6 Risk
When some information is given about the probability of occurrence of states of nature, we speak of decision-making under risk. The most straight forward technique to make a decision here is to maximize the expected outcome for each alternative given the probability of occurrence, \(p_j\).
However, the expected utility hypothesis states that the subjective value associated with an individual’s gamble is the statistical expectation of that individual’s valuations of the outcomes of that gamble, where these valuations may differ from the Euro value of those outcomes. Thus, you should better look on the utility of a respective outcome rather than on the outcome itself because the utility and outcome do not have to be linked in a linear way. The St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis.
Infinite St. Petersburg lotteries
Suppose a casino offers a game of chance for a single player, where a fair coin is tossed at each stage. The first time head appears the player gets $1. From then onwards, every time a head appears, the stake is doubled. The game continues until the first tails appears, at which point the player receives \(\$ 2^{k-1}\), where k is the number of tosses (number of heads) plus one (for the final tails). For instance, if tails appears on the first toss, the player wins $0. If tails appears on the second toss, the player wins $2. If tails appears on the third toss, the player wins $4, and so on. The extensive form of the game is given in Figure 6.1.
Given the rules of the game, what would be a fair price for the player to pay the casino in order to enter the game?
To answer this question, one needs to consider the expected payout: The player has a 1/2 probability of winning $1, a 1/4 probability of winning $2, a 1/8 probability of winning $4, and so on. Thus, the overall expected value can be calculated as follows: \[ E = \frac{1}{2} \cdot 1 + \frac{1}{4} \cdot 2 + \frac{1}{8} \cdot 4+ \frac{1}{16} \cdot 8 + \dots \] This can be simplified as: \[ E = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \dots = + \infty. \] That means the expected win for playing this game is an infinite amount of money. Based on the expected value, a risk-neutral individual should be willing to play the game at any price if given the opportunity. The willingness to pay of most people who have given the opportunity to play the game deviates dramatically from the objectively calculable expected payout of the lottery. This describes the apparent paradox.
In the context of the St. Petersburg Paradox, it becomes evident that relying solely on expected values is inadequate for certain games and for making well-informed decisions. Expected utility, on the other hand, has been the prevailing concept used to reconcile actual behavior with the notion of rationality thus far.
Finite St. Petersburg lotteries
Let us assume that at the beginning, the casino and the player agrees upon how many times the coin will be tossed. So we have a finite number I of lotteries with \(1 \leq I \leq \infty\).
To calculate the expected value of the game, the probability \(p(i)\) of throwing any number \(i\) of consecutive head is crucial. This probability is given by \[ p(i)=\underbrace{\frac{1}{2} \cdot \frac{1}{2} \cdot \cdots \frac{1}{2}}_{i \text { factors}}=\frac{1}{2^{i}} \] The payoff \(W(I)\) is, if head appears \(I\)-times in a row by \[ W(I)=2^{I-1} \] The expected payoff \(E(W(I))\) if the coin is flipped \(I\) times is then given by \[ E(W(I))=\sum_{i=1}^{I} p(i) \cdot W(i)=\sum_{i=1}^{I} \frac{1}{2^{i}} \cdot 2^{i-1}=\sum_{i=1}^{I} \frac{1}{2}=\frac{I}{2} \]
Thus, the expected payoff grows proportionally with the maximum number of rolls. This is because at any point in the game, the option to keep playing has a positive value no matter how many times head has appeared before. Thus, the expected value of the game is infinitely high for an unlimited number of tosses but not so for a limited number of tosses. Even with a very limited maximum number of tosses of, for example, \(I = 100\), only a few players would be willing to pay $50 for participation. The relatively high probability to leave the game with no or very low winnings leads in general to a subjective rather low evaluation that is below the expected value.
In the real world, we understand that money is limited and the casino offering this game also operates within a limited budget. Let’s assume, for example, that the casino’s maximum budget is $20,000,000. As a result, the game must conclude after 25 coin tosses because \(2^{25} = 33,554,432\) would exceed the casino’s financial capacity. Consequently, the expected value of the game in this scenario would be significantly reduced to just $12.50. Interestingly, if you were to ask people, most would still be willing to pay less than $12.50 to participate. How can we explain this? Well, it is not the expected outcome that matters but the utility that stems from the outcome.
The impact of output on utility matters
Daniel Bernoulli (1700 - 1782) worked on the paradox while being a professor in St. Petersburg. His solution builds on the conceptual separation of the expected payoff and its utility. He describes the basis of the paradox as follows:
``Until now scientists have usually rested their hypothesis on the assumption that all gains must be evaluated exclusively in terms of themselves, i.e., on the basis of their intrinsic qualities, and that these gains will always produce a utility directly proportionate to the gain.’’ (Bernoulli, 1954, p. 27)
The relationship between gain and utility, however, is not simply directly proportional but rather more complex. Therefore, it is important to evaluate the game based on expected utility rather than just the expected payoff. \[ E(u(W(I)))=\sum_{i=1}^{I} p(i) \cdot u(W(i))=\sum_{i=1}^{I} \frac{1}{2^{i}} \cdot u\left(2^{i-1}\right) \] Daniel Bernoulli himself proposed the following logarithmic utility function: \[ u(W)=a \cdot \ln (W), \] where \(a\) is a positive constant. Using this function in the expected utility, we get \[ E(u(W(I)))=\sum_{i=1}^{I} \frac{1}{2^{i}} \cdot a \cdot \ln \left(2^{i-1}\right)=a \cdot \sum_{i=1}^{I} \frac{i-1}{2^{i}} \ln 2=a \cdot \ln 2 \cdot \sum_{i=1}^{I} \frac{i-1}{2^{i}}. \] The infinite series, \(\sum_{i=1}^{I} \frac{i-1}{2^{i}}\), converges to 1 (\(\lim _{I \rightarrow \infty} \sum_{i=1}^{I} \frac{i-1}{2^{i}}=1\)). Thus, given an ex ante unbounded number of throws, the expected utility of the game is given by \[ E(u(W(\infty)))=a \cdot \ln 2 . \]
In experiments in which people were offered this game, their willingness to pay was roughly between 2 and 3 Euro. Thus, the suggests logarithmic utility function seems to be a pretty realistic specification. The main reason is mathematically that the increasing expected payoff has decreasing marginal utility and hence the utility function reflects the risk aversion of many people.