This week we had two posts on the foundations of rationality and two on whether there’s been a recent surge in government spending.
The government spending posts are here and here. It seems to me that the graphs in the second post are dispositive.
The rationality posts are here and here. These led to a lot of convoluted discussion, so let me give you the executive summary.
I claim these things:
Claim One: I have here a mystery pet that might be a dog and might be a cat. If you prefer dogs to cats, you’ll prefer a dog to the mystery prize — and vice versa.
Claim Two: If you prefer dogs to cats, then you’ll prefer an 11% chance at a dog to an 11% chance at a cat — and vice versa.
Claim Three: The same things are true if you replace “dog” and “cat” with any other prizes you care to think of.
Claim Four: The desirability of a lottery depends only on the prizes and the probabilities of winning.
Claim Five: If you accept Claims One through Four, then you must answer either A to both of the following questions or B to each of the following questions:
Question 1: Which would you rather have:
- A million dollars for certain
- A lottery ticket that gives you an 89% chance to win a million dollars, a 10% chance to win five million dollars, and a 1% chance to win nothing.
Question 2: Which would you rather have:
- A lottery ticket that gives you an 11% chance at a million dollars (and an 89% chance of nothing)
- A lottery ticket that gives you a 10% chance at five million dollars (and a 90% chance of nothing)
Claim Five, as I’ve explained in the earlier posts, is a logical necessity. So if you want to defend inconsistent answers to Questions 1 and 2, then you are forced to deny at least one of Claims One through Four. It would be very helpful if participants in the ongoing discussion would specify exactly which claim they dispute.
I’ll have more to say about rationality — and a variety of other topics — over the next week.
Dogs and cats are very different things. I may not want a cat at all (for example, I’m allergic).
Money, however, is the same. You’re not getting something different, you’re getting more of the same thing. So your analogy is poor.
Your original example offers large amounts of money, money that makes real differences in people’s lives. When the odds of winning that money are small, you want to increase the odds of getting some money. When the odds of getting the money are high, you want to increase the odds of a big payoff (since you’re almost certainly going to get some payoff). I don’t see the problem.
I’m assuming here you get to play the game once. If you get to play it lots, then picking the same choice on both makes perfect sense.
David Pinto: So if I replace the dog with a million dollars and the cat with a lottery ticket that gives you a 10/11 chance at five million, which of the claims #1, 2 or 3 are you disputing?
I do not believe that claim 5 follows from claims 1-4. You have some more hidden assumptions, and they may not necessarily be valid for a rational person. I guess I will tune in next week to look for an explanation.
I agree claims one to five follow from logical necessity.
However, I had the same reaction as reaction as David Pinto that Dogs and Cats are a poor comparison to money. Steve in his response to David has asked which of the claims in the post is being disputed. In response – claim 3 is being disputed that dog and cats can be replaced with one million and five million dollars respectively. Claim 3 would not be disputable if Cats were worth 5 times as dogs monetarily, or were preferred by a factor of five (assuming rational people prefer $5million over $1 million) and cats were easily tradable with no conversion transaction costs (as is mostly true for cash money) for other goods.
If there was an adequate explanation why dog and cats are analogous to money and hence no consequences in customer perception towards a decision I would be convinced claims one to five apply to the specific original rationality post over different amounts of money.
I don’t accept Claim Three. It really does depend on the prizes. The desirability of some things may be related, or conditional on each other.
Cos (and David):
Claim Three is really two claims:
Claim Three-A: If you prefer an X to a Y, then you’ll prefer a sure X to a prize that might be either an X or a Y.
Claim Three-B: If you prefer an X to a Y, then you’ll prefer an 11% chance of an X to an 11% chance of a Y.
Which of those subclaims do you dispute?
I think Claim 3B is the one that is not true in the real world, for psychological reasons.
