
So the answer to (c) seems to be... always choose to boost? I'm not sure this is right.