Dice question - expected winnings of rolling dice $2$ times

Typical trading interviews consider gambling problems such as rolling a dice and winning its face value. The expected winnings are $$\\\3.5$$, $$\\\4.25$$, $$\\\\frac{14}{3}$$ for one throw, two throws, and three throws respectively. The expected winnings of $$n$$ throws may be calculated recusively knowing the value of $$n-1$$.

However, now consider a different game, in which one obtains $$\\\10,000,000$$ for each face value of the dice. How much would you pay to enter this game?

Values of this big need us to be risk averse, in which the utility function is not linear but concave. A canonical choice is the logarithm, using which the money equivalent of the game is the geometric mean, $$(\prod_{i=1}^6 i*10^8)^{1/6} \approx 2.9938* 10^8$$. So a risk reverse investor will pay $$30M$$ to enter the game. However, what if now I can throw the dice two times? Obviously, the same strategy before cannot apply here. How should we price this game then? Using binomial tree?

• In practice, if you offer to pay USD 30 million, but you can't pay it, then your counterparty would be foolish to do this trade with you, knowing this. So let us limit ourselves to what you can pay. Suppose you can lose/gain USD 1000, this loss/gain would be more painful/pleasant for you than for some others, yet less painful/pleasant than for yet others. Often a loss is more painful than a gain having the same notional. Economists define utility functions (q.v.) to describe this preference. Generally, you'd pay the expected value of your utility of the payoff, while your counterparty would Aug 26 '21 at 2:45
• would accept the expected value of their utility of the payoff. You agree on the 'expected value' part, but the parties' utility of the same payoff may differ. Aug 26 '21 at 2:47
• Further, you may want to allow the utility (risk aversion, both yours and the counterparty's) to change after you win or lose lots of money. For example, suppose you bet a quarter in a slot machine and win a hundred-dollars. Would you rather keep your winnings and leave, or gamble some more? Aug 26 '21 at 3:53

Let's apply basic utility theory. Note that the expected utility is driven by the outcomes, their probabilities and the initial wealth $$W_0$$ of the gambler (see @Dimitri's comment to your question). Let $$\mathrm{EU}$$ denote expected utility

$$\mathrm{EU}\equiv\sum\limits_{i=1}^np_iu(x_i+W_0)$$

and $$\mathrm{CE}$$ be the certainty equivalent (or money equivalent in your question) $$\mathrm{CE}\equiv u^{-1}(\mathrm{EU})-W_0,$$ i.e. $$u(\mathrm{CE+W_0})=\mathrm{EU}$$. If the initial wealth is orders of magnitude above the gamble's outcomes, $$W_0>>x$$, then utility can be well approximated by a linear function, $$u(W_0+x_i)\approx u(W_0)+cx_i$$ with $$c$$ some constant depending on $$W_0$$ and $$u$$. This is reflected in the first part of your question.

Let's further introduce optimal decision making. With $$n$$ more rounds to go in the game, your gambler will always stop the game at some result $$x_i+W_0$$ whenever the utility of that result exceeds the continuation value,

$$u(x_i+W_0)>\max{F_n}$$

where, somewhat sloppily, $$\max F_n$$ denotes the expected utility from optimally stopping the game at some future time. In your example, this problem can be solved recursively: With one more round to go, the gambler will stop at some outcome $$x_j$$ if $$u(W_0+x_j)>\sum\limits_{i=1}^np_iu(x_i+W_0)\Leftrightarrow x_j>u^{-1}(EU)-W_0\equiv \mathrm{CE}_1$$

The gambler will continue to play at any $$x_j<\mathrm{CE}_1$$. With two more rounds to go, they know their optimal decision in the next round and they can adjust the game's expected continuation value accordingly, i.e. now they stop if

$$u(x_j+W_0)>\sum_{i:x_i < \mathrm{CE}_1}p_iu(W_0+\mathrm{CE}_1)+\sum_{i:x_i > \mathrm{CE}_1}p_iu(W_0+x_i)$$ And they will again find some certainty equivalent $$\mathrm{CE}_2$$ for this game, and so on.

At any step, the certainty equivalent (money value) of playing $$n$$ games can be calculated by calculating

\begin{align} \mathrm{CE_n}&=u^{-1}(\mathrm{EU_n})-W_0\\ &=u^{-1}\left(\sum_{i:x_i < \mathrm{CE}_{n-1}}p_iu(W_0+\mathrm{CE}_{n-1})+\sum_{i:x_i > \mathrm{CE}_{n-1}}p_iu(W_0+x_i)\right)-W_0 \end{align}

N.B.: For completeness, the price of the gamble should, of course, also be reflected in the utility and outcome... Also: @Dimitri's comment is very insightful: You are only willing to pay some utility based reservation price for such a gamble, but the gambling house would, on average, loose money at such a ticket price.

• great explanation, as always. if I may illustrate it without the use of math formulas - many people (but not everyone) are willing to pay \$1-2 for a lottery ticket, whose expected payoff is much less than its retail price. Conversely, the lottery administrator wants to earn money and is not willing to sell tickets for a lower price than much more than their payoff. So the administrators put much effort into figuring the ticket prices and payoffs that maximize their earnings. Very similarly, many trading desks' risk aversion is highly seasonal. Aug 26 '21 at 13:36
• In January, some traders are eager to go into relatively risky stuff, pushing the boundaries of their risk appetite. But once their P&L reaches their annual plan by summer/fall, the traders turn more risk averse, more interested in preserving the "good enough" P&L they already have for the year. Should any interesting but risky opportunities appear in the fall, they may need to wait until January. Aug 26 '21 at 13:41
• Hi Dimitri, yes! Your first observation (buying 'lemon tickets') is not explained by rational choice theory, but requires some behavioral economics (misapprehension of small probabilities, for example). Aug 26 '21 at 13:42
• I like that comment regarding traders' behavior :D Aug 26 '21 at 13:45
• Is the poor bloke who has zero chance of ever earning a million bucks unless he buys an almost-zero chance for one buck - acting irrationally? There's lots of cool-sounding research on (seemingly) irrational economic behavior (e.g. en.wikipedia.org/wiki/Richard_Thaler 's popular books) - I wish I had the time/enrgy to study it more. But the the original interview question, to me, a response to "how much would you pay for this riky bet" that focuses solely on the expected payoff is less complete than a response touching on both counterparties' utility and "irrationality". Aug 26 '21 at 13:58