Tversky, Kahneman, and Gili Bar-Hillel (WikiPedia). Taken by Maya Bar-Hillel at Stanford, summer 1979.
The following post was kindly contributed by Ehud Friedgut.
During the past week I’ve been reading, and greatly enjoying Daniel Kahneman’s brilliant book “Thinking fast and Slow”.
One of the most intriguing passages in the book is the description of an experiment designed by Kahneman and Tversky which exemplifies a judgmental flaw exhibited by many people, which supposedly indicates an irrational, or inconsistent behavior. I will describe their experiment shortly.
I still remember the first time I heard of this experiment, it was related to me over lunch in Princeton by Noga Alon. Returning to this problem, 15 years later, I still, as in my initial exposure to the problem, made the “inconsistent” choice made by the vast majority of the subjects of the study. In this post I wish to argue that, in fact, there is nothing wrong with this choice.
Before relating their experiment, let me suggest one of my own. Imagine, if you will, that you suffer from gangrene in one of your toes. The doctor informs you that there is a 20% chance that it is “type A” gangrene, in which case you can expect spontaneous healing. There is a 75% chance that it is type B, in which case you will have to amputate it, and a 5% chance that it is type C. In the last case there is a shot you can be given that will save your toe, but it will cost you 2000$.
What would you do? I would probably not take the shot. My guiding principle here is that I hate feeling stupid, and that there’s a pretty good chance that if I take the shot I’ll walk around for the rest of my life, not only minus one toe and 2000$, but also feeling foolish for making a desperate shot in the dark.
Now, say I declined the shot, and I return after a week, and the doctor sees that the condition has worsened and that he will have to amputate the toe. He asks me if I wish (say for no cost) that he send the amputated toe for a biopsy, to see if it was type B or C. Here my gut reaction, and I’m sure yours too, is a resounding no. But even when thinking it over more carefully I still think I would prefer not to know. The question is which is better:
Option 1) I have a 75/80 probability of having a clean conscience, and a 5/80 chance of knowing clearly for the rest of my life that I’m lacking a toe because I’m what’s known in Yiddish as an uber-chuchem (smart aleck).
Option 2) Blissful ignorance: for the rest of my life I enjoy the benefit of doubt, and know that there’s only a 5/80 chance that the missing toe was caused by my stinginess.
I prefer option 2. I’m guessing that most people would also choose this option. I’m also guessing that Kahenman and Tversky would not label this as an irrational or even an unreasonable choice. I’m almost positive they wouldn’t claim that both options are equivalent.
Now, back to the KT experiment. You are offered to participate in a two stage game. In the first stage 75% of the participants are eliminated at random. At the second stage, if you make it, you have two choices: a 100% chance of winning 30$ or an 80% chance of winning 45$. But you have to decide before stage one takes place.
What would you choose?
I’ll tell you what I, and the majority of the subjects of the study do. We choose the 30$. Here’s my reasoning: 30 $ is pretty nice, I can go for a nice lunch, 45$ would upgrade it, sure, but I would feel really bad if I ended up with nothing because I was greedy. Let’s stick to the sure thing.
Now a different experiment: you have to choose between 20% chance of gaining 45$, or a 25% chance of gaining 30$.
What do you choose?
Once again, I chose what the majority chose: I would now opt for the 45$. My reasoning? 20% sounds pretty close to 25% to me, the small difference is worthwhile for a 50% gain in the prize.
O.k., I;m sure you all see the paradox. The two games are identical. In both you choose between a 20% chance of 45$ and a 25% chance of 30$. My reference to “a sure thing” represented a miscomprehension, common to most subjects, who ignored the first stage in the first game. Right?
No, wrong. I think the two games are really different, just as the two options related to the gangrene biopsy were different.
It is perfectly reasonable that when imagining the first game you assume that you are told whether you proceed to the second stage or not, and only if you proceed you are then told, if you chose the 80% option, whether you were lucky.
In contrast, in the second game, it is reasonable to assume that no matter what your choice was, you are just told whether you won or not.
