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The Famous Game Show Problem

game show problem

Ahh, the famous game show problem (also known as The Monty Hall Problem).

This is a probability puzzle you’ve heard of:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

Marilyn Vos Savant, one of the “smartest” people in the word, offers her answer:

When you switch, you win 2/3 of the time and lose 1/3, but when you don’t switch, you only win 1/3 of the time and lose 2/3. You can try it yourself and see…..

The winning odds of 1/3 on the first choice can’t go up to 1/2 just because the host opens a losing door. To illustrate this, let’s say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you’ve chosen, we’ve learned nothing to allow us to revise the odds on the shell under your finger.

The benefits of switching are readily proven by playing through the six games that exhaust all the possibilities. For the first three games, you choose #1 and “switch” each time, for the second three games, you choose #1 and “stay” each time, and the host always opens a loser.

Warren Buffett and the Power of Nontransitive Dice

nontransitive diceIn a terrific book by William Poundstone, Fortune’s Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street, is the story of a 1968 dinner meeting between mathematician Edward Thorp and fund manager Warren Buffett.

Poundstone casually mentions that Buffett and Thorpe discussed their shared interest in nontransitive dice.

“These are,” writes Poundstone, “a mathematical curiosity, a type of ‘trick’ dice that confound most people’s ideas about probability.”

A nice description of nontransitive dice by Ivars Peterson at the Mathematical Association of America’s website:

The game involves four specially numbered dice. You let your opponent pick any one of the four dice. You choose one of the remaining three dice. Each player tosses his or her die, and the higher number wins the throw. Amazingly, in a game involving 10 or more throws, you will nearly always have more wins.

Here’s what the dice look like:

The trick is to always let your opponent pick first, and then you pick the die to the left of his selection (if he picks the die with the four 4s, then circle round to the die with the three ones). It’s just like playing Rock, Paper, Scissors — only you get to see what the other guy picks in advance.

With these dice, you always have a 2/3 probability of winning — what a great sucker’s bet!

Three Questions to Remove Ego from Decision Making

Here are three questions leaders can ask to help make better decisions.

1. How will this decision make things better for the organization?

Consider how the decision will affect the organization’s ability to fulfill its mission. Managers who push their teams to achieve “stretch goals” without providing adequate support and resources may be seeking to get noticed by their bosses rather than helping the company serve its customers. Such behavior will have another side effect — talent will exit. The answer to this question must enhance the organization, not simply the resume of the manager.

2. How will this decision affect employees?

The business case for your decision should factor in the people quotient and affects on headcount, training, and development. Employees must execute what leaders decide, so if employees perceive that their boss is only doing something to make himself look good, they’ll be reluctant to embrace the change. They may comply, but they may never commit unless they determine the benefit for themselves.

3. How will this decision affect me?

When you are involved in a project, it is easy to entangle ego with outcome. Healthy ego is necessary, but when too much ego makes a you blind to obvious problems such as lack of resources, customer disinterest, and employee morale, problems arise. As we have seen with corporate executives in the financial sector, it isn’t positive when personal interest comes before corporate and public interest. So if the answer to this question is more in favor of you rather than the company, the issue may be over-personalized and need more deliberation.

Bertrand Russell: On Avoiding Foolish Opinions

bertrand russell avoiding foolish opinion

We’d all like to avoid folly, wouldn’t we?

Avoiding stupidity is easier than seeking brilliance. And it doesn’t take a genius. Only a few simple ideas.

Philosopher Bertrand Russell (1872-1970) offers some advice, which will not keep us from all error but will help us navigate away from obvious error.

On Avoiding Foolish Opinions

Via The Basic Writings of Bertrand Russell:

If the matter is one that can be settled by observation, make the observation yourself. Aristotle could have avoided the mistake of thinking that women have fewer teeth than men, by the simple device of asking Mrs. Aristotle to keep her mouth open while he counted. He did not do so because he thought he knew. Thinking that you know when in fact you don’t is a fatal mistake, to which we are all prone. I believe myself that hedgehogs eat black beetles, because I have been told that they do; but if I were writing a book on the habits of hedgehogs, I should not commit myself until I had seen one enjoying this unappetizing diet. Aristotle, however, was less cautious. Ancient and medieval authors knew all about unicorns and salamanders; not one of them thought it necessary to avoid dogmatic statements about them because he had never seen one of them.

