That's not how it works. I don't have an "effective HP per enemy weapon type". I have an effective HP given the totality of enemy weapons.
Actually, you DO have an effective hp against each specific weapon type. That's exactly how defenses work in practice. They are bonus hp that only comes into play against a specific weapon type.
It's clear to see why your model is wrong when you consider the total amount of damage that has to be done to my ship for it to be destroyed. Since, in your example, all of my defenses will be fully depleted by the time my ship is destroyed, a total of 10+16+16+16 = 58 damage will be done to my ship. (58-10)/16 = 4 hp/dp.
If we assume he was firing 1 point of each, then yes. But let's assume he's firing 27 points of laser at you with that first shot. You're now dead, in spite of your 58 total hit points. Because 48 of them only count against specific weapons. They are conditional; they are only worth 1/3rd of a hit point.
If we assume an infinite number of battles, and that the enemy is also attempting to tailor his ships toward YOUR weaknesses (rather than just you tailoring to his), then on average each point of defense is equal to 1.3 hit points.
How does my assumption only apply in very, very specific circumstances? It applies assuming you spend some effort building defenses roughly proportionate to enemy weapon selection.
It also requires that the enemy spends NO effort building weapons to target your weaknesses. You are critiquing my position based on the randomness of the defenses, while at the same time relying on the randomness of the enemy's weapon of choice.
Essentially, you're demanding the right to always be able to select the correct defense for the attack that you're about to suffer in order to buck the statistical average, but not granting your opponent the right to select the weapon that your ship is most vulnerable to. That's the specific circumstances. If both sides are tailoring, then over time you should find that it returns to the average. If neither side is tailoring (i.e., random selection), then over time it will again return to the statistical average.
I find it somewhat awkward to evaluate the worth of defenses in an offtype scenario - because they should always be ontype. If not, then there was one or more false decision made previously (in not matching the enemy's weapons in the ship designer) and this has to take its own responsibility and mustn't mean that all defenses are automatically too weak.
As with lolpurple, you presume the enemy is just passively picking a weapon and type never changing it, so that you can always fight them with the correct defense in play. It also presumes that you won't fight 2 or more enemies at once, who have each picked different specializations.
The 1.3 ratio in the above formula only occurs if defenses are either picked totally randomly, OR if the enemy will always use all 3 types of weapons.
The first example doesn't account for a good AI/player decision or gameplay, sure it's not always possible to match but a player can match that much more often than just the mathematical average number. As for the AI, that would need to be observed over a longer time.
Again, this is only true if the player also CANNOT pick attacks correctly to target his enemies weakness at a better-than-average rate. This probably works against the AI, but if we have a MP situation, you are now demanding that two opposing players are both better at selecting the other's weakness and preventing the other from doing so - it's a contradiction in terms. You're posing a paradox, basically. If one player is better-than-average at selecting the right weapon for the job, the other player is automatically worse-than-average for picking the right defense. And the best bit? It's a zero-sum game, so once we total up the battle wins of both players, we arrive at the average again.
Remember, this is a broad average over millions upon millions of fights, involving ALL players and AIs who ever play the game, ever. For the purposes of valuation, individual player skill is irrelevant, since for every better-than-average player there is, by definition, a worse-than-average player (or, more likely, AI), and so the average is unaffected. Individual fights where you've managed to get the drop on the enemy are also irrelevant, since once again every time someone gets the drop on someone else , the other person is being a drop-ee and so their contribution to the total average is reduced by exactly the amount your has raised by.
As for the latter example, this approach actually gimps itself because it has to use much more technological resource which could have been used otherwise (for example, increasing logistics, mini etc) and I don't think the cost of this is justified only to decrease the enemies defenses.
This requires you to only be fighting enemies who have kindly all agreed to use the same attack type. Otherwise, you need to spend those tech resources anyway. Also, note in the response to lolpurple, the effectively killed an enemy ship by circumventing more than half it's total dp+hp values; that may actually justify the tech expense (particularly since in a tech-trading game, you can acquire them easily enough).
The biggest problem of defenses actually lies in that it doesn't provide a cumulative fleet-wide bonus (such as attack) and therefore, doesn't positively scale together with increased logistics. If you mathematically compare the values of an all-attack ship versus an ontype defense ship (that still retains a somewhat balanced approach to attack) then you'll notice the odds for the defense ship are best in a 1 vs. 1 scenario and become increasingly worse as larger the fleet(s) get via logistics. As a rule of thumb defenses are only worth it when when they can reload, and that is totally dependant on the magnitude of the incoming attack, and this increases via logistics.
Finally, in order to make a genuine comparison, you'd also have to take the build cost of each ship into equation, as well as the time or effort it will take to get the required technologies.
This I agree with.
I'm still interested to see some test results with this, though; while it's a given that defenses are most effective when you have equal-or-more ships to the enemy, I'd like to know how effective 'most' actually is - if they're still <50%, then no defenses are worth buying and the all-weapon ship is always superior. Hell, given that fleets are easy enough to produce, even early on, it's questionable if 50% is good enough.