image by iChaz

Hey guys, I’m (again) taking time off for a personal matter. I have a lot of ideas in the works, and I will return as soon as I can deliver a quality article. Those of you subscribed by email or RSS will get the update automatically. Thanks for understanding, and hope to be back soon.

Presh

Yes, getting rich is about saving and earning. But it is also about competition. Winning in the job, stock, and housing markets is about outsmarting opponents and thinking strategically. In such games, it is often important to make the right move before the action begins, as the next puzzle illustrates.

The puzzle

I came across an interesting puzzle in Peter Winkler’s Mathematical Puzzles called “Coins in a Row”

On a table is a row of fifty coins, of various denominations. Alice picks a coin from one of the ends and puts it in her pocket; then Bob chooses a coin from one of the (remaining) ends, and the alternation continues until Bob pockets the last coin.

Prove that Alice can play so as to guarantee as much money as Bob.

The implication of the game is mind-blowing. This is a seemingly fair game that is rigged for the first player. No matter how the second player arranges the coins, it is impossible for him to end up with more money.

Why is this the case?

Getting started

The puzzle is hard to approach because it is difficult to imagine all the distributions of fifty coins of various denominations. So let’s build up from a few observations.

Observation 1: Alice and Bob end up with the same number of coins

Since Alice and Bob alternate picking, they end up with 25 coins apiece. This is a rather trivial observation but it turns out to be useful to solving the puzzle.

Observation 2: If the coins were the same denomination, then Alice and Bob end up with the same amount of money

This directly follows from observation 1. It is easy to see that if all the coins were pennies, nickels, dimes, or quarters, then Alice and Bob would end up with equal valued sums.

This observation again is trivial, but it illustrates that Alice’s advantage comes from the precise arrangement of coins of various denominations.

Observation 3: By going first, Alice can affect which coins she picks

To see this, suppose the coins on the table are numbered 1 to 50 from left to right. The amazing part is that Alice by playing first jcan guarantee that she can pick all the odd-numbered coins 1, 3, 5, … 49, or she can all the even-numbered coins 2, 4, 6, …, 50. How is that possible?

To begin, consider what happens if Alice starts the game by picking coin 1. In this case, Bob is forced to choose among coins 2 and 50. Notice that Bob’s choices are both even-numbered coins. This means regardless of what Bob picks, he cannot grab an odd-numbered coin, and additionally he has to leave Alice with an odd-numbered coin. If Bob picks coin 2, then Alice can pick coin 3 (leaving 4 and 50 for Bob). If Bob picks coin 50, then Alice can pick 49 (leaving 2 and 48 for Bob). This logic can be extended, and consequently, Alice can collect all of the odd-numbered coins.

A similar argument demonstrates that Alice can guarantee she can pick all the even-numbered coins if she starts by picking coin 50.

Putting it all together: the solution

Now Alice’s strategy can be formulated. Before the game starts, Alice computes the sum of all of the odd-numbered coins and the even-numbered coins. One of the two sets has to at least be as big as the other (also an application of the pigeonhole principle).

Therefore, Alice can pick the set that gives her at least as much as Bob.

Bonus: what if the game has 51 coins?

If the game has 51 coins, how does that change the game?

(Hint: Winkler’s text says “However, if there are 51 coins instead of 50, it is usually Bob (the second to play) who will have the advantage, despite collecting fewer coins than Alice.”)

The current swine flu scare is a reminder that the flu is a fierce disease. New strains can develop quickly and disarm populations. One only has to remember the 1918-1919 Spanish flu which caused anywhere from 20 million to 40 million deaths.

Combating the flu is no small feat. It requires anticipating new strains, developing vaccines, and delivering medicine to masses. And ultimately the success depends on coordination of real people, which means global governments, hospitals, and individuals. Accordingly, there are many strategic issues in fighting the flu that can be better understood using the framework of game theory. Here are three strategic questions:

1. Is the swine flu panic good?

A casual observer might be cynical about the media attention. The death toll so far does not seem to warrant the extensive coverage. Is the fear meant to promote ratings and profits? Is it meant to help the government extend its power?

Perhaps the answer is something more beneficial. Jake at EconomPicData offers an interesting alternative on the swine flu scare using game theory. His matrix suggests that “induce panic” is a dominant strategy (I’ve recreating the text in a slightly more readable table):

link to original

Jake’s analysis makes a good point, but there are a few problems. One is the government plays this game repeatedly with the public over many issues. Calling too many false alarms will damage its reputation and limit its effectiveness.

