February 2023

Volunteer’s dilemma

Imagine a situation where N offenders face punishments of ten years for a crime. If one of them confesses, that person gets one year term, and the rest are released. So everybody has an incentive for confession (a reduction of nine years imprisonment) or nothing (going free if at least one of them – the volunteer – admits).

In game theory language, the payoff matrix is as follows for the two players.

Volunteer’s
Decision
Do
volunteer		Don’t
volunteer
Other’s
decision		Do
volunteer		-1,-1-1,1
Don’t
volunteer		-1,1-10,-10

There are a lot of examples in real life where the volunteer dilemma operates, albeit, to the detriment of the community. A famous one is the bystander effect – a group of people witness a crime in which someone is stabbed. Everyone hesitates to call for help, fearing some cost (questioning by police or the possibility of getting into the criminal’s watchlist). At the same time, they all sincerely hope someone volunteers and help the person getting the treatment.

It is not difficult to understand that the chance that no one calls increases as the cost of volunteering increases. But what is surprising is the lowered probability of getting help as the number of bystanders increases.

References

Andreas Diekmann, Volunteer’s Dilemma, The Journal of Conflict Resolution, 1985, 29(4), 605
William Spaniel, The Murder of Kitty Genovese (Volunteer’s Dilemma): Youtube

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CEO Puzzle

Alex, Becky and Carol are CEOs of three companies. They each have 41% voting rights in their companies. The remaining 59% is divided equally among the other board members. To implement a policy, the board requires to get a simple majority. If Alex’s board has five other members, Bcky’s six members and Carol’s seven, who among them is the most powerful CEO?

Let’s start with Alex: Each of his other members holds 59/5 = 11.8% control. So, if Alex manages to move one out of 5 towards him, he can change the policy (41 + 11.8 = 52.8 > 50). Becky also needs only one other member. So she has one in six chance to influence (41 + 9.83 = 50.83 > 50). So Becky has an easier job than Alex. Finally, Carol: one member (59/7 = 8.43) is not enough to make the majority. So, she needs two out of seven support which is harder than one in six. So Becky is more powerful.

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Bayes’ Theorem in a Flu Season

At the peak of flu season, one in a hundred gets the flu. But half of the infected show no symptoms. People with allergies or colds can also show Flu symptoms; one in twenty people who don’t have flu can show flu-like symptoms. So the question is: if a person shows signs of the flu, what is the probability that she has the flu?

We will use this example to illustrate problem-solving through Bayes’ rule. So what is the ask here? You have to tell the chance that a person has flu, given she is showing symptoms, S. Or in shorthand, P(F|S). But what do all we know about flu?

  1. If a random person is picked from the street, there is a one in a hundred chance that he has flu. In other words, P(F) = 1/100 = 0.01. It also means that 99 out of 100 random people in the street have no flu, P(nF) = 0.99.
  2. Only half of the people who have flu show any symptoms. The probability of expressing signs given the person has flu = 0.5. In shorthand, it is P(S|F).
  3. 1 in 20 people who don’t have flu can show flu-like symptoms or P(S|nF) = 0.2

Use all the above information and plug it into Bayes’ equation.

P(F|S) = P(S|F) x P(F) /[P(S|F) x P(F) + P(S|nF) x P(nF)] = 0.5 x 0.01 /[0.5 x 0.01 + 0.2 x 0.99] = 0.0246 = 2.5%

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Flow Model of Happiness

What makes us happy is a question that people have been raising over the years. Psychologist Mihaly Csikszentmihalyi has an answer. He describes the state of feeling happy by the term, flow.

He examined hundreds of people about what made them happy and found out the conditions for flow.

  1. Intense focus on a task. A person focuses on one activity that she forgets about everything else.
  2. Freedom from all self-scrutiny. The job is not imposed upon, but it’s own choice. No fear, no doubt.
  3. Get immediate feedback. An actor gets it when her opposite person reacts; a climber knows how far she must climb before the summit.
  4. Optimally challenging: neither over-challenging nor under-challenging.

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Cognitive Dissonance Theory

Cognitive dissonance – a term attributed to psychologist Leon Festinger – arises when there is an inconsistency between someone’s actions and what she thinks she must do. An often-quoted example is smoking: if the person smokes and believes that smoking is unhealthy, there exists a dissonance.

This inconsistency is a serious issue and can affect a person mentally and physically. And that calls for a solution. But how do people manage it?

Modify the thought

“Well, I do smoke; but smoking relaxes me, so it is not that bad”. Such a change of view could restore consistency.

Modify the behaviour

Another way of dealing with cognitive dissonance is to repair the inconsistency by quitting the habit.

Rationalise

The person can argue about, say, other healthy behaviour that she follows to justify a bit of smoking. “I do exercises, annual medical check-ups, not all smokers get cancer”, etc.

Ignore

The last attitude is to trivialise it and simply declare: “I don’t care!”.

Cognitive dissonance: Wiki
Cognitive Dissonance Theory: A Crash Course: Youtube

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Spoilt for Choices

Can you imagine a world without choices? The link between intrinsic happiness and the availability of choice is almost a given. But there is also the feeling of confusion and indecisiveness caused by having too many options. The well-known study by Sheena Iyengar addresses this topic in her research and uncovers some interesting facts about life.

