[Freeciv-Dev] Re: [RFC] approximations for win_chance

[Gregory Berkolaiko <Gregory.Berkolaiko@xxxxxxxxxxxx>]

On Wed, 3 Apr 2002, Raimar Falke wrote:

> On Sat, Mar 23, 2002 at 08:30:33PM +0000, Gregory Berkolaiko wrote:
> > 
> > The approximation used widely in the AI code (through kill_desire 
> > equation) is 
> > prob_to_win = 
> >   (HP_A * P_A * FP_A)^2 / ((HP_D * P_D * FP_D)^2 + (HP_A * P_A * FP_A)^2)
> > where
> > A stands for attacker, D stands for defender,
> > HP is hitpoints
> > P is the corresponding power (attack or defense)
> > FP is the firepower
> > 
> > I suspect that sometimes the values are not squared.
> > 
> > Recently I encountered an example where this approximation gives a wrong 
> > answer:
> > Attacker = Stealth Bomber(HP=20, A=21, FP=2, cost=160)
> > Defender = Partisan fortified on a mountain with a river
> >                          (HP=20, D=27, FP=1, cost=50)
> > 
> > The exact answer (win_chance) is: .8845473776
> > The approximation (with squares) is: .7075812274
> > Approximation without squares is: .6086956522
> > 
> > 0.177 absolute error might seem to be insignificant but the effect is that 
> > the bomber doesn't want to risk trying to kill the partisans:
> > kill_desire (win_chance) is: 25 shields
> > kill_desire (approx w/squares) is: -11 shields
> > 
> > Recently I played a little bit with Maple and discovered that win_chance 
> > can be very well approximated by 
> >  (HP_A * P_A * FP_A)^5 / ((HP_D * P_D * FP_D)^5 + (HP_A * P_A * FP_A)^5)
> > 
> > I don't know what is the meaning of the magic number 5, but it seems to 
> > work: I plotted numerous graphs (*.mws is available upon request) and I 
> > very rarely saw the absolute error be above 0.02.
> > 
> > In the example above my approximation gives: .9010694067 which is
> > 0.0165 greater than the exact value.  This results in kill_desire being
> > 29 shields.  I think it's not too bad.
> 
> > In future I will probably base my computations on the exact win_chace, 
> > to avoid stupid behaviour.  And when the unbelievers will see the time 
> > consumed by win_chance soaring, we can convert to my approximation.
> 
> This is ok. Have you measured how fast the approximation is?

no.
will do one day.

> 
> > By the way, it is really easy to cache the result of my
> > approximation: there are only two real parameters.
> 
> And the others?

The true parameters are
HP_D * P_D * FP_D and HP_A * P_A * FP_A

which are vulnerability and belligerence btw.

this is what physicists call "scaling laws".

> 
>       Raimar
> 
>