Category Archives: Bargaining

Park Place Is Still Worthless: The Game Theory of McDonald’s Monopoly

McDonald’s Monopoly begins again today. With that in mind, I thought I would update my explanation of the game theory behind the value of each piece, especially since my new book on bargaining connects the same mechanism to the De Beers diamond monopoly, star free agent athletes, and a shady business deal between Google and Apple. Here’s the post, mostly in its original form:

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McDonald’s Monopoly is back. As always, if you collect Park Place and Boardwalk, you win a million dollars. I just got a Park Place. That’s worth about \$500,000, right?

Actually, it is worth nothing. Not close to nothing, but absolutely, positively nothing.

It helps to know how McDonald’s structures the game. Despite the apparent value of Park Place, McDonald’s floods the market with Park Place pieces, probably to trick naive players into thinking they are close to riches. I do not have an exact number, but I would imagine there are easily tens of thousands of Park Places floating around. However, they only one or two Boardwalks available. (Again, I do not know the exact number, but it is equal to the number of million dollar prizes McDonald’s want to give out.)

Even with that disparity, you might think Park Place maintains some value. Yet, it is easy to show that this intuition is wrong. Imagine you have a Boardwalk piece and you corral two Park Place holders into a room. (This works if you gathered thousands of them as well, but you only need two of them for this to work.) You tell them that you are looking to buy a Park Place piece. Each of them must write their sell price on a piece of paper. You will complete the transaction at the lowest price. For example, if one person wrote \$500,000 and the other wrote \$400,000, you would buy it from the second at \$400,000.

Assume that sell prices are continuous and weakly positive, and that ties are broken by coin flip. How much should you expect to pay?

The proof is extremely simple. It is clear that both bidding \$0 is a Nash equilibrium. (Check out my textbook or watch my YouTube videos if you do not know what a Nash equilibrium is.) If either Park Place owner deviates to a positive amount, that deviator would lose, since the other guy is bidding 0. So neither player can profitably deviate. Thus, both bidding 0 is a Nash equilibrium.

What if one bid \$x greater than or equal to 0 and the other bid \$y > x? Then the person bidding y could profitably deviate to any amount between y and x. He still wins the piece, but he pays less for it. Thus, this is a profitable deviation and bids x and y are not an equilibrium.

The final case is when both players bid the same amount z > 0. In expectation, both earn z/2. Regardless of the tiebreaking mechanism, one player must lose at least half the time. That player can profitably deviate to 3z/8 and win outright. This sell price is larger than the expectation.

This exhausts all possibilities. So both bidding \$0 is the unique Nash equilibrium. Despite requiring another piece, your Boardwalk is worth a full million dollars.

What is going wrong for the Park Place holders? Supply simply outstrips demand. Any person with a Park Place but no Boardwalk walks away with nothing, which ultimately drives down the price of Park Place down to nothing as well.

Moral of the story: Don’t get excited if you get a Park Place piece.

Note 1: If money is discrete down to the cent, then the winning bid could be \$0 or \$0.01. (With the right tie breaker, it could also be \$0.02.) Either way, this is not good for owners of Park Place.

Note 2: In practice, we might see Park Place sell for some marginally higher value. That is because it is (slightly) costly for a Boardwalk owner to seek out and solicit bids from more Park Place holders. However, Park Place itself is not creating any value here—it’s purely the transaction cost.

Note 3: An enterprising Park Place owner could purchase all other Park Place pieces and destroy them. This would force the Boardwalk controller to split the million dollars. While that is reasonable to do when there are only two individuals like the example, good luck buying all Park Places in reality. (Transaction costs strike again!)

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Now time for an update. What might not have been clear in the original post is that McDonald’s Monopoly is a simple illustration of a matching problem. Whenever you have a situation with n individuals who need one of m partners, all of the economic benefits go to the partners if m < n. The logic is the same as above. If an individual does not obtain a partner, he receives no profit. This makes him desperate to partner with someone, even if it means drastically dropping his share of the money to be made. But then the underbidding process begins until the m partners are taking all of the revenues for themselves.

In the book, I have a more practical example involving star free agent athletes. For example, there is only one LeBron James. Every team would like to sign him to improve their chances of winning. Yet this ultimately results in the final contract price to be so high that the team doesn’t actually benefit much (or at all) from signing James.

Well, that’s how it would work if professional sports organizations were not scheming to stop this. The NBA in particular has a maximum salary. So even if LeBron James is worth \$50 million per season, he won’t be paid that much. (The exact amount a player can earn is complicated.) This ensures that the team that signs him will benefit from the transaction but takes money away from James.