Second, the entry labeled “nothing happens” for a false scare is not accurate. There is much cost to creating a false panic. Hospital emergency rooms are being flooded, schools are closing temporarily, and governments are banning imports of pork products without solid evidence. In a drastic measure, Egypt killed almost 300,000 pigs without conclusive evidence that the pigs were infected with the virus.

The appropriate answer, therefore, is to walk the middle line. Just as excessive punishment does not deter crime appropriately, excessive fear does not prepare populations cost effectively. The punishment must fit the crime, and the fear induced should fit the facts.

2. When a vaccine is developed, who needs to take it?

There is little question why the elderly and the sick are advised to get flu shots. They account for 90 percent of deaths from the flu, and so the benefit of the vaccination appears to outweigh whatever risks there are.

But what about the healthy and young? The flu is not as likely to affect them, and there are costs and risks to the vaccination. What is the right answer?

From a social perspective, it is beneficial for the healthy to get vaccinated, as explained by Robert Bazell in Slate:

That is the major reason you should get a flu shot. Even if spending a week violently sick and bedridden doesn’t worry you, by immunizing yourself you vastly lessen the chances you will spread the virus to some child or older person (family member, friend, or stranger) who might die from it.

In medicine, this concept is called “herd immunity”-that is, if enough members of a group of animals (including humans) are immunized against a disease, the entire group is more likely to escape infection.

The problem with herd immunity, however, is deciding who has to get the shots. Individually healthy people would prefer not to go through the effort of getting the shot and “free ride” on those who do. The situation is akin to the prisoner’s dilemma, which may explain the low vaccination rate in the U.S.

(Two ways to tackle this problem would be to make vaccinations mandatory and subsidizee flu shots to minimize the cost barrier).

3. Going forward, what’s the best way to select the flu vaccine?

Selecting the flu vaccine is an amazingly complex problem, and the mathematics are stunning. Consider the following simplified and slightly unrealistic math problem (obtained from these lecture slide ppt)

Suppose there are two expected strains of flu and two vaccines that can combat them. The distribution of strains is unknown and one can only guess how effective the vaccines can be.

It is believed that vaccine 1 will have an 85 percent chance of working against strain 1 but only a 70 percent chance against strain 2. Similarly, vaccine 2 has a 60 percent chance against strain 1 but a 90 percent change against strain 2. Here is the matrix to summarize the odds:

Suppose each person can only get one vaccine due to cost considerations. Clearly neither vaccine is effective against both strains. Is there a combination that can maximize the population immunity?

The answer is yes, and the solution is entirely similar to an earlier post about optimizing serves in tennis. If the strains are equally likely to occur, for instance, then give vaccine 1 to 2/3 of the population and vaccine 2 to 1/3 of the population to yield a 76.7 percent immunity rate. (The problem can also be solved for other distribution of strains, per the minimax theorem)

In practice, there are many more issues to be modeled. But even then, game theory and probability are used in both coordinating the supply chain (ppt) as well as selecting the flu vaccine the W.H.O. recommends.

Game theory is usually discussed in economics contexts, which have been the focus of this site. But the science of strategy can serve well in other contexts too. A few weeks ago my friend alerted me about how game theory was being used to predict political events. One of the prominent figures is Bruce Bueno de Mesquita, a professor at New York University and senior fellow at the Hoover Institution at Stanford. Mesquita’s predictions have had stunning accuracy, so much so that his work was detailed on a History channel show called The Next Nostradamus.

This background got me interested in learning more, and I was pleased to find that Mesquita has given a talk on TED.com giving some insight in to how he operates. In this February 2009 talk, Mesquita explains three predictions about Iran derived from his game theoretic calculations.

The 19 minute talk is available from TED.com (you can also download the video (zipped mp4 or iTunes):

Video link: Bruce Bueno de Mesquita talk on TED.com

(video subject to terms of Creative Commons license)

My reactions to the talk:

[2:20]: A non-technical explanation of game theory. I like the point he makes about understanding limitations of players. Mesquita gives the example that the mathematician Ramanujam was a great person, but limited by geography.

[3:15]: What is rational? This is a highly debated concept (see my discussion about miracles and rationality). Mesquita suggests rational simply means doing what one thinks is good for oneself. He suggests everyone except young kids and those with mental disorders like schizophrenia can be considered rational.

