When Does 50 + 50 Equal More Than 100?

Some subtle economics in ten easy steps

Introduction

We have recently gotten wrapped around the axle trying to explain one of the fundamentals behind the Brams-Taylor algorithm for dividing up a collection of assets between two parties.

The algorithm guarantees that the division will be equitable, envy-free, and efficient. It is claimed by the authors that in the worst case, each player gets exactly half of the value, which seems reasonable. They also claim that if the players value the assets differently, then both can get more than half the value as they perceive it. This has been interpreted by some people as “getting something for nothing,” or, equivalently, as “50 + 50 can’t equal more than 100.” Very intelligent people get wedged on this issue.

The rest of this page attempts to explain this paradox, so that the Brams and Taylor approach becomes more intuitive as a paradigm.

You can buy a basketball at a store for $10. Let us assume that this is the “fair market price” of a basketball; putting issues of differing quality basketballs aside, a basketball costs $10, and you can do no better. Our common language says that a basketball is “worth $10.” This is where the problem starts.

If you have ever tried to sell a basketball at a garage sale, you will quickly discover that it can be worth $10, and it can be worth a lot of other things too. For example, if it is a slow day and no basketball players show up at your sale, you will not sell the basketball, and, in terms of its cash value at the end of the day, it is worth zero. In fact, as the afternoon wears on and this possibility looms large, you will be inclined to take an offer of $8 for the ball. In this case, the ball was worth $8 to the buyer and it was worth $8 for you to sell it to him. If he offered you $2, you might decide that the ball was “worth” more than that; you could opt to sell it another day, hopefully for more than $2. So every time a deal is consummated, the “worth” of the object sold is based on what the buyer is willing to pay and the seller is willing to accept. Whatever other “intrinsic worth” we think the object might have somewhere else at some other time to some other buyer is irrelevant.

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What is a Basketball Worth?

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Sometimes it is Worth More

Note that it can work the other way too. We might be surprised when some kid offers us $12 for the ball. We jump at the chance, because we believe the ball is only worth $10. But no, our buyer is very knowledgeable. The basketball in question is a vintage Bob Cousy signature model, and in very high demand due to its scarcity. So our buyer decides that it is “worth” $12 to him and makes us that offer. We accept it. As far as that transaction goes, the basketball was worth $12. To that buyer, right then, right there. When $12 changes hands, there can be no argument about the ball being worth anything else. Ten days later at a basketball convention with very knowledgeable buyers, the ball may all of a sudden be “worth” $50. Worth is a function of time, place, and buyer.

So worth can be defined as what someone subjectively values an asset. There is no “objective” worth. There is a “market price,” which is a statistical average of a large number of like-kind assets sold in commodity markets. But we need to be careful about assigning the market price to the thing we call worth, especially where single transactions are involved.

If you think your house is worth $500,000 and I think it is worth $400,000, it is unlikely we will close a deal. When two opinions of what an asset is worth differ widely, the buyer and seller usually look elsewhere.

This gives heightened meaning to that old, time-worn expression, “What’s it worth to you?” In fact, really aggressive salesmen say, “What’s it worth to you, right here, right now?” They really do understand. Walk away and come back later, and the price may change.

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The Free Market at Work

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Collections of Assets

Now let’s consider a pool of three assets, A, B, and C. These assets may have differing values, but for the sake of argument let us agree that the collection has a market value of $100. We do this to establish a common baseline.

Now we have two people who wish to divide up this pool. Let us assume, somewhat reasonably, that they will each value the pool at $100. Although they may decide on the relative worth of each asset differently, we constrain them to agree that in their eyes the collection, in toto, is worth $100. As a proof of this belief, we ask them to each give us $50 to buy their share.

For the sake of concreteness, let us assume we are dealing with male jocks, Bob and Ray, and the items A, B, and C are an angling rod, a bowling ball, and a catcher’s mitt.

Bob, it turns out, is an avid fisherman. He looks at the angling rod and determines that it is an especially fine specimen. He values it at $60. He also likes the bowling ball and catcher’s mitt, as he both bowls and plays softball. He rates them about equally, and, given the $100 constraint, values them each at $20, as he has $40 left to “spend.”

Notice that if he gets the angling rod he will feel he got a good deal. He put in $50 and got back something he was willing to pay $60 for.

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Bob's Preferences

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Ray's Preferences

Ray looks at the same angling rod, and, as a part-time fisherman, thinks that it is only worth $40 to him. On the other hand, he bowls a lot, and, when he is not bowling, he catches for his softball team. Both the bowling ball and catcher’s mitt are great acquisitions in his eyes, and he would gladly pay $30 each for them. This is how he would “spend” his $100.

Note that if he gets the bowling ball and the catcher’s mitt, he will feel he made out like a thief. For only $50, he got items worth a total of $60 to him.

So, if I ask Bob and Ray to each individually and in secret make up their list of what each item is worth to them, I can now adjudicate the affair with the wisdom of Solomon. When I look at their lists, I give the angling rod to Bob and the bowling ball and catcher’s mitt to Ray. They are both happy, for two reasons:

1. They both got “more” than they paid for – for $50, each of them got $60 worth of goods as they perceived them;

2. They each got the same award, $60. Thus the award is equitable.

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The Wisdom of Solomon

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Why Does This Work?

No, folks, there is no sleight of hand at work here. Let us assume that the “market value” of the items is really as follows: angling rod, $50; bowling ball, $25; and catcher’s mitt, $25. The total market value of the collection is $100.

What happened here is that due to personal preferences, Bob overvalued the angling rod by $10, and Ray undervalued it by the same amount. On the other two items, Bob undervalued them each by $5, and Ray overvalued them symmetrically. There is nothing “perverse” about this; different people have different preferences.

What is crucial is that because of this asymmetry in valuation, it is possible for each of them to get “more than half the value.” That is, they get more than half as they have valued the assets. We have in no way “created” $120 of market value out of $100 of assets. The assets still have whatever market value Bob and Ray can subsequently sell the items for to buyers who have their own preferences.

What Brams and Taylor do is take this concept and generalize it, providing an algorithm that allows for this kind of division when the outcome is not quite so obvious. When the two players value the items identically, a 50/50 split will result. As the preferences diverge, each side “gets more,” in some cases as much as 75%. And the algorithm of course guarantees that both sides get the same amount of the “surplus.”

This has been a long way around the barn, but I hope this now makes the logic behind Brams and Taylor more transparent. The goods are divided equally, and each side may get more than half, based on their own subjective valuation of the assets.

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Conclusion

The Brams Taylor algorithm is protected by U.S. Patent # 5,983,205

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