Question

what is St. Petersburg paradox?

you have to show what it is?

what kind of problem it comes with?

how this problem can be solved?

how the utility could solve this problem.?

Answer 1

St. Petearhusg Pasados. The pasado known as the St Peterbury Paradox kohtained by com klip game. The rulues are simple keer tripping until you get tails. It your fixt flip in a tail tails, then you cole $2, if your first ails is on the second flip, then you win 84. if your flip tails is on the sound filip, you win re etc. More concisely, it your fact tails is on the uth fup, you will win Tablel. St. Petersburg ganae Trial to here Ist fails appears Probability Peryout on - Lit's look out how the probabilities in the table were calculated. The probability of rolling a tails on the first try to simply . If your tiret tils is on the and flip, that means your first was reade. The probability of this happenning in 2) 6 4 a. If your list first tails are on the third flip then your first t100 stipe t kap flips must have been heads and the probability of flipping heads, heade thon tails of the text etc.

Kit's calculate our expected value for the game, we have a chance of winning 82, at de chance of $4 a chance of winning of Beetc. So, our expected value is $27.1 + $49.4, +($+-.. I (2K) (3x) = 1+1+1+ --- the in the expected value for this pous infinite. Because of this it can be said a player should accept any finite price & to play this game because it will lor leat than the expected winnings But how manch should one actually pay to play this game? Most people probably would not play pay more than $20 40 pay this game because the odar of making the profite are so low (6.25%.). Ito this game was offered for $ $1000 dollars, we could spend bur whole site trying to find someone who would ay play this game for that much there have been many attrapte to try and find a resolution to the paradox or in the other words to try to find pri a fair price to offer this game for.

Posplction 1. By experiment We can calate the expected winning off the experiment, the careful when calculating the probabilition for example, the probability of rolling a tails on the first tip based off this experiment in 1861 not / , because Buffon (afrench mathematician, who simple ran the expresiment himself a total of 2048 times I got his result 1061 out of the 2048 times he did, so the expected toinning based off this experiment is (32) 1061 + (64). 194 + (48) 222 + (816) 1 3 2 2 + (864) 2 1 + $128) 2 12 + * $956) vg + $512). b. x $9-62. One should expect to pay obout $10 to play this game, but based off this experiment. Table 2: Button's St. Petersburg Experiment Trial (k) where first tails appears Frequency Payout (24) 2048 1977 2048 2048 248 2048 1061 494 232 $4 137 on AwN- 56 29 $64 $128 $250 $ $12

2. Based on uhlity some resolutions use the concept of o utility to désire a fair price of this gaune. Utility can be thought of as now much practical use it you gained. Surely going from 80 to $1,000,000 in mulh more dramatic tham going from $1,000,000 to $2,000,000 Rren though they are both increases of million dollars. In the former & coe went from nothing to millionaire, and in the letter we essentially went from rich to rich again so it can be said that it we won $2,000,000 tun we would gaire more ulility in our fiast million than we would gain in your our 2nd million. Daniel Bernoulli used the equalt equation utility = log10 (w), where on the winnings in dollars. So if we te win $2 ton them by this equation, we have wow log, , (2) & 0.301 in ulility . A new table can be constructed for the St. Petersburg, & um game now!

Table 3 Tia(k) where first St Petersburg Gaine teils appears Probability Payout (2) - $2 $4 $ 8 y $16 Uility (109, 087) 301 602 903 1. 204 Now instead of expected winning in dollars, we Calculate expected utility, which turns out to be about .602. or about $4.