In Axis and Allies, Is There A Way to Measure the Benefits of Various Techs Against Each Other? And the Cost of Discovery?
Your only hope is to give each tech a deterministic, crowd-sourced price
In practice, I think the game's tech mechanic is too complicated to yield a useful mathematical model. The average (i.e. expected) cost of successfully researching a technology--or of researching one in particular--is only part of the story. The more important metric is the variance of the cost; that is, the average may be 30 (or 180) IPCs, but how often is it going to cost 5 and how often 300?
It is this uncertainty that drives player's decisions as much as the mean because everyone's desperation and risk tolerance is different. I have a low risk tolerance and prefer to play a slow-and-steady game, believing that superior play will eventually win out and balance out all but the worst of rolls. Thus I never research tech, feeling that the times when I get it cheap will be outweighed by the certain defeat earned if I spend two or three turns researching to no avail. Some people love to take the risk knowing that if they get lucky they can almost assuredly win the game, and don't mind losing the majority of the times they don't get it.
Now to make it even more complicated, this risk calculus isn't static but is influenced by the changing game situation. For example, I've played several games where the other player succeeded in a desperation bid for heavy bombers. Although I was never willing to go for it before, as soon as that happens I know I have to match them or lose the game.
Any proposed tech model must have formulas for the expected cost and the variance (which could be done by folks better at probability math that I). The almost impossible part is that it must have a model for the value of the tech specific to each country and to the particular situation on the game board (even heavy bombers aren't that useful if the bombers aren't in range of suitable factories, and have delayed gratification if the player has very few bombers already in the arsenal).
I cannot think of a first principles method to calculate this tech value; I think the only hope would be to set the price empirically, like a bookie sets the odds on a horse race based on the bets they're receiving. Perhaps this could be done by modifying the rules to make the cost of each tech deterministic (i.e. a tech can be bought for X IPCs). Then have experienced players play a ton of games, adjusting the tech prices as necessary until whether or not a given technology is purchased has a statistically insignificant effect on the outcome of the game.
Once that data existed perhaps the model could be made more complicated by having the price be a function of the turn in which it is bought. Once those numbers were established, the mean and variance from the rolling-for-tech mechanic could be used to set the price per roll such that the median cost were equal to the market price.
Problem is you'd need a infeasible number of games as your data set.