[4:00]: Changing the world depends on understanding influence and incentives. In game theory you do not trust someone because they like you, but rather because it is in their self interest. It is important to see where the influences are coming from.

[5:33]: To predict correctly, one has to pay attention not just to big players, but also small players that influence them. This is what makes game theory interesting is the dynamic among players.

[6:00]: Modeling such small interactions gets complicated. With five players, there are 120 connections one can model among them. This is difficult though not impossible to track mentally. The complexity scales very rapidly. With 10 players, for instance, there are over 3.6 million connections. At this stage one must use computer simulations to keep track of all the data.

[8:00]: Mesquita’s shows a slide that his model works 90 percent of the time, according to a CIA study. This is quite amazing and I am interested in reading how they came up with this number.

[8:25]: The keys to the model are simple: who are the players, what they say they want, how focused the players are, and how much power each player has. Mesquita says he uses public sources and experts to gather the inputs.

[10:15]: What isn’t needed: history! How players get to their current stage has historical importance, but not modeling importance. It is sufficient to accurately model the current situation to predict reasonably well. Interesting idea, but I wonder if taking history into account may do even better.

[11:11]: Three predictions about Iran. I wasn’t impressed by this part of the presentation which goes on for a few minutes. There was not enough detail explaining the somewhat confusing graphs.

[16:00]: Mesquita claims other complicated negotiations can be predicted, like health care, mergers, etc. Interesting hope to speed

[17:40]: The host asks an amazing question: does making the prediction public lessen it’s accuracy since parties might react to it? Rational expectations mean it’s necessary to act by surprise (see article on the Federal Reserve interest rate policy.) Mesquita’s answer: he doesn’t think his model predictions will be affected since negotiations typically end up at the same point regardless, so making predictions public may even speed the process and lead everyone to an agreeable solution without “manipulation” like economic sanctions.

What are your thoughts on Mesquita’s talk?

Things can get difficult when money is tight. Less money often means going to fewer movies, passing on gourmet coffee, and delaying big ticket purchases like a new car or elective surgery. Less money means fewer options, and there is a real tension about being limited and feeling powerless.

In such times, it is useful to remember that money isn’t everything. There are many things that cannot be had for money. Here’s an inspirational passage from the writer Arne Garborg that hits the nail on the head:

It is said that for money you can have everything, but you cannot. You can buy food, but not appetite; medicine, but not health; knowledge, but not wisdom; glitter, but not beauty; fun, but not joy; acquaintances, but not friends; servants, but not faithfulness; leisure, but not peace. You can have the husk of everything for money, but not the kernel.

The passage reminds me that the essence of virtually any enjoyable activity depends on non-monetary factors. On that note, here are a few more comparisons that came to my mind:

• You can buy a college degree, but not an education
• You can buy insurance, but not safety
• You can buy jokes, but not laughter
• You can buy a bed, but not sleep
• You can buy a house, but not a home

I suspect you’ve thought about this topic at some point too…any ideas you’d add to the list?

This is a sensitive topic, and I’m not entirely sure where I stand. But there are a lot of strategic issues to it that are fun to discuss.

For now, I’ll just open the topic up. Recently Mind Your Decisions reader Tim sent me a joke that shed a new light on the strategic nature of English as a national language. Let me know what you think, and enjoy!

The joke

A foreign diplomat, looking for directions, pulls up at a bus stop where two Americans are waiting.

“Entschuldigung, koennen Sie Deutsch sprechen?” he asks. The two Americans just stare at him.

“Excusez-moi, parlez vous Francais?” he tries. The two continue to stare.

“Parlare Italiano?” No response.

“Hablan ustedes Espanol?” Still nothing.

The diplomat drives off, extremely disgusted. The first American turns to the second and says, “You know, maybe we should learn a foreign language.”

“Why?” says the other. “That guy knew four languages, and it didn’t do him any good.”

(If you’re visiting for a new game theory article, I apologize. This week went by really quick and I don’t have a new article to post. In the meantime do browse the game theory archive…Thanks and see ya next week).

In 1994, shock jock Howard Stern created a public storm and ran for the governor of New York. His campaign was regarded as a publicity stunt, but some worried his radio popularity would convert into votes and make a mockery of American politics. Luckily, the fears never materialized. Stern dropped out because he didn’t want to disclose his finances (or salary), a requirement for office.