Non-sports business scheme in similar ways. More than 100 year ago, the De Beers diamond company realized that new mine discoveries would mean that diamond supply would soon outstrip demand. This would kill diamond prices. So De Beers began purchasing tons of mines to intentionally limit production and increase price. Similarly, Apple and Google once had a “no compete” informal agreement to not poach each other’s employees. Without the outside bidder, a superstar computer engineer would not be able to increase his wage to the fair market value. Of course, this is highly illegal. Employees filed a \$9 billion anti-trust lawsuit when they learned of this. The parties eventually settled the suit outside of court for an undisclosed amount.

To sum up, matching is good for those in demand and bad for those in high supply. With that in mind, good luck finding that Boardwalk!

What Does Game Theory Say about Negotiating a Pay Raise?

A common question I get is what game theory tells us about negotiating a pay raise. Because I just published a book on bargaining, this is something I have been thinking about a lot recently. Fortunately, I can narrow the fundamentals to three simple points:

1) Virtually all of the work is done before you sit down at the table.
When you ask the average person how they negotiated their previous raise, you will commonly hear anecdotes about how that individual said some (allegedly) cunning things, (allegedly) outwitted his or her boss, and received a hefty pay hike. Drawing inferences from this is problematic for a number of reasons:

1. Anecdotal “evidence” isn’t evidence.
2. The reason for the raise might have been orthogonal to what was said.
3. Worse, the raise might have been despite what was said.
4. It assumes that the boss is more concerned about dazzling words than money, his own job performance, and institutional constraints.

The fourth point is especially concerning. Think about the people who control your salaries. They did not get their job because they are easily persuaded by rehearsed speeches. No, they are there because they are good at making smart hiring decisions and keeping salaries low. Moreover, because this is their job, they engage in this sort of bargaining frequently. It would thus be very strange for someone like that to make such a rookie mistake.

So if you think you can just be clever at the bargaining table, you are going to have a bad time. Indeed, the bargaining table is not a game of chess. It should simply be a declaration of checkmate. The real work is building your bargaining leverage ahead of time.

2) Do not be afraid to reject offers and make counteroffers.
Imagine a world where only one negotiator had the ability to make an offer, while the other could only accept or reject that proposal. Accepting implements the deal; rejecting means that neither party enjoys the benefits of mutual cooperation. What portion of the economic benefits will the proposer take? And how much of the benefits will go to the receiver?

You might guess that the proposer has the advantage here. And you’d be right. What surprises most people, however, is the extent of the advantage: the proposer reaps virtually all of the benefits of the relationship, while the receiver is barely any better off than had the parties not struck a deal.

How do we know this? Game theory allows us to study this exact scenario rigorously. Indeed, the setup has a specific name: the ultimatum game. It shows that a party with the exclusive right to make proposals has all of the bargaining power.

That might seem like a big problem if you are the one receiving the offers. Fortunately, the problem is easy to solve in practice. Few real life bargaining situations expressly prohibit parties from making counteroffers. (As I discuss in the book, return of security deposits is one such exception, and we all know that turns out poorly for the renter—i.e., the receiver of the offer.) Even the ability to make a single counteroffer drastically increases an individual’s bargaining power. And if the parties could potentially bargain back and forth without end—called Rubinstein bargaining, perhaps the most realistic of proposal structures—bargaining equitably divides the benefits.

As the section header says, the lesson here is that you should not be afraid to reject low offers and propose a more favorable division. Yet people often fail to do this. This is especially common at the time of hire. After culling through all of the applications, a hiring manager might propose a wage. The new employee, deathly afraid of losing the position, meekly accepts.

Of course, the new employee is not fully appreciating the company’s incentives. By making the proposal, the company has signaled that the individual is the best available candidate. This inevitably gives him a little bit of wiggle room with his wage. He should exercise this leverage and push for a little more—especially because starting wage is often the point of departure for all future raise negotiations.

3) Increase your value to other companies.
Your company does not pay you a lot of money to be nice to you. It pays you because it has no other choice. Although many things can force a company’s hand in this manner, competing offers is particularly important.

Imagine that your company values your work at \$50 per hour. If you can only work for them, due the back-and-forth logic from above, we might imagine that your wage will land in the neighborhood of \$40 per hour. However, suppose that a second company exists that is willing to pay you up to \$25 per hour. Now how much will you make?

The answer is no less than \$40 per hour. Why? Well, suppose not. If your current company is only paying you, say, \$30 per hour, you could go to the other company and ask for a little bit more. They would be obliged to pay you that since they value you up to \$40 per hour. But, of course, your original company values you up to \$50 per hour. So they have incentive to ultimately outbid the other company and keep you under their roof.

