Life

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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Subscribing Irrationality

We have seen the role of expected values as a rational means of making decisions. Or the expected utility in other cases. But life is not as simple as in the case of a textbook example. And life never presents situations such as betting on a number of a die or an 80% chance of $45 vs a sure-shot $30, where someone can estimate the value arithmetically. It gives options on products with price tags. But how the value of a product is visible to the decision-maker?

The author, Dan Arie, discusses this dilemma and concludes that most humans like to have a reference and use a value based on relativity. Be it the price of a meal or television – we need something to relate to before choosing an option. And the sellers know that very well and try to use it in pricing their products. Here is one possible example I encountered this morning – the subscription offers of The Atlantic magazine.

Select your plan

First, the big picture: here is what you see on the website:

There are three options: online, online + print and online + print + something else! We shall come to that something else sometime later. Imagine if the choice was between the two options, digital and digital + print:

As seen in various studies, the aspiring subscriber makes a comparison a may go for the second most expensive option. She may further justify her action for the online version as a new way of working in the digitalised world.

It is more expensive – thrice the difference between the first two
Visibly distinct – three-digit whole number vs two-digit factions with deception (e.g. 79.99 sounding 70 instead of 80)
It has repeated mentions of the word ‘free’: likely a lure for the emotional few.

Let’s do a few hypothetical calculations to demonstrate the expected value (to the seller).

Case 1: two options – 80% for option 1 and 20% for option 2. The seller’s earnings per subscription = 0.8 x 80 + 0.2 x 90 = 82.
Case 2: three options and no ‘free’ – 60% for option 1 and 40% for option 2. Earnings per subscription = 0.6 x 80 + 0.4 x 90 = 84.
Case 3: three options and ‘free’ – 60% for option 1, 30% for option 2 and 10% for option 3. Earnings per subscription = 0.6 x 80 + 0.3 x 90 + 0.1 x 120 = 87.

Dan Ariely, Predictably Irrational

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Coherent Arbitrariness

What determines the price of an object? If you are buying an asset, it could be the present value of all cash flow from it. It could also be the meeting point between supply and demand curves (or the willingness to pay and marginal cost). Well, there is another factor – human irrationality.

Ariely et al. call it coherent arbitrariness induced by the anchoring effect. In one of their studies, the experimenters selected 55 students of the Sloan School MBA program and tried a bidding game for six products. The experimental design was as follows.

The researchers described six products – wines, chocolates, books and computer accessories. The students need to do the following:
1) Write down the last two digits of their social security (SS) number on top of the paper.
2) Write down the same number (SS) against each item and indicate their choice (as accept/reject) if it was the price of the product in dollars.
3) Write down the maximum willingness to pay for each item.

The results are in the following table. The values with the dollar sign represent the average willingness to pay mentioned by the subjects.

Last 2 digits of SS –> 00-1920-3940-5960-7980-99
Cordless
trackball		$ 8.64$11.82$13.45$21.18$26.18
Cordless
keyboard		$16.09$26.82$29.27$34.55$55.64
Average
wine		$ 8.64$14.45$12.55$15.45$27.91
Rare
wine		$11.73$22.45$18.09$24.55$37.55
Design
book		$12.82$16.18$15.82$19.27$30.00
Belgian
chocolates		$ 9.55$10.64$12.45$13.27$20.64

Look at how the average willingness to pay changed with the anchor (person’s social security number)!

Dan Ariely, George Loewenstein, Drazen Prelec, The Quarterly Journal of Economics, February 2003

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Inadequate Moral Positioning on Charity

Peter Singer’s 1972 paper, “Famine, Affluence and Morality,” challenges some of the fundamental premises of our moral positioning. He argues how timely actions can reduce the sufferings of the disadvantaged and challenges the common knowledge of helping others as supererogatory rather than obligatory.

The backdrop of Singer’s paper was the suffering of the millions in East Bengal in 1971. In this view, charity and generosity are unacceptable terms to describe the act of helping people facing death due to lack of food, medicine and shelter. Because of this notion, a person who does charity is praised, but the one who avoids it is not condemned – something Singer despises severely.

