We know what the expected value theory is. The St. Petersburg paradox seriously challenges that. It is a coin-tossing game and it goes like this:
A casino makes a coin-tossing game for a single player. In the first toss, if you get a head, you win a dollar, and the game ends. If it’s a tail, the game continues but doubles the payoff (two dollars) for the next round. At the appearance of the first head, you go home collecting whatever you won. What is the price you want to pay to the casino to enter the game?
The expected value
Let’s see what the expected value of the game is.
EV = P(1 T) x V(1 T) + P(2 T) x V(2 T) + P(3 T) x V(3 T) + …
where P(1 T) is the probability of one tail and V(1 T) is the value of one tail etc.
EV = (1/2) x 2 + (1/4) x 4 + (1/8) x 8 + …
= 1 + 1 + 1 + 1 … = Infinity.
Therefore, the rational player must be willing to pay any price to get into the game!
In reality, you will not pay that amount. Think about this: what is the probability of getting a head in the first toss (and you get one dollar)? It is 50%. Similarly, the chance of ending up with 4 dollars is 25%, and so on.
This disparity between the expected value and the reality is the St. Petersburg paradox.
Bernoulli’s solution
Daniel Bernoulli suggested using utility instead of value to solve this problem. The utility is a subjective internal measure of the player towards the gain from the game. According to him, the utility of the additional amount (earned from the contest) was a logarithmic function of the money.
u(w) = k log(w), w represents the wealth. It is logarithmic, he hypothesised, as there is an inverse relation (1/w) between change in wealth and its value. Mathematically,
du(w)/dw = 1/w
With this information, let’s rework the expected utility of this game
| n | P( nT) | w | u(w) u = logw (k = 1) | Expected Utility |
| 1 | 1/2 | 2 | 0.69 | 0.35 |
| 2 | 1/4 | 4 | 1.39 | 0.35 |
| 3 | 1/8 | 8 | 2.08 | 0.26 |
| 5 | 1/32 | 32 | 3.47 | 0.11 |
| 10 | 1/1024 | 1024 | 6.93 | 0.007 |
| 1.07 |
Unlike the previous case, the sum of utilities converges.
