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Monthly Archives: November 2011

From zero to hero

New Scientist, November 2011
A concept of zero is essential for arithmetic to work smoothly – why then did the idea take so long to catch on? Richard Webb follows its convoluted path.

I used to have seven goats. I bartered three for corn; I gave one to each of my three daughters as dowry; one was stolen. How many goats do I have now?

This is not a trick question. Oddly, though, for much of human history we have not had the mathematical wherewithal to supply an answer. There is evidence of counting that stretches back five millennia in Egypt, Mesopotamia and Persia. Yet even by the most generous definition, a mathematical conception of nothing – a zero – has existed for less than half that time. Even then, the civilisations that discovered it missed its point entirely. In Europe, indifference, myopia and fear stunned its development for centuries. What is it about the zero that stopped it becoming a hero?

This is a tangled story of two zeroes: zero as a symbol to represent nothing, and zero as a number that can be used in calculations and has its own mathematical properties. It is natural to think the two are the same. History teaches us something different.

Zero the symbol was in fact the first of the two to pop up by a long chalk. This is the sort of character familiar from a number such as the next year in our calendar, 2012. Here its acts as a placeholder in our “positional” numerical notation, whose crucial feature is that a digit’s value depends on where it is in a number. Take 2012, for example: a “2” crops up twice, once to mean 2 and once to mean 2000. That’s because our positional system uses “base” 10 – so a move of one place to the lest in a number means a digit’s worth increases by a further power of 10.

It is through such machinations that the string of digits “2012” comes to have the properties of a number with the value equal to 2 x 103 + 0 x 102 + 1 x 101 + 2. Zero’s role is pivotal: were it not for its unambiguous presence, we might easily mistake 2012 for 212, or perhaps 20012, and our calculations could be out by hundreds or thousands.

The first positional number system was used to calculate the passage of the seasons and the years in Babylonia, modern-day Iraq, from around 1800 BC onwards. Its base was not 10 but 60. It didn’t have a symbol for every whole number up to the base, unlike the “dynamic” system of digits running from 1 to 9 that is the bread and butter of our base 10 system. Instead it had just two symbols, for 1 and 10, where were clumped together in groups with a maximum headcount of 59. For example, 2012 equates to 33 x 601 + 32, and so it would have been represented by two adjacent groups of symbols: one clump of three 10s and three ones; and a second clump of three 10s and two ones.

This particular number has nothing missing. Quite generally, though, for the first 15 centuries or so of the Babylonian positional numbering system the absence of any power of 60 in the transcription of any number was marked not by a symbol, but (if you were lucky) just by a gap. What changed around 300 BC we don’t know; perhaps one egregious confusion of positions too many. But it seems to have been at around this time that a third symbol, a curious confection of two left slanting arrows, started to fill missing places in the stargazers’ calculations.

This was the world’s first zero. Some seven centuries later, on the other side of the world, it was invented a second time. Mayan priest-astronomers in central America began to use a snail-shell-like symbol to fill gaps in the (almost) base 20 positional “long count” system they used to calculate their calendar.

Zero as a placeholder was clearly a useful concept, then. It is a frustration entirely typical of zero’s vexed history, though, that neither the Babylonians nor the Mayans realised quite how useful it could be.

In any dynamic, positional number system, a placeholder zero assumes almost unannounced a new guise: it becomes a mathematical “operator” that brings the full power of the system’s base to bear. This becomes obvious when we consider the result of adding a placeholder zero to the end of a decimal number string. The number 2012 becomes 20120, magically multiplied by the base of 10. We intuitively take advantage of this characteristic whenever we sum two or more numbers, and the total of a column ticks over from 9 to 10. We “carry the one” and leave a zero to ensure the right answer. The simplicity of such algorithms is the source of our system’s supple muscularity in manipulating numbers.

Facing the void

We shouldn’t blame the Babylonians or Mayans for missing out such subtlety: various blemishes in their numerical systems made it hard to spot. And so, although they found zero the symbol, they missed the zero number.

