March 2023

Stag Hunt Game

Two hunters want to go to a range which contains one stag and two rabbits. Each hunter will decide on the equipment to bring but won’t know what the other is carrying. Two people are necessary to hunt a stag, but one is sufficient to catch rabbits. And a stag catch is always a better deal as it has more meat.
The payoff matrix is:

Hunter 2
StagRabbit
Hunter 1Stag3,30,2
Rabbit2,01,1

So, if both hunters bring devices for catching stag, they get the maximum benefit (3 each). If only one brings tools for rabbits, that person will get both rabbits. If both get tools for the rabbit, they split one each.

The game has two pure-strategy Nash equilibria – rabbit-rabbit and stag-stag. And the best situation is the latter. On the one hand, this differs from the prisoner’s dilemma because it has a dominant strategy (to betray), and the stag hunt has none. What is similar in both cases is the role of cooperation among players.

In a fully cooperating regime, both players expect the other to bring in tools for stang, and both reap maximum benefit. But if one player has even a slight doubt about the other (that the other will bring an instrument for the rabbit), she has no choice but to get the same tool or end up with nothing.

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False Positive Riddle

There is a rare mineral that is available in 1% of rocks. Anna has a detector that never fails to detect that mineral. Otherwise, it is accurate 90% of the time, i.e. 10% false positive. She puts the device on a rock, and it goes off. She collects it and makes an offer to her colleague Ben to sell the rock for $100. Should Ben take that offer if the value of the mineral is $1000?

chance of the rock is rare = 1 x 0.01 / (1 x 0.01 + 0.1 x 0.99) = 0.0917

If the probability of success is 0.0917, then the expected value of the gamble is 0.0917 x 1000 = 91. So paying $100 is not worth it.

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The Raven Paradox

German philosopher Carl Gustav Hempel introduced this in the 1940s, questioning what composes evidence for a hypothesis. Here is how the paradox works:

You see a raven, and it is back. Then you see more ravens, and they all tend to be black. These observations prompt the scientist in you to form the hypothesis that all ravens are back.

So far, so good. Now it becomes a conditional (statement of the form: “if A, then B”). As per logic, a conditional is equivalent to its contrapositive:

If A, then B == If not B, then not A

For ravens, the equivalent statement is:

All ravens are black == All non-back things are non-raven, or if an object is not black, then it is not raven

Now let’s collect evidence for the hypothesis. Every black raven is a piece of evidence. Every non-black non-raven also has to be evidence! Green grass, red shirts, and yellow flowers are some examples.

So, what is the issue with this? Well, there could be rare non-black ravens which escape our sight. In the original form of the hypothesis (conditional), you only sample ravens and verify their colour. But in the contrapositive form, you can potentially collect an infinite number of objects (every non-black entity in the world) and appear to strengthen your hypothesis significantly.

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It’s Not Fair!

Inequality aversion is a concept in behavioural economics. It means humans have a notion of fairness and will reject what they consider as inequalities. This psychology is responsible for individuals refraining from targeting higher rewards if they perceive another party getting better incentives. We have seen this in the centipede game.

Kahneman et al. report results from their study in which they carried out telephonic surveys on the residents of Toronto and Vancouver. Each participant got a maximum of five questions regarding fairness in a telephonic interview.

Question 1

“A hardware store has been selling snow shovels for $15. The morning after a large snowstorm, the store raises the price to $20. Please rate this action.” 80% of the respondents thought it was unfair.

Question 2

Question 2 has two parts. 2A: An employee is working in a photocopying shop at a wage of $9/hour. Upon seeing unemployment rising in that area and noticing other smaller shops paying $7/hour for their employees, the owner reduces the employee’s wages to $7/hour.
2B: An employee is working as in question 2A. After she leaves, the employer recruits a new person at a wage of $7/hour.

To question 2A, 83% of the respondents replied as unfair, and to 2B, only 27%.

House on rent

Similar ideas of fairness exist in residential tenancy. Different rules of rent-hikes are accepted for a new tenant (higher tolerated) vs a tenant renewing the lease (lower). At the same time, people thought it was ok if the landlord sold the house and the new owner charged higher rent from the existing tenant!

