The trolley problem is a thought experiment in ethics about a scenario in which an onlooker has a chance to save five people being hit by a runaway trolley by diverting it to another track, hitting one person.
The fallacy of dilemma
It is an informal fallacy in which the proponent restricts the options to choose into a few, say, two. It is a fallacy because the framing of the premise is erroneous.
Back to the trolley
In my view, the trolley problem is a false dichotomy (two options) problem that does two things. It forces you to believe that there are only two options – kill five or kill one. It then helps you to justify killing the one as one generous act to save five. And this has been consistently practised by political leaders, especially of the oppressor types, to push their malicious agenda whilst satisfying the collective imagination of the majority.
Dealing with the trolley
The best way to deal is to resist the premise. Why are there diversions or two tracks? Why are there only two tracks? Why is the onlooker not closer to the five so she can save them (by pushing or something)? Why does only the diversion switch work and not the stopping switch?
Here is a game played between two players, player 1 and player 2. There are two piles of cash, 4 and 1, on the table. Player 1 starts the game and has a chance to stop the game by taking four or the pass to the next player. In the next round, before player 2 starts, each stack of money is doubled, i.e. they become 8 and 2. Player 2 now has the chance to take a pile and stop or pass it back to player 1. The game continues for a maximum of six rounds.
Best strategy
To find the best strategy, we need to start from the end and move backwards. As you can see, the last chance is with player 2, and she has the option to end the game by taking 128 or else the other player will get 256, leaving 64 for her to take.
Since player 1 knows that player 2 will stop the game in the sixth round, he would like to end in round five, taking 64 and avoiding the 32 if the game moved to another.
Player 2, who understands that there is an incentive to be the player who stops the game, can decide to stop earlier at fourth, and so on. So by applying backward induction, the rational player comes to the Nash equilibrium and controls the game in the first round, pocketing 4!
Irrationals earn more
On the other hand, player 1 passes the first round, signalling cooperation to the other player. Player 2 may interpret the call and let the game to the second round, trusting to bag the ultimate prize of 265. Here onwards, even if one of them decides to end the game, which is a bit of a letdown to the other, both players are better off than the original Nash equilibrium of 4.
In the previous post, we have seen a variation of the 100 prisoners problem applied to the Monty Hall problem. But what is the 100 prisoners problem?
One hundred prisoners numbered 1 to 100 are given a choice to free themselves if they all find their numbers, written on a piece of paper, randomly kept inside 100 boxes numbered 1 to 100. In other words, the bit carrying number 8 could be inside box 64, #21 inside box 99 etc. The room that keeps the boxes allows only one prisoner at a time, and he can open a maximum of 50 boxes. Even if one fails in his attempt, the whole group loses, and none will walk free. Prisoners can discuss strategies only before the game.
The Strategy
Suppose each player tries opening boxes at random. The probability of a person finding his number from 100 boxes by opening 50 is (50/100) or (1/2). For 100 people to find their numbers, the chances are (1/2) multiplied 100 times or (1/2100), a number close to zero.
Instead, they decide on a strategy; the person first opens the box that carries his number. If he finds his number, fine or else he goes to the box that corresponds to the number inside the first box. It continues until one of the two things happens. He finds his number, or he reaches the maximum of 50 attempts. This strategy can raise the probability of success to a mind-boggling 31%!
To understand this, take the case wherein the prisoner finds his number inside a box. If he continues the strategy, the number he just discovered, which is his number, will lead to the container that carries his number, which is nothing but the first one he opened. Or he always completes a circle of 50 (or lower) boxes if he is successful. Every unsuccessful attempt means the (unfinished) loops are longer than 50.
The travel of prisoner number 24
Each prisoner makes one loop
So the problem becomes the collection of such loops. The probability that such a cluster has no loop larger than 50 is 31%.
Here is a variation of the Monty Hall problem based on the 100 prisoner challenge. Three objects – a car, a key and a goat – are randomly placed behind three doors. You play the game with a partner. And each of you can open a maximum of two doors, one person at a time. In order to win the car, the first player must find the car, and the second the key.
If the players play at random, the probability of the first player to find the car in two attempts (out of the possible three doors) is 2/3. Similarly, for the second, finding the key is 2/3. The probability of winning is (2/3) x (2/3) = 4/9 = 44.4%.
