2. The St. Petersburg paradox and the dismal theorem

The well-known resolution of this paradox is to introduce the expected utility theory. If the agent has the CRRA utility function given by (1), then the expected utility is calculated as:

$$\begin{align} EU & =\frac{1}{1-\sigma }\left[\frac{1}{2}2^{1-\sigma }+\frac{1}{2^{2}} 2^{2(1-\sigma )}+\cdots +\frac{1}{2^{k}}2^{k(1-\sigma )}+\cdots \right] \\ & =\frac{1}{1-\sigma }\left[2^{-\sigma }+2^{-2\sigma }+\cdots +2^{-k\sigma }+\cdots \right] \\ & =\frac{1}{1-\sigma }\lim_{n\rightarrow \infty }\frac{2^{-\sigma }(1-2^{-n\sigma })}{1-2^{-\sigma }}.\end{align}$$

If $\sigma >1$, the value of EU is finite.Footnote ⁶ Thus, the agent avoids participating in this risky situation if the fee is sufficiently large.

Next we present a result that has a similar property to the dismal theorem, along with a small modification of the St. Petersburg paradox. Consider another agent who has one million dollars of wealth. He/she risks losing 50 per cent of the wealth with probability 1/2; 75 per cent with probability $1/(2)^{2}$; 87.5 per cent with probability $1/(2)^{3}$; and so on.Footnote ⁷ That is, the agent loses a fraction $1-1/(2)^{k}$ of the wealth with probability $1/(2)^{k}$ (we call this situation A). Then, he or she loses a very large amount of wealth with a very small probability, corresponding to a catastrophe. Here our question is: what share of his/her wealth should the agent be ready to pay to eliminate this risk? Note that the expected amount of money associated with this risk is:

$$\frac{1}{2^{2}}+\frac{1}{2^{4}}+\frac{1}{2^{6}}+\cdots =\frac{1}{3}.$$

We consider the following utility function, which is a particular case of a class of CRRA utility functions (where σ is chosen to be 2):

$$U(C)=-\frac{1}{C}.$$

The expected utility is:

(2)$$EU=\frac{1}{2}(-2)+\frac{1}{2^{2}}(-2^{2})+\frac{1}{2^{3}}(-2^{3})+\cdots + \frac{1}{2^{k}}(-2^{k})+\cdots =-\infty.$$

Therefore, (2) is the opposite of the St. Petersburg paradox. The answer to the above-mentioned question is that the agent is willing to pay 100 per cent of the wealth in order to eliminate the risk (in spite of the fact that the actual loss is almost surely smaller than 100 per cent). This is a dual version of the St. Petersburg paradox, which is in the same spirit as the dismal theorem.Footnote ⁸

The essence of the dismal theorem is obviously related to the dual version of the paradox. However, the dismal theorem is about marginal benefit and cost. That is, it is about a trade-off. On the other hand, the dual version is about absolute levels of benefit. Since this difference is crucial, we introduce a certain trade-off, which shows that classical insights from the paradox can be applied to resolving the dismal theorem. Here we consider a marginally more attractive risky situation (situation B). Take a very small number $\varepsilon >0$. Suppose that if the government pays some cost, then the situation becomes:

1. (i) $50(1+\varepsilon )$ per cent of wealth remains with probability $1/2$;

2. (ii) $25(1+\varepsilon )$ per cent of wealth remains with probability $1/2^{2}$;

3. (iii) $12.5(1+\varepsilon )$ per cent of wealth remains with probability $1/2^{3}$;

and so on. That is, $(1+\varepsilon )/2^{k}$ remains with probability $1/2^{k}$. In each event, the remaining wealth is increased by $100\varepsilon$ per cent. This means that the agent loses $100(1-(1+\varepsilon )/2^{k})$ per cent with probability $1/2^{k}$. It is easy to see that this change yields an increase of $100\varepsilon$ per cent in the expected amount of money: $(1+\varepsilon )/3$. Let $EU(\varepsilon )$ be the expected utility under this situation:

$$\begin{align} EU(\varepsilon )&=\frac{1}{2}\left(-\frac{2}{1+\varepsilon }\right)+\frac{1}{ 2^{2}}\left(-\frac{2^{2}}{1+\varepsilon }\right)+\frac{1}{2^{3}}\left(-\frac{ 2^{3}}{1+\varepsilon }\right)\\ &\quad +\cdots +\frac{1}{2^{k}}\left(-\frac{2^{k}}{ 1+\varepsilon }\right)+\cdots =-\infty.\end{align}$$