Reference

Daniel Kahneman, Jack L. Knetsch and Richard Thaler: Fairness as a Constraint on Profit Seeking: Entitlements in the Market, The American Economic Review, 1986, 76(4), 728

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What is in the Envelope

Here is a puzzle. There is a 200 EURO currency in one of the two envelopes, A and B. If you guess the right one, you can get the cash. Additionally, you can avail of a clue, which works as follows: There is a jar with five balls in it – three of them having the alphabet (A or B) of the envelope that carries the currency and two with the other alphabet. You can pick on the ball at random if you like. The price to pay for the clue is 25 EURO. The questions are:

1) Is the clue worth 25 EURO?
2) If not, what is the maximum amount you would like to pay?
3) Would you be willing to pay for a second clue and pick up another ball?

Let’s answer the first question. The expected value from the guess without taking any clues is 0.5 x 200 + 0.5 x 0 = 100 EURO. It is because there is a 50-50 chance that your guess turns right. What is the expected value of the guess with the first clue? It is 0.6 x 200 + 0.4 x 0 = 120 EURO. When you pick one ball, there is a 60% (0.6) chance that it is the right one (3 out of 5) and a 40% chance it is the wrong one.

Therefore, the maximum added value of going for the clue is 120 – 100 = 20 EURO. So, the answer to the first question is NO, and the second is 20 EURO.

What about a second pick? To answer this, we will need to perform several conditional probabilities using our favourite Bayes’ rule, which we’ll do next.

Is Extra Information Helpful? A Probability Puzzle: William Spaniel

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Money and Behaviour

Psychological experiments, despite their imperfections, still offer valuable insights into human behaviour. Well-known Imperfections are their lack of reproducibility and the nature of participants – typically originate from a narrow distribution of the society, e.g. students. The paper from Kathleen Vohs, Nicole Mead and Miranda Goode is an example we want to discuss today. They investigated how the notion of money (actual cash or a mere thought) can change one’s attitude while doing a job.

Self-sufficiency hypothesis

Before we get into the details, let’s understand what is meant by self-sufficiency. It is a concept where an individual recedes from the collective to herself and spends her effort towards achieving personal goals. The objective of the experiment was to verify the hypothesis that the concept of money can make a person self-sufficient.

Two tasks

The experiment consists of two tasks: 1) descramble 30 jumbled words, and 2) complete a challenging problem of arranging 12 disks into a square.

Three groups

The participants are arranged into three groups. Control group – gets neutral words to descramble and do the second task; the play-money group – gets neutral words in task 1 but primed with a pile of monopoly money while doing task 2; the money-prime group – gets money-related words to descramble and then do the second task. Before leaving the room, the investigator mentioned his availability to help if required. Therefore, the measurement parameter of the experiment was persistence (time before a participant seeks help).

The results

The results fulfilled the hypothesis – the money groups (money-prime and play-money) both parties took the job seriously and spent more time before approaching the investigator for help. The mean times were money prime = 314.06 s (sd = 172.79); play money = 305.22 s (sd = 162.47) and control = 186.12 s (sd =118.09).

Kathleen D. Vohs, Nicole L. Mead, Miranda R. Goode, The Psychological Consequences of Money, Science, 2006, 314.

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What Asteroid Ryugu Tells Us

I’m sure you remember Miller–Urey experiments that, in the 1950s, generated molecules of life by passing electric discharge over a mixture of methane (CH3), ammonia (NH3), water (H2O) and hydrogen (H2). The molecules reported were amino acids such as aspartic acid, glycine, alanine and alpha-amino butyric acid.

Ferus et al. in 2017 went even further. They shone electric discharge (simulating lightning) and laser (simulating asteroid plasma) on a mixture of NH3, CO and H2O, producing RNA nucleobases – uracil, cytosine, adenine, and guanine.

Straight from space

While laboratory experiments such as these demonstrated the origin of fundamental molecules from simple gaseous species present in the universe, it can never replace evidence from space, the true cradle of these building blocks of life. And that’s what happened when scientists analysed samples from an asteroid.

