Offered a coin flip that pays £100 on heads and costs £100 on tails, almost everyone says no. The bet is fair, yet refusing it looks sensible once you grant one fact: a loss hurts more than the same-sized gain pleases. Prospect theory builds a whole model of choice on that asymmetry — the behavioural topic intermediate students most often bring to an economics tutor in Edinburgh.
1 · From final wealth to gains and losses
The expected-utility model — the subject of the risk and insurance page — scores a gamble by the expected utility of your final wealth, with probabilities entering linearly. It is a clean benchmark that rests on a claim people routinely break: that only your ending position matters, not the path to it.
Prospect theory swaps the carrier of value. What you feel are changes — gains and losses measured against a reference point, usually where you stand now. The same £50,000 feels like a gain to someone who climbed from £40,000 and a loss to someone who fell from £60,000. Fixing the reference point is the first move: it decides which outcomes count as gains and which as losses.
2 · The value function: concave gains, convex losses
Value attaches to those gains and losses through a value function, v(x), that passes through the reference point at the origin. Kahneman and Tversky gave it a shape and estimated its parameters: v(x) = x0.88 over gains, v(x) = −λ(−x)0.88 over losses, with a loss-aversion coefficient λ ≈ 2.25. The 0.88 and the 2.25 are their published estimates, not universal constants.
The exponent below one makes the curve concave over gains and convex over losses — an S on its side. Concavity is diminishing sensitivity: £0 to £100 feels bigger than £900 to £1,000. Convexity over losses is the surprise — you turn risk-seeking when facing losses — the reflection effect: preferences flip when you negate the outcomes.
Work one pair. Over gains, choose a sure £50 or a 50–50 shot at £100 or nothing: the sure thing scores v(50) ≈ 31.3, the gamble ½·v(100) ≈ 28.8, so you take the £50 — risk-averse. Negate everything — a sure −£50 or a 50–50 shot at −£100 or nothing: the sure loss scores v(−50) ≈ −70.4, the gamble ½·v(−100) ≈ −64.7, so now you gamble — risk-seeking. Same odds, mirror-image stakes, opposite choice.
3 · The kink at the reference point
Look where the branches meet. The loss side falls far more steeply than the gain side climbs — the kink at the reference point, and it is loss aversion. A £100 gain is worth v(100) ≈ 57.5; a £100 loss, v(−100) ≈ −129.5 — the loss looms 2.25 times as large, exactly the λ in the formula. That one asymmetry explains the refused fair bet and why small favourable gambles get turned down. The diagram plots the whole function, both £100 points marked.
4 · Weighting, framing and ownership
Three further departures round out the model.
Probability weighting. People act not on raw probabilities but on decision weights that overweight small probabilities. That explains two habits that look opposed: you buy the lottery ticket and you insure against the rare disaster — both overweight a tiny chance of a large outcome. The gambler and the insurer are the same person.
Framing. Because value is read from a reference point, how a prospect is described can shift that point and flip the choice. Framed as saving 200 of 600 people, a programme is chosen cautiously; framed as 400 of 600 dying, the same programme is gambled on. Only the coding changed.
The endowment effect. Once something is yours, giving it up lands on the steep loss branch, so you demand more to sell it than you would have paid — the reason people want more for a mug they own than they would spend to buy one. Ownership moves the reference point; loss aversion does the rest.
5 · Nudges: defaults and libertarian paternalism
If reference points and frames shape choices, whoever sets them holds influence — the bridge to policy. A nudge changes the choice architecture without changing the options or their payoffs; the flagship is the default.
Defaults bite because opting out reads as an active loss of the status quo, and because inertia is strong. Make organ donation the default and participation runs far higher than under opt-in; auto-enrol employees in a pension, with freedom to leave, and far more save than when they must sign up. Same options, different default, different outcome.
Thaler and Sunstein called this libertarian paternalism — paternalist because it steers you toward what is likely good for you, libertarian because the alternatives stay one step away. Critics reply that a nudge is still a designer choosing for you, and who sets the default, in whose interest, is the question the framing glosses over.
Worked example — a fair coin flip you would still refuse
Price the opening bet: a 50–50 gamble, win £100 on heads, lose £100 on tails. Your reference point is current wealth, so heads is a £100 gain and tails a £100 loss, valued by v(x) = x0.88 and v(x) = −2.25(−x)0.88.
Step 1 — Value the win. v(100) = 1000.88 ≈ 57.5 value units.
Step 2 — Value the loss. v(−100) = −2.25 · 1000.88 ≈ −129.5 — 2.25 times the magnitude of the same-sized gain.
