Oh, I thought this was going to be about the old trackball arcade game. Or perhaps it is? Same sort of rules? The maths is going so far over my head I can't hear the whoosh.
Would be interesting to know how the probabilities change once all your X band radars are destroyed. And then again how they change when all your L band radars are destroyed...
The author explains that this problem is actually adversarial, in the sense that the attacker gets to observe defenses and allocate warheads and decoys accordingly.
Thinking of our current circumstances, this suggests another cost of war: our offensive capabilities, as well as our defensive capabilities become more observable. Our adversaries are studying our strengths and weaknesses in Iran, and they will have a much improved game plan for countering us in future conflicts.
"Our" adversaries, huh? There are more people in our country than pedophile billionaires, but it's this group starting the wars, murdering civilians, and producing generations of "adversaries".
On the other hand the Air force is getting the workout of their lifetime. Which could come in handy. The low bodycount among US military so far makes the whole clusterfuck just and expensive training program.
Two more sobering axes to introduce: cost and manufacturing capability.
Numbers are hard to find for obvious security reasons, but using the numbers most optimistic to the defender[0] suggests an adversary using a Fatah type hypersonic is spending 1/3rd the cost of an Arrow interceptor, and is launching missiles that are produced at a much faster rate. Interception is deeply asymmetric in favor of the attacker.
Game theory would be useful for these kinds of modeling.
Perhaps the government should have and advisory body that employs the smartest mathematicians for running these scenarios. Of course a lot of randomness needs to be modeled too. Wonder what would be a good name for such a body :)
Paradoxically, if anyone leaks unpalatable information from the inside that would be a problem for the government.
> Claude Shannon called him "the smartest person I've ever met", a common opinion.
> Von Neumann founded the field of game theory as a mathematical discipline.
> .. leading him to a large number of military consultancies and consequently his involvement in the Manhattan Project.
> In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group..
> In 1955, von Neumann became a commissioner of the Atomic Energy Commission (AEC), which at the time was the highest official position available to scientists in the government.
> In his final years before his death from cancer, von Neumann headed the United States government's top-secret ICBM committee..
> Directed energy has been proposed as a cost-effective alternative, but introduces its own scheduling constraints — dwell time, platform coverage, atmospheric degradation — with similar scaling issues
The author is doing the thing where a writer tries to bamboozle the reader into a conclusion without having to prove it by overwhelming the reader with nouns. Life is too short for shitty gosh gallops.
could use some investigation of the ukranians techniques -- the number of interceptors the U.S. used within the first four days of the war eclipsed the total amount Ukranians have had for the war
What is the steady state? Assume you have two competent superpowers, both researching missile offense and defense, over the next 1000 years. What are the asymptotics of the interception rate from 0 to 1000?
Pardon the pun but this is an arms race and the defenders are going to lose. There are broadly five classes of missiles (one isn't a missile per se):
1. Ballistic. These are traditional rockets, basically. While rockets are designed to reach orbit or leave the Earth, a ballistic missile basically goes straight up and comes down. The higher it goes, the further away it can get because of the ballistic trajectory and the rotation of the Earth.
Ballistic missiles are most vulnerable in the boost phase ie when they're just launched. Since you have little to no warning of that, that's not really helpful.
But one weakness of ballistic missiles is you pretty much know the target within a fairly narrow range as soon as they launch. That's the point of early-warning radar: to determine if a launch is a threat so defenses can be prepared.
Attackers can confuse or defeat defenses in multiple ways such as making small course corrections on approach, splitting into multiple warheads, using decoys for some of these warheads, deploying anti-radar or anti-heat seeking defenses at key points and breaking into many small munitions, sometimes called cluster munitions on the news but traditionally that's not what a cluster bomb is or was. In more sophisticated launch vehicles, the multiple warheads can be independently targeted. These are called MIRVs (multiple independently targetable reentry vehicles).
Economicallky, depending on range and capability, a ballistic missile might cost anywhere from $100k+ to $10M+.
