Paradoxes of Expected Value

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1 Paradoxes of Expected Value

2 Expected Value A central notion in game theory, economics, and rationality, is the idea of Expected Value To find the expected value of a particular event, you figure out all the different possible outcomes of the event and the probability that each outcome occurs. You then multiply the probability of an outcome times the value of that outcome and sum the different outcomes together. For instance, suppose I ask you to play a game where we flip a fair coin; if it lands heads, I give you $1, whereas if it lands tails you give me $1, the expected value of this game for you is 0.5(-$1)+ 0.5($1)=0 Find the expected value of a game where we roll a single, fair, 6-sided die and I pay you dollars equal to the number on the die (e.g. 4 =$4)

3 Expected Value Of course, expected value need not be limited to monetary bets. One can think through any decision this way. For instance, one might think of deciding one s major: EV(Philosophy)= 0.2(100h) + 0.8(30h)=20h + 24h=44h EV(Engineering)= 0.1(90h) + 0.6(60h) (40h)= 9h + 36h + 12h=57h EV(Accounting)=0.4(20h)+ 0.6(-20h)=8h + -12h=-4h Even though majoring in Philosophy has the chance to make you the most happy, majoring in Engineering has a very good chance of making you pretty happy, so overall you expect to have a happier life if you major in Engineering; hence, that is what you should do. In general, it seems like one is rational if one does something with positive expected value (so majoring in Accounting would be irrational), and one is most rational if one does the thing with the highest expected value. Call this the Expected Value Principle

4 Expected Value Expected Value Principle An action is rational in proportion to the expected value of that action. Suppose a casino offers to let you play the dice game (which pays you $ s=the number shown on a 6-sided die) for $3, what should you do? The Expected Value Principle would seem to dictate you should play, and you should play as many times as you can as fast as you can, because you will be (on average) gaining money every time you play. What about if they charge $4 for the game? If the EVP is based on money, you should not play. One could however calculate expected value in terms of happiness, in which case the expected happiness of playing may outweigh the expected loss of happiness from losing 50 cents.

5 Consider the following game: The St. Petersburg Game I am going to flip a fair coin until it comes up heads. If the first time it comes up heads is on the 1st toss, I will give you $2. If the first time it comes up heads is on the second toss, I will give you $4. If the first time it comes up heads is on the 3rd toss, I will give you $8. And in general, if the first time the coin comes up heads is on the nth toss, I will give you $2n How much should you be willing to pay to play the St. Petersburg Game? The probability that I first get a heads on the nth flip is 1/2n, which means the value of the game is given by: EV (SPG ) = 1/2($2) + 1/4($4) + 1/8($8) + = $1 + $1 + $1 + $1 + $1 + $1 + i.e. the value of the game is infinite

6 Valuing St. Petersburg The Expected Value Principle would say we should be willing to pay any finite amount of money to play the St Petersburg Game Nonetheless, many people think it would be rational to not pay (and perhaps irrational to pay) certain amounts to play the game. How can they justify this? One problem is that you can t reasonably expect to be paid off by me (or by anyone) since no one has infinite money. However, suppose you were playing it with Jeff Bezos ( $156 billion) where you would not get paid any more past the 37th flip; the expected value of the game would be $37, and yet many still would think it was rational not to pay $35 to play this game.

7 Valuing St. Petersburg Daniel Bernoulli, brother of the creator of the paradox Nicolaus Bernoulli, claimed it illustrated the phenomenon of diminishing returns, the idea that something becomes less valuable per unit as you have more of it. However, even if dollars have diminishing utility, it seems we could still have the paradox by using things other than dollars (such as days in heaven) or by increasing dollar amounts more than double so that utility continued to increase at a steady pace. A different response is to try to explain it in terms of risk aversion No doubt we are risk averse, but this still leaves open the questions of if risk aversion is rational and how we can create a principle of rationality that is consistent with being risk averse and maximizing utility?

8 A different puzzle for expected value I Suppose you get the chance to play a game where a fair coin is flipped; if it lands heads you win $2, if it lands tails you win $0.50. Should you pay $1 to play this game? I The answer is yes, but then consider this game: Two Envelopes (open) There are two envelopes which we can arbitrarily label A and B. The only thing you know about them is that they each contain money and one contains twice as much money as the other. You open one and find $10. Should you switch to the other envelope?

