How to take a chance

08 March 2012 -


Grasping probabilities and game theory is as key to success at the office as it is in the Casino, writes Ben Walker

Did you hear the amazing story of the woman who went on a long-trunk holiday and met an acquaintance by chance on the beach? Odds on, you probably did.

Statistically, it’s not in any way amazing. “A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates,” says the sharp-tongued mathematician John Allen Paulos in his book Innumeracy, which contends that many otherwise educated people are comfortable – even take pride in – their innumeracy. “If you don’t specify a predicted event precisely, there are an indeterminate number of ways for that event to take place.” Paulos takes us through the paradox of ostensibly unlikely events actually being very likely. If there are more than 23 people in a room, he writes, odds are that two of them will have the same birthday. If two American strangers meet on a plane, he adds, there’s a 99% chance they are be linked by a sequence of two intermediates. It’s certain, he says, that at least two people living in Philadelphia have exactly the same number of hairs on their head. Interesting? Perhaps. But what’s all this got to do with careers, management and leadership? A lot: because building a career, managing and leading rely in great part on your willingness to take calculated risks. And to take calculated risks, you first need to be able to calculate them. Most people can’t.

Calculating chance

There are two key variables one must consider when calculating risk: the odds and the stakes. To bring this to life, let’s consider the fictional example of Dave, a manager in the creative sector. Dave has a vacancy for a designer. Two candidates stand out – both have equal technical skills, Jim has more experience and a more urbane manner; Donna is brusquer but with brighter ideas. What should Dave do? First he must consider the risks associated with each hire, and the stakes. The main risk associated with hiring someone is that they leave, either voluntarily or by compulsion, because they are not able to do the job; because the job doesn’t suit them; or because they don’t suit the people they work with. Research shows that it is rare for people to be let go during their probationary period due to poor performance: in almost all cases a lack of skills is weaned out at the interview stage. It’s social clashes that are usually responsible for an early termination of employment. Although Donna is better skilled, Jim is almost certainly good enough to do the job. Therefore we have to see Donna as a considerably bigger risk. Given her sometimes brusque tone, we might rate her as a 30% chance of upsetting her co-workers to such an extent that her continued employment is untenable, regardless of the quality of her work. The Chartered Institute of Personnel Development says the average cost of replacing an employee is £7,750, once associated costs of staff turnover are accounted for.

Mathematicians calculate risk (r) by multiplying the stake (s) by the probability of the event occurring (p). In Donna’s case this is 30%, which is expressed as 0.3.

r = s x p

So in Donna’s case, the monetised risk of hiring her is £7,750 x 30% (0.3) = £2,325. In other words, the average cost to a business of hiring Donna (or a similar personality) would be £2,325 – because three times out of ten their employment would be terminated, and cost the organisation £7,750.

Let’s go back to Jim. We know he’s urbane, polite and charming, and that the chances of his not being able even to do the job are remote. Therefore, we might reasonably calculate that the probability his employment doesn’t work out is as low as 4%. Therefore the monetised risk of his hire is a fraction of Donna’s, at £310, hardly a king’s ransom. Easy equation then? Hire Jim.

But maybe it’s not that easy. None of the risk assessment above accounts for the possible benefits to the firm of choosing Donna. Remember that she is a better designer, so has more potential upside than Jim. If you never hire a Donna, it’s likely that your designs might be a bit safe, lack innovation, and edge. Sometimes managers do have to take risks. This is where the art of taking a chance – choosing what risks to take and when – comes into play.

Shari Peace, president of professional speaking firm Peace Talks, suggests six tests to see whether you should take a chance on a Donna-style risky. If the risk you were considering taking falls outside Peace’s blacklist, you should almost certainly take it – even if you feel uncomfortable about it. “Every day, ask yourself what you’ve done today that is daring or that is a bit of a stretch,” the psychologist and business coach Gary Leboff tells Natural Health Web. “If at the end of each day you can find just one thing, then you are moving, but if not, then you are getting more and more limited and your horizons are getting smaller and smaller.”