The team led by Yasuhiro Oba analysed samples collected in 2018 from asteroid Ryugu and found uracil, one of the four bases of RNA.

Pristine sample

The beauty of this sample is that it was uncontaminated by anything from the earth as it was collected and sealed at the asteroid surface by the Hayabusa2 mission.

Studies like these suggest that foundations of life, such as the molecules of interest, might have been formed in carbonaceous asteroids and delivered to the early earth.

Reference

Yasuhiro Oba et al.; Nature Communications, 2023, 14:1292

Asteroid sample study: The conversation

Stanley L Miller, A production of Amino Acids Under Possible Primitive Earth Conditions, Science, 1953

Formation of nucleobases in a Miller–Urey reducing atmosphere, PNAS, 2017

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The Pirate Problem

The pirate chief and his four mates found 100 gold coins and wanted to divide them among themselves. As expected, they are perfectly rational and strategic people. Here are a few rules.

  1. The leader can propose a division.
  2. If half of the team (including the leader) accepts the proposal, it becomes valid.
  3. If not, the chief will be thrown out, the next in line will become the chief, and the game will continue.

So, what should be the chief’s offer to survive?

To find the solution to this problem, we must start from the last pirate and work backwards.

If the last one becomes the chief, he doesn’t need to make any offer and can keep all 100 coins. Simple! But what happens if two pirates remain? Then, the chief can decide not to give anything to the last one as he secures the approval by voting himself.

So, moving another level up – with three pirates. The chief requires at least two votes, but he gets one, i.e., his own. Also, he doesn’t want to give away more money than he needs to. Which of the other two pirates is cheap to buy? There is no point in giving money to the next person as he will disapprove; he knows he can keep all the coins by becoming the nest chief. Therefore, the last guy will vote for the current chief if the former gets at least one coin.

Now, four. The chief needs one more vote. He looks at the three and figures out what would happen if he loses, and the second one becomes the new chief. If that happens, the third one will not get any coin. Therefore, he becomes the cheapest vote to buy.

In the last case, the original case with five pirates, the chief needs three votes to survive. One comes from him, and he needs to buy two more. There is one cheap way: we know what happens if the proposal fails and the next becomes the chief. That will lead to the third and fifth not getting any coins, and they know that. So buy those two.

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Cournot Duopoly Game 

Cournot Duopoly is an economic strategy game where two firms producing the same type of products compete for the market by controlling their output. It is a simultaneous move game, and there is no collusion.

Let the firms be 1 and 2 choosing to produce q1 and q1 quantity of goods. In this model, they only decide how much to make. The price will be determined by the market using an inverse demand curve. So, price P = a – b x Q, where Q = q1 + q2 and a and b are positive numbers.

The marginal costs of production are C1 and C2, respectively. These suggest the profit of firm 1, Profit 1 = revenue – cost = P x q1 – C1 x q1. Similarly, Profit 2 = P x q2 – C2 x q2.

Profit 1 = (a – b x (q1 + q2) )x q1 – C1 x q1
= (a – bq1 – bq2 – C1)q1
Profit 2 = (a – bq1 – bq2 – C2)q2

Nash Equilibirum

To get the Nash equilibrium, we’ll maximise the payoffs (profits) of firm 1 and firm 2 by differentiating with respect to q1 and q2 and setting them to zero.

d(Profit 1) / d(q1) = a – 2bq1 – bq2 – C1 = 0
d(Profit 2) / d(q2) = a – bq1 – 2bq2 – C2 = 0
q1 = (a – bq2 – C1)/2b
q2 = (a – bq1 – C2) / 2b

So, q1 is a function of q2 and vice versa. The first equation, a straight line, will look like the following.

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Road Safety in India – Comparison with the US

In the last three posts, we have been looking at the statistics of road accidents in India. It would be interesting at this stage to compare that with the US.

ParameterThe USIndia
Population
(mln)		3301321
Fatalities38,824131,714
Fatalities per
million population 		117.65100
Injured
Persons		2,282,015348,729
Injured per
million population 		6915263
Crashes/
Accidents		6,393,624366,138
Accidents per
million population 		19374277
Survival probability
Injured /(Injured +fatality)		0.980.72

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