The story illustrates an interesting part of American culture: most of us treat salary as personal information. Like Stern, we worry about things like being judged by peers or getting charged more by unscrupulous mechanics and plumbers.

But that’s only half of the story. We occasionally find ourselves on the other side of the table when we’re looking for jobs and we need to learn about salaries. In this case, we have to be careful to learn about salary information without coming off as nosy or rude.

So we’re left with a daunting challenge: How can we learn about salaries without asking too much personal information?

This is a question that requires some strategic thinking. If you’ve been reading this blog, you know that game theory and strategy are often best illustrated through puzzles. In the past I’ve used puzzles to explain fixed points (the monk puzzle), group coordination (the hat puzzle), Monte Carlo simulation (the dice puzzle), ESPN’s million dollar streak contest (the tennis puzzle), survival of the weakest (the truel) and even the Joker’s motivation in the Dark Knight (the pirate puzzle).

The game of salary information too can be understood better through today’s puzzle.

The salary puzzle

Three friends want to know their average salary for negotiating purposes. How can they do it without disclosing their own salaries to each other?

A hint to get started: read the article on email encryption

The answer (will be posted next week)

(Wow, thanks for all the comments while I was posting this. Scott, Dan and Ku were all thinking along the right lines and Cheryl offers a practical and direct way I didn’t even think about)

The friends can calculate the average through a clever encoding process. The idea is that each person encodes their salary by adding a random number to it. These encoded salaries can be added together and then the random numbers can be subtracted. The resulting figure is the sum of the three salaries from which the average can be obtained.

Since additions and subtractions are easy to decode, however, the tricky part is implementing a solution where no person obtains knowledge of the other two party’s random numbers, for that would reveal enough information to obtain individual salaries. To do that, one can sequence the additions and subtractions carefully.

The process is somewhat cumbersome, but its gets the job done. Here is one particular solution, with the supporting algebra as verification:

Here is a worked out example with specific values:

Strategic considerations

The solution is interesting but it begs for further analysis. There are several considerations that came to mind which are addressed below.

Can the solution be extended for more than three players?

Yes. The process can scale up, but there are a few practical problems. First, it might be hard to coordinate a larger group. Second, there is a risk that the average may get “diluted” by incorporating workers of different skill levels. Third, adding more people will likely increase the chance of lying players and collusive groups. These turn out to be tremendous problems.

What happens when someone lies?

Alas, the answer to the salary puzzle is too idealistic. It depends entirely on individuals telling the truth. In fact, from a strategy perspective, it makes a lot of sense to lie.

Why lie? It’s important to consider your action relative to what others do. If others tell the truth, then the algorithm will produce a result equal to the sum of their true salaries plus the number you report. With this information, you can still compute the group average because you know your own salary. You might as well lie since honest reporting provides no benefit but comes at the risk of others learning your salary.

If, on the other hand, someone else has lied, then you definitely don’t want to be honest in reporting your salary. You have no chance of finding the correct average, and honest reporting has the risk of your salary getting disclosed.

Here’s an example of how someone lying can take advantage of the game:

How might the game play out if people thought strategically? (What is the Nash equilibrium of the game?)

As discussed above, lying is a dominant strategy. If others are honest, it makes sense to lie since you get the same benefit without any risk. And if others lie, you have nothing to gain and honesty comes with a risk. Therefore, everyone lies and the average is meaningless.

What would happen if players collude?

Another flaw to the salary puzzle is that it’s susceptible to collusion. If a few players decide to cooperate in advance, they can con the other members of the group.

To see how, suppose players B and C simply want to learn player A’s salary. They can do this by lying and sharing their fake numbers with each other, as follows:

Final remarks: how to share salaries in practice

As this puzzle illustrates, there are many strategic reasons why it can be hard to learn salary information. People are scared of revealing information without getting anything in return, exactly because it’s easy and favorable to lie.

How can we get around this problem? First, players should be guaranteed some useful information so they want to participate. Second, players need to be given a reason to be honest, or else they will free ride on other people’s honesty. And finally, the process should be anonymous so that no one fears revealing their personal salary information to others face to face.

Amazingly, it seems there are websites coming up that deal with these issues. One that I recently came across is glassdoor.com which allows you to look at salaries for job titles after you provide a salary for a current or previous employer. I haven’t had a chance to use the site myself, but it sure seems interesting.

I’m curious has anyone tried it out? Any other sites like this?