Strategy
If they do the following strategy, they can maximise the probability to (2/3), which is equal to the chance of any of them reaching their individual goal. Here is how they do it.
Person 1 goes and opens door 1. If she finds the car, she completes her task and over to person 2. If instead, she finds a key, she will try door 2. Whereas, if she finds a goat on the first attempt, she will go for door 3.
The second person might not start 1 out of 3 times, i.e. whenever the first person was unsuccessful. If that is not, she will first open door 2. If it was a key, the team wins. If she finds a goat behind door 2, she will open door 3; if it is a car, she will open door 1. The possibilities are in the table below.
Sequence
car-key-goat
car-goat-key
key-car-goat
Player 1
door 1: car
door 1: car
door 1: key door 2: car
Player 2
door 2: key (wins)
door 2: goat door 3: key (wins)
door 2: car door 1: key (wins)
Sequence
key-goat-car
goat-key-car
goat-car-key
Player 1
door 1: key door 2: goat
door 1: goat door 3: car
door 1: goat door 3: key
Player 2
no game
door 2: key (wins)
no game
You can see that whenever the first person does her job (wins the car), the second one gets the keys.
We know what the expected value theory is. The St. Petersburg paradox seriously challenges that. It is a coin-tossing game and it goes like this:
A casino makes a coin-tossing game for a single player. In the first toss, if you get a head, you win a dollar, and the game ends. If it’s a tail, the game continues but doubles the payoff (two dollars) for the next round. At the appearance of the first head, you go home collecting whatever you won. What is the price you want to pay to the casino to enter the game?
The expected value
Let’s see what the expected value of the game is. EV = P(1 T) x V(1 T) + P(2 T) x V(2 T) + P(3 T) x V(3 T) + … where P(1 T) is the probability of one tail and V(1 T) is the value of one tail etc. EV = (1/2) x 2 + (1/4) x 4 + (1/8) x 8 + … = 1 + 1 + 1 + 1 … = Infinity.
Therefore, the rational player must be willing to pay any price to get into the game!
In reality, you will not pay that amount. Think about this: what is the probability of getting a head in the first toss (and you get one dollar)? It is 50%. Similarly, the chance of ending up with 4 dollars is 25%, and so on.
This disparity between the expected value and the reality is the St. Petersburg paradox.
Bernoulli’s solution
Daniel Bernoulli suggested using utility instead of value to solve this problem. The utility is a subjective internal measure of the player towards the gain from the game. According to him, the utility of the additional amount (earned from the contest) was a logarithmic function of the money.
u(w) = k log(w), w represents the wealth. It is logarithmic, he hypothesised, as there is an inverse relation (1/w) between change in wealth and its value. Mathematically, du(w)/dw = 1/w
With this information, let’s rework the expected utility of this game
n
P( nT)
w
u(w) u = logw (k = 1)
Expected Utility
1
1/2
2
0.69
0.35
2
1/4
4
1.39
0.35
3
1/8
8
2.08
0.26
5
1/32
32
3.47
0.11
10
1/1024
1024
6.93
0.007
1.07
Unlike the previous case, the sum of utilities converges.
A judge gives the verdict to the prisoner that he will be hanged one noon next week, on a weekday, but the day would be a surprise. The prisoner goes back to the cell and makes the following assumptions.
If the executioner doesn’t appear by Thursday noon, he will not be hanged as the last day, Friday, is no more a surprise. After eliminating Friday, he extends the logic to Thursday, which now becomes the final day. Finally, he concludes that he will not be executed as none of the days will come as a surprise.
The prisoner is happy and confident that he will not be hanged, only to find out by complete surprise that he was executed on Wednesday at noon. The judge now stands correct; it was a surprise to the convict. But what was wrong with his logic?
Our fundamental instinct to resist changes reflects well in the first-instinct fallacy of answering multiple-choice questions. However, studies have time and again suggested in favour of rechecking and updating the initial ‘gut feel’ as a test-taking strategy. One such example is the test conducted by Kruger et al. in 2005.
