AI level 4 and 5: Entropy and energy

In this video, yellown kart is using "level 3" intelligence so it score an option with N different futures with Ln(N), and it would be ok if all futures were equiprobable, but they aren't.

In the case not all futures are equipropable you have to switch to another, more complex, way of calculating entropy.

When you have N microstates but each one has a different probability of happening, call it P(i), in the instantaneous or "clasical" entropy, we use:

S = Sum. on all possible microstates(P(i)*Ln(P(i)))



If it were about a gas molecules, the Ln(P(i)) part will correspond to the momentum of the molecules, so translating it into our case, it correspond to m*v, the momentum of the kart. As m is the mass and it is a constant, we can forget about it, and only v remains, the kart velocity.

But we are talking on futures, so we need to integrate v -the kart velocity- over the path that followed the kart, and as we use constant time deltas to construct the futures, on each step v is proportional to the raced distance, call it x.

Integrating x over a path gives you r^2/2, with r = length of the path, so using r^2 seems to me like the perfect way to mimic the real entropy on the AI model (it may be wrong, not sure I really understand all the inners of the entropy at this level).

So lets compare and see!

Yellow kart: Use Level 3 with score = Ln(N).
Grey kart: Use level 4 with score = Sum.(r) (r = raced distance)
Black kart: Use level 5 with score = Sum.(r^2)

In my modest opinion, level 5 on black kart properly reflect the original paper formulaes for future entropy, and makes a really great job as a driver.

Can't we do it any better? It is a perfect AI? No, it is a nice and general algorithm, but there are still some aspect of it that can be tweaked for better, not many, but some.