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Author: Matteo Iannucci

The cake paradox

14/10/2016 /

A few days ago at work, I found myself in one of those situations where numbers behave in a counterintuitive way. A situation where an apparently logical reasoning leads to incorrect conclusions—what I call the cake paradox.

I brought up cakes because I used them to explain (with difficulty) to my colleagues where the trick was.

Let’s go over everything with a similar example (made-up data for illustrative purposes):

  • “Leggo” is an Italian publishing company active in two markets: Italy and China.
  • In Italy, “Leggo” is the market leader, with a 50% market share in 2015 (meaning, for every 100 books sold, 50 are published by “Leggo”).
  • In China, “Leggo” has a 5% market share in 2015.
  • Given the size differences of the markets, “Leggo” has a total market share of 9.1% (aggregating data from Italy and China, with calculations available in the table below).
  • In 2016, “Leggo” increases its sales and manages to steal market share from its competitors in both Italy and China.
    • In Italy, it goes from 50% to 52% market share.
    • In China, it goes from 5% to 6%.

Even though “Leggo” performed better than its competitors in both markets, the overall market share (China + Italy) decreased from 9.1% to 8.8%.
Does something not add up?

How is it possible that even though “Leggo” is doing very well, increasing its sales in both markets, stealing market share both in Italy and China, its total market share is decreasing?

Try it for yourself; all the calculations are in the table:

ItalyChinaTotal
2015 Book market 100 m1,000 m1,100 m
2015 “Leggo” sales50 m50 m100 m 
2015 Market share50%5%9.1%
2016 Book market102 m1,600 m1,703 m
2016 market growth2%60%55%
2016 “Leggo” sales53 m96 m149 m
2016 market share52%6.0%8.8%

The paradox is explained by the two different markets: Italy and China. Italy is much smaller than China, and China is growing at a much higher rate than Italy.

Forget the example of books with realistic numbers; let’s abstract the concept by thinking of the cakes mentioned earlier.

We have two cakes of similar size, one white and one black. Initially, we own 80% of the white cake and only 20% of the black cake.
In total, we own the equivalent of one entire cake (meaning our market share will be 50%, one cake out of two).

Suddenly, the black cake rises and becomes 100 times the size of the white cake.
Our total market share drastically drops from the previous 50% to a number close to 20%, the market share of the huge cake (exactly 20.6%).

Now, it should be easier to understand how our market share, which was previously an average of 20% and 80% (since the cake were the same size), will now be much closer to 20% of the black cake, which is 100 times larger than the white cake.

In practice, our dominant position in the white cake, which used to balance out the black cake (since they were of similar size), now barely matters in a situation where the black cake represents almost the entire market.

The Cake Paradox shows us how it can be detrimental for a company to hold a leadership position in a slow-growth market. Even if it outperforms its competitors in emerging markets, it can still lose market share overall!

The Gambler’s Ruin Theorem

17/02/2016 /

I’ve always been fascinated by gambling: the adrenaline generated by risk, the dream of a big win.

Since I first understood how roulette works, I’ve been struck by how “fair” this game is.

When you do the simple math, betting €1 on any number gives you a 1/37 chance of winning, and in the case of a win, the house pays 36 times the stake. Playing repeatedly, the house wins on average only 2.7% of the total amount bet (1/37). Blackjack is even more “fair.” Played by the rules, the house edge drops to as low as 0.5%.

When we think of games more familiar to us, like the Lotto, a single number has a 1 in 18 chance of being drawn and pays 10.23 times the stake (a margin of 38%). This margin doesn’t change whether the number is overdue or not, as Giovanni explained to us here.

The “Cinquina” (five-number match), however, is a real theft, with a margin of 86%. That is, for every €100 wagered, the house pays out only €14 in winnings on average (compared to €97.3 for roulette!). Similarly popular sports betting can have a margin of up to 40–50% when played as a parlay.

House Edge by Game Type

GiocoMargine del banco
BlackJack0.5-1.5%
Roulette2.7%
Slot machine3-10%
Scommessa sportiva* (in singola)3-10%
Scommessa sportiva* (in multipla)20-50%
Lotto - Estratto38%
Lotto - Cinquina86%

*Indicative margins for sports betting.

After appreciating the low margins of casino games, I discovered something else that decisively tilts the odds in the house’s favor: the so-called “Gambler’s Ruin Theorem,” which highlights the significant difference between a casino’s capital and that of the player.