Indeed, the risk of the status quo is often overlooked. Inertia is often mistaken for sanctuary, says Professor Morgan W. McCall Jr, author of Whatever It Takes, which explores managerial decision-making. “Risk can be associated with staying the same,” he tells Professional Manager. “The world is go to change around you. Given that, I don’t know how you avoid risk. The trick is to make that risk a calculated one – not a wild one.”

Return on luck

So how do you do that? “That’s the tricky question,” he admits. “There are always things you can do to lay the groundwork to be more successful.” The reverse is also true – you can also exacerbate the negative return on a piece of bad luck. In another of his books, Off the Track, McCall examined the reasons managers’ fail. “When we looked at the mistakes that derailed people we always found that they had contributed in some way to their own downfall,” he says, “even if the initial problem was just bad luck.”

This influence human will has on the outcome of random events is known as return on luck. Sport fans will recall the quote by the South African golfer Gary Player who, in response to a spectator’s gibe about his holing a “lucky” shot out of a bunker, responded: “The more I practice, the luckier I get.” Luck is by its very nature a haphazard happenstance that occurs regardless of one’s skill level – but Player articulated succinctly that the best golfers make the most of any luck they get. Hit the ball consistently well and, sometimes, you’ll make a birdie from the bunker. Prefer poker to golf? The analog is winning a large pot when you have a slightly better flush than your opponent; and folding cheaply when your pair of Queens are edged out by your adversary’s Kings.

Like great golfers and poker players, “luckier” managers are not luckier at all – they simply maximise the benefit of the good luck they do have, and minimise the costs of the bad. This important concept of return on luck is explored in detail in the management tome Great by Choice.

The book analyses the fortunes of so-called 10X companies, firms that outperform the benchmark for their industry by ten or more times. The author, Jim Collins, found that leaders of those companies were no more likely to get more luck, or luck at a more advantageous time, than those at an average organisation. “The real difference,” writes Collins, “wasn’t luck per se but what they did with the luck they got.”

Luck is outside our control. Return on luck is within it. So what is the best way to improve our return on luck? Great by Choice is clear. It’s people. More specifically, it’s working with the right ones. The book refers to a speech made by Gordon Binder, chairman of biotech specialists Amgen, a 10X firm. Binder told the audience of the defining moment in his company’s history. He didn’t choose a new territories rollout, a stellar promotional campaign or a corporate restructure. He chose the hire of a Taiwanese scientist. Fu-Kuen Lin worked tirelessly on cloning a gene, being mocked on the way by his colleagues, who thought the project doomed to failure. The outcome was rather different – Fu-Kuen’s work led to the first billion-dollar biotech blockbuster. “The best way to find a strong current of good luck is to swim with the right people,” writes Collins. The trick is to know how to find the Fu-Kuen’s. Understanding game theory helps.

Game theory

If you have ever seen the film A Beautiful Mind, you’ve been misled. The most famous scene from the film tries to bring game theory, and specifically Nash equilibria, to life by applying it to dating. Nash equilibrium is an important concept to grasp – it is an outcome whereby no player regrets their strategy given what their opponents have done. In the film, a truly stunning blonde walks into a bar accompanied by four of her friends who, while attractive, are not as beautiful as their flaxen-haired companion. Russell Crowe’s character, the mathematician John Nash, who would later win the Nobel Prize for Economics, was seated with three male friends. He argued that, rather than their all hit on the blonde, as single men might normally do, they should all ignore the blonde and chat up one of her friends each.

The film explains that, if they all went for the blonde, they’d just get in each other’s way, badger her and none of them would score. Nor would they then be able to approach one of her friends – as no woman likes to be second choice. The film is right that the ‘all chase the blonde’ strategy is not a Nash equilibrium. “But,” argues author of management book Game Theory at Work Professor James Miller, “A Beautiful Mind should be stripped of its Oscars because the outcome that Nash proposes in the movie is not a Nash equilibrium either. Each of the men would regret a strategy of ignoring the babe if everyone else ignored her too. Sure, it might be reasonable not to pursue the best-looking woman in the bar if many other men are hitting on her. If, however, everyone else ignored this stunning babe, then obviously you – assuming you like women – should go for her.”

