Yesterday, I was eating lunch with a friend when he brought up this question: "Are video games destroying our economy?"
His argument was such. Premise 1: video games tend to be addictive. Premise 2: when people purchase a video game, they tend to remove themselves from society, production, and consumption. Premise 3: that such a removal from society has negative effects on the economy. Premise 4: that the addictive-ness (is that a real word?) of video games means people will consume more and more of them at an ever increasing rate. Conclusion: Video games are destroying our economy.
I thought for a second, and said of course not. People tend to make decisions to maximize their utility. If they get more satisfaction from playing video games then, say, playing soccer or shopping, than they are better off by playing video games. His concern has to deal with the externalities that are produced by people participating in society. However, each of the externalities produced (extra guy playing soccer, extra girl shopping) can be reduced to yet another market good for a different individual (the person who wants to put together a soccer team, the store trying to sell goods). As long as people are left to their own decisions, the optimal economic situation will occur. That's pretty much Austrian Economics 101.
Then I paused and thought a little bit more. That relied on the assumption that people do rationally make decisions that maximize their utility. And this is where Premises 1 and 4 raise their ugly heads. Addiction is probably the most obvious case of human rationality failing completely. You cannot argue successfully that a heroin addict is maximizing his utility when he is doing anything he can to satisfy his fix. It short circuits the decision making process. So, a video game player may be choosing to play a video game because it's the most profitable activity he could be doing at that moment OR he could be playing to satisfy a fix, in which case he is overall experiencing a net loss of utility (when opportunity costs are taken into account).
Thus, the question stops being a theoretical question and starts becoming an empirical question. Are video game players happier than non video game players, with all other variables accounted for and controlled for? To that, I have no answer.
Thursday, January 27, 2011
Monday, January 17, 2011
Context Matters
"Past ideas may be tendentiously misrepresented. This may be done by stating/implying that earlier thinkers were trying to solve our problems, and/or by using today's terminology in describing their work. It's all too easy to project our own concerns onto ancient writings that bear some superficial resemblance to ours, in order to make a progressivist story appear more plausible." Margaret Boden, "Mind as Machine" pg 19
Any good argument will have three main parts to it: a logical structure, relevant data, and a conclusion that follows from the first two. Far too often, people tend to concentrate on the conclusion and ignore the other parts. Obviously, that is flawed. If you disagree with another person's conclusion, that means you either disagree with that person's logic or that person's data. If you wish to change someone's conclusion, then you need to change either that person's logic or that person's data. Either way, you'll need to examine both.
The logical structure of an argument tends to be straightforward. A causes B, B causes C, therefore A causes C. I do not feel any pressing need to go into logic at this time, though I may under a different occasion. If you do feel an urge to reexamine how logic works, I suggest Richard Feldman's Reason and Argument. It's an excellent introductory text.
On the other hand, data and the context we find it in is something I'd like to explore in this post. A little over two weeks ago, my friend Andrew posted this on his Facebook wall under the headline "Welcome to America":
The implication was, of course, that it is unfair that the top 1% of taxpayers pay as much in taxes as the bottom 95%. A surprisingly large number of my friends ended up posting comments, usually along the lines of: "Just cause your successful does not mean you should pay a ridiculous amount... fucking liberals" and "Such garbage."
According to the data Andrew published, 1% of the population was hauling as much weight as 95% of the population. According to my friends, that's ridiculously unfair and just another example of how the rich are being punished by our government. That's not exactly true, though.
My reply was a little different. My first comment ran at two paragraphs. I'll divide it into parts here and go into greater depth just for excessive clarification.
The first paragraph: "So... you're saying that the same people who own 58.9% of the total wealth in America (top 5%, see link below, also note, my data is 6 years old, since then the gaps have grown wider) pay roughly 55-60% of the taxes?"
Followed immediately by: "That strikes me as... actually, completely reasonable."