Many matters, however, are less easily brought to the test of experience. If, like most of mankind, you have passionate convictions on many such matters, there are ways in which you can make yourself aware of your own bias.

If an opinion contrary to your own makes you angry, that is a sign that you are subconsciously aware of having no good reason for thinking as you do. If some one maintains that two and two are five, or that Iceland is on the equator, you feel pity rather than anger, unless you know so little of arithmetic or geography that his opinion shakes your own contrary conviction. The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion. So whenever you find yourself getting angry about a difference of opinion, be on your guard; you will probably find, on examination, that your belief is going beyond what the evidence warrants.

A good way of ridding yourself of certain kinds of dogmatism is to become aware of opinions held in social circles different from your own. When I was young, I lived much outside my own country in France, Germany, Italy, and the United States. I found this very profitable in diminishing the intensity of insular prejudice. If you cannot travel, seek out people with whom you disagree, and read a newspaper belonging to a party that is not yours. If the people and the newspaper seem mad, perverse, and wicked, remind yourself that you seem so to them. In this opinion both parties may be right, but they cannot both be wrong. This reflection should generate a certain caution.

Becoming aware of foreign customs, however, does not always have a beneficial effect. In the seventeenth century, when the Manchus conquered China, it was the custom among the Chinese for the women to have small feet, and among the Manchus for the men to wear-pigtails. Instead of each dropping their own foolish custom, they each adopted the foolish custom of the other, and the Chinese continued to wear pigtails until they shook off the dominion of the Manchus in the revolution of 1911.

For those who have enough psychological imagination, it is a good plan to imagine an argument with a person having a different bias. This has one advantage, and only one, as compared with actual conversation with opponents; this one advantage is that the method is not subject to the same limitations of time or space. Mahatma Gandhi deplores railways and steamboats and machinery; he would like to undo the whole of the industrial revolution. You may never have an opportunity of actually meeting any one who holds this opinion, because in Western countries most people take the advantage of modern technique for granted. But if you want to make sure that you are right in agreeing with the prevailing opinion, you will find it a good plan to test the arguments that occur to you by considering what Gandhi might say in refutation of them. I have sometimes been led actually to change my mind as a result of this kind of imaginary dialogue, and, short of this, I have frequently found myself growing less dogmatic and cocksure through realizing the possible reasonableness of a hypothetical opponent.

Be very wary of opinions that flatter your self-esteem. Both men and women, nine times out of ten, are firmly convinced of the superior excellence of their own sex. There is abundant evidence on both sides. If you are a man, you can point out that most poets and men of science are male; if you are a woman, you can retort that so are most criminals. The question is inherently insoluble, but self esteem conceals this from most people. We are all, whatever part of the world we come from, persuaded that our own nation is superior to all others. Seeing that each nation has its characteristic merits and demerits, we adjust our standard of values so as to make out that the merits possessed by our nation are the really important ones, while its demerits are comparatively trivial. Here, again, the rational man will admit that the question is one to which there is no demonstrably right answer. It is more difficult to deal with the self esteem of man as man, because we cannot argue out the matter with some non-human mind. The only way I know of dealing with this general human conceit is to remind ourselves that man is a brief episode in the life of a small planet in a little corner of the universe, and that, for aught we know, other parts of the cosmos may contain beings as superior to ourselves as we are to jellyfish.

Still curious? Check out Bertrand Russell’s 10 Commandments of Teaching.

Unskilled and Unaware of It: How Difficulties in Recognizing One’s Own Incompetence Lead to Inflated Self-Assessments

We tend to feel we’re more able and smarter than we really are. We think we’re above average drivers, we’re above average investors, and we make better decisions than everyone else.

