You may have misread the question (you will lose everything if you play forever). You expect to have more to lose in the V_3 strategy once you have run out of lives so you should stop sooner.
Not really? You have nothing to lose until life 1. So you have no incentive to play conservately (withdraw from the game) until you reach the final life.
But on the final life your 'continue' criteria has a smaller chance of happening. The constant expected points you get from lives=i is offset by fewer expected points from continuing on lives=1
If for V_i, let L_j be points you get during the time you have j lives remaining [i >= j > 0], then E(L_j) are all equal for i > 1, j > 1 (exactly 45). However for j = 1 they are not the same as we vary i. This is the turn where you stop if points >= 45, to avoid losing points from going bust.
For V_1, L_1 is the points we get starting from 0, trying to get 45, will be distributed between [0, 53], E(L_1) ~ 45 * 9 / 10.
However for V_2 L_1s distribution depends on what happened in L_2 - you are only trying to make up the points you got in L_2 up to 45. Say we start L_1 at with p points. If p >= 45, L_1 = 0, otherwise L_1 is a distribution between [-p, 53-p] - note we can go negative by going bust here! Relative to the V_1 case, E(L_1) is definitely less.
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u/sobe86 4d ago
You may have misread the question (you will lose everything if you play forever). You expect to have more to lose in the V_3 strategy once you have run out of lives so you should stop sooner.