The war of attrition involves two players making simultaneous moves. The play moves from period to period. Each player at each period has an option to fight or quit. The game ends when at least one player quits. The player that did not quit gets the prize, V. Each period in which both fights incur a cost of C for each player. If both leave at once, then they get 0.
After the first period,
Action A
Fight
Fight
Quit
Quit
Action B
Fight
Quit
Fight
Quit
Payoff A
-C
V
0
0
Payoff B
-C
0
V
0
If both fight, then the game continues to the second period. Let’s check what can happen in the second period.
Action A
Fight
Fight
Quit
Quit
Action B
Fight
Quit
Fight
Quit
Payoff A
-2C
V-C
-C
-C
Payoff B
-2C
-C
V-C
-C
The game can go on taking, adding more costs to the players. At some stage, you realise the war of attrition is no longer about the prize – you might lose more than you can win – it’s about winning. An example is two firms fighting for a market that can hold one company. The game occurs when they compete with each other, losing money but hoping the rival will eventually withdraw.
A well-known example is the competition between British Satellite Broadcasting (BSB) and Sky Television in the 1980s over the satellite television market. By the time they ended the fight and merged into one first, they had accumulated over £1 billion in losses.
We know what is the tragedy of the commons. Any resource that has too few owners leads to overuse. Examples of this are overfishing of oceans and pollution of the environment, to name a few. A solution to the tragedy of the commons is private ownership. It assumes that the owner manages consumption and conserves a scarce resource.
Privatisation as a solution to the commons also has problems. In anticommons, there are too many owners, and each can exclude others from using, leading to underuse. It becomes a coordination failure. In other words, some resources don’t get invented or don’t reach the market.
It is a binary game similar to a coin toss. Player 1 selects a sequence (of length 3 or larger) first, followed by Player 2. The player whose sequence comes up first wins. The question is, can player 2 maximise her chance?
Apparently, Player 2 can always select a sequence based on what Player 1 has already picked that can maximise her winning odds. It is based on a simple strategy. The second player looks at Player 1’s sequence and picks the opposite of the middle one to start with, followed by the first player’s first two choices.
In the last post, we saw how CVD incident rates have increased since the start of the pandemic and the possible reasons for this. Today, we examine why the vaccine—and not COVID itself—has become the principal offender in the common belief.
Chemophobia
Blame it on the ‘silent spring’, the Bhopal tragedy, or Chornobyl; chemophobia, or the fear of chemicals, is real. We have seen how heuristics or mental shortcuts play a role in decision-making. Studies found that most of us, the non-experts of toxicology, tend to rely on heuristics when judging chemical safety. The public leans on three ‘rules of thumb’ when evaluating chemicals. Natural-is-better heuristics: People associate better confidence in dealing with natural substances than synthetic ones. It may sound incredible, but people find it more comfortable trusting a herb containing 10,000 unknown molecules than a well-researched single compound drug when dealing with a medical condition. The reason? – one is natural, and the other is made. It goes to such an extent that in one study, Siegrist and Bearth found that only 18% of the people surveyed thought the chemical structures of synthetically prepared and naturally occurring NaCl were identical. Contagion heuristics: These come from a lack of knowledge of the concept of dose. People view a chemical as either safe or toxic while missing out on the quantity. For the decision maker (the brain), this keeps the decisions simple. In the same survey, three-quarters of the people believed that a toxic substance is always dangerous irrespective of its dose. Trust heuristics: States that people rely on their trust (or lack thereof) in key stakeholders, such as chemical industries and governmental and non-governmental organisations, to evaluate the associated risk.
For ordinary people, the leading COVID-19 vaccines—Moderna, Pfizer, and Oxford—were all human-made. Therefore, they are dangerous. On top of this, thanks to the ever-vigilant regulators in the EU and the US, the side effects of vaccines—that they could cause severe blood clots or myocarditis in a few in a million people—were public within a few months of their introduction.
Affirming the consequent
Irwin, the hypochondriac: “I’m sure I have liver disease.” “That’s impossible”, replied the doctor. “If you have liver disease you’d never know it.” Irwin replies: “Those are my symptoms exactly.”
Rationality by Steven Pinker
Affirming the consequent is a formal logical fallacy of the following type. IF P, THEN Q. Q. Therefore, P.
In the case of the vaccine, the logical fallacy works this way: A. Vaccines cause myocarditis and pericarditis in some. B. The patient had a heart attack. C. It must be the vaccine.