Of course, both games can be generated by the same random process, with the same outcome (choose a random integer between 1 and 100, and observe whether it’s in [1,75], [76,95] or [96,100] ), but that doesn’t mean that when you chose the 45$ option and lose you always go home with the same feeling. In game 1 if you chose the risky route you have a 75% probability of losing and knowing that your loss has nothing to do with your choice, and a 5% chance of kicking yourself for being greedy. In game 2 you have a 80% chance of losing, but enjoying the benefit of doubt, knowing that there’s only a 5/80 chance that the loss is your fault.
Of course, my imagination regarding the design of the games is my responsibility, it’s not given explicitly by the original wording, but it certainly is implicit there.
I maintain that there is nothing irrational about trying to avoid feeling regret for your choices, and that I would really stick to the “paradoxical” combination of choices even in real life, after fully analyzing the probability space in question.
For those of you reading this blog who don’t know me, I’m a professor of mathematics, and much of my research has to do with discrete probability. That doesn’t mean that I’m not a fool, but at least it gives me the benefit of doubt, right?
O.k., now, here’s part two of my post – after finishing the book.
Vox populi, vox dei!
Today is the general election day in Israel. This is an exciting day. For me election is about participation much more than it is about influence and I try not to miss it. This is why I regard the influence-based arguments for “why it is irrational to vote,” as unconvincing, and even a little silly. Influence vs participation was also the basis for my poll on “would you decide the election if you could.” I voted ‘no’ but about two-thirds of the participants voted yes. Four years ago I brought on this day the nice story of Achnai’s oven.
Lenore Holditch is a freelance writer. Here is what she wrote to me: “I love learning about new topics, so I am confident that I can provide valuable content for your blog on any topic you wish, else I can come up with a post most relevant to your blog theme. The content would be fully yours.” Lenore only asked for a link back to her site . She sent me also a few of her articles like this one about finding jobs after graduation and this one about video games for training.
I asked Lenore to explore the following issue and she agreed:
Is the game Angry Bird becoming gradually easier with new versions so people get the false illusion of progress and satisfaction of breaking new records?
The following paradox was raised by Rann Smorodinsky:
Rann Smorodinsky’s Privacy Paradox
Suppose that you have the following one-time scenario. You want to buy a sandwich where the options are a roast beef sandwich or an avocado sandwich. Choosing the sandwich of your preference (say, the avocado sandwich) adds one to your utility, but having your private preference known to the seller, reduces by one your utility. The prior people have on your preferences is fifty-fifty.
If you choose the avocado sandwich your utility is zero, hence you can improve on this by picking each type of sandwich at random with probability 1/2. In this case your private preference remains unknown and you gain in expectation 1/2 for having the sandwich you prefer with probability 1/2.
But given this strategy, you can still improve on it by choosing the avocado sandwich.
Here is a question from last year’s exam in the course “Basic Ideas of Mathematics”:
You buy a toaster for 200 NIS ($50) and you are offered one year of insurance for 24 NIS ($6).
a) Is it worth it if the probability that damage covered by the insurance will occur during the first year is 10%? (We assume that without insurance, such damage makes the toaster a “total loss”.)
b) Is it worth it if the probability that the toaster will be damaged is unknown?
As an additional test of your imagination, can you think of reasons why buying the toaster insurance would be rational?
You are guaranteed to win one of the following five prizes, the letter says. (And it is completely free! Just 6 dollars shipping and handling.)
a) a high-definition huge-screen TV,
b) a video camera,
c) a yacht,
d) a decorative ring, and
e) a car.
Oh yeah, you think, a worthless decorative ring, and throw the letter away.
But once I got a letter with the following promise:
You are guaranteed to win two of the following five prizes, the letter said.
a) a high-definition huge-screen TV,
b) a video camera,
c) a yacht,
d) a decorative ring, and
e) a car.
Now, one prize will be a worthless decorative ring, but what will the second prize be?
Can chess be a game of luck?
