I'm going to offer some alternative answers below, but first a mini-rant / discussion.
Many answers here go with AI is a crapshoot view that all AI will turn evil and kill us all in their sleep as soon as possible. I don't buy this view, there is no more reason to expect a pure evil AI then a purely loving AI. Many answers focus on ambiguity of the stated goals and how they can lead to truly extreme and undesirable outcomes, a common trope, but I don't think it's a realistic trope because it requires hypothesizing an immensely competent, yet immensely incompetent, AI at the same time.
To have a significant impact on the world of the sort some suggest your AI must be pretty smart, human level strong AI smart. If it's going to try to control nation politics to arrange birthrate or modulate politics or control subscription outcomes it must be very smart, capable of understanding humans, their motives, and their psychology, and capable of handling complex abstract ideals. It's going to need to have human, or super-human, level reasoning and abstract thought.
However, any AI this intelligent will therefore be able to do something we all can and are doing, look at it's objectives and point out they are ambiguous, in fact impossible as described above, and ask for clarification. If you can handle understanding large sociological phenomenon you can handle a simple question to those who employ you. It's unreasonable to expect an AI to manage such complexity of planning and manipulation to fail on such basic reasoning levels.
Of course part of the problem is the objectives as written are objectives you would give to a sapient strong AI, something smart enough to reason on it's own (and thus smart enough to reason about the flaws and ambiguities of the objectives and clarify them). If you have anything less then strong AI the objectives above just don't work, they are not mathematically rigorous, far to ambiguous, and simply can't be 'understood' by your AI.
Okay, having let me mini-rant on how I deem the standard tropes to not be realistic lets try to look at some realistic response to this. I'm looking at a very capable but not sapient AI here, if your AI is sapient the question is not state objectives but how you motivate it to want to achieve your objectives, their about psychology of your sapience not writing of programing and best addressed as a separate question.
Lets start with the boring, but most realistic, outcomes.
Error segmentation fault at line 234521508 core dump
Most likely outcome, your AI doesn't work. Your asking for a pretty complex program after all, it failing to get off the floor is always possibility.
Similarly it could have issues such as being unable to connect to appropriate systems to collect data. Or unable to process data fast enough (this is a real issue with all our cloud ingest now of days, we have so much coming in we don't know how to look at it all fast enough).
Were paying how much!? shut that program down!
A program like this would be expensive. As I mentioned monitoring so much data real time would require a cloud solution, paying amazon a ton for cloud computing power every month (which is still much better then having to maintain your own servers cost rise). Your have to pay to maintain the systems that feed data to the computer, handle wires being cut or other data lose, pay for maintaining the program, fixing bugs, and doing other logic. It's quite possible that the cure is worse then the disease, that the insurance company pay more maintaining the program then it saves them.
Man the WatsonCare website sucks
People fail to adopt and use the tool because it's confusing, hard to understand, had UI bugs etc. Can you make Watson not just work, but sexy and appealing?
Moving on to the presumption it actually gets used at all. There are some conceptual issues, independent to requirements, which will prove major problems with this approach. Their mostly due to human nature, which isn't always reasonable:
I just want to eat my deep fat fried bacon pizza with nacho cheese dipping sauce in peace!
It sounds like the main goal of Watson is to keep people from doing unhealthy things that end up hurting them. Unfortunately, we like unhealthy things. If we didn't enjoy them we wouldn't do them. Sure sitting in front of my computer playing games all day isn't as healthy as going outside, but as a geek I like playing games, I don't want to have my computer shut off until I go for a walk. Even if Watson only provided polite reminders it will annoy people if they get reminder every ten minutes. For that matter people won't want to know how unhealthy that fast food is, because it will ruin their enjoyment when they inevitable eat it anyways.
Do you remember 'clippy' from word? The little paperclip everyone loved to hate! Watson could be the new clippy, only massively worse. Watson could be what would happen if scrappy doo and whesly crusher had a gay love-child raised by jar jar binks and trained by clippy! (I would call rule 34 on the first part of that statement but I'm already running close to geek critical mass already so best to leave well enough alone).