Zero is admittedly not an entirely welcome addition to the pantheon of numbers. Accepting it invites all sorts of logical wrinkles that, if not handled with due care and attention, can bring the entire number system crashing down. Adding zero to itself does not result in any increase in its size, as it does for any other number. Multiply any number, however big, by zero and it collapses down to zero. And let’s not even delve into what happens when we divide a number by zero.

Classical Greece, the next civilisation to handle the concept, was certainly not keen to tackle zero’s complexities. Greek thought was wedded to the idea that numbers expressed geometrical shapes; and what shape would correspond to something that wasn’t there? It could only be the total absence of something. the void – a concept that the dominant cosmology of the time had banished.

Largely the product of Aristotle and his disciples, this world view saw the planes and stars as embedded in a series of concentric celestial spheres of finite extent. These spheres were filled with an ethereal substance, all cantered on Earth and set in motion by an “unmoved mover”. It was a picture later eagerly co-opted by Christian philosophy, which saw in the unmoved mover a ready-made identity for God. And since there was no place for a void in this cosmology, it followed that it – and everything associated with it – was godless concept.

Eastern philosophy, rooted in ideas of eternal cycles of creation and destruction, had no such qualms. And so the next great staging post in zero’s journey was not to Babylon’s west, but to its east. It is found in Brahmasphytasiddhanta, a treatise on the relationship of mathematics to the physical world written in India in around 628 AD by the astronomer Brahmagupta.

Brahmagupta was the first person we see treating numbers as purely abstract quantities separate from any physical or geometrical reality. This allowed him to consider unauthodox questions that the Babylonians and Greeks had ignored or dismissed, such as what happens when you substract from one number a number of great size. In geometrical terms this is a nonsense: what are is left when larger area is substracted? Equally, how could I ever have sold or bartered more goats than I had in the first place? As soon as numbers become abstract entities, however, a whole new world of possibilities is opened-up the world of negative numbers.

The result was a continuous number line stretching as far as you could see in both directions, showing both positive and negative numbers. Sitting in the middle of this line, a distinct point along it at the threshold between the positive and the negative worlds, was sunya, the nothingness. Indian mathematicians had dared to look into the void – and a new number had emerged.

It was not long before they unified this new number with zero the symbol. While a Christian Syrian bishop writes in 662 that Hindu mathematicians did calculations “by means of nine signs”, an incription of dedication at a temple in the great medieval fort at Gwalior, south of Delhi in India, shows that two centuries later the nine had become ten. A zero – a squashed egg-symbol recognisably close to our own – had been incorporated into the cannon, a full member of a dynamic positional number system running from 0 to 9. It marked the birth of the purely abstract number system now used throughout the world, and soon spawned a new way of doing mathematics to go with it; algebra.

News of these innovations took a long time to filter through to Europe. It was only in 1202 that a young Italian, Leonardo of Pisa – better remembered as Fibonacci – published a book, Liber Abaci, in which he presented details of the Arabic counting system he had encountered on a journey to the Mediterranean’s southern shores, and demonstrated the superiority of this notation over the abacus for the deft performance of complex calculations.

While merchants and bankers were quickly convinced of the Hindu-Arabic system’s usefulness, the governing authorities were less enamoured. In 1299, the city of Florence, Italy, baned the use of the Hindu-Arabic numerals, including zero. They considered the ability to inflate a number’s value hugely simply by adding a digit on the end – a facility not available in the then dominant, non-positional system of Roman numerals – to be an open invitation to fraud.

Zero the number had an even harder time. Schisms, upheavals, reformation and counter-reformation in the church meant a continuing debate as to the worth of Aristotle’s ideas about the cosmos, and with it the orghodoxy or otherwise of the void. Only the Copernican revolution – the crystal-sphere-shattering revelation that Earth moves around the sun – began, slowly to shake European mathematics free of the shackles of Aristotelian cosmology from the 16th century onwards.

By the 17th century, the scene was set for zero’s final triumph. It is hard to point to a single event that marked it. Perhaps it was the advent of the coordinate system invented by the French philosopher and mathematician Rene Descartes. His Cartesian system married algebra and geometry to give every geometrical shape a new symbolic representation with zero, the unmoving heart of the coordinate system, at its centre. Zero was far from irrelevant to geometry, as the Greeks had suggested: it was essential to it. Soon afterwards, the new tool of calculus showed that you had first to appreciate how zero merged into the infinitesimally small to explain how anything in the cosmos could change its position at all – a star, a planet, a hare overtaking a tortoise. Zero was itself the prime mover.

