Master of Orion and the Lambert W function

The game Master of Orion, first released in 1995 by Microprose, entails settling the galaxy planet by planet.  To settle a planet, the player must construct a colony ship on one of her/his planets, send that ship to a new, unoccupied habitable planet, land the ship, and then send colonists from some of their planets to the new planet.  Early in the game,

  • it costs about 550 MC to build the colony ship,
  • it takes around 5 turns for the colony ship to reach a new planet,
  • it takes and additional 5 turns for additional colonists to arrive on the planet on separate (free) transport ships,
  • it takes around 20 turns for the planet to build new factories before it can build new colony ships, and
  • the new planet might generate 110 MC per turn of income to build more ships,

so 30 turns after investing 550 MC, the player’s income increases by about 110 MC per turn.

If we try to create a differential equation to model the early stage of the game, we get something like

$$y'(t) = \frac{110}{550} y(t-30) = \frac{1}{5}y(t-30) $$

where $y(t)$ is the income available for building ships on turn $t$. This differential equation has a solution of the form $$y(t) = c \exp(\alpha t).$$ We can compute the derivative and substitute into the differential equation to find $\alpha$.

$$y'(t) = c\alpha \exp(\alpha t).$$

$$\begin{aligned} c\alpha \exp(\alpha t) &= y'(t) =\frac{1}{5} y(t-30)\\c\alpha \exp(\alpha t) &=\frac{1}{5}c \exp(\alpha(t-30)) \\\alpha &=\frac{1}{5} \exp(-30 \alpha) \\\alpha \exp(30 \alpha) &=\frac{1}{5}\\30 \alpha \exp(30 \alpha) &= 6 \end{aligned}$$
According to the definition of the Lambert W function $W_0(x)$ for $x>0$,
$$ W_0(z) = x$$
if and only if
$$ x\exp(x) = z.$$
Setting $x=30\alpha$ above gives
$$\begin{aligned}W_0(6) &= 30\alpha\\ W_0(6)/30 &=\alpha\\ \alpha &= W_0(6)/30\approx 0.0477468.\end{aligned}$$
So the the economy at the start of the Master of Orion gain should grow by about 5% per year.

More generally, if

– the time between the investment in the colony ship and the maturing of the colonized planet is $T$,
– the cost of the ship is $C$,
– and the mature planets produces $i$ income per turn,
then the income on turn $t$ can be approximated by $$y(t) = y(0) \exp(\alpha t)$$ where
$$\alpha = \frac{W_0\left(\frac{i T}{C}\right)}{T}.$$

(Thanks to for helping me format the TeX.)