2. Rockets. Militarily this is different to a rocket in a civilian context. It's not much different to a hobby rocket, actually. Often these are "dumb" but some have sensors and guidance capability or might be heat-seeking.
These tend to be incredibly cheap to produce and not terribly accurate but that's not really the point. The point is they're cheap and easy to produce and the interceptors are much more expensive.
3. Cruise missiles. Rather than a ballistic trajectory, these have more sophisticated guidance and travel much closer to the ground, usually to avoid radar. The Tomahawk missile is a prime example of this. These tend to be relatively expensive and much slower than ballistic missiles.
4. Hypersonic missiles. This is a relatively new invention that's kind of like a cruise-ballistic hybrid. It flies in the atmosphere for part or all of the time and, unlike cruise missiles, will fly faster than the speed of sound, usually significantly so (eg Mach 4-10). Such high speed makes interception near-impossible currently.
The big advantage of a hypersonic missile is that it has the speed of a ballistic missiles without the predictability of the target area. Plus it can be retargeted in-flight.
5. Drones (honorable mention). Not technically a missile but they fit in this space regardless. This is basically a scaled up commercial drone with an explosive payload. These are significantly slower than cruise missiles or rockets but can be live-targeted, re-targeted and have a variety of types ranging from dropping hand grenades from a height (eg as has happened in Ukraine) to suicide-type drones that explode on impact.
Drones are typically so slow that you could shoot them down with an shotgun in some cases. But they're incredibly cheap and easy to produce.
> Note that a more complete model would multiply each term by P(track)_j — the common-mode detection-tracking-classification factor developed in the previous section — but the standard WTA formulation assumes perfect tracking.
I'm not sure that is a useful model, or more complete. I don't think you can assign interceptors to undetected missiles, so considering their effect on the value is rather pointless. It's effectively a sunk cost.
Multiplying with the probability also makes no sense from an optimisation point of view. Why would you assign lower value to a target about to be hit simply because you were unlikely to detect the missile?
The tracking probability only shows up in the meta game described at the end, where one side is trying to optimise their ability to hit valuable targets and the other is trying to optimise their ability to prevent that from happening.
>Hence, for one warhead, a defender can launch 4 interceptors and have a 96% chance of successfully intercepting the incoming warhead.
>Unfortunately, those numbers are optimistic.
This part worth stressing, ceiling for more performant missiles, i.e. faster, terminal maneuvering, decoys are geometrically harder to intercept. Past mach ~10 terminal and functionally impossible because intercept kinematics will break interceptor airframes apart.
AFAIK there hasn't been tests (i.e. FTM series) done on anything but staged/choreographed "icbm representative" targets. Iran arsenal charitably pretty shit, including high end. Hypothetical high end missile with 10%-20% single shot probability of kill requires 20-40 interceptors for 98% confidence, before decoys, i.e. 40x6=240 interceptors for 1 missile with 5 credible decoys.
The math / economics breaks HARD with offensive missile improvements.
That goes both ways. Terminal manoeuvring gets difficult as speeds increase, too. Besides, higher tier threats are intercepted outside the atmosphere, where it would be prohibitively expensive (in mass and accuracy) for RVs to manoeuvre.
Your SSPK is way WAY too low. Also, the math on interceptions changes completely with multiple KKVs per interceptor (see NGI).
Decoys don't work even at the highest end (ICBMs), see Chevaline's history and its rapid decomissioning for an example of just that.
Terminal maneuvering vs interceptor kinematics difficult don't scale same at same rate with speed increase, a few Gs on reentry is magnified on interceptor end, the physics is not fair between attacker/defender.
Higher threats are not exo atmosphere & midcourse that NGI targets, actual speculative high end threats (since we talking about speculative NGI capabilities) are hypersonic glide in upper atmosphere, which NGI can't target. It's even more extreme intercept problem, the intercept math for those class of threats is on paper even more lopsided against defender.