Following the eraser
The study followed eraser marks of 1561 exam papers for a psychology course at UIUC. The researchers categorised the changes in answers into three, viz., wrong to right, right to wrong and wrong to wrong, based on the 3291 changes they found. And here is what they found:
Answer change
Numbers
%
Wrong to right
1690
51
Right to wrong
838
25
Wrong to wrong
763
23
An important statistic is that about 79% of the students changed their answers. It is significant because, when asked separately, 75% of the students believed that the original choices were more likely to be correct in situations of uncertainty.
Switching to the wrong hurts
The level of fear or shame on a decision to shift from right to wrong overwhelms the misery of failure by sticking to the incorrect one, even though the data showed the advantages the second thinking brings. In a subsequent study, the team asked 23 students of the University of Illinois asked which of the outcomes would hurt them most – 1) you made a switch from a correct answer to a wrong and 2) you did not move away from the initial instinct after considering the eventually correct answer. The response from the majority of respondents suggested that people who were in the first situation regretted it more than the second.
We have seen what behavioural scientists had observed when carrying out the ultimatum game on their subjects. Ultimatum game also has an economic side theorised by the game theorists for the rational decision-maker. A representation of the game is below.
Unlike the simultaneous games we had seen before, where we used payoff matrices, this is a sequential game, i.e. the second person starts after the first one has made her move. The first type is a normal form game and is very static. The one shown in the tree above is an example of an extensive form game.
The game
Player A has ten dollars that she splits between her and player B. In the game design, A has to make the proposal and B can accept or reject it. If B accepts the offer, both the players get the money per the division proposed by A. If B refuses, no one gets anything.
Backward induction
Although player A starts the game by spitting 10 dollars between herself and player B, her decision gets influenced by what she assumes about B’s decision (accept/reject). In other words, A requires to begin from the ending and work backwards. Suppose player A does an unfair split 9-1 in favour of A. B can accept the 1 dollar or get nothing by rejecting. Since one is better than zero, B will probably take the offer. If A makes a fair split, then also B will accept the 5. That means B will take the offer no matter what A proposes. So player A may choose the unfair path. This is a Nash equilibrium.
What happens if player B makes a threat of rejecting the unfair offer. It may not be explicit; it could just be a feeling in A’s mind. In either case, player A believes in that and thus makes a fair division. And this is what Kahneman learned from his experiments. In-game theory language, the threat from B is known as an incredible threat as it makes no economic sense to refuse even the unfair offer (as 1 > 0)!
In this post, we discuss an article that otherwise requires no special mention in this space. Yet, we discuss it today, perhaps as an illustration of 1) the diverse objectives that scientific researchers set for their work and 2) how the ever-imaginative media, and subsequently the public, could interpret the messages. Before we examine the motivation or the results, we need to understand something about the study’s publication status.
Preprints with The Lancet
It is a non-peer-reviewed work or preprint and, therefore, is not a published article in the Lancet, at least for now. The SSRN page, the repository at which it appeared, further states that it was not even necessarily under review with a Lancet journal. So, a preprint with The Lancet is not equivalent to a publication by the Lancet.
The motivation
You may read it from the title: Randomised clinical trials of COVID-19 vaccines: do adenovirus-vector vaccines have beneficial non-specific effects? It is a review paper, and the investigators specifically wanted to understand the impact of COVID-19 vaccines on non-COVID diseases, which, I think, is a valid reason for the research. By the way, you have every right to ask why COVID-19 vaccines should impact accidents and suicides!
Motivated YouTubers
The following line from the abstract turned out to be the key attraction for the YouTuber scientist. It reads: “For overall mortality, with 74,193 participants and 61 deaths (mRNA:31; placebo:30), the relative risk (RR) for the two mRNA vaccines compared with placebo was 1.03“. Now, ignore the first three words, “For overall mortality”, add The Lancet, and you get a good title and guaranteed clicks!
