Life

Principle of indifference

The principle of indifference is a rule that helps to assign prior probabilities in Baysian-type estimations. It says if there are several alternative possibilities for an event, and there is no particular reason to choose one, the prior – the degree of belief – should be equal among all probabilities. Well, this degree of belief is known as credence.

In the case of coin flipping, the probability that a coin (we don’t know if it is fair or not) lands on the head takes the value, one out of two possibilities, 1/2. Another example is American Roulette and the probability for the ball to land on green (0 or 00). Again, we assign those two prospects equally among 38 pockets, i.e., 2/38 or 1/19.

But if the possibilities partition in different ways, the principle of indifference land in strange situations. See the ‘light switch and ball problem’. There are three balls in an urn – red, blue and green. If I pick a ball at random and it’s red, the light is turned on. If it’s blue or green, the light is off. What is the probability the light is ON?

Well, one can say 1/3 – one in three chances that the ball is red.
One can also say 1/2 because there are two possibilities – the light is ON; the light is OFF!

References

Principle of indifference: Wiki
Principle of Indifference / Insufficient Reason: Statistics How To
The Principle of Indifference: jonathanweisberg.org

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Lions and Wildebeests

Here is a puzzle I picked up from the internet: three lions and three wildebeests want to go from one side to the other side of a river. The river has a raft on which, at a time, up to two animals can travel. There is one problem, though. If the number of lions exceeds the number of wildebeest in one place, the lions would attack and kill the latter. So, how do they all reach the other side without issues?

Decision-making process

Let’s do a systematic decision tree that includes all (allowed) possibilities. What are the starting possibilities (trip 1)? They are one lion taking the raft (L), one wildebeest taking the raft (B), two lions (LL), two wildebeests (BB) or one lion and a wildebeest (LB). We will ignore L and B from the list because we know they must return (trip 2), thus nullifying trip 1. Therefore, the valid moves are:

An explanation for the shorthand: There are three options for trip 1 – LL, BB and LB. The notation in the parenthesis is the expected outcome in the end, e.g. if LL happens. And what remains on the start shore is LBB, and the endpoint (after the “/”) is LL. “X” means the move is unacceptable (three lions and one wildebeest on one side). The possible next actions (trip 2) are:

I guess the notations and the outcomes are clear by now. One of the trips (L) is not possible; it leads to 3 lions on one side. There are two possible remaining moves, and interestingly, both will lead to the same outcome (LLBBB/L). So, the next trip (trip 3) starts from a common origin (LLBBB/L).

Trip 4:

Trip 5:

The following six trips are:

Can you solve the river crossing riddle? – Lisa Winer: TED-Ed

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Lions and Lamb

Hungry lions and lamb is a classic game theory problem. It goes like this: there is a group of hungry lions on an island covered with plenty of grass and no other animals in sight. The lions are perfectly rational, which means survival is their priority, and each knows the others are rational too. The former suggests they prefer hunger over death, and the latter implies they don’t try to outsmart each other.

One day, a lamb appears from nowhere. The lamb is just big enough to become a meal for one lion. And the lion that eats the lamb becomes too full to defend itself against another hungry lion. Given these situations, what is the survival probability for the lamb?

Let’s arrange the lamb and the lions as follows. Consider the simplest case of one lamb and one lion.

Since there are no other lions around, there is no threat for the lion to eat the lamb. What happens if there are two lions?

The situation is different here: lion2 is ready to eat lion1 if the latter chooses to eat the lamb. So, lion 1, being rational, controls the impulse and spares the lamb. The lamb survives. What about three lions?

The lion3 is ready to eat lion2 if the latter becomes full. Knowing this reasoning, lion2 will spare lion1. Since the lion1 knows that, it will eat the lamb.

To conclude: if there are an odd number of lions in the pride, the first lion can eat the lamb without worrying about being eaten by other lions. If the number is even, the lamb will be spared.

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Sleeping Beauty problem

Here is the question: Sleeping Beauty has agreed to be part of an experiment. On Sunday night, she will be put to sleep, and then a fair coin will be flipped. If the coin comes up heads, she will be awakened on Monday and put back to sleep. If the coin comes up tail, she will be awakened on Monday and put back to sleep. She will again be awakened on Tuesday and put back to sleep. Once asleep, she won’t have any memory that she was ever awakened. The experiment ends on Wednesday, and she can go home.

When she wakes up during the experiment, a question is asked: what is the probability that the coin came up heads? So what is her answer? Note that she has complete knowledge (of the rules) of the experiment.

The halfer position

Since she knows the coin is fair, we can always say the probability of heads is (1/2) by disregarding the prospect of having asked this question under three possible scenarios. It is an unconditional probability.

The thirder position

In the absence of any information on the day she woke up, she could imagine three possibilities with equal chances – Monday heads, Monday tails and Tuesday tails. Out of these, one corresponds to heads. So the probability is (1/3).

So there are two possible answers to one question. Well, that is not true. The two questions are different. Asking for the probability of heads can give 1/2 as the answer. The second question was about the probability, given she is awake. To put it differently, what is the chance she is awake because the coin landed heads?

Watch veritasium’s video on this topic.

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The Easterlin Paradox

The Easterlin Paradox, named after economist Richard Easterlin, addresses the relationship between happiness and income. First, the expected part: there is a positive relationship between the real income of people and their happiness – both among countries and within countries. In other words, the richer are happier.