NGI basically ensure US keeps pace with modern penetration aids proliferation, i.e NKR tier adversaries if they ever improve RVs + decoys from shit tier to mid tier. Chevaline was 50 years ago, not really relevant example, since issue was really economics of decoys vs just spamming mirv. Regardless, NGI trying to maintain 3/4/5 interceptor ratio instead of 30/40/50/100s against exo threats, which on spectrum is low/medium end. It does 0 against actual future highend threats of glide phase interception, an entirely different category.
As an alternative formulation of the same problem, maintaining peace has linear cost, completely solvable in linear time and rewards are unbounded for all parties.
I spent some time reconstructing the current US missile defense interceptor numbers. This was done from open source DoD data, CSIS P-21 procurement exhibits, and CRS reports, that are are in depth, non partisan, objective policy analyses written by experts at the Congressional Research Service (CRS) specifically for U.S. Congress members. Also used Lockheed/RTX financial disclosures.
The numbers are pretty bad… Way worse than the headlines suggest. But anyway nowadays, investigative journalism has been decimated....For example experts like Kelly Grieco at Stimson estimated that at 12 day war consumption rates, the entire US interceptor stockpile depletes in 4 to 5 weeks. We are now in week 4...
As of December 2025, CSIS documented delivery of 534 THAAD interceptors and 414 SM-3. The 12 day War burned through around 150 THAADs (that is 28% of inventory) and about 80 SM-3s. The current war has been drawing down from that already depleted starting point for 25 days straight...
Gulf states reportedly expended around 600 to 800 PAC-3 MSE interceptors in the first 72 hours of Epic Fury alone, and that is more than the entire global 2025 production ( about 620 units).
Meanwhile THAAD production is 96 per year….with a recent Lockheed commitment to quadruple to 400 per year, but that will only deliver these additional missiles after 2027 or later. For example the sole ammonium perchlorate supplier for every US solid rocket motor runs one plant in Utah, and the sole HMX/RDX source is a WWII facility in Tennessee…
The US has procured roughly 270 PAC-3 MSE ( the Patriots ) per year since 2015, but has diverted around 600 to Ukraine over four years. The exact remaining US stockpile is not known with the same precision as THAAD/SM-3, so they could not have more than 3000 before Epic Fury...
But it is known as I said above, Gulf allies burned through 600 to 800 or more PAC-3 MSE in the first 72 hours of Epic Fury alone from their own stocks. Since they have zero domestic production capacity, and will be competing with the US for the same Lockheed production line that only makes about 600 per year, Iran really has them by the balls.
By the way, the cost so far in munitions is 20 billion ( check references…).
Then on Intelligence...
Iran has 13 satellites of their own, and it is known to be receiving intelligence from the Russians. This data allows them to know exactly how many Patriots or THAAD were fired so far. They are also probably customers of MizarVision, a Chinese AI startup, that has been cataloguing every significant American military asset in the Middle East. Every base, every carrier strike group, every F-22 deployment, every THAAD battery, every Patriot missile position, tracked, labeled, analyzed, and posted publicly.
So...
Unless the US escalates to a Ground Invasion (most likely scenario…), or negotiates a deal with Iran, if Iran can keep their industrial production of missiles, or maybe move them far up and inside tunnels in its Northern Mountains, and...if the USA does not escalate to a ground invasion due to the political risks, they can actually win this war both from the political and strategic aspects, as incredible as that might seem.
Who is truly screwed are the Gulf countries, as their stocks of US missiles get progressively depleted… And they wont get a refill soon.
Russia strategic interests are in helping Iran, since it weakens the US and strengthens their hand in Ukraine.
What might make it worst for the Iranians is the Chinese view of this. I speculate they will prefer to help the US and its economy, by forcing the US to do a great commercial interesting deal for them, then using their strong leverage on Iran to come to an agreement.
Strategically, over the next four to six months: Russian wins, Iran wins (despite all the destruction), China wins, Israel loses, the US loses.
Trump truly is the biggest loser...
"Are We Running Out of Missile Defense Interceptors?" - Teever↗
Great comment. I'm really concerned that the average American isn't taking this impending calamity seriously because they're completely unable to reason about a world where America doesn't have absolutely military supremacy due to decades of propaganda and the American identity being so tightly entwined with the American military mythos.
