Re: [computer-go] An [open] question on game tree search theory

[Daniel Hallmark <dgh@xxxxxxxxxx>]

On 19 January 2005, Antti Huima wrote:

> What is a probability of winning?
> 
> Probability that the playing algorithm will win against a hypothetical, 
> fixed opponent mechanism---which is circular, because the evaluation 
> function affects this probability? And what is the opponent model?
> 
> Or probability that the game tree node is a game-theoretic win? What 
> does this mean? One game node is either win or not, so there is no 
> stochastic experiment. You need to have a repeated experiment or a 
> population larger than one object to be able to speak of probabilities.
> 
> I'm not being rude. I think this is one of the key points. "Probability 
> of winning" is often mentioned, but what does it mean? Every node is a 
> game-theoretic win or not, so no probability here.


May I ask a question?

You have defined G(t) as consisting of only 3 discrete values:
Black wins, draw, or White wins.

If it is true that there is such a G(t) for every board state, then
I would agree that probability never enters into the equation except
for the probability that E(t) gives us an answer that directly
correlates to G(t) for that state.

However, and this is most likely my ignorance of Game Theory showing,
can it be conclusively shown that your definition for G(t) is correct
for the game of Go?

Is there not a possibility that for a given game state t, some
percentage of G(t') are unrefutable wins for white, some are
unrefutable wins for black, and some can be fought to a draw?

If this is the case, can your definition of G(t) hold for all
positions?  If not, then you may have some board positions which
do have definite discrete game theoretic values.  But perhaps there
are a class of positions for which the game theoretic value can only
be expressed as a probability distribution that depends on G(t').

If that is true, then it would seem that your statement that there 
is no probability associated with winning can only be true because
you have defined the game/problem in those terms.

Daniel Hallmark

P.S. I really am uneducated with respect to game theory.  No
disrespect is intended, and if I have misunderstood your question
please feel free to correct me if you think it is worth your time.
_______________________________________________
computer-go mailing list
computer-go@xxxxxxxxxxxxxxxxx
http://www.computer-go.org/mailman/listinfo/computer-go/