Thus the expected value is still infinite. Now we can evaluate the difference between $EU(\varepsilon )$ and EU in (2) by comparing the kth terms in the two situations:

$$\frac{1}{2^{k}}\left(-\frac{2^{k}}{1+\varepsilon }\right)-\frac{1}{2^{k}} (-2^{k})=\frac{\varepsilon }{1+\varepsilon }.$$

This implies that:

$$EU(\varepsilon )-EU=\infty.$$

Thus the expected benefit from this small improvement in terms of the expected amount of money is incredibly large; the agent is willing to pay an extremely large amount of money to obtain this small improvement from A to B. We note that for any small $\varepsilon >0$, this carries over. This negative implication of the cost-benefit analysis is roughly and intuitively similar to that of the dismal theorem. That is, by using extensions of the St. Petersburg paradox, we get a conclusion similar to the the dismal theorem.

These intriguing similarities between the St. Petersburg paradox and the dismal theorem are obtained under a general formulation of the CRRA utility function given in (1). The expected utility under situation A (before paying the cost) is:

$$\begin{align} EU & =\frac{1}{1-\sigma }\left[\frac{1}{2}2^{\sigma -1}+\frac{1}{2^{2}} 2^{2(\sigma -1)}+\cdots +\frac{1}{2^{k}}2^{k(\sigma -1)}+\cdots \right] \\ & =\frac{1}{1-\sigma }\left[2^{\sigma -2}+2^{2(\sigma -2)}+\cdots +2^{k2(\sigma -2)}+\cdots \right]. \end{align}$$

After paying some cost, the agent can receive an infinitesimal improvement. In this situation, the expected utility is given by:

$$EU(\varepsilon )=\frac{1}{1-\sigma }\left[\frac{2^{\sigma -2}}{(1+\varepsilon )^{\sigma -1}}+\frac{2^{2(\sigma -2)}}{(1+\varepsilon )^{\sigma -1}}+\cdots + \frac{2^{k2(\sigma -2)}}{(1+\varepsilon )^{\sigma -1}}+\cdots \right].$$

Focusing on the kth term in each situation, we obtain the following difference between the two situations:

$$A(k)=\frac{2^{k2(\sigma -2)}}{\sigma -1}\left[1-\frac{1}{(1+\varepsilon )^{\sigma -1}}\right].$$

Note that $EU(\varepsilon )-EU=\sum _{k=1}^{\infty }A(k)$. Furthermore, if $\sigma \geq 2$, then

$$0<A(1)\leq A(2)\leq A(3)\leq A(4)\leq \cdots \text{.} $$

This implies that $EU(\varepsilon )-EU>\infty$ if $\sigma \geq 2$. Then, for any $\varepsilon >0$ and any $\sigma \geq 2$, the value of a small improvement is incredibly large, and thus we have a result similar to the dismal theorem.Footnote ⁹

This argument illustrates that a dismal-theorem-like result can be obtained from an extension of the St. Petersburg paradox. It is helpful to see how traditional discussions on the St. Petersburg paradox are associated with recent arguments in the context of the dismal theorem. Since the work by Menger (Reference Menger1934), the idea that the boundedness of utilities is crucial for overcoming the St. Petersburg paradox has been discussed. Nordhaus (Reference Nordhaus2009) discussed the utility function employed in the dismal theorem being one of the sources of the consequence. He pointed out that a utility level with near-zero consumption is a key factor for the dismal theorem: indeed, the CRRA utility function we employed is not bounded. More importantly, to derive the dismal theorem, its marginal utility must be unbounded. Recently Millner (Reference Millner2013) pointed out that the dismal theorem does not carry over under a class of harmonic absolute risk aversion (HARA) utility functions, which is defined as:

$$U(C)=\alpha \left(\beta +\frac{C}{\sigma }\right)^{1-\sigma },$$

where $\alpha (1-\sigma )/\sigma >0$. Here, suppose that $\alpha =-1$, $\beta =1$, and $\sigma =2$. In our framework, it is easy to understand the working of this utility function and how a cost-benefit method breaks down. Note that if $\beta =0$, it corresponds to the CRRA utility with $\sigma =2$, which we used to derive our dismal-theorem-like result. Then we have $U(C)=-(1+C/2)^{-1}$. Note that

$$U\left(\frac{1}{2^{k}}\right)=-\frac{2^{k+1}}{2^{k+1}+1}\quad\mbox{and}\quad U\left(\frac{ 1+\varepsilon }{2^{k}}\right)=-\frac{2^{k+1}}{2^{k+1}+1+\varepsilon }.$$

