by Eliezer Yudkowsky, originally on LessWrong
Once upon a time, three groups of subjects were asked how much they would pay to save 2000 / 20000 / 200000 migrating birds from drowning in uncovered oil ponds. The groups respectively answered $80, $78, and $88 . This is scope insensitivity or scope neglect: the number of birds saved - the scope of the altruistic action - had little effect on willingness to pay.
Similar experiments showed that Toronto residents would pay little more to clean up all polluted lakes in Ontario than polluted lakes in a particular region of Ontario , or that residents of four western US states would pay only 28% more to protect all 57 wilderness areas in those states than to protect a single area .
People visualize "a single exhausted bird, its feathers soaked in black oil, unable to escape" . This image, or prototype, calls forth some level of emotional arousal that is primarily responsible for willingness-to-pay - and the image is the same in all cases. As for scope, it gets tossed out the window - no human can visualize 2000 birds at once, let alone 200000. The usual finding is that exponential increases in scope create linear increases in willingness-to-pay - perhaps corresponding to the linear time for our eyes to glaze over the zeroes; this small amount of affect is added, not multiplied, with the prototype affect. This hypothesis is known as "valuation by prototype".
An alternative hypothesis is "purchase of moral satisfaction". People spend enough money to create a warm glow in themselves, a sense of having done their duty. The level of spending needed to purchase a warm glow depends on personality and financial situation, but it certainly has nothing to do with the number of birds.
We are insensitive to scope even when human lives are at stake: Increasing the alleged risk of chlorinated drinking water from 0.004 to 2.43 annual deaths per 1000 - a factor of 600 - increased willingness-to-pay from $3.78 to $15.23 . Baron and Greene found no effect from varying lives saved by a factor of 10 .
A paper entitled Insensitivity to the value of human life: A study of psychophysical numbing collected evidence that our perception of human deaths follows Weber's Law - obeys a logarithmic scale where the "just noticeable difference" is a constant fraction of the whole. A proposed health program to save the lives of Rwandan refugees garnered far higher support when it promised to save 4,500 lives in a camp of 11,000 refugees, rather than 4,500 in a camp of 250,000. A potential disease cure had to promise to save far more lives in order to be judged worthy of funding, if the disease was originally stated to have killed 290,000 rather than 160,000 or 15,000 people per year. 
The moral: If you want to be an effective altruist, you have to think it through with the part of your brain that processes those unexciting inky zeroes on paper, not just the part that gets real worked up about that poor struggling oil-soaked bird.
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