There are many games that have an investment phase followed by an exploitation phase including the card game Dominion, Terraforming Mars, Slay the Spire (during a combat), Saint Petersburg, Roll for the Galaxy, Master of Orion, Hansa Teutonica, Century Spice Road, Cash Flow and many more games.
One of the basic ideas is that you should not invest in something if there is not enough time for the investment to pay for itself.
A VERY SIMPLE HYPOTHETICAL GAME
Suppose that there are $T$ turns left in a game excluding the current turn, your current income level is I dollars per turn, and you have two choices:
- (invest) Invest this turn to increase your income from $I$ dollars per turn to $I +i$ dollars per turn. You would then receive $I+i$ dollars per turn for the remaining $T$ turns for a total of $(I+i)\cdot T$ dollars.
- (don’t invest) Receive $I$ dollars this turn and the remaining $T$ turns for a total of $(T+1)\cdot I$ dollars.
EXAMPLE
For example, assume that
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it is currently turn 5,
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at the start of turn 5 you have an income of $I$ = \$2 per turn,
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the game ends on turn 9, and
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you have the choice between two options:
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Invest this turn to increase your income from \$2 per turn to \$3 dollars per turn. You would then receive \$3 dollars per turn for turns 6,7,8, and 9 for a total of \$3 * 4 = \$12 dollars.
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Do not invest and you receive \$2 on this turn, turn 5, and the remaining turns 6,7,8, and 9, for a total of \$$2\cdot5$ = \$10 dollars.
For this example, $T=4$ because there are 4 remaining turns after the current turn. You have the option of increasing your income from \$2 to \$3, so the investment increases your income by $i$ = \$1 per turn.
The Correct Strategy
If you choose option 1, then your total earnings will be
invest_option_returns = $T\cdot (I+i)$ dollars.
If you choose option 2, then your total earnings will be
no_investment_option_returns = $(T+1)\cdot I$ dollars.
So, you should invest if
$$\begin{aligned}T\cdot (I+i) &> (T+1)\cdot I \\T\cdot I+T\cdot i &> T\cdot I + I \\T\cdot i &> I \\T&>\frac{I}{i}. \\\end{aligned}$$
So, according to the math above, you should not invest in something unless there are more than $$\frac{I}{i}= \frac1{r}$$ turns left in the game after the current turn where $$r=\frac{i}{I}$$ is sometimes called the return on investment (ROI).
When you invest $I$ dollars to get an increase of $i$ dollars per turn, it takes $I/i$ turns for the investment to “pay for itself”.
The Gain
If there are more than $\frac1{r}=\frac{I}{i}$ turns in the game, then you will gain
gain = invest_option_returns – no_investment_option_returns
$$\begin{aligned}&=T\cdot (I+i) – (T+1)\cdot I\\&=T I+T i – T i – I\\&= T i – I \\&= T r I – I \\&= (Tr -1) I \\&= I r \left(T -\frac1{r}\right) \end{aligned}$$
dollars by choosing to invest. The investment pays for itself after $1/r$ turns and every turn after that gives you $I r=i$ dollars.
APPLYING THE CORRECT STRATEGY TO THE EXAMPLE
For the example, $T=4, I=2,$ and $i=1$, so $r = i/I =0.5$. The correct strategy is to invest if
$$T > I/i,$$
but T=4 which is greater then I/i = 2, so the best option is the investment option.
I am hoping to write about applying this idea to several games over the next few weeks.