Then, the expected utility under situation A is:

$$EU=-\left(\frac{4}{5}+\frac{8}{9}+\frac{16}{17}+\cdots +\frac{2^{k+1}}{ 2^{k+1}+1}+\cdots \right)=-\infty.$$

This implies that the expected utility is not defined. Moreover, it is easy to see that $EU(\varepsilon )$ (expected utility under situation B) is also infinite. However, we can see that

$$\begin{align} B(k)&=\frac{2^{k+1}}{2^{k+1}+1}-\frac{2^{k+1}}{2^{k+1}+1+\varepsilon }=\frac{ 2^{k+1}(2^{k+1}+1+\varepsilon )-2^{k+1}(2^{k+1}+1)}{(2^{k+1}+1)(2^{k+1}+1+ \varepsilon )}\\ &=\frac{2^{k+1}\varepsilon }{(2^{k+1}+1)(2^{k+1}+1+\varepsilon ) },\end{align}$$

and $EU(\varepsilon )-EU=\sum _{k=1}^{\infty }B(k)$. Note that

$$\frac{\varepsilon }{2^{k+1}}\geq \frac{2^{k+1}\varepsilon }{ (2^{k+1}+1)(2^{k+1}+1+\varepsilon )}.$$

Thus, it follows that

$$EU(\varepsilon )-EU\leq \sum_{k=1}^{\infty }\frac{\varepsilon }{2^{k+1}} <\infty. $$

This implies that our dismal-theorem-like result disappears. This observation shows how it can be the case that both EU and $EU(\varepsilon )$ are infinite and their difference is finite.

Nordhaus (Reference Nordhaus2009) also pointed out that a fat-tailed distribution is a key factor in the dismal theorem. Weitzman (Reference Weitzman2009, Reference Weitzman2014) noted that the dismal theorem does not happen under a normal distribution. The probability distribution employed in this study can be regarded as a type of fat-tailed distribution. Moreover, in the context of the St. Petersburg paradox, it has been recognized that a combination of a utility function and a probability distribution is quite important. Seidl (Reference Seidl2013) provided a general result by utilizing d'Alembert's ratio test. We can provide a general dismal-theorem-like result by modifying Seidl's argument. Let $x:\mathbb {N} \rightarrow X\subseteq \mathbb {R}$ be such that $x_{k}>x_{k+1}$ for all $k\geq 1$. Let $U(x)$ be a non-decreasing function and $p(x)\in \lbrack 0,1]$ a probability function such that $\sum _{k=1}^{\infty }p(x_{k})=1$. Let $\varepsilon >0$ be a small real number. Then we have the following result:

1. (i) $EU(\varepsilon )-EU<\infty$ if there exists $k^{\ast }\geq 1$ such that $p(x_{k})=0$ for all $k\geq k^{\ast }$;

2. (ii) $EU(\varepsilon )-EU<\infty$ if there exists $k^{\ast }\geq 1$ such that

   $$\sup_{k\geq k^{\ast }}\frac{[U((1+\varepsilon )x_{k+1})-U(x_{k+1})]p(x_{k+1})}{[U((1+\varepsilon )x_{k})-U(x_{k})]p(x_{k})} < 1; $$

3. (iii) $EU(\varepsilon )-EU=\infty$ if there exists $k^{\ast }\geq 1$ such that

   $$\inf_{k\geq k^{\ast }}\frac{[U((1+\varepsilon )x_{k+1})-U(x_{k+1})]p(x_{k+1})}{[U((1+\varepsilon )x_{k})-U(x_{k})]p(x_{k})} \geq 1;$$

4. (iv) $EU(\varepsilon )-EU$ may converge or diverge if

   $$\lim_{k\rightarrow \infty }\frac{[U((1+\varepsilon )x_{k+1})-U(x_{k+1})]p(x_{k+1})}{[U((1+\varepsilon )x_{k})-U(x_{k})]p(x_{k})} =1,$$
   where
   $$EU=\sum_{k=1}^{\infty }p(x_{k})U(x_{k}),$$
   and
   $$EU(\varepsilon )=\sum_{k=1}^{\infty }p(x_{k})U((1+\varepsilon )x_{k}).$$

Note that case (iii) corresponds to our dismal-theorem-like result. The key is comparing the growth rate of improvement in consumption and the shrinking rate of probability. As demonstrated by Seidl (Reference Seidl2013), his result from the ratio test unifies many observations associated with the St. Petersburg paradox.