My data in this case comes from Fairfield University. According to that data, in 2004 the top 1% owned 34.3% of total wealth and the next 4% owned 24.6% of total wealth for a combined top 5% owning 58.9% of total wealth. According to the data Andrew himself posted, that same group paid roughly 55%-60% of taxes in 2010. A group paying taxes in proportion to what they own does seem reasonable. However, a debate could arise over whether it is fairer to tax people in accordance with their wealth or their income. In order to head off this debate, I included this second paragraph in my first comment:
"And just to preempt the inevitable wealth v. income debate, I turn to Alan Greenspan, former chairman of the Federal Reserve Bank, making the case for wealth:
"Ultimately, we are interested in the question of relative standards of living and economic well-being. We need to examine trends in the distribution of wealth, which, more fundamentally than earnings or income, represents a measure of the ability of households to consume."
Context, bitches. It matters."
My friend Ethan was the next to reply, posting this:
"I actually think Jack has a point. Most of those sort of websites are propoganda for one political part or another. You rarely see a comparason of tax burden relative to overall income. Though I do think some sort of flat tax would be the best solution."
To which I replied:
"I'd have to disagree, Ethan. The flat tax is inherently regressive because the marginal dollar is worth more to the low income household than the high income.
I propose a graduated sales tax, where the most expensive items (e.g., a $20 mil. yacht) are taxed at high rates, and the cheapest, most necessary items (ramen noodles and other food stuffs), are taxed at either nothing or something close to it.
I'd feel really clever right now, except that it turns out Prof. Steven Landsburg proposed the same damn thing nearly three years ago:
http://www.slate.com/id/2181833/"
At this point, Andrew weighed in again:
"[...] flat taxes are not a good idea. A non-graduated consumption tax would be my vote. It still would end up being progressive, but not nearly as severe as today's setup. It's nice because if you should decide to spend more money, you'll be taxed more."
Before I allow his quote to continue, I have to point out the above comment is fallacious. A non-graduated consumption tax implies just a flat sales tax on both goods and services. A flat tax is almost always regressive (meaning that it tends to have the poorer pay proportionally more than the wealthy). The reason is that every person usually has to spend a certain lump sum every period just in order to live. This is the cost of living, and it's almost the same from person to person. We all consume roughly the same amount of food, electricity, and water per person just for our basic needs. Everything above and beyond that is to satisfy a luxury. But what this means is that the basic cost of living is a significantly larger percentage of a person's total wealth the poorer that person is. With no taxes at all, the system is inherently regressive. The wealthier a person is, the larger the wealth percentage will be that that person is able to reinvest, making that person even wealthier. The flat tax increases the proportion that everyone spends by the same percentage, or looking at it from a different angle, decreases the proportion that everyone saves by the same percentage. This means that there is no difference between a flat tax and no taxes when it comes to a re-appropriation of wealth.
Now, in the interest of covering my ass so that this post won't come back to haunt me later, I have heard arguments on the graduated income tax that state: " I’d be curious as to what grounds one might use to support a graduated tax? It fails on economic efficiency grounds. It fails on moral grounds. It fails on the principles of liberty. It certainly fails on property rights grounds. It has failed (certainly with steep escalation) to even achieve any of its distributional goals, so that even if liberty were not important, or morals were not important or efficiency were not important, it has been demonstrated time and again to not achieve its stated objective" - my Economics Professor
I have not read the data behind those claims, so I'm unable to say what my opinion is on them. Which brings me to a critical point on context and data: If you don't know anything about the data at hand, stick to what you do and admit what you don't. But at the same time, that has little to do with the thrust of this debate, which is: Is our current tax system unfair to the wealthy?
Unfortunately, Andrew seems to have not bothered to read what I wrote before, especially when he commented:
"Jack, your argument is interesting... looking at taxes based on wealth instead of income could almost be seen as wanting to unfairly tax those who save their money."