Thus a better understanding of zero became the fuse of the scientific revolution that followed. Subsequent events had confirmed just how essential zero is to mathematics and all that builds on it. Looking at zero sitting quietly in a number today and primed with the concept from a young age, it is equally hard to see how it could ever have caused so much confusion and distress. A case, most definitely or much ado about nothing.

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Posted by on November 26, 2011 in Science

 

Customer Loyalty Isn’t Enough. Grow Your Share of Wallet

Harvard Business Review, October 2011
By Timothy L. Keiningham, Lerzan Aksoy, Alexander Buoye and Bruce Cooil.

Companies spend a great deal of time and money trying to improve customer loyalty by measuring and managing metrics like satisfaction and Net Promoter Scores. But traditional gauges of loyalty correlate poorly with what matters most: share of wallet. This is the percentage of a customer’s spending within a category that’s captured by a given brand, or store or firm. Customers may be very satisfied with your brand and happily recommend it to others—but if they like your competitors just as much (or more), you’re losing sales. Making changes to increase satisfaction won’t necessarily help. This doesn’t mean traditional metrics aren’t valuable; it can be very useful to know whether your customers are satisfied and would recommend you to their friends and colleagues. But these measures in themselves can’t tell you how your customers will divide their spending among you and your competitors.

Walmart had a rude awakening in this regard. In 2008, guided by extensive customer feedback, it launched Project Impact, a remodelling initiative designed to improve customers’ experiences. It removed unsightly stacks of pallets from the aisles, trimmed distracting endcap displays, and thinned out overstuffed shelves. As expected, satisfaction scores rose. But same-store sales entered their longest decline in the company’s history. “The customers, for the most part, are still in the store shopping,” Charles Holley, Walmart’s chief financial officer, recently observed, “but they’ve started doing some more shopping elsewhere.” Even as satisfaction increased, share of wallet fell.

If traditional loyalty metrics don’t link to share of wallet, what does? To find out, we undertook a two-year longitudinal study of more than 17,000 consumers, looking at purchasing in more than a dozen industries and in nine countries. We asked a broad array of questions and collected on-going purchase histories and satisfaction and loyalty ratings. Our analysis—to our knowledge the largest and most rigorous of its kind—revealed an elegant correlation: The rank that consumers assign to a brand relative to the other brands they use predicts share of wallet according to a simple, previously unknown formula, which we’ve named the Wallet Allocation Rule. From company to company and industry to industry, the correlation between a brand’s Wallet Allocation Rule score and its share of wallet was remarkably consistent—the average was greater than 0.9 (a perfect correlation is 1.0). Even more important, the correlation between changes in the Wallet Allocation Rule score and in a customer’s share of wallet was a robust 0.8. The correlation between changes in satisfaction or intention to recommend and in share of wallet was very weak—only 0.1.

The essential distinction of the Wallet Allocation Rule is that it takes into account both rank—Is your brand a customer’s first choice? Second?—and the number of brands in the set the consumer uses. Knowing these two values allows you to confidently predict share of wallet. (For a step-by-step demonstration of the calculation, see the exhibit “Using the Wallet Allocation Rule.”) For example, if your brand is one of only two a customer uses for a given purpose, the rule shows that the difference between being her first choice and being her second can have a major financial impact. In such a situation, even being tied has grave consequences: Half of each dollar you could be collecting from the customer is going to your competitor instead. The flip side is that the negative impact of being second diminishes as the consumer’s choice set increases.

HOW TO IMPROVE YOUR RANK
Boosting your Brand’s rank means minimizing the reasons your customers turn to your competitors. Below is a simple process you can implement right away.
FOLLOW the Wallet Allocation Rule to establish the share of wallet to each competitor your customers use. PRIORITIZE your opportunities to improve your share of wallet: Estimate the costs of addressing each reason your customers choose a competitor and weigh those costs against your potential financial return in each case. Remember to take into account the cumulative impact of addressing issues that apply to multiple competitors.
DETERMINE how many of your customers use each competitor.
CALCULATE the revenue that goes from your customers to each competitor
IDENTIFY the primary reasons your customers use your competitors.