9 Two Envelopes (open) There are two envelopes. The only thing you know about them is that they each contain money and one contains twice as much money as the other. You open one and find $10. Should you switch to the other envelope? Suppose you reason as follows: I There is 50% chance you took the envelope with more money, and a 50% chance you took the envelope with less money. I If you took the envelope with more, then there is $5 in the other envelope I If you took the envelope with less, there is $20 in the other envelope. I Thus, there is a 50% chance there is $20 in the other envelope, and a 50% chance there is $5 in the other envelope. I Thus, EV(switching)=0.5($20)+0.5($5)=$12.50 I Since EV(not switching)=$10, you should switch.

10 I At first, one might think that the situation is somewhat like the Monty Hall problem we learned some new information and that new information made switching better but consider the following: Two Envelopes (closed) There are two envelopes. The only thing you know about them is that they each contain money and one contains twice as much money as the other. You select one, but before you open it, you are given the chance to switch? Should you switch to the other envelope? I One could then reason as before, that there is a 50% chance you picked the envelope with one more and 50% you picked the envelope with less. I Call money in the envelope you took A. One could then say EV(switching)=0.5(2A)+0.5(0.5A)=1.25A I Since EV(not switching)=a, you should switch

11 Two Envelopes (closed) There are two envelopes. The only thing you know about them is that they each contain money and one contains twice as much money as the other. You select one, but before you open it, you are given the chance to switch? Should you switch to the other envelope? I Something has gone wrong. You chose freely and have gained no information, so there should not be a strong mathematical reason to switch. I Furthermore, if you call the contents of the envelope you switched to B, then you know there is a 50% chance that B is the higher envelope and a 50% chance that B is the lower envelope, so you could again reason EV(switching back to A)=0.5(2B)+0.5(0.5B)=1.25B I Since EV(not switching back to A)=B, you should switch back to A

12 Lest one trade envelopes ad infinitum we can provide other calculations of the value of switching. For instance, if x is the smaller amount of money, then we have equal chance to gain or lose x by switching, so EV(switching)=0.5(x)+0.5(-x)=0 One could alternatively calculate EV(A)=0.5(x)+0.5(2x)=1.5x=EV(B) The key to the paradox is in figuring out what went wrong in the switching argument. Suggestions include: It assumes there is no upper or lower bound on money It assume we think all amounts of money are equally likely to be in an envelope, but we clearly thing (for some good reasons) that certain amounts are morel likely that others. The calculation requires two inconsistent values for the other envelope (either $5 and $20 or 2A and 0.5A)

13 The Principle of Indifference Given a range of mutually exclusive outcomes of an event, absent any other information, one should assign equal probability to each outcome. The Principle of Indifference is crucial for probability assessments. When we say that a die or coin is fair, we are saying that one can use the principle of indifference and assign a 1/2 probability to each side of the coin and a 1/6 probability to each side of the die. In the two envelope examples, we could use it to say that each envelope has a 1/2 chance of containing x and a 1/2 chance of containing 2x, and thereby calculate the expected value of each envelope to be 3/2x Alternatively, we could use it to say that there was a 1/2 chance you picked the higher envelope and a 1/2 chance you picked the lower, and thereby calculate that the other envelope contains 1.25 times as much as the present envelope.

14 Bertrand Paradox A train leaves at noon from Akron, OH to travel 300 miles to South Bend. It will travel at a constant speed between 100mph-300mph. What is the probability that it will arrive before 2pm? I It seems we can calculate this in two ways I First, we could determine that to get here by 2pm it has to travel 150mph. If we use the Principle of Indifference, we can treat all its speeds from 100mph to 300mph as equally likely, so there is a 3/4 chance it arrives by 2pm. I Second, we could determine that it has to arrive sometime between 1pm and 3pm. If we use the Principle of Indifference we can treat all arrival times as equally likely, so there is a 1/2 chance it arrives before 2pm. I These cannot both be right, so what is the difference and which should we accept? I The lesson many take from this paradox is that the Principle of Indifference cannot be applied unless we know the mechanism by which the thing is coming about.

15 Summing Up There seems to be a certain parallel between Bertrand s paradox and the Two Envelope problem On the one hand, we can be indifferent between which envelope has more (when we are choosing), and on the other hand we can be indifferent between the other envelope having half as much or twice as much as the envelope we chose. The first, and not the second, seems consistent with the mechanism we were given, and somehow we are led astray by a misapplication of the principle of indifference