Economics has a very important theorem about expected utility - basically how people decide what's the best choice when faced with uncertainty. As I study for finals I've been reading about some of the choices people make that violate this theorem, which is based on some fairly innocent assumptions. Here's one of the "tests" where a lot of people make choices not in line with the theory.
You're on a game show where you can win one of the three possible prizes: (A)$2,500,000, (B) $500,000, or (C) nothing (not much of a prize, is it?). You get to decide which of the following lotteries (probability sets) you'd prefer.
First, you can decide between the following two: (I) Walk away with the $500,000 as a sure thing OR (II) Have a 10% chance of getting the $2.5 million, a 89% chance of getting the $500,000, and a 1% chance of getting nothing at all.
Which do you choose?
Well, after you've made your decision on that, you fail to name the 27 members of the European Union quickly enough, and the announcer tells you that you won't be able to keep the odds you chose earlier. Instead you have to choose again with not-as-great odds. Now the choices are: (1) No chance of getting the $2.5 million, an 11% chance of getting the $500,000, and an 89% chance of getting nothing OR (2) a 10% chance of getting the $2.5 million and a 90% chance of getting nothing.
Which do you choose now?
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I'll post the "answer" (what economists expect - or hope - most people would choose) in a little while. Note that it doesn't just depend on whichever has the highest expected return; it's okay to be risk adverse & want the sure thing. What this situation gets at is consistency in people's choices.
Posted by Elmer's Glue (Member # 9313) on :
Someone's been watching Deal or No Deal..
Posted by Jhai (Member # 5633) on :
No, this is straight out of Chapter 6 of Mas-Colell, Whinston, & Green, the holy book of Microeconomics. Well, except for the stuff about the European Union.
I actually can't stand watching Deal or No Deal.
Posted by mr_porteiro_head (Member # 4644) on :
If you're interested in reading more about this, let me recommend The Paradox of Choice: Why More Is Less by Barry Schwartz.
Posted by Mucus (Member # 9735) on :
II) and 2) Are the probabilities on the second one correct?
Posted by cmc (Member # 9549) on :
I picked (I) the first time...
I'd sort of want to know what subject the question was going to be from... I'd probably go with (1) regardless.
Posted by MrSquicky (Member # 1802) on :
cmc, I don't think you understand the problem. You don't get to walk away if you choose (I). That's just the payoff matrix if you get the question right. In the problem presented, you get the question wrong and still have to choose between 1 and 2.
Posted by El JT de Spang (Member # 7742) on :
I'd take (II) and then (2).
ETA: Of course, I don't happen to believe money == happiness, though.
Posted by Mucus (Member # 9735) on :
Yeah, I wasn't sure how the question interacts with the odds. I just assumed that the odds were completely correct and the question was a complete surprise or something.
Posted by ketchupqueen (Member # 6877) on :
I went with (I) and (2).
Posted by MrSquicky (Member # 1802) on :
I'd do II - 2
Posted by cmc (Member # 9549) on :
MrSquicky... I understood. I was just giving both choices.
edit: Mucus - I wasn't thinking that the odds were contingent upon the question... more that the topic could influence my willingness to take a risk. : )
Posted by MrSquicky (Member # 1802) on :
Oh, jeez, I totally read you wrong. My bad. I read the 1 as an I.
Posted by cmc (Member # 9549) on :
No biggie...
Posted by Kama (Member # 3022) on :
but I KNOW the 27 member states...
Posted by cmc (Member # 9549) on :
Could you say them fast enough, though? (said with a smile)
Posted by Kama (Member # 3022) on :
yes. it's my job. Posted by Jhai (Member # 5633) on :
The probabilities on the second one are correct. Like I said, it's not about the expected return. And sorry if it's unclear. You need to make both decisions for the theorem to come into effect, that is, you need to choose between I & II AND between 1 & 2. (And you don't have a choice about answering the European Union question - which you're fated to get wrong no matter what )
Edit: and for Kama, the time limit is 20 seconds. Posted by erosomniac (Member # 6834) on :
(I) and (1).
To expand: this is a gutshot reaction, without calculating diminishing returns, etc. It's also biased by the fact that $500,000 is basically enough money for me to accomplish everything I want in life.