Singer argues that humans are obliged to prevent a wrong from happening, whether it’s in the neighbourhood or an unknown land. To quote his famous example of a drowning child,

if I am walking past a shallow pond and see a child drowning in it, I ought to wade in and pull the child out. This will mean getting my clothes muddy, but this is insignificant, while the death of the child would presumably be a very bad thing.

Peter Singer, Famine, Affluence, and Morality, Philosophy & Public Affairs 1 (3), 1972, 229.

This act of saving the child is not just praiseworthy; it is required.

To summarise, Singer challenges our moral positioning about charity. His idea, one way or another, paves the foundation of genuine altruism (as a moral requirement) in society. His views are twofold: 1) it recognises contributions of affluent people as mandatory, and 2) it rejects the lack of proximity of the needy as an excuse not to help.

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Missing Jimmy Stewart and SVB’s Crisis

We have seen coordination failure and its consequence in bank runs and what might have happened at Silicon Valley Bank last week. Two videos on YouTube (A and B) have prompted me to write this post. Video A is about a CEO who just managed to pull out her money from the bank before the collapse, an event partly transpired by actions such as hers. The second video shows the pivotal moment from the movie, “It’s A Wonderful Life” (1946), where the hero James Stewart single-handedly prevented a bank from collapsing. Real heroism!

Bank’s decision making

So what happened at the 2023 bank scene? SVB held large quantities (in the order of $200B) of deposits from start-up companies. The bank keeps the required minimum cash or fractional reserve banking, typically about 10%, in their vaults; the rest is turned around to make profits (earning from the investing – the interest paid to the depositor). SVB has invested ca. $90B of its cash in what is known as held-to-maturity (bonds). There is nothing wrong so far, as these instruments are pretty risk-free, but not this time! The bank invested its money at ca. 2% return for about four years in 2021, and the Federal Reserve raised the interest rate a year later, making a heavy dent in the current market value of the 2021 investment.

Meanwhile the investors

Two things happened at the investor’s end. The depositors (the technology companies) wanted to take out more money from the bank as the funding started declining for the firms. The news of the declining fair value of the 90 billion bonds became public with the annual report. The second news made the depositors and their seed investors nervous; they wanted to withdraw all their money.

Perfect storm

The end result was a perfect bank run. On March 10, the bank announced they had failed to raise capital and were looking for a buyer. A few hours later, the bank was shut down by the regulator.

The math behind the trouble

Imagine the bank had bought treasury bonds worth $100 in 2021 for four years at a rate of return of 2%, and the Fed raised the interest rate to 5% immediately after that. If the bank waits for four years, it will get 100 x (1.02)4 = 108.2 at 2% returns. If the bank wants to encash before, it must go to the secondary market to sell. The buyer at the secondary market, who can now get 5% returns on a bond, therefore, will value the bank’s bonds at $88 (108/(1.05)4).

The psychology behind the trouble

But the math is just a catalyst to the trouble. The broader issues are the decision-making by the bank that invested significant cash in long-term bonds (duration risk). And the depositors, triggered by their investors, wanted to withdraw their money all at once (irrationality). And alas, the Jimmy Stewarts, who could charm the depositors from carrying extreme actions, exist only in movies and textbooks.

Further Watch

A) CEO describes pulling money from bank hours before collapse: CNN
B) Bank Run Scene from “It’s A Wonderful Life” (1946): Ian Broff
Why Banks Are Collapsing: Graham Stephan

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Bank Runs and Pareto Efficiency

Let’s play a new game. Imagine there is a group of people. The players have two choices and payoffs: 1) Invest nothing and get nothing, 2) Invest $100, and there are there two outcomes: if more than 90% of the group invests, there is a net profit of $50, and if fewer than 90% invests, then the investor loses the money (-$100).

There are two Nash equilibria possible here. In the ‘good’ scenario, all invest and get profited. In the other case, no one invests; therefore, nobody loses. If the game is played for the first time, two things can happen: more than 90% invest and get a profit or fail to meet the 90% mark and lose money.

