Expected Utility Theory

[phk]

As a slight departure from the macro heavy focus of this board. I'v decided to post about a micro/behavioral concept called expected utility theory.

It was first developed by Daniel Bernoulli sometime in the 1700s and further expanded upon by John Von Neumann and Oskar Morgenstern who famously wrote the seminal work "Theory of Games" sometime in the 1950s.

Now the question is that how do economists understand individuals preferences in the presence of 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.  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).

The foundational underpinnings of the theory are based on 4 axioms: 1) Completeness 2) Transitivity 3) Mixture Continuity and 4) Independence

In order for people to make decisions according to the EUT framework, the 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. Mixture 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? Do people actually make decisions according to these rules?

[ag]

A few comments.

1. Bernoulli is a precursor. He predates aything like the formal development of the utility theory (expected or otherwise). Overall, his insight is too far removed to be properly considered the part of the modern theory.

2. You fail to highlight the main feature of vNM approach (as distinct, say, from the Savage approach, etc.): probabilities (beliefs) are viewed as known.

3. The "dollars" here are just an example, not the real money. The proper way of putting it is: let's suppose, for simplicity, that there is only one good and let's call it dollars.

4. Completeness and transitivity (continuity is purely technical and not worth talking about) are necessary for any utility function representation of preferences, expected or non-expected. This is what economists call rationality. Since this is a lot more fundamental then just the EU, let's leave it aside for the moment.

5. Independence is the key assumption here. It's really a linearity assumption: since expectations are linear in probabilities, only linear preferences may be represented w/ expected utilities. Independence provides a nice behavioral interpretation of the mathematical condition.

6. Of course, there are cases when people do not satisfy independence (or any other axiom, for that matter). Look up the Machina or Allais paradoxes, for instance, for intuitive examples of such violations. The theory works pretty well for a whole bunch of applications - and less well for others. Like any model, it's just an approximation. An approximation that allows to use pretty neat technical tools, though: if you can't take expectations, life is harder.