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At one level, we can measure the cost of discovering tech using IPCs. Each die roll costs 5 IPCs, and one discovery will be made every six rolls, on average, yielding an average cost of 30 IPCs. Some discoveries may be worth that much or more during the course of a game, others probably not.

If we could measure the various techs in terms of IPCs, then we could compare them against each other. We could also weigh them against the 30 IPCs it takes to find them. (In the Classic Game, one needs to think in terms of 180 IPCs for a "package" of all techs; the first one might not be heavy bombers.)

Everyone "knows" that heavy bombers carry a disproportionate value. Can we now construct a tool to find out how disproportionate, and whether reducing the value of the tech from three to two die rolls is adequate? And the reverse question for some others: do they need to be upgraded to make them worth researching?

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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.

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Could your "auction" mechanism work for tech? That is, can we (over time) estimate the value of tech by the number of IPCs people are willing to pay to try to get it? Maybe heavy bomber research (the TWO roll version) should cost 15 IPCs a roll, and super subs 2 IPCs, etc. –  Tom Au Jun 3 '11 at 14:37
@Tom: Yes, setting prices by demand is probably the only way to find their true value, although your first-principles method probably provides a decent first guess, until real data tells us how to adjust the guesses up or down. You make another important point not in my answer: any implementation like should probably set a different price for each tech. That eases the burden of ensuring the techs are all balanced in ability, because if one's a little worse than the others it would just be cheaper. Now the only reason to try and balance techs is so they all get bought at some point or another. –  Adam Wuerl Jun 3 '11 at 14:56

I'm going to take a crack at answering my own question. My answer has certain limitations, insofar as it is a purely mathematical, mechanistic construct. It also applies to starting "endowments" of military and territory, and doesn't allow for the fact that new techs may affect strategies (e.g. jet fighters make you build more fighters than otherwise). Nevertheless, something is better than nothing.

The least useful tech is super subs. It improves subs' combined combat factors (attack plus defense) by 25%, (five versus four). Applied to an IPC value of 8, that's a gain of 2 IPCs. And most countries have only on sub. I can't see spending 30 IPCs to research this. Others pay for themselves in about four turns (or less).

Likewise, the combined combat factors of jet fighters is improved from seven to eight. Applied to their 12 IPCs, the gain is about 2 IPCs. Most. countries have three or four fighters so the gain is 6-8 IPCs.

Economic relief reduces IPC costs of units by 1. The gain is 50% for infantry and 4% for bombers. Arbitrarily, I said that the "average" unit is a tank, for a 25% gain. That's a pick up of 6 IPCs (out of 24) for Russia, and 9 IPCs for the US.

Rockets give (one) anti-aircraft gun the value of a bomber. Essentially, your 5 IPC gun has been upgraded to a 15 IPC bomber for a gain of 10 IPCs.

Long range aircraft? Bombers and fighters are about equal in combat capacity (the first on offense, the second on defense, seldom the other way around). But bombers cost 3 more IPCs for two more zones of range. Using this value for the upgrade, and an average of four "planes" per country, the gain is 3 x 4 or 12 IPCs.

And heavy bombers themselves? In the original version, a heavy bomber is worth THREE normal bombers. Since the latter are worth 15 IPCs, the gain is 30 IPCs, way out of proportion to the others. Using the "two roll" rule, the pickup is more like 15 IPCs, at the high end of the "pack," but no longer over the top.

My preference is to use modifications of long range aircraft (plus 1 square instead of 2) for a gain of about 6 IPCs and the 50% better heavy bomber (THREE good rolls on two dice instead of four; strategic bombing averages 5.5 hits instead of 3.5), for an IPC gain of about 8. These changes would make "rockets" (ten IPCs) the best tech to research.

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I disagree with your valuation of the techs. One thing that has to be taken into account beyond how many of that unit a country is likely to have is if they are in a position to build more of it. Subs to Russia are worth zero, they likely don't have any and only have one port city to build new subs in, and don't have anyone to with them anyway. Economic relief only provides value for new purchases, which are likely infantry so that increases buying power by 50%. US has around 30/turn, so that's 15 IPCs in one turn, or ~12 for Russia, or 18+ for the Axis. This tech is way too good. –  Adam Wuerl Jun 3 '11 at 13:49
As for the rocket tech, it's too simplistic to say it's worth 10 IPCs because an AA gun has been turned into a rocket. A bomber can redeploy at 6 spaces a turn, an AA gun at one. A bomber can participate in attacks, including attacks on ships, a rocket cannot. A bomber has more range, as it can land closer than it took off. The only benefit a rocket has is that it cannot be destroyed by the AA gun. If we try and value the rocket at all it should be the IPC damage it can inflict: an average of 3.5/turn. Pretty much useless. –  Adam Wuerl Jun 3 '11 at 13:52
@adam: Re: economic relief. Suppose we say "-1 cost" EXCEPT FOR infantry (which remains 3). So my "6-9 IPCs" per turn is a CAP, rather than an average. Is it still "too good?" –  Tom Au Jun 3 '11 at 14:50
That probably fixes it, since lots of time all that people would want to buy is infantry anyway. –  Adam Wuerl Jun 3 '11 at 14:57

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