Using the chart, it’s easy to see how, even in a fair game, a player spending an evening at the casino is almost doomed to lose.

Let’s assume the player is willing to lose €15. The house, with far greater resources, only needs to wait for fortune’s fluctuations to push the player to the point of no return (in the chart, this happens on the 430th round). While the player’s balance oscillates around zero, the game ends when the player runs out of funds, whereas the house essentially has no loss limit.

The chart demonstrates this with a zero-margin game (perfect fairness). In the presence of even a small house edge, the player will reach the point of no return even faster.

The theorem can be summarized by this formula, calculating the probability of the player’s ruin in a fair game (zero margin), assuming play continues until either the house or the player runs out of capital:

Player’s Probability of Ruin = House Capital / (Player Capital + House Capital)

Even with modestly different amounts (Player = €15, House = €40), the odds are heavily skewed, with the player going broke in 72% of cases. Considering the house usually has vastly greater capital than the player, the probability of ruin approaches 100%.

At this point, it should be clear that the low house edge in casino games does not limit the house’s profitability.

In light of this, the next time you spend an evening at the casino, either bet all your capital in one go (and nearly break even) or simply enjoy the price of an entertaining evening!

A statistical approach to terrorism

22/11/2015 /

Datastory.it is also about current events, and following the attacks in Paris, we want to share our opinion on the matter.

The series of attacks that struck the French capital on November 13, 2015, seems to have shaken public opinion and mobilized European governments. In newspapers, parliaments, and international forums, the primary focus is how to ensure safety and prevent the horrific events in Paris from happening again. Many hypotheses are being considered: stricter border controls, revising the Schengen Agreement, increased surveillance in high-risk areas, and the installation of cameras in major cities.

Additionally, there are discussions about allocating more personnel and resources to security (the press mentions €400 million in Belgium, €120 million in Italy). And then there are the bombings in Iraq and Syria, with the United States, Russia, and France taking the lead. Some estimates suggest the U.S. spends $10 million daily on these operations, while Russia spends about a third of that amount.

The caliphate in Iraq and Syria is undoubtedly a threat to the Western world. More broadly, in recent years, Islamist terrorism has caused deaths and suffering even in Europe.

From the Madrid bombings on March 11, 2004, to the Paris attacks on November 13, 2015, 411 people have died in seven Islamist terrorist attacks. The number rises to 488 if we include the July 22, 2011, attack in Norway by Breivik (which had an anti-Islamic motivation).

But how much does it cost us to protect ourselves from terrorist attacks? How far are we willing to go to prevent more lives from being lost to fanatics killing in the name of their religion?

And in a world where resources are limited, what are we willing to give up to increase our security?

Let’s reflect on these questions using some concrete examples. In Italy alone, around 1,000 people die each year in workplace accidents (source: Osservatorio sui morti del lavoro), 3,385 people in road accidents (2013, source: ISTAT), 12,004 women from breast cancer (2012, source: ISTAT), and an astonishing 83,000 deaths are attributable to smoking!

How many lives could be saved if the significant resources budgeted for defense against terrorist attacks were instead used for anti-smoking campaigns? If millions were invested in securing the road network and law enforcement increased traffic and alcohol checks? How many of the 12,000 women who die each year from breast cancer could be saved if we doubled free mammograms?

It’s difficult to answer these questions (though we will delve deeper into this topic), but let’s make a few assumptions. Let’s take the €120 million the government has declared it will allocate to protect us from terrorist attacks following the Paris events. We could decide to allocate them in three different ways, as outlined in the table below:

InvestmentImpact (Assumption)
Informational campaign on the dangers of smoking in all schools and free copies of the book “The Easy Way to Stop Smoking” for all smokers who request it1% reduction in smoking-related deaths: 830 lives saved per year
Doubling free mammograms (every year instead of every two years)10% reduction in breast cancer deaths annually: 1,200 lives saved per year
Installing speed enforcement systems on all highways and tripling alcohol-related checks5% reduction in road deaths: 170 lives saved per year

How should we decide how to allocate the €120 million? Is it wise to invest it in counterterrorism (which has caused 488 deaths across Europe in the past 15 years) rather than in breast cancer prevention, which could save 1,200 women per year in Italy alone?

It’s legitimate to think that the Paris attacks did not just cause 136 deaths but also created a sense of insecurity. But how do we determine the value of a life lost to terrorism compared to one lost in a road accident?

How can we allocate our €120 million wisely, rather than being emotionally swayed by recent events?

What do you think? Share your thoughts in the comments!

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