If the placebo effect actually worked exactly like that, then yes, you would die while the self-deluded person would do better. However, from personal experience, I highly suspect it doesn't (I have never had anything that I was told I'd be likely to die from, but I believe even minor illnesses give you some nonzero chance of dying). Here is how I would reason in the world you describe:
There is some probability I will get better from this illness, and some probability I will die.
The placebo effect isn't magic, it is a real part of the way the mind interacts with the body. It will also decrease my chances of dying.
I don't want to die.
Therefore I will activate the effect.
To activate the effect for maximum efficiency, I must believe that I will certainly recover.
I have activated the placebo effect. I will recover (Probability: 100%). Max placebo effect achieved!
The world I live in is weird.
In the real world, the above mental gymnastics are not necessary. Think about the things that would make you, personally, feel better during your illness. What makes you feel more comfortable, and less unhappy, when you are ill? For me, the answer is generally a tasty herbal tea, being warm (or cooled down if I'm overheated), and sleeping. If I am not feeling too horrible, I might be up to enjoying a good novel. What would make you feel most comfortable may differ. However, since both of us enjoy thinking rationally, I doubt spouting platitudes like "I have 100% chances of recovery! Yay!" is going to make you personally feel better. Get the benefits of pain reduction and possibly better immune response of the placebo effect by making yourself more physically and mentally comfortable. When I do these things, I don't think they help me get better because they have some magical ability in and of themselves. I think they will help me get better because of the positive associations I have for them. Hope that helps you in some way.
Well, yea obviously it's a simplified model to make the math easier, but the end result is the same. The real formula might for example look more like P=0.2+(expectation^2)/3 than P=expectation/2. In that case, the end result is both a real probability and expectation equal to 0.215377 (source: http://www.wolframalpha.com/input/?i=X%3D0.2%2B%28X^2%29%2F3 )
Also, while I used the placebo effect as a dramatic and well known example, it crops up in a myriad other places. I am uncomfortable revealing to much detail, but it has an extremely real and devastating ...
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