January 2024

Duel – Backward Induction

In the previous post, we developed the condition for the player to shoot the opponent. I.e., the distance (d*) at which the probability of the player who gets the turn is greater than or equal to the chance of the opposite missing in the next turn.

When the player, say P1, reaches d*, she can believe in two things. 1) P2 will not shoot in the next turn (d*-1) or
2) P2 will shoot in the next turn.
In the first case, P1 should not shoot. In the second case, P1 assesses that P1(d*) + P2(d*-1) = 1. So, P1 must shoot.

But what should Player 1 believe? The answer comes from backward induction.

Imagine the players came in close range (d = 0), and it’s P2’s turn. P2 will shoot and win. Taken one step back: P1 knows that in the next step, P2 will shoot, so she takes option 2. She assesses that P1(1) + P2(0) > 1, and she shoots. Go back one more step. Now, P2 knows that P1 will shoot in the next step, and if the distance is less than d*, P2 must shoot. It will continue until d* is reached.

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Duel

There are two players in the duel game, P1 and P2. Each carries a pistol loaded with a single bullet positioned a certain distance apart. Each player gets one round at a time, one after another. A player can shoot the other or step forward in her turn. Whoever hits the other wins the game. So, the player’s strategic decision is when she should shoot.

Let Pi(d) be the probability that player i will get her target (the opponent) from a distance d. The two players can have different abilities. In other words, P1(d) can be lower (or higher) than P2(d). Also, at d = 0, P1 and P2 hit the target (P1(0) = P2(0) = 1).

So, who will take the shot first, and when?

Imagine it’s P1’s turn, and she can believe in the following two things:
1) P2 will not shoot in the next turn
2) P2 will shoot in the next turn

If P1 believes in option 1 (that P2 will not shoot in the next turn), P1 will not shoot in this turn. If P1 thinks that P2 will shoot, then P1 will evaluate her options in the following manner. If P1 thinks her probability of hitting the target in this turn is greater than or equal to her opponent’s chance to miss in the next turn, she will shoot. Mathematically, that is:

P1(d) >/= 1 – P2(d-1) or
P1(d) + P2(d-1) >/= 1

The distance at which the condition is satisfied is d*. From the picture, it is clear that below d*, the probability, P1(d) + P2(d-1) > 1 and above d*, it is < 1.

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Only Trade-Offs

American economist and political commentator Thomas Sowell summarises the core of decision-making, like nobody has, in his famous quote: There Are No Solutions, Only Trade-offs.

A crucial thing in trade-off is the assessment of the consequence of each option. A common practice in investment decisions is the cost-benefit analysis.

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Logistic Regression and R

We have seen logistic regression as a means to estimate the probability of outcomes in classification problems where the response variable has two or more values. We use the ‘Default’ dataset from ‘ISLR2’ to illustrate. The data set contains 10000 observations on the following four variables:

default
: A factor with levels No and Yes indicating whether the customer defaulted on their debt

student:
 factor with levels No and Yes indicating whether the customer is a student

balance: 
The average balance that the customer has remaining on their credit card after making their monthly payment

income: Income of customer

We change YES to 1 and NO to 0 and plot default vs balance.

Let p(X) be the probability that Y = 1 given X. The simple form that fits this behaviour is:

P(X) = \frac{e^{\beta_0 + \beta_1 X}}{1 + e^{\beta_0 + \beta_1 X}}

The beta values (the intercept and slope) are obtained by the ‘generalised linear model’, glm function in R.

D_Data <- Default
D_Data$def <- ifelse(D_Data$default=="Yes",1,0)
model <- glm(def ~ balance, family = binomial(link = "logit"), data = D_Data)
summary(model)

Call:
glm(formula = def ~ balance, family = binomial(link = "logit"), 
    data = D_Data)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.2697  -0.1465  -0.0589  -0.0221   3.7589  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -1.065e+01  3.612e-01  -29.49   <2e-16 ***
balance      5.499e-03  2.204e-04   24.95   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 2920.6  on 9999  degrees of freedom
Residual deviance: 1596.5  on 9998  degrees of freedom
AIC: 1600.5

Number of Fisher Scoring iterations: 8

Now, we will test how good the model predictions are by plotting the function using the regressed beta0 ( = -10.65) and beta1 (= 0.0055).

plot(D_Data$balance, D_Data$def, xlab = "Balance", ylab = "Default")

y <- function(x){
  exp(-10.65+0.0055*x) / (1 + exp(-10.65+0.0055*x))
}

x <- seq(0, 3000)
lines(x, y(x), col = "red")

But R has an even simpler way to get this curve. 

plot(D_Data$balance, model$fitted.values)

Logistic Regression and R Read More »

Counting Humans Ever Existed

“The world’s human population is so large that the number of people alive today exceeds the number of people who died in the past!” Surely, you must have heard such claims. But how many of you have verified the truth behind such statements? Well, the Population Reference Bureau (PRB) do have estimates on it.