Technically, this is true, but I refer back to my Alan Greenspan quote for why it makes more sense to base taxes on overall wealth than on income.
"Yes, most would be in the top income brackets, but the system currently in place almost incentivizes spending every dime..."
Again, also true. The capital gains tax does decrease the marginal benefit of saving, which is why I suggested switching to a sales tax. The sales tax would incentivize saving over spending.
Andrew ended our argument on this note:
"Maybe this is simply America becoming ever more uncompetitive globally and only those who are extremely hard-working & intelligent can expect to command more of the wealth in the U.S.. Those who can't keep up are simply left behind on welfare, which is then paid for by those who became successful... Did you know for every $1 earned with income below $24,000, they get back $8.21 in benefits? Above $100,000 income you are receiving $0.41 back per year. The message to America? Don't earn more than $100,000 because you don't deserve it. "
My first reaction was to dismiss the first line. To assume "this is simply America becoming ever more uncompetitive globally" is a huge assumption, especially without any data on hand. First, what does he mean by uncompetitive? Does he mean by standard of living? By GDP? By manufacturing output? By Olympic Gold Medal wins?
Second, how is that tied in with our argument on taxes? Is he saying that our tax rate makes us more uncompetitive? Since that’s what he’s been arguing the entire time, his opening line doesn’t provide any new information. It only exists to rhetorically color the debate to make his ideas seem more sympathetic.
So, I dismiss the line as being irrelevant.
But then the second part emerges. I was aware that people who paid less in taxes tended to get back more in benefits per dollar paid than people who paid more. In fact, I assumed it was logical. Assume that benefits are uniformly distributed (this might not be exactly true, but it's probably close enough). Everyone has roughly the same access to roads, fire departments, police and military protection, justice under the law, voting, etc as everyone else. So, the government probably spends roughly the same amount providing benefits for every person. But, as heavily debated above, different people spend wildly different amounts in taxes. All that Andrew is showing by that statement is that the wealthier pay more in taxes than the poor.
So where do I stand on Andrew's conclusion? His belief is that the wealthy are unfairly punished by our current tax system. My analysis of the data that he presented, and further exploration into its context, indicates to me that the wealthy are taxed in rough proportion with their own wealth, which is about as fair of a tax system as I can think of. Does that mean that I think the tax system is perfect, or that there aren't other systems that might work better? No, and on top of that, I did briefly outline one above that I think would be better.
The trouble, I feel, is that Andrew relied very heavily on assumptions that he didn't realize he was making. Assumptions that he used in place of data and that have been handed down to him through previous sources. There were other times in American history when the tax system did unfairly punish the wealthy, and during those times, many excellent arguments were presented against them. These arguments, Andrew probably read, or read accounts of them, and assuming the conclusions for them were still correct, substituted them for his own conclusion and then cherry picked data to support it.
This is not meant to be too critical of Andrew. He is a good friend, and I'm sure that I make the mistake many times myself, and probably have a couple times in this post. But, I hope this example does illustrate the necessity of examining the conclusions we hold and the arguments that support them.
Finally, how does this all tie in with the quote by Margaret Boden that graces the top of this post? It is, I admit, a bit out of place, but it does tie in well with the overreaching thought. Too often, we borrow conclusions from others and fit them into our own prearranged patterns without examining how well those conclusions apply to our current situation. The superficial similarities between our problems and problems people have had in the past makes it all too tempting to assume they are the same, and as my little discussion with Andrew shows, will occasionally lead to fallacious claims.
The context of the data we choose is often as important as the data itself. Failing to properly examine not only the logical structure of another person's or our own arguments but the data presented and the context it is presented in undermines debate and makes rational argument impossible.
Saturday, January 8, 2011
The Paradoxes of Rational Choice Theory
Rational Choice Theory is the current basis for most of microeconomics. Its major feature is rationality, the idea that people want more than less and when given a choice between options, will inevitably choose the option that satisfies more of their preferences than the other available options. Intuitively, the theory makes a great deal of sense.
