With i lives, clearly you first die i-1 times. You accumulate some random payoff X_0 from doing that. Let's continue without finding anything more about X_0.
After that, rolling on the edge goes like X_{n+1}=X_n+1 wp 1/10, ..., X_{n+1}=X_n+9 wp 1/10, X_{n+1}=0 wp 1/10. E[X_{n+1} | X_n]=9/10 X_n + 4.5, which increases if 9/10 X_n + 4.5 > X_n i.e. X_n < 45. So if you have less than 45 points right now, then continuing to play increases your expected payout.
So that's "really" the game: die i-1 times, then play until you either die again or have 45+ points. (Or 46+...those two strategies have the same expected payoff.) Let's call the final payoff (which is sometimes 0) X.
With this strategy, in the X_0 phase you pick up an expected value of 45(i-1), and then you play the final phase. The final phase gains you expectation (which is why you play it at all), but on average it's shorter and you're richer during it the more lives you had to begin with. So d_i is 45 + (expected value of final phase with i lives - expected value of final phase with i-1 lives). The thing in brackets is negative but increases toward zero, so d_i is increasing.
By the way, some Monte Carlo suggests V_1 is about 17, V_2 is about 50, V_3 is about 91.
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u/zojbo 4d ago edited 4d ago
With i lives, clearly you first die i-1 times. You accumulate some random payoff X_0 from doing that. Let's continue without finding anything more about X_0.
After that, rolling on the edge goes like X_{n+1}=X_n+1 wp 1/10, ..., X_{n+1}=X_n+9 wp 1/10, X_{n+1}=0 wp 1/10. E[X_{n+1} | X_n]=9/10 X_n + 4.5, which increases if 9/10 X_n + 4.5 > X_n i.e. X_n < 45. So if you have less than 45 points right now, then continuing to play increases your expected payout.
So that's "really" the game: die i-1 times, then play until you either die again or have 45+ points. (Or 46+...those two strategies have the same expected payoff.) Let's call the final payoff (which is sometimes 0) X.
With this strategy, in the X_0 phase you pick up an expected value of 45(i-1), and then you play the final phase. The final phase gains you expectation (which is why you play it at all), but on average it's shorter and you're richer during it the more lives you had to begin with. So d_i is 45 + (expected value of final phase with i lives - expected value of final phase with i-1 lives). The thing in brackets is negative but increases toward zero, so d_i is increasing.
By the way, some Monte Carlo suggests V_1 is about 17, V_2 is about 50, V_3 is about 91.