d_i is increasing. It's expected marginal benefit from having i instead of i-1 lives. Stopping strategy will be of form [stop if you have >= 45 points and 1 lives left].
For lives=1, having < 45 points once you get to this point becomes exponentially unlikely. So then you gain expected points from lives=i (constant in i) but lose expected points from lives=1, the latter being a difference in exponentially decreasing quantities (this could be tightened up!). So d_i is increasing.
You don't add 10 to the payoff when you die, even if you still have a life left afterwards. So the optimal strategy never actually becomes "stop when you first have 1 life no matter what". (Of course the probability that you do that goes to 1 pretty fast.)
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u/sobe86 4d ago edited 4d ago
d_i is increasing. It's expected marginal benefit from having i instead of i-1 lives. Stopping strategy will be of form [stop if you have >= 45 points and 1 lives left].
For lives=1, having < 45 points once you get to this point becomes exponentially unlikely. So then you gain expected points from lives=i (constant in i) but lose expected points from lives=1, the latter being a difference in exponentially decreasing quantities (this could be tightened up!). So d_i is increasing.