Step 3 — Combine. ½·v(100) + ½·v(−100) ≈ ½(57.5) + ½(−129.5) = −36. Negative, so you refuse a fair bet. Equal probabilities mean any weighting scales both halves alike and cannot flip the sign.
Step 4 — Perturb. Hold the loss at £100 and raise the prize to G: you accept once ½·G0.88 + ½·v(−100) ≥ 0, i.e. G0.88 ≥ 2.25 · 1000.88.
Step 5 — Solve. G = 100 · 2.251/0.88 ≈ £251 to risk £100 on one toss.
Step 6 — Interpret. The loss looms 2.25 times larger in value; because the curve also bends, the money you demand runs to about 2.5 to 1. Treat the £100 as already gone and the same gamble can flip to accept — the bet never changed, only its coding did.
Could you price a fair coin flip from the value function alone — and show why losing £100 outweighs winning it? Deriving loss aversion and the reflection effect from v(x) and λ, rather than quoting them, is exactly what prospect-theory questions reward. A one-on-one economics tutor works the kink and the S-shaped curve with you until the refused bet is a number you compute, not a slogan you recall. Book a trial session.
Practice
Q1. Score a 50–50 gamble that wins £200 or loses £150 two ways — by prospect theory (use v) and by expected monetary value. Which says take it? (v(200) ≈ 105.9, v(−150) ≈ −184.9.)
Q2. You face £400 for certain versus a 50–50 shot at £300 or £500. Code it from a £300 reference (sure +£100 vs 50–50 +£200 or +£0), then from a £500 reference (sure −£100 vs 50–50 −£200 or −£0). Which option wins under each framing, and by how much?
Answers. Q1: PT value ≈ ½·105.9 + ½·(−184.9) = −39.5 → refuse; but expected monetary value = ½(200) + ½(−150) = +£25 → a risk-neutral agent accepts. Same gamble, opposite verdicts. Q2: Gain frame — sure v(100) ≈ 57.5 beats ½·v(200) ≈ 52.9, so take the £400, by about 4.6 units. Loss frame — ½·v(−200) ≈ −119.1 beats sure v(−100) ≈ −129.5, so gamble, by about 10.3 units. Identical prospect, reversed choice.
Key takeaways
- Value comes from changes, not levels. Gains and losses are scored against a reference point, so the same final wealth can read as a win or a loss.
- The value function is an S with a kink — concave over gains, convex over losses, so risk-averse in gains and risk-seeking in losses, with preferences reflecting when outcomes are negated.
- Losses loom larger, about 2.25 to 1 — the asymmetry behind refused fair bets, the endowment effect, and loss-framed choices.
- Small probabilities are overweighted — a separate distortion that sells both lottery tickets and insurance.
- Nudges are applied prospect theory — defaults exploit reference dependence and inertia, which makes libertarian paternalism both powerful and contested.
Why Edinburgh students choose our economics tutoring
- Models built from the value function, not slogans: sessions derive the refused fair bet and the reflection effect from v(x) and λ, so you can reproduce them under exam pressure.
- The distinctions examiners test, drilled: reference dependence versus wealth levels, loss aversion versus risk aversion, prospect theory versus expected utility.
- One-on-one and matched to your course: a tutor works from your own notation and past papers, whatever behavioural text your module follows.
FAQ
Q: What is the difference between prospect theory and expected utility?
A: Expected utility values a gamble by the utility of your final wealth, with linear probabilities. Prospect theory values gains and losses against a reference point, weights probabilities non-linearly, and makes losses loom larger.
Q: What does loss aversion actually mean?
A: A loss hurts more than an equal gain pleases — in Kahneman and Tversky’s estimates, about 2.25 to 1. That is why people refuse many fair bets.
Q: Why are people risk-seeking over losses?
A: Below the reference point the value function is convex, so a certain loss feels worse than a gamble that might avoid it. Faced with losses, people take the gamble — the reflection effect.
Q: Is a nudge just manipulation?
A: A nudge changes defaults or framing without removing options or changing payoffs. Whether that is help or manipulation depends on who designs it and in whose interest — the core criticism of libertarian paternalism.
Q: How is this different from the risk and insurance topic?
A: That topic stays inside expected utility — concave utility of wealth, certainty equivalents, risk premia. This one starts where expected utility fails to predict real choices.
Book an economics tutor in Edinburgh or online
Behavioural economics rewards students who move fluently between the picture and the algebra — the value function, loss aversion, probability weighting, and the framing and default effects that follow. One-on-one sessions build that fluency on your own past papers. Tell us your course and exam date, and we will match you with the right tutor this week.