If done wrong customers could be complaining or leaving their insurance to simply never have to interact with Watson ever again. In fact this is a very likely outcome, time and time again we have seen situations where people will complain if reminded of their own faults, even if it's part of a system to help them. Given the years of HCI study and disasters it almost seems guaranteed to be the case unless Watson is scaled back to only do 911 calls unless people explicitly activate other features; at which point he is not saving the insurance company much money because he is not preventing unhealthy habits.
Why did you break down my door? No I'm not dying I'm fine, I just took off my life alert for a shower! I'm not going to pay for that ambulance!
If Watson is going to call 911 then there will be false alarms, any program has some degree of false positives. But given the difficulty of detecting actual issues based off of the sort of detection tools we currently have available false alarms will be somewhat common.
This would be an annoyance to the one with paramedics beating down their door, but that's not the worse of it. If your calling ambulances your taking those ambulances away from the hospital. If someone else gets injured while the ambulance is at my house because I accidentally slept with my lead-lined, monitor-wifi blocking, blanket they will not be ready to go save someone who actually needs their help. People could die from these false alarms!
To give a comparison look at home security systems. These systems have a massive false alarm rating, more then 90% of their alarms are false alarms. So many in fact that police generally do not prioritize them and only send police officers over to a house if the officer has absolutely nothing else he could be handling instead (meaning when they do arrive it's often late enough that they couldn't have helped anyways, the cops aren't using sirens or speeding to get to your presumed false alarm). Some police districts refuse to even deal with home security providers any more, it was costing too much in taking their leads and so rarely helping.
However, cops, while still important, are far less likely to be needed at a life or death situation this very moment as an ambulance is. distracting them is much more likely to hurt someone else; or have them simply ignore your alarms.
My mother slowly died of a heart attack and Watson did nothing
The other PR nightmare side of the above situation. If you try to avoid false alarms you likely end up letting people die who you could have otherwise helped.
At least I feel better about the NSA spying on me compared to Watson
This was already mentioned, but this is a massive privacy issue here. People will be up in arms almost immediately. Watson would have to collect far more information then the NSA confessed to be collecting with it's metadata, and you saw how people responded to them. The public is very sensitive about privacy concerns right now, even about relatively minor issues. The degree of information collection would likely offend quite a few and cause a political backfire forcing Watson to be scaled back of disabled.
In fact people would start to demand whatever systems Watson used to collect data were disabled so no one else could collect data like Watson did, which could be harmful if they are insisting on disabling important safety features out of fear of spying after people exaggerated the threat of Watson.
Imagine someone disabling the equivalent of their life alert pendent because they kept hearing conspiracy theorists claim that Watson used it to detect heart-rate so it could detect rather they were afraid, angry, or sexually excited. Most likely the monitoring system can't even do that, but people will claim it, others will believe it, and suddenly people are less protected health wise due to paranoia about privacy.
Privacy 2, premium raising electric bogaloo
Another common claim will be made that if Watson can watch your habits it can know who is healthy and who isn't and provide that to the insurance company. The insurance company could then use that to raise rates if they decide someone isn't getting enough exercise, is eating too much fast food, or otherwise is a health risk. This could cause people who are not generally as upset about data mining to be cautious or refuse to use Watson because their certain their premiums will go up.
Save 10 dollars a month on your premium, for the low low cost of installing 10,000 dollars worth of monitoring equipment!
people don't have equipment capable of monitoring them to this degree in their homes or offices, and even in the future it's unlikely they will have equipment like this for a long time; there limited advantage to them outside of this one specific edge case. This means your asking people to buy a bunch of equipment, install it, and maintain it just to be able to use Watson. Even if you presume a future where the equipment is cheap the hassle of getting it installed could be enough to make people decide it's not worth the effort. Adoption and use of Watson could die out because it's just not worth the effort.
I know you could argue that Watson is so useful people will take the time to install it, but you may give humans credit as being logically motivated when they aren't. The truth is most of us don't accept our own health as an issue (and those that do tend to excessively over stress on it), and we can be quite easily discouraged by even a little bit of effort.
