While procrastinating from writing my thesis, I came across an interesting property of coin tosses (I realise how pathetic that sounds while writing it). But bare with me, in the end it is about more than a simple coin toss experiment but also about why this experiment exemplifies a major flaw in the thinking of today's researchers. On top, it is a little coding exercise on how to visualise simple markov chains in R.

Coin Tosses

The story goes as follows: Bob and Alice are playing a game. Both have a fair coin and flip it over and over again. Bob will end the game when he gets two heads in a row. Alice will end the game when she gets heads directly followed by tails. On average Bob needs six coin tosses to end the game and Alice needs four coin tosses.

Sounds counter-intuitive, right? If a coin is fair, a sequence should be equally likely to end with Head, Head, or Head, Tail. So I used the time in which I should actually write my thesis to find out what this is about. Before I explain the experiment using markov chains, let’s quickly simulate the experiment in R:

 
	 
	# Write a function to simulate one coin toss 

	coin_toss <- function(){ 

	rbinom(1,1, 0.5) 

	} 

	
	# Write a function that simulates the game --> Bob: condition = c(1,1); Alice: condition = c(1,0) 
	
	simulate_game <- function(condition = c(1,1)){ 

	condition_met <- F 

	tosses <- c() 

	while(condition_met != T){ 

	tosses <- c(tosses, coin_toss()) 

	condition_met <- all.equal(tail(tosses, 2), condition) 

	} 

	return(tosses) 

	} 

	
	# Simulate game for 10000 times and save the number of tosses each one needed 

	
	games_bob <- c(); games_alice <- c() 

	
	for(i in 1:10000){ 

	games_bob[i] <- length(simulate_game(condition = c(1,1))) 

	games_alice[i] <- length(simulate_game(condition = c(1,0))) 

	} 

	# Mean and Distribution for Heads, Heads to end the game 
	mean(games_bob) 

	
	# Mean and Distribution for Heads, Tails to end the game 
	mean(games_alice) 

	 

	
 
	# Mean and Distribution for Heads, Heads (Bob) to end the game 
	[1] 6.007 
	




 
	# Mean and Distribution for Heads, Tails (Alice) to end the game  
[1] 3.961 
	
	




The intuitive explanation is as follows: Both Alice and Bob first need to get heads. For Alice, if she gets tails for the next toss the game is over. If she doesn't get tails but heads shows up again it is not too bad either because with the next toss she has again the chance to get tails and end the game. This is where the difference between Alice's and Bob's winning conditions are: If Bob fails to get heads after tossing heads the first time, his streak gets reset and he needs at least two more tosses to win.

Markov Chains

I found it helpful to visualise the problem with Markov Chains (after getting the inspiration from Nassim Talbes' Twitter post: @nntaleb). Markov chains are stochastic models which are useful when describing a sequence of events. A transition plot helps to visualise how the different states are connected stochastically.

For Bob there are the following three possible states: H (Heads), T (Tails), HH (Heads, Heads) . For Alice there are the following three possible states: H (Heads), T (Tails), HT (Heads, Tails)

 
	 
	
	states_bob <-c("H","T","HH") 
	states_alice <- c("H", "T", "HT") 
	
	 
	

Then we first have to build the transition matrices for both Bob and Alice. After that we create markov chain objects so that we can plot them.

 
	 
	
	library(markovchain) 
	
	#Transition matrix for Bob 
	tm_bob <- matrix(c(0,.5,.5, 
	0.5,0.5,0, 
	0,0,1), 
	nrow=3, byrow=TRUE) 
	row.names(tm_bob) <- states_bob; colnames(tm_bob) <- states_bob 
	tm_bob 
	
	#Transition Matrix for Alice 
	tm_alice <- matrix(c(.5,0,0.5, 
	0.5,.5,0, 
	0,0,1), 
	nrow=3, byrow=TRUE) 
	row.names(tm_alice) <- states_alice; colnames(tm_alice) <- states_alice 
	tm_alice 
	
	
	#Layout for Transition Plot 
	layout_tp <- matrix(c(0,0,0,1,1,1), ncol = 2, byrow = TRUE) 
	
	# Plot transition for Bob 
	mc_bob <- new("markovchain", states=states_bob, transitionMatrix=tm_bob) 
	plot(mc_bob,package="diagram", 
	layout = layout_tp, 
	edge.curved = -0.1) 
	
	#Plot transition for Alice 
	mc_alice <- new("markovchain", states=states_alice, transitionMatrix=tm_alice) 
	plot(mc_alice,package="diagram", 
	layout = layout_tp, 
	edge.curved = -0.1) 
	
	 
	

The expectation for Alice's markov chain (MC) to converge to the state of "HT" are 4 iterations, and the expectation for Bob's MC to converge to the state of "HH" are six iterations. You immediately see, why: Bob has an additional arrow (from "H" to "T").







Existential Crisis

I came across this problem first in a Tweet by David Robinson (@drob) and two hours later I saw Nassim Nicholas Taleb tweeting about it (I mean, at this point I just had to write a blog post about it).

The reason why people (especially people familiar with probability theory) first think that it is counterintuitive is "because effects like linearity of expected value spoil us in being able to treat events as independent even if they aren't" (David Robinson). Nassim Taleb goes even further and argues "It is ONLY counterintuitive then you are trained/selected to think in Static terms".

In his new book "Skin in the Game - Hidden Asymmetries in Daily Life" Taleb explains why the thinking in static terms is one of the core problems of social scientists and economists today. He argues that most social scientists are not able to think in dynamics; we are not able of "thinking in second steps and unaware of the need for them [...] while every car operator in real life knows that real life happens to have second, third and n-th steps" (p. 9). In the book he explains why "static thinking" is a major flaw in the work of accomplished researchers like Thaler (Behavioural Economics), Piketty (Inequality), and Krugman (Trade). Static thinking is an attribute of what he calls the "Intellectuals Yet Idiots (IYI)".

Wow, a simple coin toss experiment turned into an existential academic crisis really fast...