In order to apply this result, let us consider a risky situation with a Poisson distribution. Suppose that $100/2^{k}$ per cent of wealth remains with probability $e^{-\lambda }\lambda ^{k}/k!$, where $\lambda >0$. That is, the agent loses a fraction $1-2^{k}$ of wealth with probability $e^{-\lambda }\lambda ^{k}/k!$. His/her utility function is assumed to be $U(C)=-1/C$. Note that

$$\frac{\lbrack U((1+\varepsilon )x_{k+1})-U(x_{k+1})]p(x_{k+1})}{ [U((1+\varepsilon )x_{k})-U(x_{k})]p(x_{k})}=\frac{2\lambda }{k}.$$

Thus, case (ii) can be applied. The dismal-theorem-like result disappears, and then a standard cost-benefit method works for the Poisson distribution.

Consider another case. Suppose that the agent earns $1/{d}^k$ ($d \ge 1$) dollars with probability $- ({1}/{\log (1-p)}) ({p^k}/{k})$, where $0< p < 1$. This is the logarithmic distribution. His/her utility function is assumed to be $U(C)=-1/C$. Note that

$$\frac{[U((1+\varepsilon)x_{k+1})-U(x_{k+1})] p(x_{k+1})}{[U((1+ \varepsilon)x_k)-U(x_k)] p(x_k)}=\frac{k}{k+1} p d.$$

Thus, if pd > 1, then the dismal-theorem-like result appears; if pd < 1, then the dismal theorem disappears.

3. Expression with environmental damages

We now consider another way of obtaining some dismal-theorem-like result by revisiting the standard St. Petersburg paradox. Consider a society that faces the risk of environmental damage. The society suffers from: (i) 2 units of environmental damage with probability 1/2; (ii) $2^{2}$ units with probability $1/(2)^{2}$; (iii) $2^{3}$ units with probability $1/(2)^{3}$; and so on. That is, the environmental damage is $2^{k}$ units with probability $1/(2)^{k}$. The social welfare function is:

$$u(c,d)=\sqrt{c}-d,$$

where c is the consumption level and d is the environmental damage. Here, c is positive and fixed. It is easy to see that $EU=-\infty$ under this utility function. This is the same as the St. Petersburg paradox.

We suppose that the society puts a resource g to make the situation better. The following constraint is imposed:

$$c+g=\bar{c}.$$

Given g, we can obtain the following situation instead of the original one: (i) $2/(1+\varepsilon g)$ units of environmental damage with probability 1/2; (ii) $2^{2}/(1+\varepsilon g)$ units of environmental damage with probability $1/(2)^{2}$; (iii) $2^{3}/(1+\varepsilon g)$ units of environmental damage with probability $1/(2)^{3}$; and so on. Here, ɛ is strictly positive. The resulting expected social welfare is:

$$EU(\varepsilon, g)=\sqrt{\bar{c}-g}-\frac{1}{(1+\varepsilon g)}\left[\frac{1}{ 2}2+\frac{1}{2^{2}}2^{2}+\frac{1}{2^{3}}2^{3}+\cdots +\frac{1}{2^{k}} 2^{k}+\cdots \right]=-\infty.$$

In spite of this prescription, the expected social welfare is still negative infinity. However, we have

$$\frac{dEU(\varepsilon, g)}{dg}=\infty,$$

for any $\varepsilon >0$ and $g\geq 0$. Thus, it is optimal to put all resources $\bar {c}$ into this possibility of improvements. That is, $g=\bar {c }$. Under this choice of the society, the expected social welfare becomes

$$EU(\varepsilon, \bar{c})=-\frac{1}{(1+\varepsilon \bar{c})}\left[\frac{1}{2}2+ \frac{1}{2^{2}}2^{2}+\frac{1}{2^{3}}2^{3}+\cdots +\frac{1}{2^{k}} 2^{k}+\cdots \right]=-\infty.$$

We now take the difference between $EU(\varepsilon, \bar {c})$ and the expected utility under the original situation,

$$EU(\varepsilon, \bar{c})-EU(\varepsilon, 0)=\infty,$$

where we use the same argument as that in the previous section. This implies that the government is willing to spend an extremely large amount of money to make a very small improvement. We note that a difference is that catastrophe is represented by a very small level of consumption in the original version, while it is represented by a very large level of environmental damage.

4. Concluding remarks

This paper examined Weitzman's dismal theorem, which has attracted attention in the field of climate economics and environmental economics. We showed that a conclusion similar to the dismal theorem can be obtained from a variation of Menger's version of the St. Petersburg paradox. In spite of its long history, there is no absolute resolution of the St. Petersburg paradox.Footnote ¹⁰ Although many authors have argued about the meaning and significance of the dismal theorem, it is still controversial. We believe that this controversy is due to the nature of the St. Petersburg paradox, whose resolution is still controversial.