Look at Betamax, it was in almost every way better then VHS. Yet the awesome Betamax died a said forgotten death. Why did that happen? Because originally it didn't allow 2 hours of recording at once. It totally could, but the original betamax taps were made smaller. That meant to watch a movie you had to get up and swap tapes mid way through, and no one wanted to do that. SO we settled for lower quality, more expensive, easier broken VHS because we didn't want to get up off the couch rather then just waiting for Betamax to start making the 2 hour tapes. That goes to show even a great product can fail over minor hassles making people not bother.
Of course the other reason VHS did so well is they allowed selling of pornography which was not allowed on betamax, and porn is always the first adopter of new media. It's theorized this was a major issue for proliferation of VHS. So...if you can find a way that the monitoring equipment can also be used as part of some bizarre sexual kink you may have a chance :P
My hacking cough can be detected by a Hacker
Again, privacy concerns could make people afraid to use this system for fear that they will be hacked and their data shared.
However, hacking becomes a (slight) actual issue. Data sent through SSL is pretty secure and is no more a hacking concern then any of the other, far more life critical, electronic activities we have. However, to monitor our health there would likely have to be a number of wireless detectors available and running at all times, to collect data on us wherever we are and route the information through the internet. These wireless detectors may be more vulnerable to local hacking attempts, mostly because the people building them likely won't be thinking about the hacking issue and won't secure them as well as they should.
Family of unsupervised boy who tried to fly by setting off gasoline soaked fireworks he tapping to shoes, before jumping off cliff suing Watson for not warning child of risk
If Watson is suppose to be extending life of insurances individuals people are going to start taking this for granted, especially since your likely have to advertise this as a perk of your insurance to get increased enrollment. Once you do this people will expect Watson to protect them, no matter what.
Now any time someone get's hurt or does something idiotic they will point out that Watson should have protected them and it's a contract violation that he didn't. My fireworks example, while hyperbole, is actually not that unlikely to come up at some point, though the case would likely be instantly dismissed without cause.
However, other cases can come up. What if watson doesn't warn that combining two medications could have adverse affects? What if Watson fails t notice a subtle irregular heartbeat that could have warned someone of a health condition? What if someone doesn't speak to the doctor about that odd tingling in their arm because they figure if anything was wrong Watson would tell them, and then is shocked that Watson can't detect tingliness and thus didn't have any way to realize that there was a major health problem?
People will sue, liability issues will come up. These can be handled in court and with carefully warded contracts, not enough to singlehandedly destroy watson, but they will be a common occurrence that will add to expenses for the agency, expenses watson has to help them save to even break even.
Okay so now lets get to your actual question, what issues could come out of the stated objectives. These are the less likely issues in my mind, all the above seem quite a bit more probably (except maybe the hacking one). However here are more likely issues with bad objectives, presuming a non sapient but well written weak AI. Lets look at the sort of likely programming mistakes that could come out of it's objectives.
ConflictingObjectivesException: 'extend membership' conflicts with objective 'cut expenses'
We went bankrupt in the first week because we payed all our money to pay for a machine to keep one 120 year old man breathing for 5 more minutes
This is directly related to the above rant, but objectives as they are stated don't work. People buy insurance for payouts, if your cutting payouts your going to make people not want to buy your insurance. There are tricks you can do, mostly social engineering and quoting manipulative statistics, to help cut payouts while lowering lost customers, but ultimately these two are partially exclusive. Thus you need to better articulate importance, how many customers can I lose per thousands of dollars of payout I cut, how many more payouts can I make if it increases enrollment.
Similarly preserving lives is likely to to lead to making more payments, for a similar conflict.
Frankly a far far more extensive set of requirements, prioritization between exclusive requirements etc is required to be implemented.
Lets assume that a programmer set down and wrote such through requirements designed to emulate the general concept of the above...
We were saving lives and had great enrollment until we ran out of money
Even if write more detailed requirements you still need to set priority levels on them. If you pick poor priority, for instance you value enrollment numbers too high and minimizing payouts too low, your still have a problem. A human being is still making these prioritization decisions and could make bad ones. Though really this is a moot point, because the requirements as written have more significant issues...
Sign up for new WatsonCare lite insurance, only 99 cents a decade!