Personal note: this is the first post since my hiatus a few weeks ago…Postings will still be slow in the next week. Also, if you’re emailed me I’m still working on replying back–but hope to be back to full speed soon. Thanks for understanding during this transition.

Why do markets misbehave? How should you measure market risk? And what’s wrong with academic finance?

These are a few questions that polymath Benoit Mandelbrot addresses in the fascinating book The Misbehavior of Markets. Mandelbrot suggests all of these questions can be properly understood by rejecting the standard assumptions of academic finance and instead using a “fractal view” of risk and markets.

Fractals are at the heart of this book. Fractal geometry is a form of mathematics developed by Mandelbrot that deals with rough but highly self-similar structures like trees, coastlines, and mountains. Fractals have helped explain a wide range of natural phenomena and revolutionized computer graphics, influencing movies like Star Wars Episode III. There is room for more applications in this early science, and fractals may help explain the jagged but predictably irrational patterns in the stock market, claims Mandelbrot.

In this book, Mandelbrot contends that fractals are the key to modeling the market. The interesting part is that Mandelbrot does not merely explain why he’s right but he goes to great length to explain why others-those using the standard theories of academic finance-are wrong. Mandelbrot offers interesting history, anecdotes, trivia, and beautiful illustrations to make his case. The stock market does not act like a random walk, he says, but rather it’s like the flight of an arrow down an infinite hallway. It sounds a bit abstract at first, but this is exactly where the book shines. There are stories and illustrations that make such abstract concepts easily understandable. I literally felt smarter after reading each chapter…

But back to the subject at hand. What do fractals offer to finance? First, fractal math can help generate realistic stock price series. Mandelbrot graphically illustrates that his fractal-generated prices resemble actual price data more closely than the standard (geometric Brownian-motion) generated prices. Not only will this help for valuation and understanding risk, but it could also help one estimate damages in securities fraud cases.

Second, fractals can help explain why bubbles form and how prices are dependent over time. These are phenomena that every stock trader understands, but amazingly are classified as impossible by standard academic theory. Clearly something is wrong when experience contradicts the sacred cows of the random walk, Efficient Markets, CAPM, and Black Scholes, etc. Although this fractal theory doesn’t offer a complete answer yet, it at least allows for a theory consistent with practical experience.

But Mandelbrot’s fractal view has not taken hold in academia, the author explains, and it is this conflict that drives the narrative. The arguments in the book are about the battle between Mandelbrot’s ideas and standard finance. Mandelbrot elaborates on this in many ways, and after reading the book here are some useful comparisons between the two theories:

If you have qualms with academic finance, or agree with even a few of Mandelbrot’s revisions, I think you’ll enjoy this book.

There are but two reservations I have to this book. The first is that it touches on a lot of different ideas so it doesn’t elaborate on how to put the ideas into practice. Mandelbrot talks about the concepts like alpha (α) for measuring volatility and the H coefficient as an exponent of price dependence, but the book doesn’t offer enough for my liking. This appears to be intentional, however, as the authors admit their motives early in the book:

So caveat emptor. This book will not make you rich…If it fits any genre, it is that of popular science. It explains a new, and important, way of looking at the world-in this case, the financial world. It attempts to do so using common English, with as few formulae as possible-or at least, with no jargon unexplained. (p 23)

The second reservation is that the Mandelbrot makes everything into a battle between himself and academia. It reminded me a lot of the way Taleb wrote The Black Swan. (Though to be honest, as much as it annoyed me, there is something fun about reading someone with so much conviction.)

In short, check this book out and prepare to learn a new way of looking at risk and the markets. At a minimum you’ll be more skeptical about the frequently quoted statistics of risk so you can make better investment decisions.

The Misbehavior of Markets

My apologies as I didn’t have time to write my usual game theory column. I hope to return to full speed by next week.

In the meantime, I’ve been reading a lot of game theory articles and I’ve assembled a four of my favorites. Hope you find them as enjoyable as I have.

1. Scientists prove that younger siblings get less discipline (Newsweek)

It’s considered a “fact” of sibling relations: the baby of the family always gets away with murder. If the oldest brother had a curfew of 11 p.m. on weekends, his baby sister just has to call if she’s staying out all night. But can it be proven that parents are always stricter with their first born?

Researchers from the University of Maryland, Duke University and The Johns Hopkins University say yes, if there are younger siblings in the family, out of concern for the example that is being set for them.

