Another paradox

Today, I would like to describe the St. Petersburg paradox. It is important because as it has been studied by economists and psychologists who often use it as an example of showing how people unconsciously calculate utility. Most important however, is the fact that humans' "way" of doing it is very much different from what should be expected from the classical expected value theory.

Consider the following game by Daniel Bernoulli:

If I toss a coin and it lands on heads, you win $1.00. If it doesn't, I toss a coin again. If it lands on heads after the second toss, you win $2.00. If I have to toss a coin three times to finally get heads, you will get $4.00 and so on. The game is run once, that is it ends when the coin lands on heads and you win money.

As you can see, you can't loose any money in this game.

Now the question to you is:

How much would you be willing to pay me in order to take part in this game?

The answer, as calculated by the expected value theory, should be: "anything". That is, you should be willing to pay any amount of money as possible benefit of this game is infinite!

It is not hard to see that no one really thinks like this. In fact, we are very well aware that the prospect of larger benefit diminishes in value as the probability of winning decreases. In other words, we know, for example, that winning $32.00 (heads on the 6th toss) is very unlikely.

This observation is extremely interesting for psychologists as it explains why people differ in their preference for the same gamble. The utility- wealth function derived from St Petersburg paradox shows that utility is strongly affected by one's current wealth in a non-linear manner.

Conclusively, the more wealth one owns, the less attractive the gamble will appear to be and vice versa.

Hardman, D. (2009). Judgment and Decision Making: Psychological Perspective.

Oxford: Blackwell Publishing

Problem Solving

Hey guys! Just a quick post.

This is a link for a very interesting little game which I was learning about during my first class in "Thinking and Individual Differences" unit.

http://www.learn4good.com/games/puzzle/boat.htm

How many moves do you need to complete it?

Allais Paradox

The Allais paradox is an interesting phenomenon which proves that people fail to make decisions in a way that would maximize their ulitity/pay off/benefit.
Consider the following two problems:

Problem 1
Which of the following situations would you prefer:

Situation A

$100.00 for certain

Situation B

A 10% chance of winning $500.00
An 89% chance of winning $100.00
A 1% chance of winning nothing


Problem 2
Which of the following situations would you prefer:

Situation C

An 11% chance of winning $100.00
An 89% chance of winning nothing

Situation D

A 10% chance of winning $500.00
A 90% chance of winning nothing



Many studies presented participants with this problem and obtained identical results. Majority of people prefer A to B in Problem 1, and D to C in Problem 2.
For us to decide whether these choices are appropriate, we are required to calculate the utility of each situation. This is very simple to compute, we just need to multiply a pay off and its probability for each situation.

Problem 1
Which of the following situations would you prefer:

Situation A

Utility= $100.00 * 100% = $100.00

Situation B

Utility= $500.00 * 10% = $50.00
Utility (2)= $100.00 * 89% = $89.00
Utility (3)= 0$ * 1% = $0

Total Utility= $50.00 + $89.00 + $0 = $139.00

Since $100.00 < $139.00, Situation A is worse than B.

Problem 2
Which of the following situations would you prefer:

Situation C

Utility= $100.00 * 11% = $11.00
Utility (2)= $0 * 89% = $0

Situation D

Utility= $500,00 * 10% = $50.00
Utility (2)= $0 * 90% = $0

Since $11,00 < $50.00, Situation C is worse than D

Although people make a right choice in Problem 2, they fail in Problem 1...

Now comes the interesting part. Two utilities to chose from in Problem 1 are: $100.00 and $139.00. Therefore, the latter is better by $39.00 ($139,00-$100.00).
Now consider the second problem, where two of the utilities are: $11.00 and $50.00. The latter is better by $39.00 as well!!!! ($50.00-$11.00)

What happens here is that two problems are in fact identical in terms of utlitity. In both, the second situation is better by $39.00. People however, fail to see it when they make their choice and wrongly chose the worse option in problem 1.

This is known as Allais Paradox and is very interesting for psychologists who study effects of certainty and uncertainty on people's choice behaviour.

Hardman, D. (2009). Judgment and Decision Making: Psychological Perspective.

Oxford: Blackwell

Cancer

In my previous post, I talked about the situation in which facts about risk are wrongly communicated to the public. In the following post, I would like to elaborate upon some of the possible issues that can result from biased media messages. With the HIV story, we can imagine that not much harm has been done when the risk was inadequately presented. Since the vaccination is not yet available, risky behaviour won't be affected by the perception of its effect. The worst that can happen is that people will have high hopes for the cure to be developed soon. Of course, it doesn't have to be such a bad thing as it can motivate people to donate and approve HIV research even more when they see some progress.