I lied above, your two objectives, increase subscriptions and decrease payouts, aren't actually mutually exclusive if read literally. There is a way I can get lots of people to sign up without paying out often, just make the cost of signing up trivial enough. If Watson offered some nearly free, or fully free, insurance then it would be worth enrolling for even if it rarely made payments, because if cost nothing then any payments you get are a net gain.
This obviously means that the insurance company is loosing out, because even though they pay out 1/3 of the time compared to their other policies if they still aren't taking in nearly enough in insurance premiums to cover those payouts they do make.
Or there is the other solution...
**The Watson lotto-surance is now up to $500,000! sign up today. The lucky hospitalized winner could be you!*
The objectives only stated that few payments had to be made, not that payment amounts had to be kept low. One solution is to make fewer payments, but drastically increase the amount payed out when one of those payments are made. From a purely reasonable point of view if I can sign up for an insurance that is 5% less likely to make a payment on a claim, but pays me twice as much when it does, it is optimal for me to do so because odds are I'll get more from the insurance on average. Thus I'm encouraged to sign up for this program even as Watson has lowered it's payouts.
In fact it's worse then that, because the scary thing is I could almost see this idea working. People are quite irrational with their money, done right one could probably tempt some individuals into a lottery style insurance by bragging about they high payments even if they end up paying less money out in total...no one suggest this to insurance agents!!!
Now one may question why this and the previous example don't fall into my "not giving AI enough credit" rant above, but there is an important difference. I'm now assuming a less sophisticated AI, which could still fall into these sort of solutions with a genetic algorithm style approach which simply tried possible outcomes until these turned out to work. Such a less advanced AI is more prone to mistakes in objectives resulting in odd results. However, more then that, these are mistakes I can see a programmer make!
The problem to these mistakes is that the programmer didn't actually state his real objectives. He picked specific details that usually lead to his desired outcome and presumed if those criteria were met his outcome would occur; this is a very common logical mistake I have seen in plenty of programs. The truth is that the insurance agency's real goal is to save money, possibly in an ethical way that also protects lives, but as a corporation saving money is still high on their priority list. They assumed more people enrolled while paying out fewer grants meant more money coming in and less going out and thus a net savings, but that isn't always the case. They should not have the stated objectives, and instead have an objective of "maximize profits" combined with whatever appropriate ethics objectives ensure it's done in a way they can live with. They should not specify the exact method of making money is done. It may be that increasing payouts, but also insurance premiums, to cater to certain high-income folks results in more income, even if it lowers net enrollment and thus violates both the stated goals. this is a believable multiprogramming of objectives, not a flaw in how a computer interprets ambiguous objectives, but religiously written objectives (presuming they were more rigors to clear up that ObjectiveConflictError) that simply were poorly chosen objectives.
Of course even if we redefine our objectives to be help the company profit while being moral and preserving lives we could still have issues...
*Sign up today and one of our staff of watson-cares doctors will move in down the street to service your every boo boo!**
If you employee enough people and offer enough non-payout benefits your insurance can be desirable even with fewer payouts, because of the other perks. of course it may turn out those other perks an unfathomably large amount of money to maintain, bankrupting the company in a few months, but hey, you never said anything about cost of employees.
Yes I'm assuming we redefined our objectives to conserving money, yes I still consider this a very plausible issue (well okay the above is hyperbole, but similar issues are likely). You see our non-hard AI can only consider variables it's programmed to know how to handle. If I forgot to write an algorithm to calculate the cost of employment of a new policy then the AI will never consider this, no matter how hard it's run, because it's not actually sapient. It needs me to define what variables it should consider ahead of time.
Sure I will likely have a CalculateStaffSalaries() algorithm, that's pretty obvious. but will I have a calculateHROverhead() algorithm, which considers the extra expense for managing the extra staff? will I have a updateSalaryToAnticipatedMarketDemand() method, which recognizes that employing so many doctors will mean there are fewer doctors out there, and thus the remaining doctors will ask for higher salaries and eventually I'll have artificial inflated the doctors salary to an unplayable high degree, or will my program still assume doctor salaries are the same as they were when it started running? What if I do write my updateSalary() method, but my understanding of economic supply and demand isn't perfect because I'm not an economist, and so I get my math wrong and don't anticipate the salary inflation accurately? for that matter what if I emulate all our economic knowledge perfectly, the market may still become inflated because economics are not able to reliable predict market influences that well.