This understanding of the Nash equilibrium is as useful for businesspeople as it is for single men, because it can help you make predictions. Remember that the single biggest determinant of your return on luck is who you work with. Nash, and game theory more widely, helps you find, and retain the best people.

Let’s imagine your company faces a budget shortfall and needs make cuts to its salary bill. The finance director offers you two options: cut all salaries by 10 per cent or make 10 per cent of your workforce redundant. In his book, Miller advises that you take the latter strategy, as hard as it may be to stomach. Why? Because were you to choose the salary cut option, those staffers that could find better pay elsewhere would take another job, leaving you with the weaker employees. Given the likely outcome, you would be unsatisfied with a strategy of slashing pay – you’d lose your best staff – so game theory tells us instead to lay off 10 per cent of your staff, retaining the stronger ones. Of course, your morals and legal duties will influence your eventual policy, but it is important you understand how game theory helps solves a dilemma in its rawest form before you account for other factors.

There’s more to this dark mathematics, says Miller. When you want to hire a key executive do you state a salary and, if so, what level do you set it at? Game theory suggests that, even if it feels counterintuitive, you should set the salary at the highest you can afford, rather than the least you feel would be enough to fill the vacancy. Let’s say you want an IT team leader. Your maximum budget is £40,000 but, to keep some in reserve, you advertise the job at circa £35,000. Three good candidates earning £36,000-plus decide not to even bother applying. “On average,” writes Miller, “the job candidates who would be happiest with your salary would be the candidates who are of the lowest quality – since the marketplace values them the least.” Advertise the job at below budget and you have played the game poorly because you have not considered how the other players – the candidates – are likely to react.

You have alienated strong candidates even though you may have had the resources to poach them. Better just to advertise the job at £40,000 and see where it takes you. You may or may not be able to attract your favourite candidate, but at least you will have had the chance to meet her and to win her over, so you can be satisfied with your strategy. Alienating her with a poor initial salary proposal is as futile as all four single men ignoring the gorgeous blonde in the bar. It is simply bad science.

Don’t be like Bob

Bad science is everywhere. In his entertaining 1967 book How to Take a Chance, mathematician Darrell Huff recalls the story of the Morton family, who had two daughters. Like many other 1960s families, the Mortons considered two boys and two girls the ideal family. Bob Morton, the father, longed for some boys to balance his brood. “After the girls were born,” he told Huff, “we began to doubt that we’d come out two and two in the end. But I have figured it out – our chances were fifty-fifty in the beginning, so of course they still are. Because boys are just as likely as girls.” In fact, the only part Bob was right about was the last bit – barring minute differences in the birth and survival rate boys are as likely as girls. Yet he was too optimistic at the beginning – the chances of four offspring splitting evenly is only 3 in 8. And there’s worse news for Bob: the chances of the two unborn offspring both being boys is just 1 in 4, even assuming that Bob is not genetically predisposed to generating girls, which his record so far suggests he might be.

Odd odds

Bob thought the chances of his four children being evenly split between boys and girls was 1 in 2. Right? Wrong. In fact it’s only 3 in 8. Here’s why:

Possible combinations of four children

Combinations kids

That there are so many Bobs in the world is, on one hand, depressing. Yet it can be used to your advantage. If your competitors for jobs and contracts have no understanding of probabilities, return on luck and game theory, you need only a rudimentary grasp to command a massive advantage over them. As Professor McCall remarks, some sensible risk taking is sensible: “People who just hunker down wake up one day and see that the world has moved somewhere else.” Given the benefits knowledge can bring, ignorance isn’t a chance worth taking.

When to keep the dice in your pocket

International business speaker Shari Peace has six tests for risk taking. If any apply, don’t roll the dice.

1. Double or quits: There’s a good chance you could lose everything.

2. Runt returns: You have to bet a lot to win a little.

3. At God’s mercy: There are too many factors outside your control.

4. Losing battle: You feel the odds are against you.

5. No second chance: It impossible to repair the damage if the punt doesn’t come off.

6. Blind bet: You have to take the risk before having a chance to prepare for and/or evaluate it.


Game Theory at Work by James Miller

Innumeracy by John Allen Paulos

How to Take a Chance by Darrell Huff

Great by Choice by Jim Collins and Morten T Hansen

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