Planning the conversion of research units into patents

[tympakie]

As explained in the Research Guide, converting a Research Unit (RU) into a patent has a random outcome. If a company has no Chief Technical Officer (CTO), the probability of success is 1:8, that is the same as for getting 3 heads in a row, while tossing a coin.
The Achievement guides written by players suggest that a novice player builds a research laboratory in order to win scientist achievements.
But most of the times we haven’t got or don’t want to build the suitable Research Laboratory. Thus we are faced with the problem of buying RU’s from the exchange or from other players. RU’s are expensive and we don’t want to buy too many. On the other hand we may meet the chance of buying some from the exchange at a lower price than usual and don’t want to miss it. How many should we buy?
The following plots and tables would give a help to solve the problem, or at least a reference.

First case: a company has no CTO and starts from quality 0 for a given item. How many trials are needed in order to convert research units into patents to reach quality 1, or quality 2, or quality 3?

As for prospecting mines, quarries or rigs, there is no definite answer to this question, because the number of occurring trials is completely random. But it has a known probability distribution (called negative binomial, or Pascal distribution), well centered around a mean value, as we can see from the following plots.







Back to our question: how many trials shall we plan?
A reasonable answer is: a number in the range of the green or blue ribbons in the plots. This range is bounded by the values corresponding to 25% and 75% probabilities in the following table. For example, to get Q1, half of the times we'll need from 77 to 112 trials.
The table also gives:

• the median, corresponding to the probability of 50% in the distribution, which is a value locating the center of the distribution, plotted as the separation among green and blue ribbons;
• the mean, plotted as a black vertical line;
• the 0.135 percentile and the 99.865 percentile, that are two values that bound typical runs from extremely rare runs. (They correspond to 3 sigma deviations for a Gaussian variable)

Quality	Patents	-3 sigma	1st quartile	Median	3rd quartile	+3 sigma	Mean
		0.135%	25%	50%	75%	99.865%
From 0 to 1	12	38	77	94	112	194	96
From 0 to 2	62	339	455	494	534	693	496
From 0 to 3	562	3984	4375	4494	4614	5098	4496

A very conservative approach:

It would be somewhat unusual (less than once in four runs) that you needed less trials than the value listed as the first quartile. Thus start buying as many. After converting these, buy just as many new RU’s as the number of patents still missing to reach the target quality, if you can without missing a good price, and go on this way until you reach your goal. End up in buying research units and spending them one by one. But this is certainly too conservative.

Second case: we already own a quality for a given item and want to obtain the higher one.

For example we have quality 1 and want to rise to quality 2.

Plots and tables follow, with the same conventions as before.

Quality	Patents	-3 sigma	1st quartile	Median	3rd quartile	+3 sigma	Mean
		0.135%	25%	50%	75%	99.865%
From 1 to 2	50	261	363	398	434	579	400
From 2 to 3	500	3518	3886	3998	4111	4522	4000
From 3 to 4	2000	15016	15773	15998	16224	17024	16000

Third case. From quality 3 to quality 4, without or with a Chief Technical Officer.

In order to get quality 4, it would be more convenient to hire a CTO, as explained in the Research Guide.

Let us compare plots and numbers of the trials needed to go from quality 3 to quality 4 in the cases that our company has no CTO, or it has a CTO with 10 or 20 skills.

Skills of CTO	-3 sigma	1st quartile	Median	3rd quartile	+3 sigma	Mean
	0.135%	25%	50%	75%	99.865%
0	15016	15773	15998	16224	17024	16000
10	13657	14341	14543	14748	15470	14545
20	12525	13147	13331	13518	14174	13333

Finally, let us remark that hiring a CTO who raises the probability of converting a research unit into a patent, also has the effect of decreasing the width of the distribution; i.e. it becomes less probable that a run to get a higher quality turns out into a rarely long one.