In one of the studies, the research team set up two booths outside a grocery shop, one at a time. A collection of six flavours of jams was on display in the first instance, and there were twenty-four in the second. The researchers then recorded two aspects – the initial motivation to taste and subsequent purchase – that the customer showed.

jams, marmalades, farmers market-997593.jpg

The first observation confirmed the existing belief. The number of options displayed affected the customer’s attention. The 24-flavour booth attracted 60% (145) of the total (242) customers who passed by. Whereas to the limited booth, it was 40% (104 out of 260). Then something strange happened: only 4 of the 145 (3%) purchased a product from the 24-booth, and 31 of the 104 (30%) purchased from the limited booth.

The paradox of choice

Greater availability to choose from indeed makes us happier. At the same time, the greater the choice, the higher our expectations, and we struggle to make decisions.

Iyengar and Lepper, “When Choice is Demotivating: Can One Desire Too Much of a Good Thing?, Journal of Personality and Social Psychology”, 2000, 79(6), 995-1006

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The Planning Fallacy

The term planning fallacy is closely related to something we have seen earlier – the Dunning–Kruger effect. If the latter is all about overestimating own abilities, the former is about underestimating the amount of time it takes to complete tasks. The story of the Sydney Opera House is a classic example. Located at the foreshores of Sydney Harbour, the iconic building is a venue for performing arts.

sydney, opera, house-4078776.jpg

As per the initial planning in 1957, the cost was about 7 mln dollars and was expected to finish by 1963. Eventually, the building was completed 10 years later than the plan (1973), after spending about 102 mln dollars!

History is full of stories of such mega projects running late and over budget. Yet planners keep missing deadlines and have optimistic estimates of costs. Theoretical analyses propose people who focus on singular information (personal experience) and concentrate on how to complete their task are bound to make errors compared to those who study distributional information (others’ experience).

Three features are typical among people who commit the planning fallacy: 1) too much focus on forward prediction, 2) discounting of issues with personal performances and 3) failure to incorporate relevant knowledge or ‘base rates’ from other projects. Another two personal characteristics associated with poor planners are conditionality and anchoring. In the former case, the assessor focuses on normal operating conditions and ignores the possibility of adverse situations, e.g. war, depression or societal interventions. Anchoring happens when the person sticks to the best guess ignoring the extreme probabilities.

References

Sydney Opera House: Wiki
Buehler, R.; Griffin, D.; Ross, M., Exploring the “Planning Fallacy”: Why People Underestimate Their Task Completion Times, Journal of Personality and Social Psychology, 1994, 67(3), 366-381.
Kahneman, D.; Tversky, A., Intuitive Prediction: Biases and Corrective Procedures, TIMS Studies in Management Science, 1979, 12, 313-327.

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Probability of Green Balls – Final Episode

Five types of ball-picking problems are introduced here. Welcome to another episode of arrangements (and more confusion). A bag contains 4 green balls and 4 red balls. You like to pick four balls at random, in which:

One green ball

Probability to choose one green AND 3 reds = [4C1 x 4C3] / [8C4]. Remember, the multiplication in the numerator happened because of AND.

At least one green ball

At least one green = P([1 green AND rest red] OR [2 green AND rest red] OR [3 green AND rest red] OR [4 green]). Applying multiplication for AND and summation for OR,

Probability to choose at least one green = [4C1 x 4C3 + 4C2 x 4C2 + 4C3 x 4C1 + 4C4] / [8C4]

At most one green ball

At most one green = P([0 green AND rest red] OR [1 green AND rest red].
Probability to choose at most one green = [4C4 + 4C1 x 4C3] / [8C4]

No green ball

This one is easy; just pick red balls, 4C4/8C4

One green or two greens

What is the probability to pick one green AND three reds or two greens AND two reds?

[4C1 x 4C3 + 4C2 x 4C2] / [8C4]

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Chance of dissimilar Cards

Suppose there is a well-shuffled deck of cards, and you randomly select one card. Then what is the chance that you pick a second card that is not of the same suit or number? For example, if the first card is an ace of spades, the second can’t be an ace or a spade.

cards, ace, spades-1762500.jpg

Such problems are better solved using the complementary rule, i.e., estimate the probability of choosing the same number or same suit and then subtract it from 1. Pick the first card. The probability of picking the second card carrying the same number = remaining number of cards displaying the same number / total remaining cards = 3/51. Similarly, the chance of picking a card of the same suit = the remaining number of the same suit / total remaining cards = 12/51.

The probability of taking out either the same number OR the same suit is the sum of the probabilities (the OR rule), i.e., 15/51. The required probability is 1- 15/61 = 36/51 = 0.706

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Distributing Chocolates

Now that we know how to arrange balls into bins: let’s solve the original question: the number of ways a person can distribute 36 chocolates among five kids, and no one gets fewer than 5.

First, distribute five chocolates each to five kids so that everybody gets the minimum. That leaves 11, which may be distributed among five. The problem is identical to 11 indistinguishable balls into five separate bins or rearranging 11 similar balls + four similar partitions. That is given by:

15C4 = 1365

Note that it is identical to 15C11

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