The obvious objection is that people don't bother to calculate all the costs and benefits of every situation. This is explained away by the "pool table idea;" in the same way that you don't need to be a physicist to play pool yet the balls will still follow the laws of physics, you don't need to be an economist to make rational choices.
The other typical objection is that some people seem to make decisions that, no matter how you cut it, goes against their best wishes. The explanation for this is that since we can't possibly know a person's individual preferences, we must assume that they are rationally pursuing them. By this point, Rational Choice Theory becomes a tautology - after all, if people are all rationally pursuing their own preferences, preferences which can't be known, it's impossible to empirically test whether or not they are actually doing so, and Rational Choice Theory is therefore true because it's defined to be.
That said, there are a few paradoxes which point us towards weaknesses in Rational Choice Theory.
St. Petersburg Paradox
Imagine you are offered the following bet: A coin will be flipped. If it is heads, you get nothing. If it is tails, you get $2 and the opportunity to flip again. If on the second flip, you get a heads, you keep the $2. If you get tails, you get $4. Next flip, $8, then $16, then $32, and so on.
How much pay to take part in this gamble?
According to Rational Choice Theory, you would pay anything less than the expected value of the gamble. After all, everyone desires more than less and if you keep playing the game over and over again, your average winnings will be the expected value.
To calculate the expected value, all you need to do is multiply the percentage chance of winning a certain amount by that amount and then sum all possibilities. The result will be your average winnings.
So, we know we have a a 50% chance of getting heads on the first flip. So that's 50% times nothing. Easy so far. 50% chance of getting tails. But wait. 50% of the times you get tails, you'll go on to win again, so, we can only add 50% of that initial 50% times the $2 winnings. Easy enough, .5 x .5 x $2 = $0.50. But then there's the next set:
.5 x .5 x .5 x $4 = .125 x $4 = $0.50
and the next:
.5 x .5 x .5 x .5 x $8 = .0625 x $8 = $0.50
and so on:
5 x .5 x .5 x .5 x .5 x $16 = .03125 x $16 = $0.50
5 x .5 x .5 x .5 x .5 x .5 x $32 = .015625 x $32 = $0.50
5 x .5 x .5 x .5 x .5 x .5 x .5 x $64 = .00778125 x $64 = $0.50
and so on ad infinitum. The series diverges. The expected value equals infinity x $0.50.
So, the rational person would bet literally any amount of money to take part in this gamble.
But people don't bet any amount of money on this. They bet quite a bit less.
There are several other explanations for this, among them utility curves, risk aversion, people's awareness that there is a limited amount of money in the world, etc.
Perhaps the most convincing theory I've read that explains this paradox is this paper by Benjamin Hayden on the "median heuristic." The paper sets forth the idea that people don't always rely on Rational Choice Theory (or perhaps never rely on it) and instead suggests that people make their decision in this particular paradox by picking a bet close to what the median return from the bet is. This is very different from betting on the expected value. The expected value is roughly the mean return from the bet and is significantly larger than the median due to the skew created by the occasional absurdly long streak of tails. The other noticeable thing about the median is that it is far easier to accurately estimate. Finding the mean requires to sum all results and then divide by the total count. Finding the median just requires taking a stab at the middle number, and with larger samples, if you're off, it's not by much. Considering that the mind often seems to prize efficiency over accuracy and precision, this heuristic seems especially likely.
This new median heuristic predicts that people will make a bet of about $1.70 some odd, which they found widely predicts what people actually do.
Allais Paradox
Now, having lost all of your money making bad bets on the St. Petersburg Paradox, you are offered a new game.
There are two urns, A and B.
A contains 99 black marbles and 1 red marble.
B contains 90 black marbles, 5 white marbles, and 5 red marbles.
If you pull a black marble, you get $1 million. If you pull a white marble, you get $5 million. If you pull a red marble, you get nothing.