Any time that I've raised this issue of unsufficient stockpiles and poor preparadness for a major conflict for about a year now on HN and the response is pretty poor and my comments are often downvoted and/or ignored.[0][1][2][3]
There doesn't seem to be a way out of this without a substantial and undeniable US military loss that wakens people to the severely dysfunctional state of military stockpiles and planning, but with that said I'm highly skeptical that this incarnation of America can rise to the occasion like previous generations did in WW1 and WW2.
52 comments
[ 3.9 ms ] story [ 69.5 ms ] threadThinking of our current circumstances, this suggests another cost of war: our offensive capabilities, as well as our defensive capabilities become more observable. Our adversaries are studying our strengths and weaknesses in Iran, and they will have a much improved game plan for countering us in future conflicts.
That’s why StarCraft players sends “scouts” into enemy bases in the early game.
Specifically: don't start wars thousand of miles away of your borders.
Numbers are hard to find for obvious security reasons, but using the numbers most optimistic to the defender[0] suggests an adversary using a Fatah type hypersonic is spending 1/3rd the cost of an Arrow interceptor, and is launching missiles that are produced at a much faster rate. Interception is deeply asymmetric in favor of the attacker.
[0] https://www.jpost.com/israel-news/defense-news/article-82314...
Perhaps the government should have and advisory body that employs the smartest mathematicians for running these scenarios. Of course a lot of randomness needs to be modeled too. Wonder what would be a good name for such a body :)
Paradoxically, if anyone leaks unpalatable information from the inside that would be a problem for the government.
https://en.wikipedia.org/wiki/Daniel_Ellsberg
https://en.wikipedia.org/wiki/Ellsberg_paradox
https://en.wikipedia.org/wiki/RAND_Corporation
https://en.wikipedia.org/wiki/John_von_Neumann
> Claude Shannon called him "the smartest person I've ever met", a common opinion.
> Von Neumann founded the field of game theory as a mathematical discipline.
> .. leading him to a large number of military consultancies and consequently his involvement in the Manhattan Project.
> In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group..
> In 1955, von Neumann became a commissioner of the Atomic Energy Commission (AEC), which at the time was the highest official position available to scientists in the government.
> In his final years before his death from cancer, von Neumann headed the United States government's top-secret ICBM committee..
> Directed energy has been proposed as a cost-effective alternative, but introduces its own scheduling constraints — dwell time, platform coverage, atmospheric degradation — with similar scaling issues
The author is doing the thing where a writer tries to bamboozle the reader into a conclusion without having to prove it by overwhelming the reader with nouns. Life is too short for shitty gosh gallops.
1. Ballistic. These are traditional rockets, basically. While rockets are designed to reach orbit or leave the Earth, a ballistic missile basically goes straight up and comes down. The higher it goes, the further away it can get because of the ballistic trajectory and the rotation of the Earth.
Ballistic missiles are most vulnerable in the boost phase ie when they're just launched. Since you have little to no warning of that, that's not really helpful.
But one weakness of ballistic missiles is you pretty much know the target within a fairly narrow range as soon as they launch. That's the point of early-warning radar: to determine if a launch is a threat so defenses can be prepared.
Attackers can confuse or defeat defenses in multiple ways such as making small course corrections on approach, splitting into multiple warheads, using decoys for some of these warheads, deploying anti-radar or anti-heat seeking defenses at key points and breaking into many small munitions, sometimes called cluster munitions on the news but traditionally that's not what a cluster bomb is or was. In more sophisticated launch vehicles, the multiple warheads can be independently targeted. These are called MIRVs (multiple independently targetable reentry vehicles).
Economicallky, depending on range and capability, a ballistic missile might cost anywhere from $100k+ to $10M+.
2. Rockets. Militarily this is different to a rocket in a civilian context. It's not much different to a hobby rocket, actually. Often these are "dumb" but some have sensors and guidance capability or might be heat-seeking.