Unfortunately, not always "risk miscommunication" is so harmless. I would like to use another example, the news about the death of Natalie Morton. As you may remember, the very first coverage about this case was presented with headlines like:

-"Cancer jab alert after girl dies" BBC

-"Cancer jabs suspended after death" BBC

-"Schoolgirl, 14, dies after being given cervical cancer jab" Daily Mail

-"Schoolgirl Natalie Morton, 14, Dies After Having Cervical Cancer Jab" Sky News

At the time when these were published, I was truly shocked with the complete lack of responsibility shown by the media. It was only in the first paragraph of each of the articles where the information about the complete lack of evidence for the link between the death and jabs was presented. In other words, the information from headlines was in-congruent with the content of the actual article in all cases.

Following the introduction of the cervical cancer vaccinations, occurrences of the disease dropped by 70%. This means that from around 3 000 cases in the whole of UK before the vaccine was available, only 682 were registered after the vaccine was introduced. Bear in mind that we are talking about the relative risk reduction here. Since about 1 out of 3 women die from the cancer when she gets it, we can expect that 227 (682/3) will die instead of a 1000, thanks to the vaccinations in UK.

Now, following the publications of the media many parents decided to withdraw their consent to have their children vaccinated. The probability of dying from a jab is .000001% (1 in a million or maybe even less), while the chance of dying from the cancer if not vaccinated is .00003% (1000 out of around 30 million women in UK), that is about 33 women per 1 million... Since the vaccination lowers deaths by 60-70%, for each life lost due to the adverse reaction to the vaccination, 25 lives are saved from the cancer's death toll.

Although it is obvious that everyone should still be vaccinated, harm was already done as some parents withdrew their consent.

To sum up in harsh but true words: The information presented in news headlines was untrue and imagined. It created an "out of proportion" belief that a cancer jab can lead to death which in turn resulted in less people getting vaccinated. At the end of the day we can expect slightly more deaths from the cancer thanks to the media's miscommunication of risk...

Personally, I would be much more concerned with the risk of dying in a car crash on my way to get a vaccination...

Gigerenzer, G. (2002). Reckoning with Risk.

London: Penguin Group

HIV vaccination

For my first entry I would like to comment on a very recent news story which drew my attention and made me think about risk and psychology. The story I want to talk about relates to the way in which information about risk is communicated to the wide public.

You may remember that in the last week a large amount of coverage has been devoted to HIV vaccination research due to a report of one, very successful, experiment conducted in Thailand. Accordingly to the majority of newspapers and TV news channels, a prominent breakthrough has been achieved following a very lengthy and costly struggle to manufacture a vaccination against the deadly virus. The significance of this event is even stronger if we consider the record of the last forty years of fight against HIV. Since 1981, 25 million people have died from AIDS.

Following a very positive feeling associated with this success of science and the prospect of possible benefits, I began to analyze the information presented to me with scrutiny. What really triggered my suspicion was the content of the dominant number of headlines which stated that (more or less) “chances of catching the virus were 30% less for those who had taken a vaccine”. Consequently I began to wonder, Am I 30% less likely to catch HIV if vaccinated? Or maybe out of 1000 people we can expect 300 less to become infected? As a condition to understand the actual effect of a vaccination we need to comprehend that 30% is a “relative risk reduction”. What it means is that the reduction of incidences relates to the difference between people who got HIV and were not vaccinated and people who also got HIV but were vaccinated. Important here is that no information about the prevalence of HIV within a population is given as the relative risk reduction has nothing to do with it.

Let’s use the following example: if 1% of the population gets HIV during their lifetime (real probability is much lower but would make computations unnecessarily complex) we can expect 10 people out of 1000 on average to be infected. Now imagine that the new vaccination has been administered to another 1000 people and a total number of 7 people were infected. The difference of 3 people between the two conditions constitutes for the risk reduction of 30%! I hope that you agree that the effect of the vaccination suddenly doesn’t seem so great. If we add to it that the difference is actually not statistically significant and that only one experiment has been conducted with little control, then this situation becomes quite grim.

Let me present here, the effect of this vaccination in a more informative way, using “absolute risk reduction”. We said that the prevalence in the population is 1% and that it is represented by 10 cases out of 1000 random people. Now, 7 people out of 1000 is equal to .7%. Therefore the prevalence has been reduced by .3% (1% minus .7%). In other words, .3% less people will be infected with HIV if vaccinated.

It is quiet obvious that by following these very simple computations we now have a much more accurate and realistic view of a scale of change that the vaccination produces.

My question is: Why would news stations provide us with a much more confusing and misleading relative risk reduction? The headlines could read .3% risk reduction as well. I am afraid that the only reasonable answer is a very upsetting one indeed. The bigger number, effect or magnitude of an event produces stronger emotional responses, consequently drawing more attention to the news. This is an example of putting two simple psychological findings into practice. Both statistical innumeracy and human’s tendency to respond with emotions rather than reason are used to increase the attractiveness of an event.

At the end I need to state that BBC news presented the explanation of the risk reduction on an example from the study itself. It doesn’t explain however, why the headlines were still using relative risk reduction. It should be obvious that majority of people will be biased by the information that is presented to them in the first place. Much less people will be willing to bother to try and understand the meaning of the presented percentages…