Bernoulli performed the following experiment while his petersburg’s stay: The participants iteratively tossed the coin until getting head. If the head fell in the k-th attempt, the participant was paid by \(2^k\) Rubbles.
Expected payoff of the experiment participant is
\[E = 2^1 (\frac{1}{2})^1 \dots + 2^i (\frac{1}{2})^i + \dots + \dots\] \[E = 1 \dots + 1 + \dots = \infty\]
If asking the participants how much are they willing to pay for having chance to participate, the answer was that the max amount of money is about 10 Rubbles. It does not correspond with the Expected payoff (slide above), which is infinity! (Paradox)
Expected payoff is not a good predictor for the game price;
And what is the good predictor?
Answer: Individuals make decisions based on the expected utility of an outcome, not just the expected financial value. This theory accounts for the diminishing marginal utility of money—each additional Rubble is worth less in terms of utility to a person than the previous one.
People take the decisions based on the Expected utility of the payoff \(E(u(x))\), which, respecting the law of diminishing marginal utility implies the following:
\[E(u(x)) \approx E\left[u(\overline{x}) + u'(\overline x)(x - \overline x) + ... + \frac{1}{i!}u^(i)(\overline x)(x - \overline x)^i + \dots \right]\]
If taking into account first three members of the series and neglecting the rest, and using the Expecting operator property on sum, than
\[E(u(x)) \approx E\left[u(\overline{x})\right] + E\left[u'(\overline x)(x - \overline x)\right] + E\left[\frac{1}{2}u''(\overline x)(x - \overline x)^2\right]\]
resp.
\[E(u(x)) \approx \underset{(\frac{du}{d \overline x}>0)}{u(\overline{x})} + u'(\overline x)E\underset{(=0)}{\left[(x - \overline x)\right]} + \frac{1}{2}\underset{(-)}{u''(\overline x)}E\left[(x - \overline x)^2\right]\] resp.
\[E(u(x)) \approx u(\overline{x}) + \frac{1}{2}\underset{(-)}{u''(\overline x)}E\left[(x - \overline x)^2\right]\] which means, the Expected utility is growing with the mean value of the stochastic payoff \(\overline x\) and falling with its dispersion \(E\left[(x - \overline x)^2\right]\)
Pratt’s risk aversion measures are foundational in understanding an individual’s aversion to risk within the context of expected utility theory. These measures are derived to quantify the level of risk aversion exhibited by an individual when making decisions under uncertainty. There are two primary measures of risk aversion according to Pratt: the Absolute Risk Aversion (ARA) and the Relative Risk Aversion (RRA). Both are derived from the utility function of stochastic payoff.
The Absolute Risk Aversion (ARA) is defined as:
\[ARA(w) = -\frac{u''(x)}{u'(x)}\]
Where: 1. \(U(x)\) is the utility function of stochastic payoff \(x\), 2. \(U'(x)\) is the first derivative of the utility function with respect to stochastic payoff, representing the marginal utility of stochastic payoff, 3. \(U''(x)\) is the second derivative of the utility function with respect to stochastic payoff, indicating the curvature of the utility function or the change in marginal utility.
The Relative Risk Aversion (RRA) is defined as:
\[RRA(x) = -\frac{x \cdot U''(x)}{U'(x)}\]
Where the variables are the same as those in the ARA formula. The key difference here is the multiplication of the second derivative by stochastic payoff \(w\), making the measure relative to the size of the stochastic payoff.
Both measures provide insights into how risk-averse an individual is, with higher values indicating greater aversion to risk. The ARA gives an absolute measure of this aversion, while the RRA adjusts the measure relative to the individual’s stochastic payoff level, providing context on how risk aversion changes with stochastic payoff.
Nobel prize winner H. Markowitz proposed to use the quadratic or logarithmic functions as the appropriate representants of the utility function \(u\).