You Don t Have To Settle For Less Than The Biggest Piece of Pie A. M. Fink You are a lawyer or a mediator and a client comes to you about a disputed piece of property. He thinks he is entitled to half of it. If you had a risk free way to guarantee that he gets at least half of it, would you use it? It isn t always used. Take the high profile case of Barry Bonds 73 rd home run ball. On October 6, 2001, Barry Bonds hit home run numbers 71 and 72 in Pac-Bell park in San Francisco, breaking Mark McGwire s two year old season record of 70. McGwire s 70 th home run ball was sold for 3.2 million dollars. Bonds 72 nd ball bounced back onto the playing field so no fan obtained it. If Bonds hit another (and last of the season) home run on the last day of the season (October 7), the ball was estimated to be worth at least a million dollars. Alex Popov had a seat behind home plate for that last game. He wasn t going to catch the home run ball there so he traded for a ticket that allowed him to choose a place on the walkway beyond the outfield fence. Incredibly, Bonds did hit his 73 rd and last home run of the season into the glove of Popov. Unfortunately for him, it didn t stay there. In the ensuing scramble, the ball fell or was jostled out of his glove and ended up in the hands of Patrick Hayashi. Several bystanders told Popov that they would testify that he had caught the ball. In addition, someone had recorded the scramble on a video camera. Encouraged by this, Popov decided to sue for the possession of the ball. After pre-trial meetings did not resolve the issue, the case went before Superior Court Judge Kevin McCarthy, who heard the case without a jury. He determined that it was not clear that Popov had caught the ball and declared that Popov and Hayashi had joint and equal
ownership of the ball. When they could not agree on a resolution, Judge McCarthy ruled that a third party would determine the worth of the ball. Judge McCarthy would have been the wisest judge since Solomon had he not made the mistake of having a third party decide the worth of the ball. Why is it a mistake to introduce the third party? Here is why. There are now three numbers on the table, what each interested party thinks the ball is worth (evidently different) and the third party s determination. Ranking these by magnitude from largest to smallest, there are six scenarios. Party A* Party A* Party B* Party B* Third party Third party Third Party Party B* Third Party Party A* Party A Party B Party B Third Party Party A Third Party Party B Party A If the third party estimate is lower than yours, you are unhappy. So in four of the six scenarios, at least one party is unsatisfied (here those indicated by a *). In the last two, they are satisfied but those scenarios seem to be the most unlikely. Furthermore, each party is dissatisfied half of the time. This is the prognosis in the Bonds case. It could be worse. If one of the parties is to be the buyer and the price is determined by a third party then the scenarios are: Third Party Third Party Non-buyer * Non-buyer* Buyer Buyer Buyer * Non-buyer Third Party Buyer Non-buyer* Third Party Non-buyer Buyer * Buyer* Third Party Third Party Non-buyer Only the last scenario is satisfactory to both the buyer and non-buyer. In each of the above, one can see that each party is satisfied in only half of the scenarios. The odds are not good. Why be satisfied with this? Especially since it can be arranged that both parties can be guaranteed that they will be satisfied.
In fact, if the parties disagree then the judge can guarantee that each party gets more than half of the ball (it is fair), and each party will get more than the other party (envy free). This claim sounds outrageous but all you have to do is follow The Rule. The Rule Each party submits a sealed (and secret) bid for the full value of the disputed object. The highest bidder gets physical possession of the object and pays the other on the basis of the average bid. A tie is broken by an agreed upon random device. To see how this might work suppose the ball has been valued at $400,000 by Party A and $600,000 by Party B. By The Rule, B gets the ball and pays on the basis of the average bid, $500.000, ie. pays A $250,000. Party B wanted a rise in equity of $300,000 but actually got a $600,000 ball less the $250,000 payment which is $350,000. This is $50,000 more than half and $100,000 more than Party A got. Meanwhile Party A who wanted $200,000 got $50,000 more than that and calculates what he thinks Party B got. He thinks that Party B got a $400,000 ball and paid out $250,000 so his rise in equity is $150,000. That is $100,000 less than what Party A got. The outrageous claim is verified. What is the guarantee? As soon as Party B claims a value of $600,000, then he is guaranteed to get a value of $300,000 regardless of what the other bid will be. If the other bid is less, then he gets the object and pays out on the basis of an average that is less than $600,000 so will be less than $300,000 and he gets a bonus. If the other bid is more than $600,000, then he will be the recipient of a cash payment based upon an average that is more than $600,000. The payment is thus larger than $300,000. A similar guarantee is made to Party A. Note the paradox. If the parties disagree on the value then each party gets more than half. If they agree they get exactly one half. So disagreement is better than agreement.
