We actually have such a system these days: Distributed Denial of Service
As more and more of our world becomes dependent on the Internet functioning at high speeds, the ability to disrupt this communication can be catastrophic. Imagine a world shortly in the future where cars are navigating and talking with each other over the Internet; where airplanes fly pilotless remote controlled by command and control systems over the Internet. Bitcoin is the main currency of the world.
The reason DDOS can act as a MAD device is because there are MILLIONS of machines just waiting to fire their payloads, on every network, in every ISP. Most houses in America probably have at least one compromised device on their network. Antivirus and firewalls don't help, because any machine can become a cannon spewing forth network noise.
You don't know which machines will cause trouble until the "fire" command is sent, at which point your entire Internet is being flooded with nonsense garbage.
I had a rack of web servers on a 10GB link go down because it was receiving 10GB of packets containing nothing but the string "SO I HERD U LEIK MUDKIPZ". Seriously. We did what we could, but basically we just had to weather the storm.
What are you going to do? Filter the packets out? You'll spend more CPU power trying to filter the packets than you possibly could have. Firewalls are useless, because they have to know who to block, and you're getting traffic from a million addresses. You can't blackhole blocks of addresses, because they're randomly distributed over the entire address space. You can't whack specific addresses because new addresses will just pop up to take their place (and now you're spending MORE CPU trying to filter out addresses). Literally the only thing you can do is SHUT DOWN THE NETWORK and wait for the DDOS to stop.
The attacker, on the other hand, has no such problems. They have command and control over the botnet, so they can instruct the botnet to avoid their specific address space. Routers will handle intra-country traffic, and just ignore the rest of the world. It's a MAD situation, so I'm not talking firewalls, I'm talking literally unplugging the country's network from the rest of the world. Even if the rest of the world retaliates, there's no route back to your country.
Won't the enemy just cut off their network? Sure, but it's too late: you've sent the command to DDOS and infect. They'll have to play whack-a-mole, tracking down individual machines and disconnecting them from the source. Every IP camera, every cell phone, every car WIFI access point can be an potential DDOS transmitter. The Internet is dead, everywhere, except the specific places you took offline beforehand. Self-replicating self-healing botnets reinforce themselves, scramble addresses, go dormant when anyone gets too close to finding them.
This is mutually assured because either side can pull the trigger. The Capitalists and the Commies probably both have botnets in each other's country. The first strike involves activating the enemy network, then pulling your network cables before they can send a retalitory command to THEIR network. Or at least that's the theory. There's probably backup command servers, canary heartbeat servers which would discover that they're disconnected from NORAD and start firing... but that's the Mutually Assured part. It's each country's job to keep their botnets from unintentionally destroying the other country's Internet, while technically allowing a non-nuclear option.
EDIT: What about 'Mutually'?
There's a point to be made that script kiddies can do DDOSes. I've been DDOS'd by 4chan, and it's not cool. But modern infrastructure has terrabit pipes, you'd need to be a state-actor level in hacking and network building. 4chan machines generally come from a pool of machines used for criminal activities using command and control software based on DNS records or IRC chat rooms. This is fine if you want to swindle a few credit cards or knock a colo server off the grid. MAD level DDOSes will depend on almost-sterile computers with just the enemy state's code on it. It'd probably actively immunize the OS against other virus type activity. It must be undetectable except under the highest scrutiny.
Ironically, a MAD DDOS system would likely increase the security and stability of the enemy state's computers up until the moment the button is pressed.
EDIT: What about the "Assured"?
Neither side actually expects a nuclear strike to be the end of the war. Maybe, at the very beginning, the idea of a 10 minute war was viable, but both sides have so much redundancy and readiness built in to the Nuclear MAD that we focus now on a First Strike; the hope being to hit so hard right away we can stop a retaliation and force a surrender before the other side gets reorganized.
"We've got enough nukes to destroy the entire earth 12 times" (or whatever) isn't realistic, because we expect most of those missiles to miss, or burn out, or fail to launch, or refuse arming commands, or be jammed, or intercepted... that part of the "Assurance" for Nuclear MAD; so many missiles you can't stop them all.
The reality is, a Nuclear War would involve a first strike followed by the remnants of each country fighting for survival and victory, with hopes that enough of the other side's infrastructure is destroyed that they can't fight longer than we ca.
This is why a State Actor DDOS can be MAD. It's "Assured" that the first strike on the enemy WILL scramble their communications, crash their infrastructure, cause power outages and civilian panic, disrupt transportation channels. You hope you can scramble the enemy long enough to get conventional bombers or marines or whatever to destroy or capture the command and control of the enemy while they're disorganized and force them into defeat.
EDIT: But what about the "Destruction"?
My intention was that this shortly future future is way more reliant on Internet than we are. I'm not talking "you get to work and can't log in to email," I'm going for "nobody's cars will leave their driveways because they can't establish a link to the traffic router" and "an undetected flaw in the control systems for that nuclear power plant allowed an attack in, but nobody can log into the computer because it's too busy throwing away packets."
I agree, this disaster isn't as obvious as nuclear weapons, but if you've ever worked in a datacenter, you've seen how the phones will light up and alarms will go off when 4chan's Orbital Laser Canon turns on them. And that's for people's cat websites. Servers don't just get their network card jammed; they shut down, overheat, cause rolling failures as the stress backs up into further systems.
I'm not talking Netflix going down; I'm talking planes falling out of the sky because they ran out of fuel waiting for clearance to land because the control tower computers crashed. I'm talking hospitals backing up because none of their high-falutin' test equipment is responding to commands, MRI machines have exhausted their supplies due to crashing, and the entire medical records system is locked up nation-wide, so everyone's medical histories, allergies, everything is essentially gone. The central communication systems of the government probably still work, but they've just been thrown back to Early World War II as every computer around them is either spewing crap or being buried by crap.
If nothing else, it'll cause panic, confusion, and fear in the enemy state, as everyone suddenly finds themselves trapped in the 50s. Many in this generation have never used a rotary telephone or been without an internet connection for more than a day or two. The initial panic, confusion, and fear could be enough to destabilize the other side.