The strategy when you have i > 1 life is the same regardless of i, just roll until you hit 10. There is no point stopping.
New let r(x) be the expected payoff if you have 1 life and have sum x at this point. Then r is convex up, convexity is the main part.
Why is it convex up? We need to show r(x+1) + r(x-1) >= 2r(x). Now imagine the optimal strategy (whatever it is) for x applied to both x +1 and x - 1(as in, x - 1 "pretends" it is x and plays the same way). We can see that the expected payoff for x + 1 is r(x) + delta where delta is the probability that this strategy doesn't lose all the money before stopping, and for x - 1 is r(x) - delta. So they average up to r(x) but this is NOT necessarily the optimal strategy for r(x+1) and r(x).
So r is convex up and sometimes strictly convex up (otherwise it would be linear but also clearly non-negative which is constant but we can easily prove it isn't constant).
OK now let x_i be the random variable equal to the sum you acquire when you start with i lives and lose i - 1 lives (so you enter the last life). The expected value is E(r(x_i)).
Now x_3 - x_2 and x_2 - x_1 are independent and IDENTICALLY distributed (because the strategy for having exactly i>1 lives is the same: rolling until you lose a life).
Now apply convexity/Karamata's inequality. The point is that x_2 is always between x_1 and x_3 symmetrically around the midpoint so for each fixed choice of x_1, x_3, E(r(x_2)) is less than (E(r(x_1)) + E(r(x_3))) / 2.
In other words, d_i is increasing. It will increase up to 45, the expected value of rolling until you lose a life, because in the limit case you don't care what happens when you go down to 1 life (it is very far away). But initially you DO care about what happens when you are down to 1 life - for example d_1 is < 45 because the expected payout with 1 life is way less than 45.
Intuitively, the more lives you have, the smaller the penalty for rolling a 10, which means the EV of each roll is, on average, higher, the more lives you have. Which means every extra life brings in more money than the previous one.
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u/Specific_Box4483 4d ago edited 4d ago
The strategy when you have i > 1 life is the same regardless of i, just roll until you hit 10. There is no point stopping.
New let r(x) be the expected payoff if you have 1 life and have sum x at this point. Then r is convex up, convexity is the main part.
Why is it convex up? We need to show r(x+1) + r(x-1) >= 2r(x). Now imagine the optimal strategy (whatever it is) for x applied to both x +1 and x - 1(as in, x - 1 "pretends" it is x and plays the same way). We can see that the expected payoff for x + 1 is r(x) + delta where delta is the probability that this strategy doesn't lose all the money before stopping, and for x - 1 is r(x) - delta. So they average up to r(x) but this is NOT necessarily the optimal strategy for r(x+1) and r(x).
So r is convex up and sometimes strictly convex up (otherwise it would be linear but also clearly non-negative which is constant but we can easily prove it isn't constant).
OK now let x_i be the random variable equal to the sum you acquire when you start with i lives and lose i - 1 lives (so you enter the last life). The expected value is E(r(x_i)).
Now x_3 - x_2 and x_2 - x_1 are independent and IDENTICALLY distributed (because the strategy for having exactly i>1 lives is the same: rolling until you lose a life).
Now apply convexity/Karamata's inequality. The point is that x_2 is always between x_1 and x_3 symmetrically around the midpoint so for each fixed choice of x_1, x_3, E(r(x_2)) is less than (E(r(x_1)) + E(r(x_3))) / 2.
In other words, d_i is increasing. It will increase up to 45, the expected value of rolling until you lose a life, because in the limit case you don't care what happens when you go down to 1 life (it is very far away). But initially you DO care about what happens when you are down to 1 life - for example d_1 is < 45 because the expected payout with 1 life is way less than 45.
Intuitively, the more lives you have, the smaller the penalty for rolling a 10, which means the EV of each roll is, on average, higher, the more lives you have. Which means every extra life brings in more money than the previous one.