2. Online retailers play pricing games (cnet)

Think you’ve found the lowest price online? Better double check.

That’s the advice of economists who research why a plethora of online price comparison systems haven’t succeeded in leveling prices on the Internet.

…There is still considerable price dispersion online.

And the prices don’t just fluctuate from merchant to merchant–they can vary from day to day on the same site.

3. The eligible-bachelor paradox (Slate)

The problem of the eligible bachelor is one of the great riddles of social life. Shouldn’t there be about as many highly eligible and appealing men as there are attractive, eligible women?

Actually, no—and here’s why.

4. Why the NFL should replace the overtime coin toss with an auction system (Slate)

If the Super Bowl goes into overtime for the first time ever, it’s fairly certain who will be victorious: the team that wins the coin toss…Since 2002, the team that’s gotten the toss has won more than 60 percent of overtime games…

With a little ingenuity, there is a way for overtime to be both fair and fast.

5. Bonus: a few articles from the game theory archive

I’ve been getting a few requests from new visitors who are interested in reading more from game theory archive (which now has 80+ articles) but would like my guidance on worthwhile topics. This serves as a good opportunity to highlight some of the “hidden gems”–good articles that were published before the game theory column took off in popularity. Here are a few of them:

–The game theory of surprises

What is a surprise? How is it possible to surprise people who think ahead? Can you keep surprising people?

–Game theory and airport security

A brief discussion of ARMOR, a security system used at LAX airport to combat terrorism using game theory research from the University of Southern California.

–The Shapley-Shubik index: one way to measure voting power

Voting rules can often create mind-boggling strategic considerations. This article discusses voting power in the Israeli Supreme court and how a proposed rule change is less useful than it sounds.

–Heart disease and used cars

Why are so many people on cholesterol lowering medications? Perhaps it is because the market for medical advice is similar to the market for used cars.

While most companies are cutting benefits in this economy, there are a few that are expanding. “Jim” is preparing for his annual review and wants to know how to negotiate his pay raise. Jim works in a medium size company that consults to law firms. He’s performed very well this year and he wants to know (1) how to make his case for a raise and (2) how much to ask for.

These are tough questions to answer. There are few hard and fast rules to salary negotiations. Raises depend not only on individual performance, but also on company performance and the subtle nuances of office politics. Some offices, for instance, make it a point to be fair and offer raises based mainly on years of experience to avoid ill-will and jealousy among peers.

That said, there are a couple guidelines you can use for reference. I came across two interesting points in the entertaining book Automatic Wealth for Grads by Michael Masterson.

The first point concerns what to emphasize when asking for a raise. Masterson suggests it’s very important not to focus on what you think is important, but on what your boss thinks. His specific tip is:

So list your boss’s top 10 priorities. Then identify the 2 that will have the greatest impact on his or her success…Those are the priorities you should focus on. (page 107)

This advice is great in a sneaky way. Not only does it highlight that businesses work on group bottom-line thinking (very different than college where grades are individual bottom-line thinking), but it also hints that employees need to connect with their bosses. How many times have you asked your boss what his or her priorities are? Communication can go a long way to building good will and a sizable raise.

So if you do perform well, how much should you ask for? The guideline here is that your raise will be dependent on how much you benefit the company’s bottom line. Typically, your raise will be a portion of the additional profits you can bring in for the company. Masterson offers a 10-for-1 guideline when asking for raises:

If you want to get a raise that’s $1,000 more than ordinary, make sure you’ve been a major contributor to an idea that will generate at least $10,000 in additional net profits for your company. So if you want to increase your current income by, say, $25,000, you are going to have to find a way to increase business profits by about a quarter of a million dollars. And that’s not just $250,000 once. It’s $250,000 this year and next year and every year thereafter that you expect to maintain that much-higher-than-average-salary. (page 118)

Masterson’s advice can sound daunting, especially for new grads, but I think it illustrates some good points. First, many companies are structured so that profits flow to the top. If you want to keep more of the profits of your labor, you have to get higher up (or start your own business). Second, raises are based not just on a single good year but rather an expectation that you will continue to perform at a high level. This is why companies look for employees with consistent track records.

What are your thoughts?

I told Jim I would post his situation to get your wonderful insights. How have you negotiated big raises? Do you agree with the 10-for-1 rule? Have you found useful websites or books on salary negotiation? What are mistakes to avoid during negotiation? Is your company offering raises in this economy?