Another example is: people might prefer $500 to a free skydiving lesson, but they’d prefer an 11% chance of winning a free skydiving lesson to an 11% chance of $500 (if you asked them straightforwardly they would not admit to this, but I bet you could design a survey to show somewhat of a discrepancy here). This is because, in isolation, $500 is superior to a skydiving lesson, and the responsible father, say, recognizes this, and he’d face ridicule if he chose a skydiving lesson over $500. But as far as future storytelling opportunities are concerned, it sounds cooler to say “I once won a skydiving lesson in a lottery where I faced long odds,” than “I once won $500 in a lottery where I faced long odds.” In cases where the odds are long, people do not as intensely fear future ridicule for irrational decisions, and they care more about storytelling opportunities. Stories told about lotteries won are more exciting when the lottery odds were long, and when the prize was quirky.
As I stated before I agree with the logical necessity of Claim one to Claim five. I also agree with claim three-A and three-B.
However, I am disputing claim three-B in the context of the original post because it does not accurately reflect the choices presented in Q2 which were $1 million@11 % and $5 million @10%. Claim three-B would be relevant in the context of the original post if Y was not the same good as X the only difference being that it is a greater quantity by a factor of 5. Assuming a consumer prefers X, surely he would prefer 5X (where 5X=Y). Note I have not included the 1% difference between X and 5X in this reasoning even though it further accentuates the choice of A and B in Q1 and Q2 respectively in the original post and also because claim three-B posits similar probabilities for X and Y.
The above stated differently is that an individual may prefer X when it is a choice between X@100% over nothing@1%( and X@89%,Y@10%), but prefers Y@10% over X@11% when Y=5X. Why is this irrational?
The reason for the discontent in the comments seem to be due to the fact that most people do not seem to think Y is really 5X (ignoring the 1% for now) while in case of prizes apart from money, for e.g. Dogs are not really equal to 5 cats for most people. The Dog/Cat analogy woul be relevant if you stated that the prize winner could exchange a cat for 5 dogs or for an equivalent value preferred by the winner (monetary or otherwise). In general people treat an exchange of value diffrently from a commodity itself worth a similar value.
Here is a question for the people who pick 1.A and 2.B:
Question 3: For what values of X would you pick option A instead of option B:
A. A lottery ticket that gives you an X% chance of nothing and a (100-X)% chance at a million dollars.
B. A lottery ticket that gives you an (X+1)% chance of nothing, a 10% chance for five million dollars, and a (89-X)% chance at a million dollars.
As X increases from 0 to 89, at what value do you switch from 3.A to 3.B? By doing this, you are denying Claim Four.
I really would be interested in learning what values of X you pick, and what rationale you use to do so.
Jonathan Campbell:
“. . . people might prefer $500 to a free skydiving lesson, but they’d prefer an 11% chance of winning a free skydiving lesson to an 11% chance of $500 . . . ”
(To make some small numbers work out, I’m changing your 11% to 10%. The general point remains the same, though.)
How about this?
A box contains 10 note cards. 9 of them say “sorry, you didn’t win anything”; 1 of them says “choose one: $500 or a free skydiving lesson”. A card is drawn for the person you are thinking of in the quote above. If they ‘win’, what do they choose?
Another: 10 people are in a room. One is selected at random and offered a choice between $500 or a free skydiving lesson. If the person you are thinking of is selected, what do they choose?
David:
As an aside, if you accept all of the Claims, and you place zero value on $0, then $5M only needs to be something more than 1.1 times as valuable as $1M for a person to select answer B in both questions. If you value $5M the same as $1M, or anything less that 1.1 times as much, then you should select answer A.
Thomas Bayes – good examples showing why the behavior I described is, in a sense, ultimately irrational. I bet that in both of those cases, $500 would be chosen, since in each case the decision point comes when the chances of success are already 100%. I think that if a person were forced to make the decision when the chances of success were still 10%, he’d more likely choose the quirkier, yet objectively less valuable, prize.
Some thoughts on Claim Four, which, I believe, is implicitly rejected by many people.
Imagine that you must play a game of ‘rock, paper, scissors’.
Question 1: Which would you prefer?
A: A piece of paper for certain.
B: A lottery ticket that gives you an 89% chance to win a piece of paper, a 10% chance to win a pair of scissors, and a 1% chance to win a rock.
