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Four Axioms needed for Expected Utility Theory

Written By: Jason Shafrin - Jul• 30•07

How do economists understand individuals preferences when there is risk? Without risk, economists generally believe that individuals have a utility function which can convert ordinal preferences into a real-valued function. This real valued function is the utility function.

When risk enters into the picture, the expected utility theory (EUT) is used. This theory was developed by Daniel Bernoulli (1738) and expanded by John von Neumann and Oskar Morgenstern (1947). The EUT implies that utility functions have the following functional form:

  • U=Σi piu(xi)

Here there are i states of the world. In each state of the world, i, the individual receives xi dollars. The probability of receiving xi is pi. An individual will prefer one risky lottery over another if their utility is higher in the first lottery compared to the second.

For example, let us assume that there are two lotteries. In lottery A you receive $100 for sure. In lottery B you have a 60% chance of receiving $200 and a 40% chance of receiving $0. Thus your utility in each case would be:

  • UA= 1*u(100)
  • UB= .6*u(200)+.4*u(0)

The lottery you choose will be based on your expected utility. Risk neutral individuals have linear utility functions, risk averse individuals have concave utility functions (u”<0) and risk loving individuals have convex utility functions (u”>0).
Do people actually make decisions according to these rules?

4 axioms

In order for people to make decisions according to the EUT framework, 4 axioms must hold. Let q, r, and s, be defined as the following lotteries: q=(x1,p1; x2,p2;…xn,pn), r=(y1,q1; y2,q2;…yn,qn) and s=(z1,w1; z2,w2;…zn,wn). Also, define aWb to mean that ‘a’ is weakly preferred to ‘b’.

  1. Completeness. This entails that for all q, r: either qWr or rWq or both. If the answer is both, then I am indifferent between q and r.
  2. Transitivity. If qWr, rWs then qWs.
  3. Continuity. If qWr, rWs then there exists some p such that (q,p; s,1-p)~r.
  4. Independence. This requires that if qWr, then (q,p; s,1-p)W(r,p; s,1-p) This means that I prefer tacos to hamburgers for lunch, I will not change my preferences between tacos and hamburgers if I am offered a salad as well. This is the axiom most commonly relaxed when alternatives to EUT are examined.

Are these axioms realistic? In the next post, I will review an article which describes “Developments in Non-Expected Utility Theory” where some of these axioms are violated.

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