Posted by Kama (Member # 3022) on :
hey that's not fair
II and 2 then
Posted by cmc (Member # 9549) on :
Jhai - were you talking to me about getting them wrong?!! How'd you know I'd for sure freeze up on that one?! and Kama - I think I want you on my team for when I need answers to stuff like that which I have NO clue about.
Posted by swbarnes2 (Member # 10225) on :
2 and II are mathemetically better, but I bet that people treat the difference between 100% and 99% differently than they treat the difference betweeen 89% and 90%. Many people will pick I, because that offers them the choice between risk and no risk. (Even though the risk is tiny compared to the increased reward.
1 and 2 are both risky, and almost equally so, and I bet that people will see that, and pick the option that might yield them a better payout, which is 2.
Posted by HollowEarth (Member # 2586) on :
This is probably obvious, but there should be no expectations that people behave rationally when involved in 1) games of chance, 2) dealing with large sums of money. Both of these should be obvious given the number of lottery tickets sold and the popularity of slot machines.
Posted by erosomniac (Member # 6834) on :
quote:Many people will pick I, because that offers them the choice between risk and no risk. (Even though the risk is tiny compared to the increased reward.
Yeah. I mean, I play poker pretty routinely; pot odds say you hvea to pick (II) no matter what. But $500,000 is enough money that I am willing to sacrifice a small shot at five times that much to rule out the possibility of getting nothing at all.
Posted by Foust (Member # 3043) on :
The question is based on a false premise, isn't it? Has anyone ever really looked back on their life and said "I am SO GLAD I calculated those odds so well and won all that stuff."
Posted by Xavier (Member # 405) on :
quote:The question is based on a false premise, isn't it? Has anyone ever really looked back on their life and said "I am SO GLAD I calculated those odds so well and won all that stuff."
While winning 2.5M dollars is not guaranteed to improve any individual's happiness, I can say with great certainty that it would improve my own.
Most people blow their money within a few years of when they win it in a game show. I am not most people.
Posted by erosomniac (Member # 6834) on :
quote:(I'd bet most people taking option 1 are doing so because they suspect a trick answer, and its the less obvious choice.)
Not me. A 1% increase in the odds of winning an already ludicrous sum of money is very much worth it to me.
Posted by Amanecer (Member # 4068) on :
I would take I and 2. Like Eros, I can't imagine walking away from a guaranteed $500,000. If I hit the 1% of getting nothing, it would be a very, very sore spot. On the other hand, I could certainly console myself with only getting $500,000 over $2.5 million. I'd consider that rational even though statistically, it is the wrong choice.
Posted by Fusiachi (Member # 7376) on :
I take II and 2.
Reasoning: In the first choice, I view 1% as trivially small.
Likewise, the difference between the 89% and 90% chance of losing it all in the second choice is equally trivial.
Posted by Jhai (Member # 5633) on :
So it's been over 90 minutes & 30 posts, which seems a reasonable time to post the "answer." Don't continue to read if you don't want to know yet.
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Like I said with the original post, consistency in the answers is what matters, not whether you prefer to play it safe or to take risks. To be consistent, you need to either pick I & 1, or II & 2.
The reason for why this is consistent is the following: Suppose you get X oodles of utility (happiness, pleasure, good feelings, whatever) from $2,500,000, Y oodles from $500,000, and Z oodles from nothing. Now suppose you're one of the people who chose I over II in the first choice.
That means that you have shown that you prefer 1*Y over .1*X + .89*Y + .01*Z, or, to put it a bit more mathematically: Y > .1X + .89Y + .01Z. For people who chose II over I, you'd just reverse the inequality. Note again that this payoff is in oodles of utility, not dollar terms, which allows us to deal with problems like some people valuing $2.5 million far more than others do.
Now add to both sides of the inequality the oodles of utility equivalent to (.89Z - .89Y). That makes the inequality .11Y + .89Z > .1X + .90Z This is the expected payoff, in oodles, of the second choice, with the right hand side being (1) and the left hand side being (2). So, again, if you prefer I to II, you should prefer 1 to 2.
