I think the relation of information to power, historically, will be too complicated and too intertwined with other variables for you to discover. Even if you count those other variables as part of the starting conditions technology has a tendency to make different aspects of the starting conditions salient such that you can't evaluate the effect starting conditions have on power until you know the path of technology. Moreover, power is a function of memetic technologies as much as physical technologies and the former will be even more difficult to quantify. You'd be better off making the function of information to power a variable in your models.
But if you persist you should keep in mind the distinction between offense-dominant systems and defense dominant systems. Offense-dominant systems occur when technology is good at invading/destroying enemies but bad at protecting you from them. Defense-dominant is the reverse. Knowing whether a technology is defensive or offensive is crucial for understanding its impact. Some of history's biggest military blunders occurred when one side misapprehended their system. For example, in WWI the Germans thought they were still in a offense-dominant system so they thought they executed the Schlieffen Plan and tried to beat France quickly and then shift their forces east to avoid a two-front war. But the invention of the machine gun/trench war technology meant that it was much, much harder to take territory than anyone had expected and so the Germans couldn't get take France fast enough (there were other reasons). Germany wasn't alone in making this mistake, lots of countries thought the war would finish quickly. Then in WWII much of Europe was still thinking in terms of the defense-dominant system of WWI and so were shocked at the speed of German progress... but such progress was inevitable given advances in tank and aircraft that rendered trench warfare tactics useless.
The more agents perceive that their system is offense-dominant the more unstable the system is since agents estimate the benefits of doing well in a conflict, and the costs of doing poorly, to be high. Mutual second-strike nuclear capabilities is actually an extremely stable system for this reason. And mutual first-strike capabilities is about as bad as it gets. Anyway, it seems this distinction would be important for any modeling of power.
Interesting point about offensive/defensive power.
You'd be better off making the function of information to power a variable in your models.
Given an amount of information, I need to compute a corresponding amount of power. "Make it a variable" doesn't help. It's already a variable. I have too many variables. That's why I want to make it a function.
I'm thinking about how to model an ecosystem of recursively self-improving computer programs. The model I have in mind assumes finite CPU cycles/second and finite memory as resources, and that these resources are already allocated at time zero. It models the rate of production of new information by a program given its current resources of information, CPU cycles, and memory; the conversion of information into power to take resources from other programs; and a decision rule by which a program chooses which other program to take resources from. The objective is to study the system dynamics, in particular looking for attractors and bifurcations/catastrophes, and to see what range of initial conditions don't lead to a singleton.
(A more elaborate model would also represent the fraction of ownership one program had of another program, that being a weight to use to blend the decision rules of the owning programs with the decision rule of the owned program. It may also be desirable to model trade of information. I think that modeling Moore's law wrt CPU speed and memory size would make little difference, if we assume the technologies developed would be equally available to all agents. I'm interested in the shapes of the attractors, not the rate of convergence.)
Problem: I don't know how to model power as a function of information.
I have a rough model of how information grows over time; so I can estimate the relative amounts of information in a single real historical society at two points in time. If I can say that society X had tech level T at time A, and society Y had tech level T at time B, I can use this model to estimate what tech level society Y had at time A.
Therefore, I can gather historical data about military conflicts between societies at different tech levels, estimate the information ratio between those societies, and relate it to the manpower ratios between the armies involved and the outcome of the conflict, giving a system of inequalities.
You can help me in 3 ways:
If you choose the last option, choose a historical conflict between sides of uneven tech level, and post here as many as you can find of the following details:
For example:
Using the two dates 1415 and 1346 leads to some tech-level (or information) ratio R. For example, under a simple model assuming that tech level doubled every 70 years in this era, we would give the English a tech-level ratio over the French of 2, and then say that the tech-level ratio enjoyed by the English produced a power multiplier greater than the manpower ratio enjoyed by the french: P(2) > 30000/5900. This ignores the many advances shared by the English and French between 1346 and 1415; but most of them were not relevant to the battle. It also ignores the claim that the main factor was that the French had heavy armour, which was a disadvantage rather than an advantage in the deep mud on that rainy day. Oh well. (Let's hope for enough data that the law of large numbers kicks in.)
After gathering a few dozen datapoints, it may be possible to discern a shape for the function P. (Making P a multiplying force that is a function of a ratio assumes P is linear, since eg. P(8) = P(8/4)*P(4/2)*P(2/1) = 4*P(2); the data can reject this assumption.) There may be a way to factor the battle duration and the casualty outcome into the equation as well; or at least to see if they correlate with the distance of the datapoint's manpower ratio from the estimated value of P(information ratio) for that datapoint.
(I tried to construct another example from the Battle of Little Bighorn to show a case where the lower-level technology won, but found that the Indians had more rifles than the Army did, and that there is no agreement as to whether the Indians' repeating rifles or the Army's longer-ranged single-shot Springfield rifles were better.)