# Expected value in a dice game

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?

• yes, you are right. – Hat Oct 16 '18 at 17:45
• "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. – David K Oct 16 '18 at 17:48
• After $B$ rerolls, $A$ is allowed to reroll. – Hat Oct 16 '18 at 17:57
• It also matters whether $A$ is allowed to re-roll if $B$ chooses not to re-roll. – David K Oct 16 '18 at 18:17

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 $$10.5$$ ...
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.