From: "Mircea Pauca" <mpauca@fx.ro>
Subject: Review: Old ARNHEM comp. game, nice non-linear formula

    I recently replayed the old ARNHEM of ZX Spectrum fame
((c)R.T.Smith, 1985, CCS) which I disassembled by hand
some years ago: 14 K of masterfully written machine code,
and 25 K of data, tables, maps etc.
    Covers the whole Market-Garden operation, battalion
level, 3 turns/day (17 Afternoon-25 Night), 1km/square.
Units are 2x2 squares, stacking 2 in any square. This
renders nicely the traffic jams which occured.
    DESERT RATS (WW2 African campaign) also uses
similar mechanics, more emphasis on limited supply.
    Very nice games for their time !
    The artificial opponent (only for Germans) is passable
with standard plans and some reaction rules.

    Each turn is divided into phases for movement:
Phase 1: mechanized+recon
Phase 2: infantry+recon
Phase 3: mechanized+recon
    each subdivided, in order, in a British, US and German
sub-phase. In each, units can move 4 km (8km in Travel
on roads) and then fire (in one phase per turn only).
    The model is fully deterministic, no random elements.

Each unit has a strength at beginning of turn (STR) and a
effective, current strength (EFF, displayed as % of STR).
The firing uses usual (linear) formulas, modified for terrain.
Enemy EFF is reduced by a fixed percentage of own EFF.

The nice part is the end-of-turn recovery formula.
    STRnew=EFFnew=2*EFF-(EFF^2/STR)
which makes the final losses, the _square_ of initial losses,
both as proportions of original strength. So if 1/3 of forces
are affected, 1/9 are definitively gone, the rest comes back
( 2/9 : stragglers, disorganized groups etc.)
    This makes 1:1 skirmishes of small forces go on
indefinitely, but 4:1 assaults result in a definite victory.

        Examples:

a) 1 Panzergrenadier (STR=50) vs. 1 Para (STR=50)
PzG fires: 50*0.31(Mech.Inf.)*0.75(rough)=11.6 , Para EFF=39
Para fires: 39*0.25 (Para) = 9.75 , PzG Eff = 41
    After recovery:
PzG = 2*41 - 41*41/50 = 49
Para = 2*39 - 39*39/50 = 48
    This skirmish can go on nearly indefinitely without result.

b) 2 Panzergrenadiers (50) vs. 1 Para (50)
Each PzG fires: 2*11 = 22, Para EFF = 28
Para fires: 28*0.25 = 7 effect, divided by 2 opponents = 3,
PzG EFF = 2*47
After recovery: PzG =2*50 (unharmed)  Para=41
    The defense will collapse in 3-4 turns.

c) 1 Panzer (60) + 2 PzG (50) + 1 artillery (50) vs. 1 Para
    Instant destruction, no chance of return fire.

    Mircea-Valer Pauca (mpauca@fx.ro)