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Fermat Pascal Dice Game Fairness

Imagine you play a dice rolling game with a friend. If you roll a '6' in 8 rolls, you win the entire stake, otherwise you lose. Suppose you have rolled 3 times and haven't won yet. Your friend now suggests to skip the 4th roll, but he will compensate you fairly for it. You will still get to roll your 5th, 6th, 7th, and 8th roll. For the compensation of skipping the 4th roll, he will take out a fair amount from the stake and give it to you, keeping the rest of the stake at play. What would be considered a fair compensation?

Discussion
I talked to a genius and she gave me the following argument "I think of it as two options, to skip or not to skip. I will only skip if the option gives me an expectation that is higher than, or at least the same as, if I don't skip."

Expectation if I don't skip: 
E1= 1/6 * S + 5/6 * (1 - (5/6)^4) * S
where S is the stake

Expectation if I skip:
E2 = c + (1 - (5/6)^4) * (S - c)
where c is the compensation

We want E1 <= E2. We'll get c >= 1/6 * S.

This means we will be willing to skip if we are compensated by at least 1/6 of the stake.

If we argue similarly from the other person's perspective, we are only willing to give out options that can improve our own expectation or at least keep it the same, and we will conclude that the compensation should be equal to 1/6 S.

Then what is fairness? This genius girl tells me that fairness should be, if we present a few options to a person, and if both of us are smart enough, the person would still be willing to pick any one of the options, and we are still willing to offer any one of the options.

We even have a side discussion on expectation and values: That the value of something may not be the same for different people, in real life, each of us have a function V() that maps a value to another value, and the expectation is actually of E(V(X)), not E(X). That is why given a bet of E(X) = 0 where have a fair chance of losing 1 million or winning 1 million, some people might take it and some may not, because V(1 million) and V(-1 million) could be different for different people. As Warren Buffett said one time time earning more money has no value to him after certain amount, then V(100 million) is just about the same as V(1 billion), and that would skew the expectation. The genius girl even gives an example of why there are many old people who love to gamble, where V(winning 50 dollars) is much higher than 50, while V(losing 10 dollars) is almost 0, which makes their expectation to be very positive even when they lose money on average. What a genius.



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