The results
First, the results from mRNA vaccines (Pfizer and Moderna):
Cause of death
Death/total Vaccine group
Death/total Placebo group
Relative Risk (RR)
Overall mortality
31/37110
30/37083
1.03
Covid-19 mortality
2/37110
5/37083
0.4
CVD mortality
16/37110
11/37083
1.45
Other non-Covid-19 mortality
11/37110
12/37083
0.92
Accidents
2/37110
2/37083
1.00
Non-accidents, Non-Covid-19
27/37110
23/37083
1.17
In my opinion, the key messages from the table are: 1) The number of deaths due to Covid-19 is too small to make any meaningful inference 2) The deaths due to other causes show no clear trends upon vaccination
Results from adenovirus-vector vaccines (several studies combined):
Cause of death
Death/total Vaccine group
Death/total Placebo group
Relative Risk (RR)
Overall mortality
16/72138
30/50026
1.03
Covid-19 mortality
2/72138
8/50026
0.4
CVD mortality
0/72138
5/50026
1.45
Other non-Covid-19 mortality
8/72138
11/50026
0.92
Accidents
6/72138
6/50026
1.00
Non-accidents, Non-Covid-19
8/72138
16/50026
1.17
My messages are: Accidental accumulation of non-Covid-19-related deaths (five of them coming from cardiovascular) gives an edge to the vaccine group and, therefore, “saves” people immunised with Adenovirus-vector vaccines from dying from other causes, including accidents, in some countries! The statistical significance of the number of cases is dubious.
Lessons learned
1) Be extremely careful before accepting commentaries about scientific work (including this post) 2) As much as possible, find out and read the original paper after being enlightened by YouTube teachers.
Randomised clinical trials of COVID-19 vaccines: do adenovirus-vector vaccines have beneficial non-specific effects?: Benn et al.
You may read this post as the continuation of the one I made last year. Evaluate the risk caused by an action by comparing it with situations without that action. That is the core of the risk-benefit trade-off in decision-making. A third factor is missing in the equation, namely, the cost.
A new study published in The Lancet is the basis for this post. The report compiles the incidents of myocarditis and pericarditis, two well-known side effects linked to the mRNA vaccines against COVID-19. The data covered four health claim databases in the US and more than 15 million individuals.
The results
First, the overall summary: the data from four Data Partners (DP) indicate 411 events out of the 15 million studied who received the vaccine. Details of what is provided by each of the DPs are,
Data Partner (DP)
Total vaccinated
Total Observed myocarditis or pericarditis events (O)
Expected events (E) (based on 2019)
O/E
DP1
6,245,406
154
N/A
–
DP2
2,169,398
64
24.96
2.56
DP3
3,573,097
94
40.08
2.35
DP4
3,160,468
99
44.61
2.22
I don’t think you will demand a chi-squared test to get convinced that the two mRNA vaccines have an adverse effect on heart health. Age-wise split of the data gives further insights into the story.
Age-group
Observed Events
Total vaccinated
Incident Rate (per 100,000)
Expected Rate (per 100,000)
18-25
153
1,972,410
7.76
0.99
26-35
62
2,587,814
2.40
0.95
36-45
63
3,226,022
1.95
1.11
46-55
62
3,597,292
1.72
1.3
56-64
71
3,764,831
1.89
1.63
The relative risk is much higher for younger – 18 to 35 – age groups. But the absolute risk of the event is still in the single digits per hundred thousand. And this is where we should look at the risk-benefit-cost trade-off of decision-making.
The risk
First and foremost, don’t assume all those 411 individuals died from myocarditis or pericarditis; > 99% recover. To know that, you need to read another study published in December 2021 that reported the total number of deaths to just 8! So, there is a risk, but the absolute value is low. The awareness of the risk should alert the recipients that any discomfort after the vaccination warrants a medical checkup.
The benefit
It would be a crime to forget the unimaginable calamity that disease has brought to the US, with more than a million people dying from it. A significant portion of those deaths happened prior to the introduction of the vaccines, and even after, the casualties were disproportionately harder on the unvaccinated vs the vaccinated.
The cost
At least, in this case, the cost is a non-factor. Vaccine price, be it one dollar or 10 dollars, is way lower than the cost of the alternate choices, buying medicines, hospitalisation or death.
Managing trade-off
Different countries manage this trade-off differently. Since the risk of complications due to COVID-19 is much lower for children and the youth, some allocate a lower priority to the younger age groups or assign a different vaccine. However, it is recognised that avoiding their vaccination altogether, due to their low-risk status, is also not an answer to the problem. It can elevate the prevalence of illness in the system and jeopardise the elders with extra exposure to the virus.
References
Risk of myocarditis and pericarditis after the COVID-19 mRNA vaccination in the USA: The Lancet