The number of people who ever lived is related to three factors:

  1. The duration of human existence
  2. The average number of people at different periods in history
  3. The birth rate during each period.

The duration of human existence

The assumption was made based on the current evidence that modern Homo sapiens appeared around 190,000 B.C.E., originated in Africa.

The average number of people

Available information/estimations about the world populations in different eras of human history are collected.

YEARPOPULATION
190,000 B.C.E2
50,000 B.C.E.2,000,000
8000 B.C.E.5,000,000
1 C.E.300,000,000
1200450,000,000
1650500,000,000
19001,656,000,000
20006,149,000,000
20227,963,500,000

Birth rate (BR)

Humans’ average life expectancy (LE) has stayed at about 12 years for most of history. Twelve years translates to 1000/12 ~ 80 births per year per 1,000 population. The number has since improved to the present value of 17. This equation (BR = 1000/LE) works in a stable population.

Once the birthrates are available at the different periods, one can assume a stable population between intervals and estimate the births each year (in between those intervals) as
the number of years (between two periods) x BR x average population /1000

The estimated cumulative number of humans as of 2022 was 117,020,448,575. In other words, today’s world population of 7,963,500,000 (2022) is 6.8% of all homo sapiens ever lived.

Reference

How Many People Have Ever Lived on Earth?: prb.org

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Survival in a Truel

Anna, Bob and Claire are doing a three-way duel (a truel). Anna has an accuracy of 1/3, Bob has 2/3, and Claire is an excellent marksman with 1 (100%) accuracy. They must take turns shooting. Since each player knows others’ skills and all are rational, what would Anna do in the competition?

If Anna starts the duel and aims at Bob, there is a 1/3 chance she will shoot Bob. In that case, Claire has the next turn and will end Anna. On the other hand, if Anna aims at Claire, there is a 1/3 chance that Bob gets a chance to aim at Anna and end her with a 2/3 chance. So, it is clear that missing shots is a better option for Anna. So, she must shoot in the air first.

In the next chance, Bob will aim at the more prominent threat, i.e., Claire. Then there is a 2/3 chance of a showdown with Anna and a 1/3 chance that the showdown will be between Anna and Claire. Let P(A) be the probability that Anna survives, P(A, B) be the chance Alice wins the duel with Bob and P(A, C) Alice wins against Claire.

P(A) = 2/3 x P(A, B) + 1/3 x P(A, C)
P(A) = 2/3 x 3/7 + 1/3 x 1/3 = 25/63

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The sum of Dice – 5 before 7

The game is to roll a pair of dice until a sum of 5 or 7 is reached. Then, what is the probability that a sum of 5 comes before a sum of 7?

We use a few handly R commands to answer the problem.

1. expand.grid: gives all combinations of vectors – in our case, two dice.

rolls <- expand.grid(seq(1,6), seq(1,6))

2. do.call: executes a function on a list of arguments, e.g. to perform + (sum) to the earlier list.

sum_rolls <- do.call(`+`, rolls)
2  3  4  5  6  7  3  4  5  6  7  8  4  5  6  7  8  9  5  6  7  8  9 10  6  7  8  9 10 11  7  8  9 10 11 12

3. table function to tabulate each categorical variable (2 to 12) with its frequencies.  

tab_rolls <- table(sum_rolls)
sum_rolls
 2  3  4  5  6  7  8  9 10 11 12 
 1  2  3  4  5  6  5  4  3  2  1

The table shows that the sum 5 occurs 4 times, and 7 can occur 6 times (out of the 36 possibilities). So, the probability of getting a five before seven (as the sum) is 4/(4+6) = 40%.

Combining steps into 2 lines.

dice_roll <- as.data.frame.table(prop.table(table(do.call(`+`, expand.grid(seq(1,6), seq(1,6))))))
dice_roll$Freq[which(dice_roll$Var1 == 5)] / (dice_roll$Freq[which(dice_roll$Var1 == 5)] + dice_roll$Freq[which(dice_roll$Var1 == 7)])

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Another Dice Puzzle

A regular die has six sides, and the probability of getting a six is 1/6. If the die is modified so that the number 6 now appears to be half than usual (the frequency is halved), what is the probability of rolling a 6?

dice, roll the dice, to play-2031512.jpg

The question implies that the probability of getting 6 is half that of the other probabilities (1 to 5). Let p6 be the probability of rolling a six. Let p be the probability of rolling any other side.