(This same mechanism means that Park Place is worthless in McDonald’s monopoly.)

Game theorists call such alternatives “outside options”; the better your outside options are, the more attractive the offers your bargaining partner has to make to keep you around. Consequently, being attractive to other companies can get you a raise with your current company even if you have no serious intention to leave. Rather, you can diplomatically point out to your boss that a person with your particular skill set typically makes \$X per year and that your wage should be commensurate with that amount. Your boss will see this as a thinly veiled threat that you might leave the company. Still, if the company values your work, she will have no choice but to bump you to that level. And if she doesn’t…well, you are valuable to other companies, so you can go make that amount of money elsewhere.

Conclusion
Bargaining can be a scary process. Unfortunately, this fear blinds us to some of the critical facets of the process. Negotiations are strategic; only thinking about your worries and concerns means you are ignoring your employer’s worries and concerns. Yet you can use those opposing worries and concerns to coerce a better deal for yourself. Employers do not hold all of the power. Once you realize this, you can take advantage of the opposing weakness at the bargaining table.

I talk about all of these issues in greater length in my book, Game Theory 101: Bargaining. I also cover a bunch of real world applications to these and a whole bunch of other theories. If this stuff seems interesting to you, you should check it out!

Park Place Is Worthless: The Game Theory of McDonald’s Monopoly

McDonald’s Monopoly is back. As always, if you collect Park Place and Boardwalk, you win a million dollars. I just got a Park Place. That’s worth about \$500,000, right?

Actually, as I show in my book on bargaining, it is worth nothing. Not close to nothing, but absolutely, positively nothing.

It helps to know how McDonald’s structures the game. Despite the apparent value of Park Place, McDonald’s floods the market with Park Place pieces, probably to trick naive players into thinking they are close to riches. I do not have an exact number, but I would imagine there are easily tens of thousands of Park Places floating around. However, they only one or two Boardwalks available. (Again, I do not know the exact number, but it is equal to the number of million dollar prizes McDonald’s want to give out.)

Even with that disparity, you might think Park Place maintains some value. Yet, it is easy to show that this intuition is wrong. Imagine you have a Boardwalk piece and you corral two Park Place holders into a room. (This works if you gathered thousands of them as well, but you only need two of them for this to work.) You tell them that you are looking to buy a Park Place piece. Each of them must write their sell price on a piece of paper. You will complete the transaction at the lowest price. For example, if one person wrote \$500,000 and the other wrote \$400,000, you would buy it from the second at \$400,000.

Assume that sell prices are continuous and weakly positive, and that ties are broken by coin flip. How much should you expect to pay?

The proof is extremely simple. It is clear that both bidding \$0 is a Nash equilibrium. (Check out my textbook or watch my YouTube videos if you do not know what a Nash equilibrium is.) If either Park Place owner deviates to a positive amount, that deviator would lose, since the other guy is bidding 0. So neither player can profitably deviate. Thus, both bidding 0 is a Nash equilibrium.

What if one bid \$x greater than or equal to 0 and the other bid \$y > x? Then the person bidding y could profitably deviate to any amount between y and x. He still wins the piece, but he pays less for it. Thus, this is a profitable deviation and bids x and y are not an equilibrium.

The final case is when both players bid the same amount z > 0. In expectation, both earn z/2. Regardless of the tiebreaking mechanism, one player must lose at least half the time. That player can profitably deviate to 3z/8 and win outright. This sell price is larger than the expectation.

This exhausts all possibilities. So both bidding \$0 is the unique Nash equilibrium. Despite requiring another piece, your Boardwalk is worth a full million dollars.

What is going wrong for the Park Place holders? Supply simply outstrips demand. Any person with a Park Place but no Boardwalk walks away with nothing, which ultimately drives down the price of Park Place down to nothing as well.

Moral of the story: Don’t get excited if you get a Park Place piece.

Note 1: If money is discrete down to the cent, then the winning bid could be \$0 or \$0.01. (With the right tie breaker, it could also be \$0.02.) Either way, this is not good for owners of Park Place.

Note 2: In practice, we might see Park Place sell for some marginally higher value. That is because it is (slightly) costly for a Boardwalk owner to seek out and solicit bids from more Park Place holders. However, Park Place itself is not creating any value here—it’s purely the transaction cost.

Note 3: An enterprising Park Place owner could purchase all other Park Place pieces and destroy them. This would force the Boardwalk controller to split the million dollars. While that is reasonable to do when there are only two individuals like the example, good luck buying all Park Places in reality. (Transaction costs strike again!)