You can only play once, so going for the expected value doesn't make sense in this case. It's solely a matter of personal preference. Which do you choose?
Now, onto a completely new game.
There are two new urns, C and D.
C contains 9 black marbles and 91 red marbles.
D contains 5 white marbles and 95 red marbles.
Again, if you pull a black marble, you get $1 million. If you pull a white marble, you get $5 million. If you pull a red marble, you get nothing.
Which do you choose?
One of my economics professors posed this question for a class I was in. If I remember correctly, A and D was the most popular choice, followed by B and D and B and C. A and C was the least popular.
But Rational Choice Theory predicts that everyone will choose A and C or B and D, and that no one will choose A and D or B and C. Let me show why.
90% of A and B are identical. In both cases, 90% of the time you get $1 million. The only real question is: for the remaining 10%, do you want a 50% chance of $5 million or a 90% chance of $1 million?
Likewise, 90% of C and D are identical. In both cases, 90% of the time you get nothing. The only real question is: for the remaining 10%, do you want a 50% chance of $5 million or a 90% chance of $1 million?
If you prefer a 50% chance of $5 million, you should always prefer a 50% chance of $5 million. If you prefer a 90% chance of $1 million, you should always prefer a 90% chance of $1 million. The fact that these preferences don't hold shows that Rational Choice Theory is flawed some way.
I haven't read an explanation for this paradox that I have found satisfactory. One idea that I think might explain it somewhat is that people think less in terms of probabilities and more in terms of "is or is not." Under that hypothesis, people would choose A over B, because A is a sure a thing while B is less sure. Meanwhile, people would choose D over C, because while both are almost certain not to happen, D gives a larger benefit if it actually does happen. For those of you who pointed out that this is really similar to rank-dependent expected utility, please hold your fire until after the end of the Ellsberg Paradox, where I readdress this issue.
I would like to emphasize that the above stated hypothesis is completely untested.
Ellsberg Paradox
Interesting side note: the Ellsberg Paradox is named after Daniel Ellsberg, who wrote about it in his Economics Phd dissertation. Ellsberg is far more famously known, however, for being the military analyst who released the Pentagon Papers in 1971.
For this example, imagine an urn. The urn contains 90 marbles, 30 marbles are black and the other 60 are yellow and red, but the ratio between the two is unknown. You can choose one of two wagers. Either a million dollars if a black marble is pulled or a million dollars if a yellow marble is pulled.
Now imagine the same urn. 90 marbles, 30 black, other 60 red and yellow. You still don't know the proportion, but it's the same as the time before. You can choose of two wagers. Either a million dollars if a black or red is pulled or a million dollars if yellow or red is pulled.
Which did you choose?
As you can probably imagine by now, what people usually choose does not reflect what rational choice theory predicts. Rational Choice Theory* says that if you pick black in the first example, then you must think there are fewer than 30 yellow marbles, so that in the next example you would pick black or red. Likewise, if you picked yellow, you must think there are more than 30 yellow marbles, and so you'd pick yellow or red.
*Technically, it's Expected Utility Theory. That said, they operate under the same basic considerations, and for the purposes of this blog post, will be assumed to be the same. Real economists, please don't chew my head off for this.
What you would not pick is black for the first choice and yellow and red for the second choice, since that assumes that there are more red than yellow in the first instance but more yellow than red in the second.
Here's the explanation I have: People prefer the sure choice against the unknown. So, most people will choose black in the first set because they know that the chances of getting one million dollars is exactly 1 in 3, while if they choose yellow it could be anywhere between zero and 2 in 3.
Likewise, if people choose yellow and red in the second set, then they have a 2 in 3 chance, while if they choose black and red it could be anywhere between 1 in 3 and certain.