Not familiar with the risk-benefit trade-off
No decision is risk-free, and medication is no exception. The important thing is to evaluate the risk caused by an action compared to a situation without that action. That is the core of the risk-benefit trade-off in decision-making. And the risks due to vaccination must be viewed that way. I will end with the scheme we developed at the peak of the pandemic.
Death due to Infection (red) vs Death by Vaccine (green)
References
[1] Siegrist, M., Bearth, A. Chemophobia in Europe and reasons for biased risk perceptions. Nat. Chem. 11, 1071–1072 (2019). https://doi.org/10.1038/s41557-019-0377-8 [2] Steven Pinker, Rationality, Penguin Random House
The World’s leading cause of death is cardiovascular diseases (CVDs) – heart attacks and strokes. Globally, the estimated number of deaths due to CVDs increased from around 12.1 million in 1990 to 18.6 million in 2019. Note that the age-standardised death rate has declined from 354.5 deaths per 100,000 people in 1990 to 239.9 deaths per 100,000 people in 2019. While pollution, unhealthy diet, alcohol and tobacco are the leading root causes, the increase in the absolute number of CVD deaths is primarily due to growth in population and life expectancy.
Against this backdrop, we examine the anomalies in death rates in the last five years. According to CDC data, heart diseases accounted for 702,880 deaths in the US in 2022. Here is the figure representing the trend from 2018 to 2022.
Contrary to trends in the last few decades, the death rates jumped from 200 to 211 from 2019 to 2020. Notably, 2020 also marked the start of the global pandemic, COVID-19. The story was no different for the rate of mortality from Coronary Heart Disease (CHD) in England.
Hypothesis on test
Let’s examine the two hypotheses to explain this rise in deaths due to the pandemic. 1) Covid-19 played a role, and 2) Covid vaccine played a role. We will start with the easier one – the vaccine.
The authorisation of leading vaccines – Moderna, Pfizer and AstraZeneca – for first use happened in December 2020, and the active vaccination program only started months later. Note that the ‘jump’ occurred from 2019 to 2020, a year earlier than the start of vaccination.
Now, the impact of COVID-19 on heart disease. Again, there are two possibilities: the virus directly causes heart disease, or the virus is part of the causal chain (VIRUS—MEDIATOR—CVD). Data suggest that there is evidence for the first possibility. While COVID-19 is a risk modifier—something that worsens pre-existing CVD risk factors such as hypertension—heart attacks are only the fourth or fifth cause of death in COVID-19 patients, respiratory failure being the leading cause.
The elephant in the room
The British Heart Foundation published a report in 2022 that summarises their investigation of the excess deaths due to CVD after the pandemic breakout. They found that COVID-19 infection alone was not sufficient to explain the 14% increase in ischaemic heart disease (IHD) compared to the pre-pandemic period. Instead, the breakdown of the healthcare system was the likely cause. The team surveyed and found
43% of patients who needed medical treatment for their heart condition have put off seeking NHS help due to ongoing fears of catching Covid or burdening NHS services.
20% of heart patients reported having had an appointment for their heart condition cancelled over the last year.
The proportion of patients with diagnosed hypertension who had their BP checked fell from 89% in March 2020 to 64% by March 2021.
Two million fewer people were recorded as having controlled hypertension in 2021 compared to the previous year.
Modelling from NHSE shows that this reduction in blood pressure control could lead to an estimated 11,190 additional heart attacks and 16,702 additional strokes over three years.
Here is a trend of the number of patients waiting for treatment (source: NHS England)
The picture is no different for heart procedures. (Source: NHS England (2022) Consultant-led Referral to Treatment Waiting Times (number of incomplete pathways)):
Another study published in Nature Medicine used monthly counts of prevalent and incident medications dispensed and found a systematic trend of decline, especially during the lockdown periods.
In summary
Managing cardiovascular diseases requires constant action via public health agencies. These include detection, consultations, medications, and procedures. The COVID pandemic has temporarily affected the flow of this machinery, and the result was an increase in CVD mortality. Yet, the public perception focused on vaccines. Why did that happen? We’ll see that next.
[3] Elezkurtaj, S., Greuel, S., Ihlow, J.hospitalisedes of death and comorbidities in hospitalised patients with COVID-19. Sci Rep11, 4263 (2021). https://doihospitalised/s41598-021-82862-5
[4]Dale, C.E., Takhar, R., Carragher, R. et al. The impact of the COVID-19 pandemic on cardiovascular disease prevention and management. Nat Med29, 219–225 (2023). https://doi.org/10.1038/s41591-022-02158-7
[5] Vosko, I., Zirlik, A., Bugger, H., Impact of COVID-19 on Cardiovascular Disease, Viruses, 15(2), 508 (2023).
