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The problem with generalizing it is that the value of money isn't constant. I would much rather have a million dollars free and clear than a 1/10,000 chance at ten billion dollars.
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When is it ever rational to enter a sweepstakes where you may have a 1/10,000 chance of winning?
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on
When is it ever rational to enter a sweepstakes where you may have a 1/10,000 chance of winning?
The technical term that behavior is risk aversion: http://en.wikipedia.org/wiki/Risk_aversion
Furthermore, the behavior you are describing can be modeled with Utility Theory: http://en.wikipedia.org/wiki/Utility
Utility theory explains many human quirks, such as loss aversion: http://en.wikipedia.org/wiki/Loss_aversion
It also explains why we're willing to pay for insurance, when often times insurance is more expensive than they payouts we receive.
*edited to a more neutral tone
In response to
When is it ever rational to enter a sweepstakes where you may have a 1/10,000 chance of winning?
Relevant reading:
http://en.wikipedia.org/wiki/Expected_value
http://en.wikipedia.org/wiki/Decision_tree
Try to quantify the cost of entering the sweepstakes. Sure, entry is free, but as you point out there are other less tangible costs of time and energy. But let's try. Let's say it takes 1 hour to enter, and another 1 hour in a couple weeks to follow up. Let's say you're a reasonably successful person, and your time is worth $50 per hour. So it will cost you roughly $100 to enter the sweepstakes.
Next we need to determine the expected value of winning the sweepstakes. Let's say the prize is $1 million. If the chance of winning is truly 1 in 10,000, then our expected value is ($1 million) x (1/10,000) = $100.
So... in this contrived little example, we are exactly neutral. It costs us approximately $100 to enter and our expected value is approximately $100. This is equal to the $0 of not messing with the sweepstakes at all.
When would it be rational to enter? We can tweak the variables on either side. For example, if your time is only worth $20 per hour and it only takes 5 minutes to enter and 5 minutes to follow-up, your total cost is $3.33. It costs you $3.33, but the expected value is $100, so the net benefit of entering is $96.67. That is better than the $0 of ignoring the contest, so rationally you should enter.
Or, consider if the sweepstakes prize was worth $1 billion. Now the expected value is ($1 billion) x (1/10,000) = $100,000. Pretty good prize! It is clearly worth your time.
We could generalize this...
For any sweepstakes where you have a 1/10,000 chance of winning, it is rational to enter if the prize is 10,000 times more valuable than the cost of entering.
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Money is not linear in utility. Even granting that risk-neutrality in utility is the only rational approach, risk-neutrality in money does not follow.
Indeed, you are correct.
In my finance education, professors always argued that your money curve should be as close to 1:1 with your utility curve as possible. Granted, that's a dubious setting. Probably correct for money managers, but not for humans in daily life.