The trouble is then, why do people sometimes not choose black for the first instance and yellow and red for the second instance? I'm stumped. I guess there is some compromise between the risk aversion and the desire for optimal output (what Rational Choice Theory) predicts, but I can't think of anything that would accurately predict what people do.
Note: the following is information poorly understood by the author. It's validity and accuracy is questionable.
Now, there was a paper written in 1986 that attempts to explain the Ellsberg Paradox and, less directly, the Allais Paradox by Uzi Segal, then an Assistant Professor at the University of Toronto, now a Professor at Boston College.
I have read the paper, entitled "The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach," but I must admit that I don't yet understand the math behind it. Therefore, I apologize in advance for the inevitable mistakes that follow. The layman's version goes something like this:
Risk aversion is where people prefer a certain value, even if it is lower than an expected value, to a higher expected, but uncertain value. Ambiguity aversion is the preference for known risks to unknown risks. According to Segal, ambiguity aversion and risk aversion are essentially the same thing within the realm of Anticipated Utility Theory*.
From my understanding of the paper, the theory is not dissimilar to my poorly stated hypothesis for the Allais Paradox and the other hypothesis I gave for the Ellsberg Paradox. Simply put, people tend to regard chance less in terms of probability and more in terms of certain and uncertain. I realize this is something of a cop out, and sometime in the future after I have had understood the math behind Segal's paper, I intend to write another post outlining rank-dependent expected utility. Until then, I hope this post has done an adequate job of outlining some of the sketchy parts of Rational Choice Theory.
*Anticipated Utility Theory, now known as rank-dependent expected utility, provides an explanation for why people engage in the seemingly contradictory behavior shown above in the Allais Paradox and in the Ellsberg Paradox. Its addition to Prospect Theory resulted in Tsverky's and Kahneman's 1992 paper on Cumulative Prospect Theory. The development of the theory, an important advancement in Behavioral Economics and a strong alternative to Rational Choice Theory, resulted in Kahneman winning the 2002 Nobel Prize. Tsverky would likely have also won had he not died in 1996. If you ever get the chance, watch his nobel lecture.
The obvious objection is that people don't bother to calculate all the costs and benefits of every situation. This is explained away by the "pool table idea;" in the same way that you don't need to be a physicist to play pool yet the balls will still follow the laws of physics, you don't need to be an economist to make rational choices.
The other typical objection is that some people seem to make decisions that, no matter how you cut it, goes against their best wishes. The explanation for this is that since we can't possibly know a person's individual preferences, we must assume that they are rationally pursuing them. By this point, Rational Choice Theory becomes a tautology - after all, if people are all rationally pursuing their own preferences, preferences which can't be known, it's impossible to empirically test whether or not they are actually doing so, and Rational Choice Theory is therefore true because it's defined to be.
That said, there are a few paradoxes which point us towards weaknesses in Rational Choice Theory.
St. Petersburg Paradox
Imagine you are offered the following bet: A coin will be flipped. If it is heads, you get nothing. If it is tails, you get $2 and the opportunity to flip again. If on the second flip, you get a heads, you keep the $2. If you get tails, you get $4. Next flip, $8, then $16, then $32, and so on.
How much pay to take part in this gamble?
According to Rational Choice Theory, you would pay anything less than the expected value of the gamble. After all, everyone desires more than less and if you keep playing the game over and over again, your average winnings will be the expected value.
To calculate the expected value, all you need to do is multiply the percentage chance of winning a certain amount by that amount and then sum all possibilities. The result will be your average winnings.
So, we know we have a a 50% chance of getting heads on the first flip. So that's 50% times nothing. Easy so far. 50% chance of getting tails. But wait. 50% of the times you get tails, you'll go on to win again, so, we can only add 50% of that initial 50% times the $2 winnings. Easy enough, .5 x .5 x $2 = $0.50. But then there's the next set:
.5 x .5 x .5 x $4 = .125 x $4 = $0.50
and the next:
.5 x .5 x .5 x .5 x $8 = .0625 x $8 = $0.50
and so on:
5 x .5 x .5 x .5 x .5 x $16 = .03125 x $16 = $0.50
5 x .5 x .5 x .5 x .5 x .5 x $32 = .015625 x $32 = $0.50
5 x .5 x .5 x .5 x .5 x .5 x .5 x $64 = .00778125 x $64 = $0.50
and so on ad infinitum. The series diverges. The expected value equals infinity x $0.50.