In short, it's quite possible that the company still goes bankrupt because the AI wasn't able to anticipate expenses or costs.
I would have gotten away with it too, if it wasn't for you meddling kids and your smallpox-ridden dog!
What if the algorithm is wonderfully written to consider every variable and equation known to man and after massive number crunching it comes up with a payout scheme designed to save the insurance agency the most money by offering lass payouts for the likely issues but offering very tempting payouts for unlikely things. This would be a great trick, tempt people to sign up because it looks great without having to pay much, Your computerized munchkin has perfectly min-maxed your system.
And then the supposedly obliterated small pox turns out to have been incubating somewhere and a breakout occurs. Everyone calls their insurance agency and it turns out our policy offered payout 40 times as high as our usual for anyone that caught small pox, because we took the "Smallpox Bankruptcy" flaw"to help pay for that "conveniently 'misfiled' claim" perk. The bane of every min-maxer, you must have accidentally upset the GM because today's encounter hits all all your unlikely flaws.
This is a hyperbole example, but the issue is not as minor or ignorable as you may imagine. We don't yet have the ability to predict the future, which means even the best AI will be unable to predict potential outcomes. This means there is always a chance of bad luck leading to an unplanned for outcome that ends badly for our AI driven company.
But the problem is a little worse then this in a way. Unlike with humans a computer will try to min-max fully, if I didn't write special code to avoid it the program will pick the situation that gives the best average case, even if it means creating a situation where bad luck doesn't just cause the company money, but completely bankrupt it. It's worried about the average case only. Since it can't predict everything it thus may choose to create a policy that has a 20% of making the company go bankrupt if the average case makes the most money.
However, people aren't interested in average case, we generally would prefer to be more cautious here. If I'm CEO of WatsonCorp I would rather settle for 'only' 600,000 annual salary (the actual average for CEO of healthcare, thank you google) then take a 10% chance of the being fired after my company goes bankrupt to in order to have a 90% chance of making 660,000 salary. The extra profit won't help me nearly as much as unemployment will hurt me and I don't want to take that risk.
In programming world we have learned to be very aware of this. When we inspect algorithms now we discuss the big O of a program, which is (sort of) how fast it runs in the worst case, how do things happen when everything goes wrong. This is because we have learned that when I run a program every day for years that 'unlikely' worse case scenario ends up coming up pretty often, and it's not uncommon for the worst case to be disproportionately bad. Well...that and it's not uncommon for either malicious people to intentional design a worst case scenario or for the worst case scenario to be far more likely then random chance would expect. Good programmers would thus write an AI with an understanding of what worst-case scenario is acceptable...but this is an easy thing to forget to add. If you don't there is a very decent chance that your end up with a small, but disastrous, potential weakness in whatever solution your AI creates.
Next up on Scandal News, we learn Watson-international refuse life saving procedure for chronically on welfare, Their uncanny-valley ridden robotic overlord explains chronic illness required too many payments for one life, new cost cutting measures allow paying for 10 times as many breast enhancements
Closest to the evil-AI doomsday scenario were likely to get, and also a bit more questionable, to presume an AI would be sophisticated to calculate this without human involvement. However, this scenario could be eached by soft AI through genetic algorithm or certain well written inference engine and calculation processes.
the idea is simple, chronically ill require constant payouts over a long length of time, and thus can be quite expensive to support. However, refusing to pay for one life saving surgery could remove all future payments for that individual-you know, him being dead and all, saving a significant amount of money. A genetic algorithm approach may notice that not paying for some surgery results in far less money payed out on average without the AI realizing that's because people died without the surgery for instance; the AI only knows saying NO to this surgery means less money payed and that's all it's calculating.
This actually is a result in flaws of the above examples. It's a mis-tunning of priority levels, the AI not valuing life preservation as high as it should have compared to cost cutting. However, this doesn't have to be life saving operations. Perhaps not paying for certain surgery results in someone going into a vegetative state and ultimately costing less to provide for that way. It could be that other moral standards, beyond lives served, are required. For instance a "treat everyone equally" standard may avoid intentional infractions upon certain categories of people which would lead to outrage if learned otherwise.