The major assumption here, which is called the independence axiom, is that if you add the same something to both sides of the equation it will not affect the way you prefer the items. So if your happiness level is 100 oodles (however you define that - maybe it's equivalent to the joy you get out of petting a happy dog), and you prefer that over 80 oodles (petting a sleepy dog), the independence axiom says you'll prefer 120 ooodles (two happy dogs!) over 100 oodles. You have to personally decide whether that's a reasonable assumption to make or not - there's still plenty of quibbles about it in economic literature.
To address a concern with happiness = money: no good economist would ever say this. However, money is a useful thing to deal with because it doesn't require descriptors other than how many units there are - if I had set up the problem with bread there would probably be someone who would want to know if it were wheat or white, homemade or store-bought, etc.
Each person can translate the amount of money being offered into their own personal utility. For example, from what erosomniac has been saying in this thread, I'd imagine the utility for him of $2.5 million is not as far apart from the utility of $500,000 as it is for other people here.
Posted by ketchupqueen (Member # 6877) on :
I'd like to see you explain me, then. Posted by Jhai (Member # 5633) on :
Also, lab experiments have found that a lot of people end up choosing I & 2. There's a number of explanations floating out there in the literature. The main four are: (a) People are just bad with small percentage differences, and tend to round off or ignore small chances, or something along those lines. But this doesn't matter in real life, since the type of situation shown in this game is very uncommon. (b) The problem is the regret that a person feels if he ends up with that 1% chance of nothing in II. This has led to the development of regret theory, which attempts to incorporate expectations of regret into these type of problems. (c) People just don't realize that they ought to be following the Expected Utility Theorem. Once they've learned about it, they'll probably start following it better. (d) The independence axiom needs to be given up in favor of a weaker assumption.
edit: I ended up posting this before I saw your comment, kq. Here's you explanation. I originally made this thread 'cause I was interested in seeing how the people on Hatrack answered. Sadly, my textbook doesn't have percentages for the general population (read: undergraduate students wanting to make a few bucks by participating in a psych study), so I can't compare numbers.
Posted by King of Men (Member # 6684) on :
quote:Originally posted by ketchupqueen: I'd like to see you explain me, then.
Well, I think that's just the point - people are not behaving as the theory predicts. Bad ketchupqueen!
Posted by Xavier (Member # 405) on :
Edit: I knew I shouldn't have tried to do math while distracted by work! My numbers were way off.
[ December 10, 2007, 07:11 PM: Message edited by: Xavier ]
Posted by Pegasus (Member # 10464) on :
quote:Originally posted by Fusiachi: I take II and 2.
Reasoning: In the first choice, I view 1% as trivially small.
Likewise, the difference between the 89% and 90% chance of losing it all in the second choice is equally trivial.
Agreed. For the same reasons given.
1% to me is trivial. not sure where the line is but 10% would definitly be non-trivial.
I guess because I view nearly all things mathematically, these sort of axioms make me fairly predictable.
Posted by Jhai (Member # 5633) on :
Edit: it's cool, Xavier. I'll leave the post up in case the different wording/explanation is helpful to anyone else.
By selecting I over II you're saying something about how much you value the different possible prizes. Namely, you're saying that you prefer the utility gained from $500,000 over the utility gained from a 10% chance of getting $2.5 million plus the utility gained from a 89% chance of $500,000, plus the utility gained from a 1% chance of getting nothing.
If you add a certain amount of utility (or it might be subtract, depending on how you value Z & Y) to both sides of the equation, then you're faced with the second choice. Unless the adding of the same amount of utility to both sides alters your feelings about that choice (independence axiom), then to be rational - i.e. consistent in your valuation of the prizes, then you need to go with I & 1 or II & 2.
There are also the explanations I listed above that might explain why a person could chose I & 2. From your statement, I'd guess that the regret theory option would be the one to go with. You know that if you end up unlucky, and get the 1% chance of nothing in (II), you're going to be angry with yourself for not going with (I). That might be enough to tip the scales towards going with (I), even though, without that regret, you'd prefer to go with (II). In (1) & (2) there's a high chance of not getting anything, so you're unlikely to feel regret no matter what happens. This seems perfectly reasonable to me, at least for some individuals' personalities, which is why more complicated models of these situations take into account regret theory.