But when it comes to changes in income or living standard as a function of time, happiness tends to remain flat. For example, the real (inflation-adjusted) per capita GDP of the United States was $25,083 in 1972, which became more than doubled ($56,000) by 2018. Yet, the Percentage of people who responded “very happy” was 30% in 1972 and 32% in 2918, and “pretty happy” moved from 53% to 57% in the General Social Survey. Also, the proportion of people who felt their financial situation changed for better or worse remained the same.

The contradiction in the relationship at a point in time vs as a function of time is the essence of this paradox. A possible explanation lies in the argument that the happiness of one arises from comparison with the other. So, even when the whole society moves up richer, if the individual sees no differential growth of their wealth compared to the neighbour, she could feel no increase in happiness. Another explanation is how fast people get used to wealth, and marginal values no longer give the same thrill – some form of diminishing marginal utility.

References

The General Social Survey: GSS
Real gross domestic product per capita: FRED
Easterlin paradox: Wiki

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Attacker-Defender Game

We have seen the rock-paper-scissors game in one of the previous posts. Let’s see an application of this in real life. Imagine there are several roads leading to the city in a trouble-hit area, and the police (or military) want to check for potential weapons carried by the criminals.

In such situations, the authorities have a limited scope to implement a deterministic strategy to block one road and check. The perpetrator will soon figure out the idea and prepare plans to avoid the threat. So, what works for the police is to confuse the attackers by choosing checkpoints randomly. In game theory language, this is called a mixed strategy, where the player has multiple options with positive probability. This is contrasted with pure strategy, in which the player has only one action with a positive probability.

Mixed Nash: Game Theory Online

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Illusory Correlation Bias

The illusory correlation refers to the error by the human mind that finds correlations between two events that are not correlated. These are sweeping generalisations by the brain that wants to conclude an event quickly (and efficiently). While we all are victims of this bias – fear of strangers, patterns of momentum in share prices, trust in lucky charms – it is not fun to know that psycho-diagnosticians, too, make these errors.

DAP and Rorschach

The pioneering works of Chapman and Chapman analysed popular psychodiagnostic tests such as Draw-a-Person (DAP) and Rorschach in the 60s. In one such study, the researchers presented drawings and descriptions of random symptom statements to a batch of practising clinicians and naive observers. You may know that psycho-diagnosticians have standard checklists correlating the behavioural characteristics of patients with their sketches. And it was no surprise that the observers fell for the stereotypes, just as the clinicians, correlations between random drawings of broad shoulders with manliness, large heads with intelligence etc.

Chapman, L. J.; Chapman, J. P., 1967, Genesis of popular but erroneous psychodiagnostic observations, J Abnorm Psychol, 72(3), 193

Chapman, L. J.; Chapman, J. P., 1969. Illusory correlation as an obstacle to the use of valid psychodiagnostic signs, J Abnorm Psychol, 74(3), 271

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The Dollar Auction Game

An auctioneer auctions a dollar bill at a starting bid of 5 cents, with a ‘simple’ condition – i.e., is both the highest and the second highest bidders pay. What then happens? It is a game devised by Martin Shubik of Yale University.

Imagine the first person bids 10 cents, thinking of pocketing a profit of 90 cents. But she is among a group of strangers, and another person, seeing the opportunity for an 85 gain, bids 15. Now, the first person can do two things – give up 10 cents and quit or continue bidding (perhaps 20 cents).

If she persists and the story repeats, a few interesting things can happen: 1) the auctioneer gets more than $1 (when the person bids for 55 cents and the previous one was 50 cents), 2) the highest bidder loses money (when she bids $1.05; more than the value of the bill).

A few more things can happen. One of them is the case when nobody bids. This is the best outcome in which no one loses (and gains!). The second possibility is for the first person to start at 5 cents and then form a coalition with the rest so that no one else bids. This results in a loss of 95 cents to the auctioneer. A third possibility is to raise the bid to 95 cents by the first person. This makes any subsequent offers unattractive.

Martin Shubik, The Dollar Auction game: a paradox in noncooperative behaviour and escalation. Journal of Conflict Resolution, 1971, 15 (1), 109-111

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Volunteer’s dilemma – part 2: It’s Her Job

An extension of the volunteer’s dilemma is the promotability problem at the workplace. It roughly translates to employees letting others take up tasks that lead to low employability in favour of the ones that impact performance evaluations. E.g., a survey of Carnegie Mellon faculty revealed that an overwhelming majority (~ 90%) thought research papers and conferences were more important than offering services to the curriculum committee or faculty senate.

Going a level deeper, studies also found that women staff, on average, spend more hours on committee work than men. Babcock et al. report results from one such study, where the team examined whether men and women differ in their preference to take up tasks of low profitability.

In study 1, the researchers caught hold of data from e-mail invitations for the service for the senate committee for the 2012-13 academic year. Of the total 3271 faculty members (24.7% female), only 3.7% volunteered. Genderwise, it was 2.6% men and 7.0% women.

The second experiment was in the classical volunteer’s dilemma mould. The participants were randomly assigned to groups of 3 to complete a task in two minutes. Each team should make a volunteering decision. The volunteer gets $1.25, and the others get $2. If no one volunteers, they all get $1. There were ten rounds; women dominated from round 1, with two-thirds of the investment made in the last two seconds.

The next experiment was a single-sex version of the previous. The volunteering rates, as well as the timing, were similar to experiment 2. And the researchers, this time, found no evidence for gender preference on the probability of investing.

The fourth experiment was a modification of the second. Here a fourth member is added who can request the three players to invest. The incentives are similar to the previous. The results showed a remarkable difference: on average, women revived 2.5 more requests than men over ten rounds.

Babcock, L.; Recalde, M. P.; Vesterlund, L.; Weingart, L., American Economic Review 2017, 107(3): 714–747

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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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