St. Petersburg Paradox

Original article was published on Artificial Intelligence on Medium

The determination of the value of an item must not be based on the price, but rather on the utility it yields. There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man.” — Daniel Bernoulli

Index

0. Dear Statistics Journal
1. Introduction to Probability
2. The Monty Hall Problem
3. Gambler’s Fallacy
4. St. Petersburg Paradox

Before we get to the good stuff, let’s knock out a few useful concepts:
. Random Variables
. Expected Value
. Expected Utility

Random Variables

A random variable is a function that maps the outcome of a random process to a real number.

Fig 1

E.g., say we have the random process of flipping a fair coin. X is essentially a function that maps values of the sample space ({Heads, Tails}) to a real number. In a sense, we are quantifying the state space of outcomes. As seen in Fig 1, we assign the values Heads=0 & Tails=1 and X is our random variable for this particular process.

Expected Value

Expected value is the average value of a random variable over a large number of experiments

# X is the random variableEV(X) = P(Xi) * Xi
= 1/6*(1) + 1/6*(2) + 1/6*(3) + 1/6*(4) + 1/6*(5) + 1/6*(6)
= 21/6
EV(X) = 3.5
# Expected value of rolling a die is 3.5
# X is the random variableEV(X) = P(Xi) * Xi
= 1/2*(1) + 1/2*(0)
= 1/2
EV(X) = 0.5
# Expected value of flipping a coin is 0.5

Expected Utility

‘Utility’ is a term in economics that refers to the total satisfaction received from consuming a good or service. Expected utility is a summarization of utility that an entity is expected to reach under any given circumstances. The expected utility of a random variable can be thought of as a specialized formulation of the expected value.

The expected utility of a random variable is the probability-weighted average of all it’s utility values.

🗒 There are different methodologies, studied in decision theory, to calculate the expected utility of a process basis the aptitude of your risk. If you'd like to learn more about that, see references

The utility value of any given outcome is self-defined. Let’s take the previous example of tossing a coin, with added complexity. You’re still playing the coin flip game, if the outcome is Heads, you get 20 bucks, if its Tails you lose 10 bucks but you choose to skip your turn and you will be guaranteed 5 bucks. As per your personal preferences, you have assigned a utility value to different outcomes and now you can choose the scenario which (in this case) maximizes your gain.

# X is the random variable# Expected Utility# Branch 1
EU(X)_1 = 1/2*(20) + 1/2*(-10)
= 10 - 5
EU(X)_1 = 5
# Branch 2
EU(X)_2 = 1*(6)
EU(X)_2 = 6
# Therefore, as per Expected Utility, to maximize your gain you should choose to not flip.# Expected Value# Branch 1
EV(X)_1 = 1/2*(1) + 1/2*(0)
= 1/2
EV(X)_1 = 0.5
# Branch 2
EV(X)_2 = 1*(0)
EV(X)_2 = 0
# Therefore, as per Expected Value, to maximize your gain you should choose to flip.

Now you could just change the utility value to get different results. True, so let me define the distinction a bit more tightly to gain more clarity.

Expected Utility vs Expected Value

Alright, let’s say a mathematician at your university opens an ice cream shop (it’s called Infinity Ice-cream). Here’s how his shop works — a customer, say you, walks in. He greets you and asks you about your favourite flavour (chocolate?). He then gives you a coin and says: “I’ll give you 2^x ice-cream sticks, where x is the number of consecutive Heads you can flip. The game starts when you flip the first Head, so how much would you be willing to pay to play the game?“. In short, if you flip Tails on your second toss, you get 2^1 = 1 ice-cream stick, if you flip 3 consecutive Heads, you get 2^3 = 8 ice-cream sticks! For convenience, let’s assume 1 ice-cream stick costs 1 buck.

# X is the random variable# Expected ValueEV(X) = 1/2*(2) + 1/4*(4) + 1/8*(8) + 1/16*(16) + ... + 1/n*(n)
= 1 + 1 + 1 + 1 + ... + 1
EV(X) = ∞
# Expected UtilityEU(X) = 1/2*(2) + ... + 1/64*(32) + ... + 1/n*(32)
= 1 + 1 + 1 + 1 + 1 + 0.5 + ... + 1/(2^(n-5))
EU(X) ≈ 6

The Expected Value of such an experiment is infinite! So ideally you should be willing to pay any finite sum of money to play the game. Right? If rationality forces us to liquidate all our assets for a single opportunity to play the game, then it seems unappealing to be rational. The St. Petersburg Paradox consists in the fact that our best theory of rational choice seems to entail that it would be rational to pay any finite fee for a single opportunity to play such a game, even though it is almost certain that the player will win a very modest reward.

Expected Utility gives a more practical value based on your preferences and practical conditions on the gains. As in this case, we can say that the total Expected Utility, for you, from this game is approximately about 6 ice-cream sticks, so you should ideally pay less than 6 bucks!

Resolution

Daniel Bernoulli devised a resolution to the paradox using utility function, expected utility hypothesis and the presumption of diminishing marginal utility of money.

For each possible event, the change in utility ln(wealth after the event) − ln(wealth before the event) will be weighted by the probability of that event occurring. Let c be the cost charged to enter the game. The expected incremental utility of the lottery now converges to a finite value. This formula gives an implicit relationship between the gambler’s wealth and how much he should be willing to pay to play (specifically, any c that gives a positive change in expected utility). For example, with natural log utility, a millionaire ($1,000,000) should be willing to pay up to $20.88, a person with $1,000 should pay up to $10.95, a person with $2 should borrow $1.35 and pay up to $3.35.

Conclusion

If a mathematician presents a proposition too good to be true, then it usually is. But on a more serious note, you can use these principles as complementary instruments for decision making. E.g., should I travel to a less infected place amid an epidemic — the probability of getting infected during your travel vs probability of getting affected at your current location if you wait out the epidemic?

Quality References

️◀ ️Previously — Gambler’s Fallacy