Utility Functions

Utility functions $u(y)$ are different for different persons and organizations. An individual utility function is defined by a lottery
$L(A,B,p)=\{pA+(1-p)B\}$. Here $p$ is the probability to win the best event $A$.
$(1-p)$ is the probability to get the worst one $B$. Denote by $C$ the "ticket price" of this lottery. There are two possible decisions:

• keep the ticket money $C$,
• bay a ticket and risk losing this money while hoping to win a greater wealth $A$ with probability $p$.

Denote by $p(C)$ a "hesitation" probability, when one cannot decide which decision to prefer. One defines the "hesitation" probability $p(C)$ by this condition

$$L(A,B,C,p(C))=[C \approx \{p(C) A + (1-p(C)) B\}].$$ (42)

Here the symbol $\approx$ denotes the "hesitation." If utilities $u(A)=1$ and $u(B)=0$, the utility of the "ticket" $C$ is equal to the hesitation probability $u(C)=p(C)$ [1].

Suppose, for example, that event $C$ is to keep all the investment capital, $y=1$, in a safe; no risk, no profit. Assume that the event $A$ means doubling the capital, $y=2$. The event $B$ means losing all the capital, $y=0$.

Denote by $p(1)$ the hesitation probability. Then
$u(1)= u(0)+p(1)(u(2)-u(0))$. If $u(0)=0$ and $u(2)=1$ then the utility of the capital $u(1)=p(1)$. Here one obtained capital utilities at three points: $y=0,\ y=1$, and $y=2$.

To define a reasonable approximation of the utility function $u(y)$, we need at least two additional points. For example, points $y=0.5$ and $y=1.5$. One defines the corresponding utilities by the hesitation probabilities $p(0.5)$ and $p(1.5)$. These are obtained by two hesitation lotteries

$$L(1.0,0.0,0.5,p(0.5))=$$ (43)

$$[(y=0.5) \approx \{p(0.5)(y=1) +(1-p(0.5))(y= 0) \}] \nonumber$$

and

$$L(2.0,1.0,1.5,p(1.5))=$$ (44)

$$[(y=1.0) \approx \{p(1.5)(y=2.0) +(1-p(1.5))(y=1) \}]. \nonumber$$

Here one obtains utility values
$u(0)=0,\ u(0.5)=p(0.5), \ u(1)=p(1),\ u(1.5)=u(1) + p(1.5) (u(2)-u(1))\ u(2)=1$. The remaining capital utility values are defined by the linear interpolation

$$u(y)=u(y_i) + p(y_i) (u(y_{i+1}) - u(y_i)),\ \ y_i \le y <y_{i+1},$$ (45)

$$i=0,1,...,4.$$

In consulting offices, the "psychological tests" defining capital utilities are not always convenient. Then one of the four "typical" utility functions can be selected. The typical utility functions represent the risky, the average, the rich and the prudent persons. The selection depends on observable personal traits.

The same capital utility function (14) could be used for all customers. Then one defines customer differences by different border points, namely:
$r_{prudent} < r_{average} < r_{rich} <r_{risky}$ (see Figure ).