For the die:
5 x p + p6 = 1
given p6 = p/2
5 x p + p/2 = 1
11p = 2
p = 2/11
p6 = p/2 = 1/11

The probability of rolling a six is not 1/12 but 1/11.

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Borda count Method

Border Count is another method to determine the winner of a voting scheme. As per Wiki, the technique was first proposed in 1435 by Nicholas of Cusa but is named after the French mathematician Jean-Charles de Borda, who developed the system in 1770.

In this method, points are given to candidates based on their ranking:

  1. The last one gets 1 point
  2. The second-to-last gets 2 points, and so on.
  3. The points are totalled, and the one who gets the maximum points is the winner.

Here is our previous example:

# Voters412513263015
1st ChoiceLLFFCC
2nd ChoiceFCLCFL
3rd ChoiceCFCLLF

Points are given as follows:
First place: 3 points
Second Place: 2 points
Third place: 1 point

# Voters412513263015
1st Choice
(3 pts)		L
41×3
123		L
25×3
75		F
13×3
39		F
26×3
78		C
30×3
90		C
15×3
45
2nd Choice
(2 pts)		F
41×2
82		C
25×2
50		L
13×2
26		C
26×2
52		F
30×2
60		L
15×2
30
3rd Choice
(1 pt)		C
41×1
41		F
25×1
25		C
13×1
13		L
26×1
26		L
30×1
30		F
15×1
15

L: 123 + 75 + 26 + 30 + 26 + 30 = 310 points
F: 39 + 78 + 82 + 60 + 25 + 15 = 299 points
C: 90 + 45 + 50 + 52 + 41 + 13 = 291 points

Under the Borda count method, L is the winner.

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Charlie Munger’s Wisdom – Continued

We saw the first 10 of the 24 Cognitive Biases from Charlie Munger’s 1995 speech – Human Misjudgement. In this post, we will go through the rest.

11. Bias from deprival super-reaction syndrome: This is akin to the loss aversion bias, illustrating how we react when possessions, even trivial ones, are taken away. Consider, for instance, the intensity of employee-management negotiations where every inch is fiercely contested.

12. Bias from envy/jealousy:

13. Bias from chemical dependency, 14. Bias from gambling dependency: These biases don’t need a lot of explanation. They trigger attentional bias, leading individuals to allocate disproportionally more attention to addiction-relevant cues than to neutral stimuli. Businesses, whether pubs or casinos, are well-versed in exploiting these vulnerabilities. Being aware of these tactics can help you navigate such situations more effectively.

15. Bias from liking distortion: It’s about liking oneself, one’s ideas, and community. And making stupid ideas only because they came from someone you liked.

16. Bias from disliking distortion: Opposite of liking distortion. In this case, you dismiss ideas from people who you don’t like.

17. Bias from the non-mathematical nature of the human brain: Human brains in their natural state (i.e., untrained state) are notoriously inefficient when dealing with probabilities. Within this entity, Munger conveniently folds various fallacies of the human mind – crude heuristics, availability, base-rate neglect, hindsight – into one.

18. Bias from fear of scarcity: The fear of scarcity can bring out pure dumbness in otherwise perfectly normal people. A familiar example is the toilet paper rush during the early days of the Covid pandemic.

19. Bias from sympathy: It’s about leaders keeping employees with dubious personal qualities. Often, this happens out of pity for the person or her family. Munger says that while paying them the proper severance is essential, keeping such people in jobs can make the whole organisation poor.

20. Bias from over-influence and extra evidence:

21. Bias caused by confusion due to information not being properly processed by the mind: Munger stresses the need to understand the reasons (answer to the question – why?) for the information to be properly registered in the brain. Like for individuals themselves, it is also important for people to explain the reasoning clearly while communicating the key decisions and proposals to their stakeholders.

22. Stress-induced mental changes: What later happened to the Pavlovian dogs (conditioned for certain behaviours) after their cages were flooded was a good example of what stress can do. The canines forgot all the training and responses that they had acquired.

23. Common mental declines:

24. Say-something syndrome: It’s a habit of many individuals to do the talk irrespective of their expertise and capacities to impact the decision-making process. They remain just soundbites, and Munger cautions to watch out for those quiet selves that eventually add quality.

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