[ December 10, 2007, 07:31 PM: Message edited by: Jhai ]
Posted by Xavier (Member # 405) on :
Sorry Jhai, I had my math way off. Saw it while you were posting, most likely.
Posted by ketchupqueen (Member # 6877) on :
So, now that you've tried to explain me, I'll tell you why I really chose what I chose.
I'm the kind of person who would usually always pick a sure thing over a risk. Even a tiny risk. I would tend to be a I and 1 person, in other words.
But once the risk of nothing becomes so significantly larger than the chance of getting anything, I go for the big prize, because why not?
Posted by cmc (Member # 9549) on :
That was pretty neat, Jhai.
Thanks for the great explanation, too. : )
Posted by rivka (Member # 4859) on :
quote:Originally posted by Jhai: (c) People just don't realize that they ought to be following the Expected Utility Theorem. Once they've learned about it, they'll probably start following it better.
And if babies started reading child development books, more of them would be "on schedule."
Posted by Jhai (Member # 5633) on :
Yes, well, I just report the facts, ma'am.
The direct quote is:
quote:[it] goes back to the normative interpretation of the theory. It argues that choosing under uncertainty is a reflective activity in which one should be ready to correct mistakes if they are proven inconsistent with the basic principles of choice embodied in the independence axiom (much as one corrects arithmetic mistakes.
I guess you could say that it's similar to someone making an argument, and, without intent, making a fallacious argument. Like Descartes' problematic argument in the The Meditations. Once someone points out the problem in your reasoning, you correct it, 'cause you recognize that things aren't logically following.
Posted by Kwea (Member # 2199) on :
My dad says that people are idiots....and that Deal or No Deal just proves his point.
I agree. Posted by Qaz (Member # 10298) on :
II and 2.
Posted by rivka (Member # 4859) on :
quote:Originally posted by Jhai: Yes, well, I just report the facts, ma'am.
I realize it's not your theory.
It's pretty funny, though. Posted by Enigmatic (Member # 7785) on :
I know I didn't get in before Jhai posted the "answer" but my choice was II and 2. My reasons are the same as what Fusiachi and Pegasus posted.
However, I was kind of tempted to go with I & 2 for reason b:
quote: (b) The problem is the regret that a person feels if he ends up with that 1% chance of nothing in II. This has led to the development of regret theory, which attempts to incorporate expectations of regret into these type of problems.
If I get nothing on the 89% chance I'll be upset that I didn't know the answer to the question that hurt my odds. If I get nothing on a 1% chance I'd curse my luck and accuse the game of being rigged. I think the set of answers that I don't understand choosing would be if someone picked II and 1.
--Enigmatic
Posted by Jhai (Member # 5633) on :
quote:Originally posted by Kwea: My dad says that people are idiots....and that Deal or No Deal just proves his point.
I agree.
I can understand playing on the show - I mean, they're offering a chance at free money. Watching the show though... You're just watching some guy who makes random guesses and get offered deals by the house that are never actuarially fair.
Posted by scholar (Member # 9232) on :
The difference between 0% risk and 1% risk is infinite (or indeterminate or something like that). But the difference between 10% and 11% is only a ten percent increase in risk.
edit to add- my husband and I watch Deal or No Deal and he likes to try to figure out what would be actuarially fair.
Posted by dantesparadigm (Member # 8756) on :
Ok, I'm ready, I memorized the countries in the European Union in alphabetical order. Thanks for helping me procrastinate on my school work.
Posted by Kwea (Member # 2199) on :
quote:Originally posted by Jhai:
quote:Originally posted by Kwea: My dad says that people are idiots....and that Deal or No Deal just proves his point.
I agree.
I can understand playing on the show - I mean, they're offering a chance at free money. Watching the show though... You're just watching some guy who makes random guesses and get offered deals by the house that are never actuarially fair.
That isn't what he means, though. He is talking about the risks people take while playing.
My agreeing with him considers both the players actions and the fact that it made it to TN (and is fairly popular.) . Posted by scholar (Member # 9232) on :
We get annoyed with how the family members tell the contestants that they should keep going because they deserve good things, as if that actually affects what is in the suitcase.