So, the rational person would bet literally any amount of money to take part in this gamble.
But people don't bet any amount of money on this. They bet quite a bit less.
There are several other explanations for this, among them utility curves, risk aversion, people's awareness that there is a limited amount of money in the world, etc.
Perhaps the most convincing theory I've read that explains this paradox is this paper by Benjamin Hayden on the "median heuristic." The paper sets forth the idea that people don't always rely on Rational Choice Theory (or perhaps never rely on it) and instead suggests that people make their decision in this particular paradox by picking a bet close to what the median return from the bet is. This is very different from betting on the expected value. The expected value is roughly the mean return from the bet and is significantly larger than the median due to the skew created by the occasional absurdly long streak of tails. The other noticeable thing about the median is that it is far easier to accurately estimate. Finding the mean requires to sum all results and then divide by the total count. Finding the median just requires taking a stab at the middle number, and with larger samples, if you're off, it's not by much. Considering that the mind often seems to prize efficiency over accuracy and precision, this heuristic seems especially likely.
This new median heuristic predicts that people will make a bet of about $1.70 some odd, which they found widely predicts what people actually do.
Allais Paradox
Now, having lost all of your money making bad bets on the St. Petersburg Paradox, you are offered a new game.
There are two urns, A and B.
A contains 99 black marbles and 1 red marble.
B contains 90 black marbles, 5 white marbles, and 5 red marbles.
If you pull a black marble, you get $1 million. If you pull a white marble, you get $5 million. If you pull a red marble, you get nothing.
You can only play once, so going for the expected value doesn't make sense in this case. It's solely a matter of personal preference. Which do you choose?
Now, onto a completely new game.
There are two new urns, C and D.
C contains 9 black marbles and 91 red marbles.
D contains 5 white marbles and 95 red marbles.
Again, if you pull a black marble, you get $1 million. If you pull a white marble, you get $5 million. If you pull a red marble, you get nothing.
Which do you choose?
One of my economics professors posed this question for a class I was in. If I remember correctly, A and D was the most popular choice, followed by B and D and B and C. A and C was the least popular.
But Rational Choice Theory predicts that everyone will choose A and C or B and D, and that no one will choose A and D or B and C. Let me show why.
90% of A and B are identical. In both cases, 90% of the time you get $1 million. The only real question is: for the remaining 10%, do you want a 50% chance of $5 million or a 90% chance of $1 million?
Likewise, 90% of C and D are identical. In both cases, 90% of the time you get nothing. The only real question is: for the remaining 10%, do you want a 50% chance of $5 million or a 90% chance of $1 million?
If you prefer a 50% chance of $5 million, you should always prefer a 50% chance of $5 million. If you prefer a 90% chance of $1 million, you should always prefer a 90% chance of $1 million. The fact that these preferences don't hold shows that Rational Choice Theory is flawed some way.
I haven't read an explanation for this paradox that I have found satisfactory. One idea that I think might explain it somewhat is that people think less in terms of probabilities and more in terms of "is or is not." Under that hypothesis, people would choose A over B, because A is a sure a thing while B is less sure. Meanwhile, people would choose D over C, because while both are almost certain not to happen, D gives a larger benefit if it actually does happen. For those of you who pointed out that this is really similar to rank-dependent expected utility, please hold your fire until after the end of the Ellsberg Paradox, where I readdress this issue.
I would like to emphasize that the above stated hypothesis is completely untested.
