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Ideas for Leaders #685

How Linear Thinking in a Non-Linear World Leads to Wrong Decisions

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Key Concept

Our brains prefer to think in straight lines: if one bag of oranges costs $5, then two bags cost $10 and three cost $15. However, this bias toward linear thinking often traps unwary business decision-makers who fail to recognize the non-linear relationships they are dealing with (e.g. increasing retention rates from 10% to 30% or from 60% to 80% does not have an equal 20% impact on customer lifetime value).

Idea Summary

When it comes to business – and most areas of life – we tend to think in straight lines. For example, if one shelf holds 50 books, two shelves will hold 100 books and three shelves will hold 150 books. Graph the number of shelves on an x-axis and number of books on a y-axis and you’ll see a straight line shooting toward the northeast corner – proof that the relationship between shelves and is a linear relationship.

Linear calculations are easy and intuitive, and that’s why we get into trouble, according to Bart de Langhe of Esade Business School in Barcelona, Stefano Puntoni of the Rotterdam School of Management, and Richard Larrick of Duke University’s Fuqua School of Business: we often expect linear relationships between variables when in actuality the relationships are non-linear. 

For example, in a recent Harvard Business Review article, de Langhe, Puntoni and Larrick present a case study about a fleet manager looking for more fuel-efficient replacements for his cars — a decision that rests on a simple question: what is the better gas mileage deal, trading in a 10 mile-per-gallon car for a 20 mile-per-gallon car, or trading in a 20 mile-per-gallon car for a 50 mile-per-gallon car? 

Linear thinking would tell us that the second option is better: You’re increasing your efficiency by 30 miles per gallon instead of 10. Doing the math, however, reveals the error in this ‘gut’ assumption: upgrading a 10 mpg car for a 20 mpg car saves 500 gallons of gas on a 10,000-mile trip compared to just 300 gallons saved on the same trip if you trade in a 20 mpg car for a 50 mpg car.

If you graph the example above with gas mileage on the x-axis and gas used on the y-axis, you would have a line that starts high in the northwest corner, dips down precipitously as you move from 10 mpg to 20 mpg gas mileage, then flattens out more and more between 20 mpg and 50 mpg: the curved line of a non-linear relationship.

In short, a fleet manager replacing cars in his fleet would make an expensive mistake if he let intuitive linear thinking guide his decision.

The fact that the wrong answer in this example seemed so obvious is exactly why the linear thinking bias is a devious trap. In their article, de Langhe, Puntoni and Larrick detail how ‘linear bias’ entraps marketers, business executives and even consumers into making the wrong decisions because of the enticing (but irrelevant) numbers it focuses on (30 mpg savings per car!).

For example, marketers cut prices and marvel at the volume increase (a 40% price cut led to an 80% volume increase!). As the authors demonstrate in their article, however, if your price cut explodes your profit per unit, an eye-popping 80% volume increase is not good enough (in their example, the company would to increase volume by 133% just to reach their original pre-sale profits).

In sum, the next time the numbers point to an obvious decision, make sure you’re not making a linear assumption for a non-linear situation.

Business Application

How can you help your people avoid the linear bias trap? de Langhe, Puntoni and Larrick offer several suggestions:

Increase awareness of linear bias. Explicitly warn decision makers about how linear thinking leads to erroneous assumptions. Present them with puzzles and problems that open their eyes to false linear assumptions (e.g. do you consume more pizza eating one 12-inch pizza or two 8-inch pizzas?).

Focus on outcomes, not indicators. The problem is that indicators and outcomes often have non-linear relationships. While the progress of your website’s organic search position may be an indicator of rising sales, the relationship between sales and search position is non-linear: moving from 2nd to 1st in search position leads to a much larger increase in sales than moving from 25th to 20th despite the five-step jump of the second situation.

Discover the type of non-linearity you are dealing with. For example, the non-linear relationship between customer satisfaction and customer retention varies across industries: in some industries, satisfied customer retention rises gradually and then shoots up; in other industries, retention rises sharply and then flattens out. Field experiments help uncover which type of curve applies to your company.

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Idea conceived

  • June 2016

Idea posted

  • December 2017

DOI number



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