Expected utility & risk types 2.0
About this app
Rebuild 2.0 (preview): St. Petersburg paradox, utility x^γ with certainty equivalent, prospect theory with loss aversion.
Prospect theory in formulas
Kahneman and Tversky’s prospect theory values changes Δ relative to a reference point. In the cumulative version of 1992 the value function is v(Δ) = Δ^α for gains and −λ·(−Δ)^β for losses, with median estimates α = β = 0.88 and λ = 2.25. On top comes probability weighting w(p): small probabilities are overweighted, medium and large ones underweighted. Together this gives the fourfold pattern: risk-seeking for unlikely gains and likely losses, risk-averse for likely gains and unlikely losses.
What the app shows
There are three topics. “St. Petersburg” plays up to a million rounds and shows that the average grows only roughly like log₂ n, while the median stays at 2 €. “Risk types” works through concrete decisions with u(x) = x^γ: a coin toss, fire insurance, a lottery ticket or a lottery of your own, each with certainty equivalent, risk premium and relative risk aversion 1 − γ. “Prospect theory” shows the value function and the weighting curve on a fixed scale.
Try the fourfold pattern
In the prospect theory topic, pick one of four bets: 5 % on +100, 5 % on −100, 95 % on +100 or 95 % on −100. For each, the app compares the bet with the sure expected value p · Δ and shows which side is preferred. Then turn λ up and notice that only the loss branch gets steeper. In the risk types topic the insurance case is worth a look: with concave utility you buy cover even though its premium exceeds the expected loss.
Frequently asked questions
- What is the fourfold pattern in prospect theory?
- The fourfold pattern describes when people seek or avoid risk. With small chances of a gain they gamble, as in a lottery; with small chances of a loss they buy insurance. With likely gains they prefer the sure amount, and with likely losses they tend to take a risk to avoid a sure loss.
- What is probability weighting?
- Probability weighting means that people factor probabilities into decisions in a distorted way. The weighting function w(p) lies above the diagonal for small p and below it for medium and large p. A probability of 5 % therefore feels larger, and one of 95 % smaller, than it is.
- Why do people buy both insurance and lottery tickets?
- Both cost money in expectation. Prospect theory explains this through the overweighting of small probabilities: the rare large loss is overrated just like the rare large win. Classical expected utility theory with a single concave utility function struggles to explain both at once.
- What is different in this version?
- It adds probability weighting and the fourfold pattern with four prepared bets. It also includes concrete insurance and lottery decisions, relative risk aversion and a simulation of the St. Petersburg game with up to a million rounds. Sources are listed in the info panel.
Subject: Data & decisions
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- The three-cowboy duel 2.0
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