MMP.8.SD:5.3 - Retain accumulated progress when the terminal goal changes

In a two-round game, all awarded points are observed immediately. The first move is either A, giving 2 points with probability 0.5 and 0 otherwise, or B, giving 1 point certainly. In the last round, the player chooses Safe, adding 1 point certainly, or Gamble, adding 3 with probability 0.4 and 0 otherwise. The gamble’s outcome is independent of the first round.

The objective is initially to maximize the probability of finishing with at least 3 points. Let c be points already earned. The terminal contribution is g(c)=1 when c≥3 and 0 otherwise. With zero intermediate contribution, the continuation compares terminal attainment:

Therefore first move A gives 0.5*1+0.5*0.4=0.7, while B gives 0.4. Choose A and condition the last move on earned points. Retain the state (round, c). The round number alone is insufficient: histories with c=0 and c=2 require different continuations.

If the objective were expected final points, Gamble’s expected increment 1.2 exceeds Safe’s 1 at every c. Both first moves have expected increment 1, so both then give 2.2 expected final points. That calculation answers a different question from attainment probability.

Now change the requested terminal target from 3 to 4 points. From c=2, Gamble attains it with probability 0.4 and Safe fails. From c=0, neither last move can attain it. Thus A gives 0.5*0.4+0.5*0=0.2. B followed by Gamble gives 0.4 and becomes preferable. The transition probabilities remain unchanged; the changed terminal criterion alters both the continuation and the first choice.