Ellsberg Paradox
Interesting side note: the Ellsberg Paradox is named after Daniel Ellsberg, who wrote about it in his Economics Phd dissertation. Ellsberg is far more famously known, however, for being the military analyst who released the Pentagon Papers in 1971.
For this example, imagine an urn. The urn contains 90 marbles, 30 marbles are black and the other 60 are yellow and red, but the ratio between the two is unknown. You can choose one of two wagers. Either a million dollars if a black marble is pulled or a million dollars if a yellow marble is pulled.
Now imagine the same urn. 90 marbles, 30 black, other 60 red and yellow. You still don't know the proportion, but it's the same as the time before. You can choose of two wagers. Either a million dollars if a black or red is pulled or a million dollars if yellow or red is pulled.
Which did you choose?
As you can probably imagine by now, what people usually choose does not reflect what rational choice theory predicts. Rational Choice Theory* says that if you pick black in the first example, then you must think there are fewer than 30 yellow marbles, so that in the next example you would pick black or red. Likewise, if you picked yellow, you must think there are more than 30 yellow marbles, and so you'd pick yellow or red.
*Technically, it's Expected Utility Theory. That said, they operate under the same basic considerations, and for the purposes of this blog post, will be assumed to be the same. Real economists, please don't chew my head off for this.
What you would not pick is black for the first choice and yellow and red for the second choice, since that assumes that there are more red than yellow in the first instance but more yellow than red in the second.
Here's the explanation I have: People prefer the sure choice against the unknown. So, most people will choose black in the first set because they know that the chances of getting one million dollars is exactly 1 in 3, while if they choose yellow it could be anywhere between zero and 2 in 3.
Likewise, if people choose yellow and red in the second set, then they have a 2 in 3 chance, while if they choose black and red it could be anywhere between 1 in 3 and certain.
The trouble is then, why do people sometimes not choose black for the first instance and yellow and red for the second instance? I'm stumped. I guess there is some compromise between the risk aversion and the desire for optimal output (what Rational Choice Theory) predicts, but I can't think of anything that would accurately predict what people do.
Note: the following is information poorly understood by the author. It's validity and accuracy is questionable.
Now, there was a paper written in 1986 that attempts to explain the Ellsberg Paradox and, less directly, the Allais Paradox by Uzi Segal, then an Assistant Professor at the University of Toronto, now a Professor at Boston College.
I have read the paper, entitled "The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach," but I must admit that I don't yet understand the math behind it. Therefore, I apologize in advance for the inevitable mistakes that follow. The layman's version goes something like this:
Risk aversion is where people prefer a certain value, even if it is lower than an expected value, to a higher expected, but uncertain value. Ambiguity aversion is the preference for known risks to unknown risks. According to Segal, ambiguity aversion and risk aversion are essentially the same thing within the realm of Anticipated Utility Theory*.
From my understanding of the paper, the theory is not dissimilar to my poorly stated hypothesis for the Allais Paradox and the other hypothesis I gave for the Ellsberg Paradox. Simply put, people tend to regard chance less in terms of probability and more in terms of certain and uncertain. I realize this is something of a cop out, and sometime in the future after I have had understood the math behind Segal's paper, I intend to write another post outlining rank-dependent expected utility. Until then, I hope this post has done an adequate job of outlining some of the sketchy parts of Rational Choice Theory.
*Anticipated Utility Theory, now known as rank-dependent expected utility, provides an explanation for why people engage in the seemingly contradictory behavior shown above in the Allais Paradox and in the Ellsberg Paradox. Its addition to Prospect Theory resulted in Tsverky's and Kahneman's 1992 paper on Cumulative Prospect Theory. The development of the theory, an important advancement in Behavioral Economics and a strong alternative to Rational Choice Theory, resulted in Kahneman winning the 2002 Nobel Prize. Tsverky would likely have also won had he not died in 1996. If you ever get the chance, watch his nobel lecture.
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