[Consim-l] Re: working backwards (ex: GO TO rules vs COME FROMprediction)

[Mircea Pauca]

[Mike]
> None of which sounds like any profound discovery, and I've completely lost 
> the connection to COME FROM programming, but, what the heck, it has been 
> fun kicking this around one more time. :)


    I really wanted to deduce final probabilities for a certain (class of)
states, from a given initial state. But contrary to this forward-going
description, the calculation must go backwards, summing all posible
states where the currently calculated state might COME FROM.
    Say we analyze the very simple Axis&Allies, with only one troop
type per side (or a fixed order of losses between different types, which
is not much heavier). To calculate the probability of "Attacker Wins"
one component of it is, say, "Attacker 5 Defender 0". To get that,
one needs to sum all the probabilities of:
P(5:1) * P (0:1+ loss given 5:1) +
P(5:2) * P (0:2+ loss given 5:2) + ...
...
P(6:1) * P(1:1+ loss given 6:1) +
...
    And if rule modifications or different decisions may apply, they
also cumulate in that end result, but the final state may COME
FROM the same initial state through any of these... That was difficult,
in early versions the probs didn't sum to 100% because there were
more ways for the path to COME FROM, not yet analyzed.
    Optimization and decisions also generates problems, particularly
if one doesn't know what objective is perceived by the other side
to optimize, or if that fluctuates (with errors or just craziness).
    Thomas Schelling said about that in the 1960's book "Strategy of
Conflict" - in extremely simplified game theory - about nukes, of course.
Could the West have known Hitler wanted to Bulge, and what German
troops, preparing it as secretly as possible, could have done instead ?

    It gets really hairy if the state space is not discrete and 
2-dimensional
(representable on a matrix as above) but spatial, on a map.
    To get the historical example: suppose Ike on June 7th, 1944
wanted to know the probability of "Allied victory" until May, 1945.
That depended very much on the definition of the class of states
called "victory" (e.g. Russians reaching the coast of Spain was not
what Western Allies preferred). But to get it *rigorously*, one
needed to conceive every intermediary state between the known
now and the analyzed future, and every possible transition, and
every possible tech or tactical advance that changed the rules...
    For continuous states, I'm just overwhelmed. Even ONE real
number (if known with any resolution desired) contains infinitely more
information than any finite collection of discrete states... (proof: the
infinite string in decimal fraction representation). One dimension can
be studied, say with Cumulative Density functions and convolutions.
For several dimensions, or where the 'addressing' of the state space is
itself continuous (think miniatures placed anywhere on a map) I don't
even know the names of the math tools required. Functional spaces ?

    That's why no one does prediction right on, because it's so hard.
Instead, sides can calculate generic "Potentials" and general kinds
of preparations against any opposing counter-measure, and let the
game of Nature be its own simulator... [Answer is: 42!]
    And surprises can lead tautologically to victory, if defined like 
Dupuy...
only that I hated that definition, exactly because it's just hindsight.

    Thank you for thinking about this,
    Mircea Pauca, Bucuresti, Romania