Research in the original Master of Orion Game
In the original Master of Orion, each technology has a cost C. After you spend a certain amount of money on research (usually less than C), you will have a 1% chance per turn of discovering the technology. If you spend more money, then the probability of discovery increases.
Some technologies give you an immediate boost in income—for example, improved eco restoration and reduce industrial waste 80%. If you only have a 1% chance per turn of discovering these technologies, it would seem to be a good idea to spend a little more on research to improve the probability of discovery thereby decreasing the average time to discovery.
The optimal probability of discovery can be deduced from the cost of increasing the probability of discovery and the amount of income the new discovery will produce. Suppose that it costs C/100 dollars to increase your discovery probability by 1%. Further suppose that upon discovery, your income will increase by i dollars per turn. Then, invest more money if
sqrt( i / C) > p
where p is the current probability of discovery.
The way to “prove" this is to look at the income “lost” while waiting for discovery. If your current probability of discovery is p, then it will take “on average" 1/p turns to discovery the tech. On average, you will “lose” i/p dollars if you just wait. If instead you invest $1 more into research, then your probability of discovery will increase to p_2 = p + 1/C and you will now “only” lose i/p_2 dollars while waiting. So you should spend $1 if
i/p > 1 + i/p_2
i/p > 1 + i/(p + 1/C)
i/p > 1 + i(p-1/C)/(p^2 - 1/C^2). (using the fact that (p+1/C)*(p-1/C) = p^2 - 1/C^2)
Let’s assume p^2 is significantly larger than 1/C^2 and drop that term from the denominator. Now, you should invest if
i/p > 1 + i(p-1/C)/p^2
p i > p^2 + i (p-1/C)
i p > p^2 + i p-i/C
0 > p^2 - i/C
i/C > p^2
sqrt(i/C) > p.
The same thing can be proven with calculus.
Cheers,
Hein
Some technologies give you an immediate boost in income—for example, improved eco restoration and reduce industrial waste 80%. If you only have a 1% chance per turn of discovering these technologies, it would seem to be a good idea to spend a little more on research to improve the probability of discovery thereby decreasing the average time to discovery.
The optimal probability of discovery can be deduced from the cost of increasing the probability of discovery and the amount of income the new discovery will produce. Suppose that it costs C/100 dollars to increase your discovery probability by 1%. Further suppose that upon discovery, your income will increase by i dollars per turn. Then, invest more money if
sqrt( i / C) > p
where p is the current probability of discovery.
The way to “prove" this is to look at the income “lost” while waiting for discovery. If your current probability of discovery is p, then it will take “on average" 1/p turns to discovery the tech. On average, you will “lose” i/p dollars if you just wait. If instead you invest $1 more into research, then your probability of discovery will increase to p_2 = p + 1/C and you will now “only” lose i/p_2 dollars while waiting. So you should spend $1 if
i/p > 1 + i/p_2
i/p > 1 + i/(p + 1/C)
i/p > 1 + i(p-1/C)/(p^2 - 1/C^2). (using the fact that (p+1/C)*(p-1/C) = p^2 - 1/C^2)
Let’s assume p^2 is significantly larger than 1/C^2 and drop that term from the denominator. Now, you should invest if
i/p > 1 + i(p-1/C)/p^2
p i > p^2 + i (p-1/C)
i p > p^2 + i p-i/C
0 > p^2 - i/C
i/C > p^2
sqrt(i/C) > p.
The same thing can be proven with calculus.
Cheers,
Hein
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