Consider a weapon which can fire once per time period T has a 50% chance of hitting the target per shot, and a weapon which fires twice as fast with the same chance of hitting on any given shot. Over any given time interval T, the weapon which fires more slowly has a 50% chance of hitting the target and a 50% chance of missing the target. Over the same time interval T, the weapon which fires faster has a 25% chance of doing nothing, a 50% chance of hitting once, and a 25% chance of hitting twice. The faster-firing weapon therefore has a 75% chance of damaging the target over a time interval T whereas the slower-firing weapon has only a 50% chance of dealing damage to the target over the same period.
This is a common mistake.
flip a quarter and the odds are 50/50 that you get heads or tails.
flip a quarter twice and the odds are 50/50 each flip
flip a quarter 10 times and the odds are still 50/50 each flip
the odds of getting a given result on a coin flip are not influenced by the result of the prior coin flip.... you could get any combination between 5h:5t to 10h:0t or 0h:10t
There is a term for it, but I don't recall the name
End of Tetrasodium's quote
If you have a 50% probability of producing a given result on any given attempt and the probability of producing this result is independent of the success or failure of previous and successive attempts, then the probability that you will succeed at least once in N attempts is (1 - 0.5^N)*100%. For N = 2, that's a 75% chance of producing at least one event. For N = 1, it's a 50% chance of producing a result. There is absolutely nothing wrong with the example I provided. To use your example of coinflips, if you flip an unbiased coin twice, the results you can produce are [0, 0], [0, 1], [1, 0], and [1, 1]. If you flip an unbiased coin twice, you will produce at least one heads 75% of the time.
The example given only becomes incorrect when you start assuming things which were not stated - say, that the weapons have equal maximum DPS. If weapon A can fire twice in the same time that weapon B can fire once, then yes, if weapon A and weapon B have the same maximum DPS the expected damage over some time interval T which is sufficiently long that A can fire twice as many times as B is the same. However, it is not the case within either the game or the example given that weapons A and B must necessarily have the same maximum DPS. To claim that my example is wrong is either a misreading of the example (adding assumptions which were not stated; remember, equal DPS is a special case, not the general case, and so should not be assumed unless it is stated to be the case) or a failure to understand the concepts involved.
Can we quantify this somehow?
I used the example above, reduced the number of kill shots to 5 and 4 for damage boost (to make the effect of the variance more prominent) and calculated the probability of killing the target before the expected kill time. Below hit chances of 0.6 these probabilities are within 1% or 2% (absolute probability) of each other for damage and as and below 0.3 hitchance they acc is also in that range.
Interestingly enough, acc is allways the variable with highest the highest probability to kill before the expected kill time (but with no chance to for early kills, since its minimum kill time is 25% higher).
If you take the probability at 75% expected kill time, the damage boost usually has the highest probability (by a few %).
End of zuPloed's quote
Notice that the model with the higher damage per shot also has higher chances of failing to kill the target within the expected kill time, and that the probability that a given target will take a certain time to kill drops off less rapidly as the time taken to kill increases than it does for the high rate of fire model. In other words, the high damage per shot model is a high-risk model. If you have a high-risk and a low-risk model with the same expected performance, you take the low-risk model, because risk minimization is a good thing.
Also, take some time and consider the problem under discussion. It is a three-variable problem, with one variable being a random variable and the expected DPS being linear in any one of the variables. If we want to compare how the expected DPS and its distribution behave for changes in the three variables, we have a couple options. We can hold two of the variables constant while varying the third, and compare the behavior in each case, though we will need to be careful not to neglect to consider how changes in one variable impact the other variables in practice when we consider what this implies for the problem. We can apply a constraint to the problem and see how variations in the variables within that constraint modify the behavior, though we need to be careful not to choose a constraint that biases the comparison for one or more variables. We can do case studies, though these can be difficult to generalize.
You appear to have chosen to apply a constraint to the problem and see how the behavior changes as the variables are adjusted within this constraint, but the constraint you appear to have chosen biases the comparison in favor of increasing accuracy, because the constraint you have chosen to enforce is constant maximum DPS, which is independent of accuracy. Unfortunately, in the situation we are modeling, maximum DPS is not independent of accuracy. I will use a case study to show this to be the case*: Consider a ship which has a hull capacity of 75 which belongs to a faction which has no component capacity requirement reductions and has access to disruptors, rapid rechargers, and targeting scanners. Such a ship can carry up to five disruptors (25 damage per shot, 5 max DPS), 4 disruptors and either a rapid recharger (17 damage per shot, 4.86 max DPS) or targeting scanner (20 damage per shot, 4 max DPS), or 3 disruptors and both a targeting scanner and a rapid recharger (12.75 damage per shot, 3.64 max DPS). Clearly, maximum DPS has a dependency on rated accuracy, and declines as rated accuracy increases. By holding maximum DPS constant, you favor accuracy because you are not penalizing either rate of fire or damage per shot for increasing accuracy.
If you do impose a constraint, such as constant expected DPS, which penalizes at least one of damage per shot and rate of fire for increasing accuracy, you see that increasing rate of fire while holding accuracy constant causes the distribution to converge to the distribution of expected DPS for a weapon with the same expected DPS and a 100% hit rate, just like increasing accuracy would, whereas increasing damage per shot while holding accuracy constant causes the distribution to converge on the distribution of expected DPS for a weapon which can kill the target in a single shot and has the same hit chance. The distribution behaves the same way for increasing rate of fire as it does for increasing accuracy; therefore if increasing accuracy is a counter to evasion, so is increasing rate of fire.
*It's true that a case study cannot necessarily be generalized, especially to bonuses from technologies, but few ways of increasing accuracy come at no cost to something else, and in many cases that something else is at least indirectly applicable to the problem of countering evasion. The empire trait for accuracy comes at a cost to some other empire trait (the hull capacity bonus is directly applicable, while the income, manufacturing, research, food, and maintenance modifiers are all indirectly applicable; a few others, like growth, speed, and sensor range, are even more indirectly applicable), either directly because you need more points to get the accuracy trait or indirectly because getting the accuracy trait prevents you from spending points elsewhere; the accuracy-boosting specializations come at a cost to whatever the other specialization options could have given you (some of which, such as component capacity requirement reductions and component effect magnitude bonuses, are directly applicable to the problem and others of which, such as manufacturing cost reductions, are in theory indirectly applicable to the problem); the fleet-wide accuracy boosters consume strategic resources which could go towards something else and require either that your fleet includes a support role ship (and even if you were already going to have at least one support role ship, adding a new component to it presumably comes at a cost to some other feature of the ship, though since support-role ships are not generally actively engaged this is somewhat easily ignored) or space on one of your line ships, and so on.
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