According to a recent study, this occurs, in part, because we “suffer a dual burden: Not only do these people reach erroneous conclusions and make unfortunate choices, but their incompetence robs them of the metacognitive ability to realize it.”

“Ignorance more frequently begets confidence than does knowledge.”

The study goes on to make several key points:

  • In many domains in life, success and satisfaction depend on knowledge, wisdom, or savvy in knowing which rules to follow and which strategies to pursue.
  • People differ widely in the knowledge and strategies they apply in these domains with varying levels of success. Some of the knowledge and theories that people apply to their actions are sound and meet with favorable results.
  • When people are incompetent in the strategies they adopt to achieve success and satisfaction, they suffer a dual burden: Not only do they reach erroneous conclusions and make unfortunate choices, but their incompetence robs them of the ability to realize it.

The authors come to the conclusions that the skills we need to have competence in any domain are often the same skills we need to accurately evaluate competence. The better we are at something, the better we’re able to judge ourselves. Because of this, incompetent individuals often exaggerate their ability more than competent ones.

Social Dilemmas: When to Defect and When to Cooperate

Social dilemmas arise when an individual receives a higher payoff for defecting than cooperating when everyone else cooperates. When everyone defects they are worse off. That is, each member has a clear and unambiguous incentive to make a choice, which if made by all members provides a worse outcome.

A great example of a social dilemma is to imagine yourself out with a group of your friends for dinner. Before the meal, you all agree to share the cost equally. Looking at the menu you see a lot of items that appeal to you but are outside of your budget.

Pondering on this, you realize that you’re only on the hook for 1/(number of friends at the dinner) of the bill. Now you can enjoy yourself without having to pay the full cost.

But what if everyone at the table realized the same thing? My guess is you’d all be stunned by the bill, even the tragedy of the commons.

This is a very simple example but you can map this to the business word by thinking about healthcare and insurance.

If that sounds a lot like game theory, you’re on the right track.

I came across an excellent paper[1] by Robyn Dawes and David Messick, which takes a closer look at social dilemmas.

A Psychological Analysis of Social Dilemmas

In the case of the public good, one strategy that has been employed is to create a moral sense of duty to support it—for instance, the public television station that one watches. The attempt is to reframe the decision as doing one’s duty rather than making a difference—again, in the wellbeing of the station watched. The injection of a moral element changes the calculation from “Will I make a difference” to “I must pay for the benefit I get.”

The final illustration, the shared meal and its more serious counterparts, requires yet another approach. Here there is no hierarchy, as in the organizational example, that can be relied upon to solve the problem. With the shared meal, all the diners need to be aware of the temptation that they have and there need to be mutually agreed-upon limits to constrain the diners. Alternatively, the rule needs to be changed so that everyone pays for what they ordered. The latter arrangement creates responsibility in that all know that they will pay for what they order. Such voluntary arrangements may be difficult to arrange in some cases. With the medical insurance, the insurance company may recognize the risk and insist on a principle of co-payments for medical services. This is a step in the direction of paying for one’s own meal, but it allows part of the “meal’ ‘ to be shared and part of it to be paid for by the one who ordered it.

The fishing version is more difficult. To make those harvesting the fish pay for some of the costs of the catch would require some sort of taxation to deter the unbridled exploitation of the fishery. Taxation, however, leads to tax avoidance or evasion. But those who harvest the fish would have no incentive to report their catches accurately or at all, especially if they were particularly successful, which simultaneously means particularly successfully—compared to others at least—in contributing to the problem of a subsequently reduced yield. Voluntary self-restraint would be punished as those with less of that personal quality would thrive while those with more would suffer. Conscience, as Hardin (1968) noted, would be self-eliminating. …

Relatively minor changes in the social environment can induce major changes in decision making because these minor changes can change the perceived appropriateness of a situation. One variable that has been shown to make such a difference is whether the decision maker sees herself as an individual or as a part of a group.

Footnotes
  • 1

    Dawes RM, Messick M (2000) Social Dilemmas. Int J Psychol 35(2):111–116