How to negotiate at your job using game theory

My underpaid friend used every trick during salary negotiations to no avail. This year she got the raise she deserved. The trick that finally worked was getting sick.

My friend joked the company would only realize her value after she left. By chance, it happened much sooner. My friend became very ill for one week and her projects suffered. Upon her return, she quickly fixed the problems. It was this incident that impressed managers at review time.

The peculiar part of the story is my friend didn’t really change herself. She didn’t take a class, or learn a new skill, or even demonstrate a hidden talent–she had cleaned up messes at the office before. What she did inadvertently do by being absent was illustrate her co-workers’ incompetence. Negotiating isn’t always about what you can do but rather what others can’t do.

One of the few credible ways to demonstrate what others can’t do is by withholding supply. This means strategically working less or producing less if you are truly in a position of power. Contrary to popular opinion, working longer and harder isn’t always the best negotiating tactic.

Withholding supply is a powerful negotiating strategy. It has been used controversially by big players such as Microsoft, Nintendo, and the Oakland Athletics. But before we get into these examples, let’s explore the theory through a simple card game.

The card game

The game comes from Co-Opetition and is quite interesting:

It’s a slow day at Harvard, and Adam and twenty-six of his M.B.A. students are playing a card game. Adam keeps the twenty-six black cards and distributes one red card to each of the students. The dean is feeling generous and agrees to put up $2,600 in prize money. He offers to pay $100 to anyone–either Adam or a student–who turns in a pair of cards, one black and one red.

That’s the game. It’s a free-form negotiation between Adam and the students. The only stipulation is that the students can’t get together and bargain as a group with Adam. They have to bargain on an individual basis. Where would you expect the negotiations to end up?

At the outset, it would appear Adam has a tremendous advantage. He has all the black cards so all pairs have to be made by trading with him. Adam can therefore exert his power by selling his cards at a premium or buying red cards at a discount. So what price will prevail during the trades?

We can make a guess using game theory. One way to proceed is by calculating how important each player is to the game. The relevant concept is a player’s “added value.” In this game, the added value is how much money a player’s presence contributes to the game. The examples below will make the idea clear.

Let’s calculate Adam’s added value. Adam has all the black cards. When Adam is in the game, he can contribute to each of the 26 pairs worth a total of $2,600. If Adam were not in the game, then there would be no black cards, and hence no chance for prize money. The difference between the two scenarios is Adam’s added value–it is $2,600.

What about each student’s added value? Each student has exactly one red card. When that student is in the game, he allows for one pair to be made worth $100 of prize money. If that student were not in the game, then there would be exactly one fewer pair. Each student’s added value is $100. The 26 students combined sum up to an added value of $2,600.

The symmetry in added values of the two sides suggests an even split of power. To make any particular pair, Adam’s black card is as important as a student’s red card. Adam cannot lowball students because they can hold out until a fair offer is made.

Here’s why. Suppose that every pair had been made except the last one. At this stage, both Adam and the student understand there is $100 up for grabs. The student’s card is as vital to the prize as Adam’s card. Neither side would likely agree to anything less than half of the $100. The result is an even split. Since each student could think this way, every student ends up with half of the prize money for the pair. The prize money is therefore split evenly between Adam and the students.

Can Adam do better? Yes-withhold supply

Adam’s monopoly on black cards surprisingly yields him no more than half of the money. But he can do better, if he withholds supply.

The trick is creating an artificial scarcity of black cards. Imagine Adam burns one of his black cards in front of the students. Now there are only 25 black cards. While burning a card will lower the total prize money by $100, it will provide Adam with incredible negotiating power.

We can see this by calculating the added values. In this setup, Adam still has all the black cards and is necessary in creating each pair of cards. Adam’s added value is still the whole pot, which is now $2,500.

What about the students? It is here that things change dramatically. There are 26 red cards but now only 25 black cards. This means one red card is in surplus and will not be paired in the end. If we removed any one student from the game, the total prize money would stay the same. Consequently, each student has an added value of zero. Suddenly, no one student is essential!

Now, the asymmetry in added values of the two sides suggests Adam has much power. No single student’s card is essential for making a pair and inevitably one student will be left out at the end. Since any student could end up with nothing, those that end up with any money-even $1-could consider themselves better off.

Another way to see this is by considering the end-game. Suppose 25 of 26 students would have ended up with $50, similar to the outcome of the first scenario. Then there is one student that gets nothing and is left out. It would be in that person’s interest to sell the card for less–say $49. When Adam accepts that offer, it will put some other student out of game. Now that person will sell for even less, say $48, rather than get nothing. In essence, the student that is left out drives down the selling price. Since every student fears getting left out, every student would settle for any money rather than get nothing. And so the price of red cards will drop.

The end result is that Adam can buy the red cards at a steep discount and end up with almost all of $2,500. This is a better individual outcome than getting half of $2,600. The trick was withholding supply to increase negotiating power.

Examples: Microsoft, Nintendo, and the Oakland A’s

If done right, big companies can similarly withhold supply for individual benefit. The actions are always controversial because withholding supply comes at an expense to society (notice that Adam destroyed $100 of value by burning a card).

It’s best to learn from real examples. Here are three ways companies have been alleged to withheld supply.

• Microsoft

Microsoft has long been charged of withholding supply. Randal Picker, at the University of Chicago Law School, describes some tactics of Microsoft. Check out the examples in this paper that directly compares some of Microsoft’s actions to the card game described above (pdf, search for “scarcity”).

• Nintendo

Nintendo also been charged with creating artificial scarcity when it introduced the Wii and stores had shortages of it. If Nintendo did withhold supply, what might have been its reason?

Nintendo may have been combating the buying power of big retailers. Stores could not simply order all the Wii units they wanted, but they had to wait for a limited quota. The retailers were turned into students hoping they would not be left out. This would shift the power and profits to Nintendo.

Interestingly, this is not the first time Nintendo has been alleged of withholding supply. Some economists have wondered this about Nintendo since 1997 (abstract, download the full paper and search for “Nintendo”).

• Oakland Athletics
  

A final example comes from baseball. We can think about selling tickets to a game as a kind of matching game. Each occupied seat is a pair of a seat plus a fan and is worth a certain surplus. The fans and owners split the surplus depending on negotiating power, just like in the card game. The owners have all the seats (like Adam with the black cards). When demand is poor, people may only buy the cheap seats. Owners can counter by withholding supply and restricting seating.

This is what the Oakland Athletics did a few years ago in the Coliseum by cutting seating capacity by 22 percent. The result was unpopular, but it did increase revenue.

In conclusion

Withholding supply can dramatically increase negotiating power but it has risks. It comes at an expense to society and can possibly be illegal. Also, the tactic fails if you lack sufficient leverage. If you were a student in the card game, burning your red card would have done you no good.

In summary, use with care and caution.