Is it ever a good idea to research weapons development?
Ok, so I asked a question over at math.stackexchange.com to get some advise on how to analyze the various weapon development options (see How to analyze risk vs. reward for spending on research and development work). Mike Spivey suggested using a decision tree, which I had some familiarity with from past classes in probability and statistics. In short, the decision tree calculates the expected value for spending 0 through 30 IPC on a turn for weapons development research.

To construct this decision tree (see below), I made some assumptions:


Heavy bombers are the only weapons development "worth" anything. In short, any money spent that resulted in discovering a tech other than heavy bombers was considered wasted.
Discovering the heavy bombers tech is worth some amount of IPCs. Because this value may differ per player (that is, I may be willing to pay, say, 50 IPC to get heavy bombers while someone else may be willing to pay 100), I created an Excel spreadsheet where this "worth" could be plugged in and the expected value would be computed for spending 0 to 30 IPC on a turn for rolling for techs.


Here is the decision tree I constructed to compute the expected values for spending 0 to 30 IPC in a turn, meaning paying for 0 to 6 rolls of the die.



The expected value of one of the rolls is computed by adding together the value (the amount in the bottommost boxes) times the probability of getting there for a particular branch in the tree.  I presume the player has a purse of 30 IPC at the start of their turn.

For instance, if you do zero rolls (the far left branch) then you have an expected outcome of 100% * 30, or 30 IPC. In other words, you'll always end up with a purse of 30 IPC after the rolls. However, if you decide to spend money to do one roll then you have a ~2.75%  chance of getting Heavy Bombers (1/36). Doing so yields a purse of 25 + X, where X is the "worth" of having heavy bombers. There is ~97.25% chance that you will not get heavy bombers, in which case you have a purse of 25 IPC. So the expected value of one roll is 2.75% * (25 + X) + 97.25% * 25. Of course, the actual hard number depends on X, which is what heavy bombers is worth to you.

I created a very simple Excel spreadsheet that calculated all seven expected values for a given value of X, where X was a number in a particular cell. Using this spreadsheet I could play with X to see at what point it actually makes sense to roll for weapons development.

The chart below shows how the expected value changes as the value of X grows. The X-axis shows the expected outcome for zero to six rolls. The series show how the expected value changes as the value of the heavy bombers tech (X) increases. Here I show the expected value for six different values: 45, 90, 135, 180, 225, and 270.



Note how at a "worth" of 180, heavy bombers have the same expected value regardless of how many times you roll the die. If the value is lower, however, it makes sense to abstain from weapons development altogether; if its higher to you, spend every last penny you have on development! 

Personally, I think heavy bombers are not worth this much. 180 IPC is the combined budget of 5-10 turns. It is equivalent to purchasing 12 standard bombers. Forgoing any sort of unit procurement for that many turns, just to get heavy bombers, is a suicidal thought. 

In closing, keep in mind that this is a very simple model. It presumes heavy bombers are the only worthwhile tech. It does not factor in that if you discover other technologies, that the odds of discovering heavy bombers actually increase since if you roll an initial 6 you get to keep rolling for your discovered tech until you discover a new one. Also, the analysis presumes the player is in a vacuum of sorts and does not factor in that he must spend certain amounts of IPCs to hold his territories from the enemy, not to mention the monies needed to go on the offensive.