Let us consider the following two scenarios:
A) We have a chess tournament where each of forty chess players pay 50 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.
B) We have a chess tournament where each of forty chess players pay 20,000 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.
Before dealing with these two rather realistic scenarios let us consider the following more hypothetical situations.
C) Suppose that chess players have a quality measure that allows us to determine the probability that any one player will beat the other. Two players play and bet. The strong player bets 10 dollars and the waek player bets according to the probability he will win. (So the expected gain of both player is zero.)
D) Suppose again that chess players have a quality measure that allows us to determine the probability that any one players will beat the other. Two players play and bet. The strong player bets 100,000 dollars and the weak player bets according to the probability he will wins. (Again, the expected gain of both players is zero.)
When we analyze scenarios C and D the first question to ask is “What is the game?” In my opinion we need to consider the entire setting, so the “game” consists of both the chess itself and the betting around it. In cases C and D the betting aspects of the game are completely separated from the chess itself. We can suppose that the higher the stakes are, the higher the ingredient of luck of the combined game. It is reasonable to assume that version C) is mainly a game of skill and version D) is mainly a game of luck.
Now what about the following scenarios:
Here the main ingredient is skill; the bet only adds a little spice to the game.
F) Two players play chess and bet 100,000 dollars.
Well, to the extent that such a game takes place at all, I would expect that the luck factor will be dominant. (Note that scenario F is not equivalent to the scenario where two players play, the winner gets 300,000 dollars and the loser gets 100,000 dollars.)
Let us go back to the original scenarios A) and B). Here too, I would consider the ingredients of luck and skill to be strongly dependant on the stakes. The setting of scenario A) can be quite compatible with a game of skill where the prizes give some extra incentives to participants (and rewards for the organizers), while in scenario B) it stands to reason that the luck/gambling factor will be dominant.
One critique against my opinion is: What about tennis tournaments where professional tennis players are playing on large amounts of prize money? Are professional tennis tournaments games of luck? There is one major difference between this example and examples A and B above. In tennis tournaments there are very large prizes but the expected gain for a player is positive, all (or at least most) players can make a living by participating. This changes entirely the incentives. This is also the case for various high level professional chess tournaments.
For mathematicians there are a few things that sound strange in this analysis. The luck ingredient is not invariant under multiplying the stakes by a constant, and it is not invariant under giving (or taking) a fixed sum of money to the participants before the game starts. However, these aspects are crucial when we try to analyze the incentives and motives of players and, in my opinion, it is a mistake to ignore them.
So my answer is: yes, chess can be a game of luck.
Now, what about poker? Continue reading
I took part in a workshop celebrating the publication of a new book on Social Choice by Shmuel Nitzan which took place at the Open University. (The book is in Hebrew, and an English version is forthcoming from Cambridge University Press.) It was a very interesting event and all the lectures were excellent. I thought of blogging about my lecture.
The main part of the lecture was about the four old theorems in the table above and about what should replace the two question marks. The left side of the table deals with properties of the majority voting rule for binary preferences. The right side of the table is about general voting rules. On the top tight is the famous Arrow Impossibility Theorem. The table is filled by two theorems I proved in 2002 (in this paper) and it now looks like this: Continue reading
During World War II, many fighter planes returned from bombing missions in Japan full of bullet holes. The decision was made to reinforce the planes, and their natural tendency was to bolster the hardest-hit sections in the body of the plane. However, the mathematician George Dantzig suggested that it was precisely the parts that were hit less that needed to be armored. Was he right?
Months after all the commentators described Hillary Clinton’s chances as so slim she was bound to lose her campaign for the Democratic presidential nomination, she continued to fight for her candidacy, saying she believed she would win and keeping up her attack on her rival. Did she act rationally? And did Benjamin Netanyahu and Tzipi Livni act rationally when each declared victory on election night? Did Meretz supporters who voted for Kadima act rationally? Is there an election method in which it would be rational for all voters to vote in accordance with their genuine preferences?
My conclusion is: