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Consider two people, person $A$ and person $B$. Person $A$ has a $30$-sided dice and Person $B$ has a $20$-sided dice. Both players roll and the person with the highest roll wins. The loser pays the winner the value of the winner's dice outcome. In the case of a tie, player $B$ wins.

What is the expected value of this game for player $A$?

So, this part was easy. I just split it up into two cases. If $A$'s dice roll is above $20$ they are guaranteed to win. If it's below $20$, by symmetry, they have a $50\%$ change of winning. Then, finally, there are $20$ ways for ties, so the expected value is just

$$E[\text{Player A}] = 1/3*51/2 - 20/600*10.5 = 8.15$$

Suppose player $B$ is allowed to reroll the dice once. Now, what is the expected value for player $A$?

I'm not too sure how to do this part. Obviously, player $B$ should reroll if player $A$ rolls less than a $20$ and he loses OR if he wins with a dice roll less than $10.5$ (because then, he can increase the amount he wins). I think the way to do this is first compute the expected value of player $B$'s roll with the reroll. Before, it was just $10.5$, but with the reroll, it becomes

$$1/2(10.5) + 1/2(15.5) = 13$$

So, I think now we have to account for the part I just canceled out before because of symmetry. I am not sure about how to do this. Any help is appreciated.

After that, I'd eventually like to calculate

In this scenario, how much is it worth for player $A$ to get a re-roll option?

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  • $\begingroup$ yes, you are right. $\endgroup$ – Hat Oct 16 '18 at 17:45
  • $\begingroup$ "In this scenario, how much is it worth for player $A$ to get a re-roll option?" Which scenario? The one where $B$ has a re-roll option? When does $A$ choose whether to re-roll: before $B$ chooses, after $B$ chooses but before $B$ rolls, after $B$ rolls, or simultaneously with $B$ choosing (each has to answer without knowing what the other's answer is)? If $A$ chooses to re-roll, when does $A$ re-roll the die? The answers to those questions may affect $B$'s strategy as well. $\endgroup$ – David K Oct 16 '18 at 17:48
  • $\begingroup$ After $B$ rerolls, $A$ is allowed to reroll. $\endgroup$ – Hat Oct 16 '18 at 17:57
  • $\begingroup$ It also matters whether $A$ is allowed to re-roll if $B$ chooses not to re-roll. $\endgroup$ – David K Oct 16 '18 at 18:17
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Obviously, player $B$ should reroll if player A rolls less than a $20$ and he loses OR if he wins with a dice roll less than $10.5$ ...

The "if he loses" part is obvious. The other part is not.

For example, suppose $B$ and $A$ both roll $10.$ In that case, if $B$ does not re-roll, $B$ wins $10.$ Then $B$ has won with a die roll less than $10.5.$

If $B$ rolls again, $B$ has a $0.55$ chance to roll $10$ or greater, in which case $B$ wins (on average) $15$ from $A.$ But $B$ has a $0.45$ chance to roll $9$ or less, in which case $B$ has to pay $10$ to $A.$ So $B$'s expected winnings if $B$ re-rolls in that case are $$ 0.55(15) - 0.45(10) = 3.75.$$ This is less than $10,$ so the better strategy in that case is not to re-roll.

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