These tend to be incredibly cheap to produce and not terribly accurate but that's not really the point. The point is they're cheap and easy to produce and the interceptors are much more expensive.
3. Cruise missiles. Rather than a ballistic trajectory, these have more sophisticated guidance and travel much closer to the ground, usually to avoid radar. The Tomahawk missile is a prime example of this. These tend to be relatively expensive and much slower than ballistic missiles.
4. Hypersonic missiles. This is a relatively new invention that's kind of like a cruise-ballistic hybrid. It flies in the atmosphere for part or all of the time and, unlike cruise missiles, will fly faster than the speed of sound, usually significantly so (eg Mach 4-10). Such high speed makes interception near-impossible currently.
The big advantage of a hypersonic missile is that it has the speed of a ballistic missiles without the predictability of the target area. Plus it can be retargeted in-flight.
5. Drones (honorable mention). Not technically a missile but they fit in this space regardless. This is basically a scaled up commercial drone with an explosive payload. These are significantly slower than cruise missiles or rockets but can be live-targeted, re-targeted and have a variety of types ranging from dropping hand grenades from a height (eg as has happened in Ukraine) to suicide-type drones that explode on impact.
Drones are typically so slow that you could shoot them down with an shotgun in some cases. But they're incredibly cheap and easy to produce.
"The loser of a knife fight dies in the street. The winner dies in the hospital."
Your so-called missile defense does nothing at all against a real missile like Iran's supersonic ICBMs which can exceed 24,000+ km/h.
I'm not sure that is a useful model, or more complete. I don't think you can assign interceptors to undetected missiles, so considering their effect on the value is rather pointless. It's effectively a sunk cost.
Multiplying with the probability also makes no sense from an optimisation point of view. Why would you assign lower value to a target about to be hit simply because you were unlikely to detect the missile?
The tracking probability only shows up in the meta game described at the end, where one side is trying to optimise their ability to hit valuable targets and the other is trying to optimise their ability to prevent that from happening.
This part worth stressing, ceiling for more performant missiles, i.e. faster, terminal maneuvering, decoys are geometrically harder to intercept. Past mach ~10 terminal and functionally impossible because intercept kinematics will break interceptor airframes apart.
AFAIK there hasn't been tests (i.e. FTM series) done on anything but staged/choreographed "icbm representative" targets. Iran arsenal charitably pretty shit, including high end. Hypothetical high end missile with 10%-20% single shot probability of kill requires 20-40 interceptors for 98% confidence, before decoys, i.e. 40x6=240 interceptors for 1 missile with 5 credible decoys.
The math / economics breaks HARD with offensive missile improvements.
Your SSPK is way WAY too low. Also, the math on interceptions changes completely with multiple KKVs per interceptor (see NGI).
Decoys don't work even at the highest end (ICBMs), see Chevaline's history and its rapid decomissioning for an example of just that.
Higher threats are not exo atmosphere & midcourse that NGI targets, actual speculative high end threats (since we talking about speculative NGI capabilities) are hypersonic glide in upper atmosphere, which NGI can't target. It's even more extreme intercept problem, the intercept math for those class of threats is on paper even more lopsided against defender.
NGI basically ensure US keeps pace with modern penetration aids proliferation, i.e NKR tier adversaries if they ever improve RVs + decoys from shit tier to mid tier. Chevaline was 50 years ago, not really relevant example, since issue was really economics of decoys vs just spamming mirv. Regardless, NGI trying to maintain 3/4/5 interceptor ratio instead of 30/40/50/100s against exo threats, which on spectrum is low/medium end. It does 0 against actual future highend threats of glide phase interception, an entirely different category.
Forgive my ignorance, but I thought Israel's "iron dome" offered a very effective defense.
Is this just from short distance missiles from neighbouring countries?
This article seems to indicate it's very difficult to achieve a high success rate against multiple missiles.
Admittedly I probably need to read up on this more.
Because, boy, do I think you'll be missing out.
