Editing Cardinal utility (section)

===Construction of the utility function===

Suppose that certain outcomes are associated with three states of nature, so that ''x''<sub>3</sub> is preferred over ''x''<sub>2</sub> which in turn is preferred over ''x''<sub>1</sub>; this set of outcomes, {{math|''X''}}, can be assumed to be a calculable money-prize in a controlled game of chance, unique up to one positive proportionality factor depending on the currency unit.

Let {{math|''L''<sub>1</sub>}} and {{math|''L''<sub>2</sub>}} be two lotteries with probabilities ''p''<sub>1</sub>, ''p''<sub>2</sub>, and ''p''<sub>3</sub> of ''x''<sub>1</sub>, ''x''<sub>2</sub>, and ''x''<sub>3</sub> respectively being

:<math>L_1 =(0.6, 0, 0.4),</math>
:<math>L_2 =(0,1,0)\ .</math>

Assume that someone has the following preference structure under risk:

:<math>L_{1} \succ L_{2},</math>

meaning that {{math|''L''<sub>1</sub>}} is preferred over {{math|''L''<sub>2</sub>}}. By modifying the values of {{math|''p''<sub>1</sub>}} and {{math|''p''<sub>3</sub>}} in {{math|''L''<sub>1</sub>}}, eventually there will be some appropriate values ({{math|''L''<sub>1'</sub>}}) for which she is found to be indifferent between it and {{math|''L''<sub>2</sub>}}&mdash;for example

:<math>L_{1}' =(0.5, 0, 0.5).</math>

Expected utility theory tells us that

:<math>EU(L_{1}') = EU(L_2)</math>

and so

:<math>(0.5) \times u(x_1)+(0.5) \times u(x_{3}) = 1 \times u(x_{2}).</math>

In this example from Majumdar<ref>{{cite journal |last1=Majumdar |first1=Tapas |date=February 1958 |title=Behaviourist Cardinalism in Utility Theory |journal=Economica |volume=25 |issue=97 |pages=26–33 |doi=10.2307/2550691 |jstor=2550691}}</ref> fixing the zero value of the utility index such that the utility of {{math|''x''<sub>1</sub>}} is 0, and by choosing the scale so that the utility of {{math|''x''<sub>2</sub>}} equals 1, gives

:<math>(0.5) \times u(x_{3})=1.</math>
:<math>u(x_{3}) = 2.</math>