Posted by erosomniac (Member # 6834) on :
quote:Originally posted by scholar: We get annoyed with how the family members tell the contestants that they should keep going because they deserve good things, as if that actually affects what is in the suitcase.
Clearly not a gambler. These things matter! Posted by Tresopax (Member # 1063) on :
The tricky thing about money is to remember that the more money you have, the less each additional unit of money is worth.
Posted by erosomniac (Member # 6834) on :
quote:Originally posted by Tresopax: The tricky thing about money is to remember that the more money you have, the less each additional unit of money is worth.
Uh...my gas is $3.49/gallon, whether I'm rich or poor.
Posted by Fusiachi (Member # 7376) on :
quote:Originally posted by erosomniac:
quote:Originally posted by Tresopax: The tricky thing about money is to remember that the more money you have, the less each additional unit of money is worth.
Uh...my gas is $3.49/gallon, whether I'm rich or poor.
Right, but we're (presumably) talking about value with respect to "utility" here, and not in terms of quantity of dollars possessed. If you're loaded, $3.49 means less to you than it would if you were working from paycheck to paycheck.
It's a case of diminishing marginal utility.
Posted by Jhai (Member # 5633) on :
I think what Tres might be referring to is the idea that there's decreasing marginal utility to money - generally people tend to care less about getting that 101th dollar than they did about getting the 1st dollar or even the 100th dollar. This is equivalent to the idea of people being risk adverse (rather have $499 than a 50% chance of $1000 & a 50% chance of nothing).
People also tend to show decreasing absolute risk aversion - the more money you have, the more risks you're willing to take. There's also nonincreasing relative risk aversion - people are willing to take higher risks in proportion to their wealth as their wealth increase. So you'd be more willing to take a gamble with $100 if you have a $1000 than you would be willing to take a gamble with $5 if you have only have $50.
Posted by scholar (Member # 9232) on :
If you would not do something for a million dollars, there is very little chance you would do it for ten million. But if you wouldn't do something for 10,000 you may do it for 100,000.
Posted by mr_porteiro_head (Member # 4644) on :
quote:If you would not do something for a million dollars, there is very little chance you would do it for ten million.
I'm not sure I agree. I am confident that if I had ten million dollars, I'd be set for life. I am far less confident that I could safely retire right now if I only had one million dollars.
I would be much more likely to do something that could keep me from ever being able to work again for ten million dollars than for one million.
Posted by scholar (Member # 9232) on :
mph- maybe $10 million vs $100 million is your threshold?
Posted by mr_porteiro_head (Member # 4644) on :
Yes.
One million dollars just isn't that much.
Posted by Strider (Member # 1807) on :
The answer to the question in the thread title doesn't really have anything to do with my answer to the question posed in the post. My expected return on choosing option I is $500,000, and option II is $695,000. So in any game of chance I would choose II if I was going strictly by the numbers. But like Eros and others have mentioned, the utility of having that much money outweighs the chance of having much more. So I may be inclined to choose I. Though knowing me I'd probably go with II anyway.
This is similar to how in a poker tournament I may conceivably make a decision to not play in a pot where the numbers say I should stay in. This happens if I'm on the cusp of "getting in the money". Since winning the pot may only make me marginally better off on the table while losing the pot could cost me any chance of getting paid off at all(which is actually why the time just before the cutoff for who makes it in the money is a great time to be aggressive and steal pots from people who are wary of risking not getting paid off).
going back to the original question though, I think it is rational. Assuming happiness does not equal money. And given that assumption, taking the utility value of that money into account(and whatever other factors you can think of) is the only rational way to go about it. Though from a completely monetary perspective, yes, II and 2 are the obvious correct answers.
Posted by Tresopax (Member # 1063) on :
quote:Uh...my gas is $3.49/gallon, whether I'm rich or poor.
For a very rich peson, paying $3.49 for gas might just mean he has $3.49 less to invest in stocks. For a very poor person, paying $3.49 for gas could mean he doesn't have money to buy food for his family that day. So, I would say the gas literally costs more for the very poor person than for the very rich person.
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I originally made this thread 'cause I was interested in seeing how the people on Hatrack answered. Sadly, my textbook doesn't have percentages for the general population (read: undergraduate students wanting to make a few bucks by participating in a psych study), so I can't compare numbers.
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