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More on value functions

The emphasis on changes as the carriers of value does not mean that the value of a particular change is independent of the initial position. Value functions are likely to become more linear with increases in assets. A change from $100 to $200 is likely to have a much higher value than a change from $1100 to $1200. The value function is then concave above the reference point (v"(x) < 0 for x > 0), and convex below it (v"(x) > 0 for x < 0). Meaning, it is concave for gains and convex for losses.

Most people dislike symmetric gambles of the sort (50,0.5; -50,0.5). In fact, if x > y 0, then (y,0.5; -y,0.5) is preferred to (x,0.5; -x,0.5).
This means that v(y) + v(-y) > v(x) + v(-x). Setting y = 0 gives us v(x) < -v(-x), and letting y approach x gives us v'(x) < v'(-x), as long as v is differentiable. So the value function for losses is steeper than that for gains.

More on the weighting function

The weighting function π, which relates decision weights to stated probabilities, is an increasing function of p, with π(0) = 0 and π(1) = 1. However, people tend to overweight very small probabilities, like 0.001, so that π(p) > p for very small p.

Next: Cumulative prospect theory

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