The assumption that a VNM expected utility function is linear in probabilities, though not in payoffs, allows us to create an indifference-curve representation.

To create a simple two-dimensional representation, let's assume we have only 3 possible outcomes, x_{1}, x_{2}, and x_{3}, such that x_{1} x_{2} x_{3}. Assume they occur with probabilities p_{1}, p_{2}, and p_{3} respectively, where p_{i} = 1.

Now, since p_{2} = 1 - p_{1} - p_{3}, we can represent these lotteries by points in a unit triangle in the (p_{1}, p_{3}) plane, known as the Machina triangle.

Upward movements in the triangle increase p_{3} at the expense of p_{2}, shifting probability from x_{2} to x_{3}. Leftward movements reduce p_{1} to the benefit of p_{2}, shifting probability from from x_{1} to x_{2}. As a result, both of these, and more generally, all northwest movements, lead to stochastically dominating lotteries that will be preferred.

### Representing risk attitudes using indifference curves

Since the indifference curves are given by solutions to the linear equation:

u* = U(x_{1})p_{1} + U(x_{2})(1 - p_{1} - p_{3}) + U(x_{3})p_{3} = constant,

they consist of parallel straight lines of slope [U(x_{2}) - U(x_{1})]/[U(x_{3}) - U(x_{2})]. As a result, we can use this type of diagram to illustrate attitudes to risk, as under:

The blue lines in both panels are iso-expected value lines, while the black lines are indifference curves. The iso-expected value lines are, of course, the solutions to:

x* = x_{1}p_{1} + x_{2}(1 - p_{1} - p_{3}) + x_{3}p_{3} = constant

Northeast movements along the blue lines do not change the expected value of the prospect, but they increase the probabilities of of the tail outcomes x_{1} and x_{3}. They are *mean-preserving spreads*, or pure increases in risk. The left-hand panel contains indifference curves much steeper than the the iso-expected value lines, so they are indifference curves formed by the concave utility function of a risk-averse individual. In the right-hand panel, the indifference curves are much flatter than the iso-expected value lines. This implies that they are representations of the risk-loving preferences of an individual with a convex utility function. Similarly, a risk-neutral individual would have indifference curves that coincide with the iso-expected value lines.

The blue lines in both panels are iso-expected value lines, while the black lines are indifference curves. The iso-expected value lines are, of course, the solutions to:

x* = x_{1}p_{1} + x_{2}(1 - p_{1} - p_{3}) + x_{3}p_{3} = constant

Northeast movements along the blue lines do not change the expected value of the prospect, but they increase the probabilities of of the tail outcomes x_{1} and x_{3}. They are *mean-preserving spreads*, or pure increases in risk. The left-hand panel contains indifference curves much steeper than the the iso-expected value lines, so they are indifference curves formed by the concave utility function of a risk-averse individual. In the right-hand panel, the indifference curves are much flatter than the iso-expected value lines. This implies that they are representations of the risk-loving preferences of an individual with a convex utility function. Similarly, a risk-neutral individual would have indifference curves that coincide with the iso-expected value lines.

### Violations of linearity in probabilities

Linearity in probabilities implies and is implied by the substitution or independence axiom. The famed Allais Paradox is the best example of systematic violations of this axiom.

Recall that subjects choose from the following:

**Gamble A:** A 100% chance of receiving $1 million

**Gamble B:** A 10% chance of receiving $5 million, an 89% chance of receiving $1 million, and a 1% chance of receiving nothing.

They then choose between:

**Gamble C:** An 11% chance of receiving $1 million, and an 89% chance of receiving nothing

**Gamble D:** A 10% chance of receiving $5 million, and a 90% chance of receiving nothing.

Most subjects choose A in the first round and D in the second. However, preferring A in the first round to B implies steep indifference curves, so the independence axiom would require that one choose C in the second round, not D. If one chooses D in the second round, the implication is that one's indifference curves are flat, and B should be chosen in the first round. This is illustrated below:

We can see that the 4 gambles form a parallelogram in the Machina triangle, where outcome x_{1} is a payoff of 0, outcome x_{2} is a payoff of $1 million, and outcome x_{3} is a payoff of $5 million.

We can see that a person's indifference curves appear to be steep in the first case, and flat in the second - they *fan out*, as under, and are not parallel.

We can see that the 4 gambles form a parallelogram in the Machina triangle, where outcome x_{1} is a payoff of 0, outcome x_{2} is a payoff of $1 million, and outcome x_{3} is a payoff of $5 million.

We can see that a person's indifference curves appear to be steep in the first case, and flat in the second - they *fan out*, as under, and are not parallel.

### More evidence of fanning out: The Common Consequence effect

Originally, the Allais Paradox was considered an isolated example, but further experimental testing has shown it to be part of a general empirical pattern, called the *common consequence* effect. The effect involves any pairs of probability mixtures of the form:

A: ad_{x} + (1-a)P**, versus B: aP + (1-a)P**;

and

C: ad_{x} + (1-a)P*, versus D: aP + (1-a)P*.

Here, d_{x} denotes the lottery yielding x with certainty. P is a lottery involving outcomes both greater and less than x. P** *stochastically dominates* P*, i.e. the more preferred outcome has a higher probability, and the less preferred outcome has a lower probability, in P** than in P*. When the distributions are over a common outcome set, say x_{1}, x_{2}, and x_{3}, the prospects will form a parallelogram in the probability triangle, as above. Any choice of A and D will once again imply indifference curves which fan out.

### References

Machina, Mark J. (1987), "Choice Under Uncertainty: Problems Solved and Unsolved," *Journal of Economic Perspectives*, Vol. 1, No. 1