The numbers are pretty bad… Way worse than the headlines suggest. But anyway nowadays, investigative journalism has been decimated....For example experts like Kelly Grieco at Stimson estimated that at 12 day war consumption rates, the entire US interceptor stockpile depletes in 4 to 5 weeks. We are now in week 4...
As of December 2025, CSIS documented delivery of 534 THAAD interceptors and 414 SM-3. The 12 day War burned through around 150 THAADs (that is 28% of inventory) and about 80 SM-3s. The current war has been drawing down from that already depleted starting point for 25 days straight...
Gulf states reportedly expended around 600 to 800 PAC-3 MSE interceptors in the first 72 hours of Epic Fury alone, and that is more than the entire global 2025 production ( about 620 units).
Meanwhile THAAD production is 96 per year….with a recent Lockheed commitment to quadruple to 400 per year, but that will only deliver these additional missiles after 2027 or later. For example the sole ammonium perchlorate supplier for every US solid rocket motor runs one plant in Utah, and the sole HMX/RDX source is a WWII facility in Tennessee…
The US has procured roughly 270 PAC-3 MSE ( the Patriots ) per year since 2015, but has diverted around 600 to Ukraine over four years. The exact remaining US stockpile is not known with the same precision as THAAD/SM-3, so they could not have more than 3000 before Epic Fury...
But it is known as I said above, Gulf allies burned through 600 to 800 or more PAC-3 MSE in the first 72 hours of Epic Fury alone from their own stocks. Since they have zero domestic production capacity, and will be competing with the US for the same Lockheed production line that only makes about 600 per year, Iran really has them by the balls.
By the way, the cost so far in munitions is 20 billion ( check references…).
Then on Intelligence...
Iran has 13 satellites of their own, and it is known to be receiving intelligence from the Russians. This data allows them to know exactly how many Patriots or THAAD were fired so far. They are also probably customers of MizarVision, a Chinese AI startup, that has been cataloguing every significant American military asset in the Middle East. Every base, every carrier strike group, every F-22 deployment, every THAAD battery, every Patriot missile position, tracked, labeled, analyzed, and posted publicly.
So...
Unless the US escalates to a Ground Invasion (most likely scenario…), or negotiates a deal with Iran, if Iran can keep their industrial production of missiles, or maybe move them far up and inside tunnels in its Northern Mountains, and...if the USA does not escalate to a ground invasion due to the political risks, they can actually win this war both from the political and strategic aspects, as incredible as that might seem.
Who is truly screwed are the Gulf countries, as their stocks of US missiles get progressively depleted… And they wont get a refill soon.
Russia strategic interests are in helping Iran, since it weakens the US and strengthens their hand in Ukraine.
What might make it worst for the Iranians is the Chinese view of this. I speculate they will prefer to help the US and its economy, by forcing the US to do a great commercial interesting deal for them, then using their strong leverage on Iran to come to an agreement.
Strategically, over the next four to six months: Russian wins, Iran wins (despite all the destruction), China wins, Israel loses, the US loses. Trump truly is the biggest loser...
"Are We Running Out of Missile Defense Interceptors?" - Teever ↗ Great comment. I'm really concerned that the average American isn't taking this impending calamity seriously because they're completely unable to reason about a world where America doesn't have absolutely military supremacy due to decades of propaganda and the American identity being so tightly entwined with the American military mythos.
Any time that I've raised this issue of unsufficient stockpiles and poor preparadness for a major conflict for about a year now on HN and the response is pretty poor and my comments are often downvoted and/or ignored.[0][1][2][3]
There doesn't seem to be a way out of this without a substantial and undeniable US military loss that wakens people to the severely dysfunctional state of military stockpiles and planning, but with that said I'm highly skeptical that this incarnation of America can rise to the occasion like previous generations did in WW1 and WW2.
[0] https://news.ycombinator.com/item?id=42391816 [1] https://news.ycombinator.com/item?id=43693330 [2] https://news.ycombinator.com/item?id=44812177 [3] https://news.ycombinator.com/item?id=45054414