Two competing beach vendors offer similar products at similar prices. They can set up their shops wherever they want along the beach. Beachgoers will buy from whichever is closer to them. What is the ideal location for each?
Let the beach be a straight line from 0 to 1, and the shops are somewhere along that line. There are infinite positions for each, and let’s check one of the realisations.
Vendor 1 (V1) gets all businesses on the left side of location V1 and part of the space between V1 and V2. This is marked by the colour red. Vendor 2 (V2) receives all the right and part of the left. The colour green marks this.
Imagine V1 gets into the middle. It acquires at least 50% of the business (the space on the left of V1) and possibly more, depending on where V2 is. Note that V2 can be anywhere on the right of V1.
Therefore, V1 is always guaranteed a strategy to get at least half of the business regardless of what V2 does. The same should be true for V2, which can also guarantee half of the business by posting in the centre. Mathematically,
V1 >/= 1/2 V2 >/= 1/2 V1 + V2 = 1
These three equations imply that V1 = V2 = 1/2. This is only possible if they are in the same position or are equidistant from the centre.
If the above is a possibility, then V1 can move towards the right to increase its business. The same is possible if both are in the same location anywhere but in the centre. If they are both located at the centre, they are in Nash equilibrium. If either one moves anywhere else, it will cause some business to lose.
Amy and Becky are playing a coin-tossing game. Amy gets a point for heads, and Becky gets a point for tails. The first to reach ten points receives a $100 prize. If the game had to be stopped when Amy was leading at 8-7, how would they have split the prize?
Let’s evaluate Amy’s probability of winning the game when the score was 9-9. At 9-9, she has a 50% chance of winning.
If Amy is leading at 9-8, there is a 50% chance she wins, and 50% she reaches 9-9 (at 9-9, we know her probability of winning, i.e., 0.5). The final winning probability for 9-8 is 0.5 x 1 + 0.5 x 0.5 = 0.75.
At 9-7, Amy has a 50% chance to win and a 50% chance to reach 9-8 (at 9-8, Amy has a 0.75 chance to win). Her chance of winning from 9-7 is 0.5 x 1 + 0.5 x 0.75 = 0.875
At 8-8, Amy has a 50% chance to reach 9-8 win and a 50% chance to reach 8-9. Her chance of winning is 0.5 x 0.75 + 0.5 x 0.25 = 0.5
At 8-7, Amy has a 50% chance to reach 9-7 and a 50% chance to reach 8-8. Her chance of winning is 0.5 x 0.875 + 0.5 x 0.5 = 0.6875
So they split the prize, with Amy taking $68.75 and Becky $31.25.
Bergstrom and West, in this book, Calling Bullshit, introduce the “Principle of Proportional Ink”, as a modification of the concept that Edward Tufte developed of data visualisation,
When a shaded region is used to represent a numerical value, the area of that shaded region should be directly proportional to the corresponding value.
Here is a comparison between two entities, A and B, with values 2 and 4, respectively. In the left plot, both width and height are doubled to represent B, which is twice that of A. This magnifies the area of B to four times that of A. On the other hand, the comparison on the right has the size of B double that of A.
As per the principle of proportional ink, the right plot better represents the relative magnitudes to the reader.
We have seen one type of ultimatum game where the player who receives the offer (Annie) has another option outside the one in the game. Those options are a weak type (0.25) or a strong type (0.5) and depend on a probability p in favour of the former.
Now, imagine Becky knows Annie’s outside option (0.25 vs 0.5). In that case, Becky will make 0.25 p times and 0.5 (1-p) times to get her offer accepted.
If p < 2/3
Becky will offer 0.25 (2/3)rd of the time and 0.5 (1/3)rd of the time. Her expected value of surplus in the case of complete information is p x (0.75) + (1-p) x (0.5) Note that in the case of incomplete information, she would have offered 0.5 all the time, leading to the expected value p x (0.5) + (1-p) x 0 The difference between the two [p x (0.75) + (1-p) x (0.5) – p x (0.5)] is 0.25p. It is the value of information.
If p > 2/3
In the case of complete information, Becky will offer 0.5 (2/3)rd of the time and 0.25 (1/3)rd of the time. Her expected value of surplus is p x (0.75) + (1-p) x (0.5) Note that in the case of incomplete information, she would have offered 0.25 all the time, leading to the expected value p x (0.75) + (1-p) x 0 The difference between the two [p x (0.75) + (1-p) x (0.5) – p x (0.75)] is 0.5(1-p). It is the value of information.