If the game is played many times, and if the players are rational, they will soon realise the basic mentality of the others and converge to one of the two outcomes – nobody invests, or everybody invests.

Bank runs and irrationality

A well-known case of such coordination failure is a bank run. As I write, we are on the cusp of a crisis at SVB (Silicon Valley Bank) in California, a significant start-up lender. So, why do bank runs happen? A bank run occurs when the depositors lose their confidence in the bank and start to withdraw their deposits. It is not a viable proposition as banks do not hold all the money in their vaults but lend or invest most of it to make a profit.

Rational customers with a good memory (of several previous incidents) may decide not to panic and stay invested. But what happens more often is people try to withdraw their money in the rush, only to pull the bank to a potentially avoidable, total failure.

Nash equilibrium: bad fashion and bank runs: YaleCourses

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Natural Medicines and Fallacies

The terms nature and natural products invoke cult-like sentiments in society. They are usually used as opposites for synthetic products, chemicals, toxins, poisons, etc. Let’s look at some common irrationalities associated with ‘nature’.

Argumentum ad populum

Or appeal to the people. In simple language, it means since everybody thinks it’s true, it must be true! There are more reasons why something popular is likely wrong, especially in specialised fields of study, as the population of practitioners in topics such as medicine is negligible in society.

Post hoc ergo propter hoc

We have seen it before. It means Y happened after X; therefore, X caused Y. Almost all traditional medicines against what is now known as viral infections are examples of this fallacy. A famous example is Phyllanthus, as a cure for Hepatitis A, a water-borne viral infection (of the liver). The illness, if it’s caused by Hepatitis A or E, will go away in itself. But what happens if a person gets the same symptoms caused by Hepatitis B? Not something pleasant.

Argumentum ad antiquitatem

Appeal to tradition is often related to one’s cultural identity. It was written, so it must be true. A classical case is where people from the East think of modern medicines as Western medicines and take pride in ancient science that treated almost everything.

Absence of data as proof of absence

The presence of side effects is a common criticism directed against evidence-based, modern medicine. They consider the treatment of an ailment using a drug to be a trade-off between the risks and benefits. Naturally, this mandates the inventors to probe deep into the dangers and advantages of the given molecules used for treatment. Historically, similar scrutiny has never occurred in traditional medicines, thereby lacking data on their adverse effects.

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Parrando’s Paradox

Let’s play this game: game A) You lose a dollar every time you play one game; game B) you lose five dollars if the money at hand is odd and gain three dollars if it’s even. You have 100 bucks at the start of each game; play for 100 games.

Following are the first few results from game A, followed by the plot of the results.

#Money
at hand
start100
199
298
397
……
1000

So, you are losing everything in 100 plays. Now, the second game: The excel code is: if(isodd(B1), B1-5, B1+3); assuming the starting 100 is in the cell, B1.

#Money
at hand
start100
1103
298
3101
496
……
1000

Again, you lose everything in 100.

Play two losing games!

We have played two losing games. Now play game B and game A alternatively and see what happens. if(isodd(A2), if(isodd(B1), B1-5, B1+3), B1-1). The game number is in the A column, starting from A2, and money is in the B column, starting from B1.

#Money
at hand
start100
1103
2102
3105
4104
……
100200

Putting the outcomes of all three games in one place (back represents game A, red represents game B, and the green represents BABA game:

Where is the paradox?

An important thing to notice here is that game A influences game B (and evades the number from being odd before game B starts). The end result becomes counterintuitive, but not a paradox in the strictest sense.

Parrondo’s paradox: Wiki
The Game You Win By Losing (Parrondo’s Paradox): Vsauce2

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Salience Bias

It is a cognitive bias where you focus on certain striking items or information that catch your attention and ignore things that don’t grab the same alert. Salience originates from a contrast between the event and its surroundings. An example is the news of a shark attack on a human – a rare occurrence – that psyches people from going out to the seaside.

Salience bias is critical to be aware of and get under control. In finance, a person’s aspirations to create wealth through long-term investments can derail her reactions to daily market stories about bulls and bears.

Salience Bias: The decision lab

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