The Rule in Practice

The new rule has important implications for strategy. To understand what drives changes in share of wallet, managers need to shift their focus from drivers of satisfaction to drivers of rank.

First, you can’t assess brand performance as if it existed in a vacuum. That sounds obvious, but in reality it’s exactly what most managers do, measuring customer satisfaction or using other metrics that are based on customers’ perceptions of their brand alone. As a result, the loyalty objectives used to evaluate and compensate managers usually have to do with achieving a certain satisfaction rating (which rarely boosts share of wallet), not with improving a brand’s rank (which actually does).

Second, the rule makes it possible to craft strategies that directly affect brand performance and then measure the impact on share of wallet. Think about how a company typically tries to improve share of wallet. The effort often boils down to launching initiatives intended to make customers happier and then measuring satisfaction. As Walmart discovered, even initiatives that result in happier customers may have little or no positive impact on the top line. Instead, companies should understand exactly why their customers use each of the brands they do. If you’re not number one, you should ask your customers why they prefer your competitor and use the insights you gain to move up the ranking ladder. The Wallet Allocation Rule is clear on this point: If you can’t improve your rank, you can’t improve your share of wallet.

Let’s look at a composite case, drawn from our research, that illustrates how a full-service grocery retailer might put the rule to use. The grocer surveys its customers and finds that they are generally very happy with their experience—53% give the store a nine or 10 on a 0-to-10-point “would recommend” scale. However, despite these good scores, only 43% of customers rank the grocer as their first choice. The unpleasant implication is that 57% either prefer one or more of its competitors or consider the grocer to be tied with one of them. Using the Wallet Allocation Rule, the grocer calculates its average share of wallet and that of its three main competitors. Multiplying these estimates by its customers’ average monthly grocery spend and the number of its customers who also patronize the competing stores, the grocer determines that its top three competitors are extracting a total of $425 million from its customers’ wallets—some of which it could capture by moving up in the ranks.

Returning to the store’s customer surveys, managers learn that the top reasons its satisfied customers recommend the grocer are the superior quality of its produce and the ambience. This is not surprising; management has worked hard to differentiate the grocer on these parameters. What attracts the store’s customers to the competition? The survey indicates that for Competitor One, the primary attraction is everyday low prices. Competitor Two also competes on price, but largely through rotating deep discounts. Competitor Three’s main appeal is the convenience of its locations.

The managers immediately realize that if the grocer is to move up to first place in more of its customers’ minds, it cannot simply enhance what it already does well; stocking even better produce or improving the aesthetics might further delight customers who already rank it first but would be unlikely to change the minds of the rest, who are mainly interested in low prices and convenience.

The grocer can’t compete on price in every category, so its managers decide to drop prices on its most commonly purchased staples, reasoning that customers who are already attracted to the store for its produce and ambience will then have less reason to shop at its strongest competitor, the everyday-low-price store. Surveys after the price change find that 49% of customers now peg the grocer as their first choice (a gain of 6%) and that the number of stores customers regularly shop in had dropped from 2.5 to 2, on average. These changes, when plugged into the Wallet Allocation Rule, translate to a seven-point increase in share of wallet. Its’ the equivalent of shifting $62 million from competitors’ registers into the grocer’s own.

Many companies could see this kind of revenue jump if they decided not to pursue customer satisfaction for its own sake and focused instead on how satisfaction and other loyalty boosters could help them pull ahead of the competition. If growth is what you’re after, stop watching your scores and start paying attention to your rank. The path to winning has always been the same. It’s not just how many points you score that matters – you need to score more than your competitors do.

Timothy L. Keiningham is a global chief strategy officer and an executive vice president at Ipsos Loyalty. Lerzan Aksoy is an associate professor of marketing at Fordham University. Alexander Buoyee is the vice president of analytics at Iposos Loyalty. Bruce Cooil is the Dean Samuel B and Evelyn R. Richmond Professor of Management at Vanderbilt University’s Owen Graduate School of Management.

 
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Posted by on November 19, 2011 in Management Fundas

 
 
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