Question 2: Which would you prefer?
A: A lottery ticket that gives you an 11% chance to win a piece of paper, and an 89% chance to win a rock.
B: A lottery ticket that gives you a 10% chance to win a pair of scissors, and a 90% chance to win a rock.
If you accept Claims 1-5, then you must go for either A or B for both questions. You’ll associate a utility with each item (rock, paper, or scissors), then compute the expected utility.
But what if you know that your opponent is going to use the lottery that you don’t select? The desirability of each lottery now depends on the one with which it is competing, and, if you want to maximize your chance of winning, you’ll take 1.B and 2.A. By doing this, you’re using rational reasoning, but you’ve turned the questions into something different.
Perhaps people do this for other problems. If the lotteries have N outcomes, they imagine another person taking the lottery that they reject, assign some utility to the N*N possible combinations, then compute the expected utility. Their utility assignments depend on the value they associate with their outcome, along with the regret they assign to knowing ‘what might have been’ based on the other lottery’s outcome. We are probably encouraging people to this by asking a question for which one of the lotteries has a fixed outcome.
Thomas, claim four is being accepted hence the preference for 1 A and 2 B.
The reason for disagreement seems to be that people are willing to accept dogs, cats, paper, scissors etc as different products but have difficulty accepting that different quantities of money as separate products in the same sense. My understanding of the questions in the original post is as follows:
Q1. A – X@100%
Q1.B – X@(>100%)
Q2.A – X@(>100%)
Q2.B – 5X@(>100%), where value of (>100%) is similar in Q2.A and Q2.B
I will go a little further that most people who prefer X over nothing and also prefer more of X to less of X (i.e. X is an individually consumable positive normal good) would consider someone choosing both A’s or B’s in the above cases as irrational.
P.S – Also I think using Cats, Dogs, Paper, Scissor, opponents, lotteries s etc changes the nature of the original question and only seems to be confusing the discussion. Let’s stick to money (i.e. X=million $ above) for sake of clarity. No offence intended. Thanks.
David:
It is incorrect to say that X is one million dollars and Y is five million dollars in Steve’s example. Actually, X is one million dollars and Y is a 10/11 chance at five million and a 1/11 chance at nothing. Now, which do you prefer, X or Y?
Thomas Bayes:
Of course including strategic interactions the way you do would invalidate the assumptions. But it seems silly to assume that people have a preference over their strategies (e.g. rock, paper, or scissors) rather than their payoff (e.g. what they get if they win).
When the chance of winning is high, I want the less certain choice, since it pays more. When the chance of winning is low, I want the more certain choice, since I want to win something.
The question you are asking boils down to, do you want more money or more certain money. My answer is, it depends on the odds of winning. At high odds, 99 in 100 in your original post on the subject, I want more money. At 10% to 11% chance of winning, I want the better odds.
If the choice was a dollar or $5 million, then I would make the same choice each time, for the higher money, since $1 would not make a difference in my life. But $1 million would make a difference, so at the lower odds I’d take some money over lots of money.
John:
I think Professor Landsburg’s recent post hits on exactly the issue of strategies versus payoffs. For Question 1, many people are using a strategy that avoids the emotional distress of knowing that their choice caused them to win nothing. For Question 2, they never know with certainty that this happened, so they focus on the payoff. I don’t think that is silly.
I dispute Claim 5, because Claims 1-4 deal with two distinct prizes, whereas Question 1 concerns a lottery with three prizes, and because according to the law of marginal utility, the utility of $5M is less than five times the utility of $1M, perhaps much less, therefore simple arithmetic comparison of expected prizes is irrelevant to a rational choice.
I think your “proof” of Claim 5, using red, white and black balls, assumes (falsely) that the relative marginal utilities a person attaches to the black and white prizes is independent of the value of the red prizes. (In fact, an Austrian economist would claim that utilities cannot even be compared, but only ranked.)
Furthermore, the (marginal) utility assigned to the variously colored balls would depend on the current status (e.g., wealth) of the person; which is why some people might rantionally choose an A and a B to your questions.