A family caught my attention in a restaurant. A young kid was keeping his parents busy. He demanded an appetizer because the main course took too long. He then passed time playing video games on his cell phone. Soon he became bored again, so he took out a portable DVD player. I wanted to see his next toy, but just then the food arrived to occupy him.Times have really changed, I thought to myself. It was not so long ago that I was the kid waiting for food, but if got bored, I didn’t have toys. My parents simply told me to be patient. I hated it then, but my parents convinced me patience would pay off. If I learned how to be patient, they told me, then I’d get along with people better and have a more relaxed life.

In fact, those benefits are only half of the story. Patience has paid off for me, but less in my personal life than in my business life. The biggest payoff has been that patience provides a competitive advantage during negotiations. It’s a weapon particularly useful against impatient opponents who grew up with instant gratification.

To illustrate, I will share an experience where patience was important. I’ll then discuss the situation using a game theory model to suggest how the strategy can work for you.

A motivating story

I was moving out of state and needed to sell many of my belongings. Because I was still using the items I intended to sell, I waited until the last minute to list them online. I was an extremely desperate seller without much leverage.

I started out by listing my items at a reasonably high price and added “or best offer” to indicate my willingness to bargain. Because time was a premium, my main goal was to find buyers that could act quickly. I would have easily chopped 10 to 20 percent of the price to sell immediately rather than wait for someone willing to pay more. Despite my negotiating disadvantage, I was able to sell each and every item for full asking price.

My trick was exploiting buyers’ impatience. Since I was listing my items on the internet, I suspected the buyers would be accustomed to one-click shopping. They didn’t want long negotiations. If anyone started negotiating, I would just show I was a bit more patient.

My intuition proved correct. Most buyers didn’t even try to negotiate and paid the full price right away. For the few that did negotiate, I simply made a counteroffer of the full asking price, explaining it was a fair price. No one called my bluff, and I received full price. One buyer demonstrated his impatience in the negotiations, “Oh, you just want full price…Okay, what the heck. I am not going to argue over $20. I’ll come by today.”

It was my relative patience that provided the edge. Although I wanted a quick settlement, I took advantage that the buyers wanted an even faster one. Like the child in the restaurant, they wanted instant gratification. And they got it, but at a cost.

How can this interaction be modeled? What does game theory have to say?

The negotiating game

There are many ways to model bargaining. The method I discuss below is based on the Rubinstein bargaining solution (ppt).

The interaction can be seen as a division of a surplus. My desperation meant that 10 to 20 percent of the price was up for grabs. The question is who would get that surplus? Would I get it—and receive full asking price—or would the buyer get it—and get a price reduction?

The negotiations can be modeled as a series of alternating offers. I start out with an offer (“I want full asking price”), and the buyer can choose whether to accept. If the buyer accepts, then the game ends and we conclude the deal. Otherwise, the buyer then proposes with a counteroffer (“I want 20 percent off”). It is then my turn to choose whether to accept and conclude the deal. If I choose not to take the counteroffer, then the game starts over again with me proposing another offer. The only difference at this stage is that time has passed, which is wasteful for both parties.

On an academic note, this interaction is like the ultimatum game but with three key differences. First, the other party is allowed to counteroffer. Second, the game continues indefinitely until both sides agree on an offer. And third, there is a time component to the game: deals concluded earlier are preferred by both sides.

Time value is crucial

How can one measure time value? The idea is that we discount money promised in the future. Getting $1 in the future is the equivalent of getting some smaller amount, less than $1, today. The size of the reduction is dependent on an individual’s time preference for money. The time value can be thought of as an interest rate. For instance, a person that views getting $1 in a year as equal to getting $0.95 today has an “interest rate” or “discount rate” of 5 percent.

Each person has an individual preference. Patient players are ones that place a smaller discount on future money so they have lower implied interest rates. This is why patient people are more likely to invest for retirement or go to college—they consider the future rewards to have substantial value today. A very impatient person, by contrast, desires instant gratification. As one comic illustrates, if you offered an impatient person $1 million in retirement versus having $10,000 today, the impatient person would take the money today and buy an iPhone. The decision should not be considered right or wrong—rather the result of weighing value differently.

But there are consequences to impatience. As you might expect, impatient players do not fare well during negotiations because they are less capable of waiting for good offers. They would rather get things done quickly, as the one-click internet buyers demonstrated in my story.

The solution (intuitive)

Negotiations are a step-by-step process. In the beginning, each party wants to claim the entire surplus and get value. The seller starts at a high price and the buyer starts at a low price.

But as time passes, the surplus shrinks. Thus, players are willing to meet each other halfway, so to speak, to speed up the negotiation and get the deal done. Eventually, both offers converge to an agreement, as the following diagram illustrates:

What determines the final selling price? The key observation is that patient players give up less at each negotiating step because they see more value in future dollars. Therefore, their strategies are represented by flatter lines. Impatient players want the deal done quickly, and their strategies are represented by steeper lines.

As a consequence, the agreed-upon offer ends up closer to the original demand of the patient side. As the above diagram illustrates, the selling price is much closer to the patient seller’s starting offer than it is to the impatient buyer’s.

In Rubinstein’s model, the seller and buyer anticipate this interaction through backwards induction and make the deal right during the first stage of the game. In practice, agreement may take some time, but ultimately the patient side takes a larger share of the surplus.

A complete mathematical derivation can be found here (ppt).

A concluding example

It is only appropriate in an article about patience that I save the best for last. The advice about being patient isn’t just important for small negotiations, but even for big ones. The strategy saved my friend thousands of dollars.

It happened when my friend was negotiating an offer on a house. He placed an offer $20,000 less than asking price. Within an hour, the seller called back with a counteroffer that was only a small reduction on the asking price. The speed and casual response indicated the seller was very impatient and wanted to act quickly. My friend too wanted to close quickly, but hid that from the seller as a negotiating tactic. He countered with essentially the same bid and told the seller to sit on it for one night. He did this to demonstrate his patience to wait it out. That got the seller nervous, and sure enough, wanting instant gratification, the offer was accepted.