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	Rugged defense Information

Properties of the score formula

The score formula (voir the definition) can be written stepwise.

Let's call it NS for new score, after a new match :

NS s, w, o, z, b

function

w your win percentage so far Bn your so far bonus s your score so far, with Bn deducted o the outcome of the new match z the scenario size on the new match b the bonus for the new match NS(s, w, o, z,b) = Bn + b + (5 s + w o z /5)2 (25 s + z w2 )

Now your winnings WS for this match is NS(s,w,o,z) - ( s + B_n ) (bonus must be reintroduced, since s is your previous score with bonus deducted), this is expressed after resolution as:

WS s, w, o, z, b

function

WS(s, w, o, z) = b + w z [( 2o - w ) s + w o2 z / 25 ] (25 s + z w2 ) where s, w, o, z, b are the same as above.

The winnings function WS seen as a function of s has the following properties :

• if your score is zero, your first winnings is

  z o2 25

  + b

• the maximum winnings is achieved if your outcome o (voir Match outcome) is equal to your so far win percent (voir Win percentage) w, then winnings is equal to b + w² z / 25, whatever the win percent, whatever the score before this match.
  This maximum is of course relative to the other parameters in the function, you still win more points with a better outcome.
• otherwise, the winnings function decreases at a speed inversely proportionnal to the square of your last score, s.
  Ie the more important your (bonus deducted) score s already is, the less points you will win, if outcome and size are kept fixed (same relativity as above).
• for large scores your winnings has reached close to an asymptotic minimum and is approximately equal to

  b +	w z (2 o - w) 25

  In fact the large score effect enters the scene rather early, for scores above 4000-5000 it's fairly close to the minimum already.

  There are in sum some turbulences before 5000 points and after that your winnings is governed by the large score formula.

Last but not least, before you choose a scen, look at the average win percent for your side, and substitute it in place of the o, substitute also your own win percent in place of the w, and the scen size in place of the z.
It will give a probable winnings after a match with this scenario, should you perform as the average. You can choose the large score formula if your score is large, otherwise substitue also your score deducted from bonus for s.