We perhaps have all heard about the following coin toss experiments on people. On each experiment, people have to choose either to play Game A or Game B.

Experiment 1): Game A: the player wins $1000 if head; wins $50 if tail. Game B: the player wins $500 regardless.

Experiment 2): Game A: the player loses $1000 if head; loses $50 if tail. Game B: the player loses $500 regardless.

With Experiment 1), people like to choose Game B.

With Experiment 2), people like to choose Game A.

While I understand this from psychological terms (humans have biases, are not rational etc.), I don't quite understand naturally or fundamentally about what really makes us do this. Hope someone can help explain.

An interesting thing I want to share here is the TED talk (July 2010) below shows that monkeys do in the same way.

Phil McDonnell writes: 

These sorts of thought experiments are played on student volunteers at college campuses. Students are notoriously impecunious. Assume a net worth of $300 or so and then calculate the log of the wealth ratio for each outcome. You will find that the most popular outcomes all follow the log utility function.

The mistake the experimenters make is to assume that $1000 is worth twice as much as $500 to a poor student. In fact it is worthless than twice as much. They assume a linear utility function when a non-linear log function is how humans really value things. 

Gary Rogan writes:

To understand why any of these biases work the way they do it's best to imagine hominids barely making it in terms of survival rather than even poor college students. There were long stretches where there were more than enough food or other resources, but there were also "funnels" where there were barely enough resources to make it through. We are all descended from the ones who made it through the "funnels". The ones who made the wrong bet left no survivors (or fewer as the case may be).

So if you've got just enough or almost enough to sustain yourself for the next few days but not after that, you realize that you need to do something soon or else you'll be history after those few days. You need to take a risk, but the weather is really bad, it's cold and stormy (or exceptionally hot, or whatever) and you can't venture out. To make matters worse though, a mildly menacing counterpart shows up and "asks" to share your meager resources. Losing half of what you have with great certainty may very well mean likely death, because you will probably be not in a position to "play" again after the weather gets better. If you calculate a 50/50% chance of winning the fight with your "friend" you will probably go for it, even if it means either a complete loss of your resources (and perhaps your life, but those are now equivalent anyway) or a mild injury.

And now a few days are gone, and your are out of resources anyway. The weather is a little better and you are now faced with walking towards a distant meadow where you are certain to find enough berries to sustain yourself for a few more days or pursuing a large but elusive prey (or trying to take something from a leopard resting in a tree with a fresh catch, or trying to fight it out with a different "friend" for his resources, but all of this uncertain), you go for the berries. You take a certain smaller gain vs. an uncertain larger one when getting enough resources means the difference between life and death.

As for how it comes about, we have neural networks in our brain specifically dedicated to evaluating rewards and costs. That's as proven of a fact as anything in neurobiology. Whatever biases worked based to get our distant (and not so distant) ancestors through the "funnels", that's pretty much what we have today.

Phll McDonnell adds:

I think one point that is overlooked in this discussion is that insurance companies do not prefer big bets. Instead they prefer to spread the risk and average out on many small bets. It is also the same reason that earthquake and hurricane insurance is so overpriced. They simply do not want highly correlated bets which increases the risk of serous capital impairment. If they write a lot of earthquake insurance in an area and the big one hits all policies come due at the same time.So they either ration the insurance by charging too much or they simply refuse to sell more than a certain number of policies in a given area.

In a way managing the insurance portfolio is a lot like managing a stock portfolio. You want to avoid bets with large possible negative outcomes and you want to avoid correlated bets. Rather you should take many smaller uncorrelated positions so no one position can wipe you out.

Leo Systrader comments: 

 Great points here. Many thanks for all the thoughts.

Question to Philip (maybe to everyone) about the non-linearity of human value. It is understandable, but I wonder if there is any scientific conclusion about it.

Let's see if we can use this theory to re-construct the experiment. Let's also assume the net worth of the players is $500. For simplicity, let's just try Experiment 1).

Experiment 1): Game A: the player wins $X if head; wins $Y if tail. Game B: the player wins $500 regardless.

We hope we can come up with an X value that is attractive enough for the players to choose Game A. Also we need to make sure that Game A does not have a significant favor of probability, so we choose Y in such a way that the expected value of Game A is not much more than that of Game B, which is $500. Apparently X will need to be much larger than 1000, so that means Y will have to be a negative number to balance out.

As the theory suggests, to make log(X) = 2 * log(500), X is about 25,000. Let's make the expected value be 600, we get Y to be -23800.

So then Game A becomes: the player wins $25000 if head; loses $23800 if tail.

Would that make people choose Game A? Not to me. Anything wrong in the above analysis?

On the other hand, we know that Kahneman has a theory saying something like "a person's magnitude of pain from losing amount D equals that of his joy from gaining amount 2.5*D".

So let's use this theory to reconstruct Game A.

Let's first decide for Y to be 300. Comparing with the 500 in Game B, the player considers a loss of 200 in Game A. So we need to make X a gain of 2.5 times 200 from 500, so we get X = 2.5*200 + 500 = 1000.

So now we have a new Game A: the player wins $1000 if head; wins $300 if tail.

I guess now it is very likely people would choose Game A over Game B. But we note that the expected value of Game A now is $650

If we make Y to be 50. It is a loss of 450 from Game B. So we make X =
2.5*450 + 500 = 1650.

Game A becomes: the player wins $1650 if head; wins $50 if tail.

Would they play it? Probably, right? In this case, the expected value is 837.5. 

Phil McDonnell responds:

Assume a starting net worth of 500.

Game A analysis: ln( (500+X) / 500 ) + ln( (500-Y) / 500 ) = expected utility.

Game B analysis: ln( (500 + 500 ) / 500 is the expected utility for both outcomes.

The thing is that we need to think in terms of wealth ration of the different outcomes. Take the natural logs of the wealth ratios. The wealth ratio is the wealth you start with before the bet divided by the wealth you end up with for the given outcome.

I took all the early propositions in the Kahneman and Tversky paper and calculated the logs of the expectations and found that in every case the participants were using log based utility function and were actually choosing quite rationally and correctly. It was the learned professors who were wrongly trying to analyze the problems using binomial probability analysis instead of utility theory.

In other ares of psychology there are various log based perceptions that have bee discovered. For example there is the concept of a just noticeable difference (jnd). For example you do not notice a sound is louder or softer until it changes by a certain amount governed by a log law. Same thing goes for brightness of light. One might add boiling live lobsters to that list.


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