If Party B is guaranteed half of his announced value, why does he not lie and bid say $800,000? In that case, the average is $600,000 so he pays $300,000 which is clearly worse than when he was honest. His guarantee of at least $400,000 is reached through the possession of a ball whose value was artificially inflated by him. So honesty is the best policy. Secrecy is also important. If Party B knows that Party A will bid $400,000 then he might reduce his own bid to $400,001 get the ball and only pay $200,00.50, but he cannot deprive Party A of his $200,000. Similarly, Party A could take advantage of knowledge of Party B s bid to up his to $599,999.50 to increase his take but he cannot deprive Party B of his $300,000. If both simultaneously do the above it is a disaster for both. Secrecy is the best policy also. Note that neither party is asked to justify the value given to the object, nor are they asked to compromise their stated value. Although the presentation of The Rule is in an adversarial context, it might be used in situations where continued amiability is a goal. Had Judge McCarthy mandated use of The Rule and done a calculation of the result as we have done above, he would have created quite a stir. Of course, he was not the only one to not invoke The Rule. Either party in the dispute or their lawyers could have asked to use the rule. You have to convince the other party to do this. But it should be a strong selling point that you can guarantee the opponent more than half of the ball. Apparently, no one connected with the case knew about The Rule. Journalists predicted that the value of the ball would go down as time passed. In the actual case, settled more than a year after the homer was hit, the ball was declared to have a value of only $450,000. We have used the Barry Bonds 73 rd home run case to explain the use of The Rule for the division of a piece of property that cannot be physically divided. We have illustrated its use when a priori each party has claim to one half of the value of the object. This restriction is not
necessary. Suppose Party A has claim to a fraction r of the object while party B has a claim of 1- r. Again sealed bids are taken for the full value and the highest bidder gets the object and pays on the basis of the average bid. But if the recipient of the cash is party A, then the payment is r times the average but if party B is the recipient, he is paid (1-r) times the average. Again the distribution is fair and envy free. What happens when there are more than two parties? Payment on the basis of the average does not work (see the example below). Here is something that does. We introduce the (fictitious) case of Grandma s Lamborghini. Grandma s will awarded her 1968 red Lamborghini equally to three of her granddaughters. Emily has a husband who would give an arm and a leg to have such a car with only 30,000 miles on it. Mary could see herself driving to the beach in such a fancy car, but Joan would be uncomfortable driving such an expensive car. They each would have trouble finding hard facts to back up their claims of the car s worth. Here is a preliminary rule assuming that each of the women has claim to one third of the value of the car. Preliminary Rule Each person gives a sealed bid for the full value of the car. We make up a pot consisting of the car and cash in the amount of 2/3 of the highest bid, supplied by the highest bidder. Then each person withdraws exactly what she wants. The high bidder gets the car and the others 1/3 of their announced value of the car. There is enough cash to do this because their bids were lower. If there is money left in the pot, divide it evenly. Ties are broken by some agreed upon random device. We give an example. Emily bids $60,000, Mary $57,000 and Joan $36,000. The average bid is $51,000 so Mary would be unhappy if payments were made on the basis of the average. According to the rule, Emily puts $40,000 into the pot and takes the car. Mary withdraws
$19,000 and Joan $12,000. Each has exactly 1/3 of their announced value. There is $9,000 remaining in the pot which is divided evenly. Each gets a $3,000 bonus. It is fair but not envy free since clearly Mary got more than Joan. This may be agreeable to all but we can make the distribution envy free in at least two ways. Option 1 Rule We make up the pot as in the preliminary rule, the high bidder takes the object and the rest withdraw amounts based on the second high bid. Any remaining cash is divided evenly. In the above example, Emily gets the car and puts $40,000 in the pot. Mary and Joan each take out $19,000. The pot now contains $2,000 which is divided evenly. This is now envy free but Joan gets a bigger bonus. Option 2 Rule Sealed bids are submitted. The high bidder gets the object and pays the others based upon the second high bid. In the above example Emily gets the car and pays $19,000 to Mary and to Joan. In this case only Joan gets a bonus, but it is envy free. In all of the above options, each woman is guaranteed at least 1/3 of the car, even if the other two are in a coalition to try to prevent it. The idea would be that if such a situation arises where the three have to share the car, the three women do not have to hammer out a shared value of the car. Instead they decide which rule to apply. The reader will have no difficulty in modifying the above rules when there are any number of participants, or if the claims are some other fraction then equal. The idea to use the second high bid is due to William Vickery, a winner of the 1996 Swedish Parliament prize in Economics in memory of Nobel. He argues that since at least two people think that the object is worth that amount, it comes closest to a community value. Here we give one more application of this idea.
Vickery Auction and Two-stage bidding. Here each person makes a sealed bid. All of the bids are announced but who made each bid is not revealed. Then a second round of bidding is done with the restriction that the only bids allowed are those in the amounts of the first round bids. Then the rules for the distribution are the following: 1) The highest bidder in the second stage is awarded the object, unless there is a tie. 2) If there is a tie for the highest bid in the second stage, then those who are tied in the pool and we go back to the first stage bid. The highest first stage bid in the pool obtains the object unless there is still a tie. 3)If there is still a tie, use some lottery device to break the tie. 4) The distribution of payment is based on the second high stage two bid. If there was a tie for the high bid, then that bid is also considered to be the second high bid. Here is an example. Suppose there are 6 people with bids 60,55,50,50,40,and 30. The bids on the table for the second round are 60,55,50,40, and 30. Say the second round bids are respectively 55,55, 50,50,50,50. The person who bid 60 and then 55 gets the object by 2) above and the payment is 55/6 to each of the other 5 participants. If the second stage bids had been 55,60,55,50,55, and 50, then the second participant gets the object and pays on the basis of 55 as above. In the preceding scenarios we have divided an object that could not be physically divided. We now turn to cases where objects to be divided are physically divisible. For example a fence can be built anywhere on a piece of land. The objects of a kitchen all put on the table are essentially of this type. In general if enough items are considered, we might consider the aggregate as divisible. In keeping with the title of this paper, we will divide a pie, where it is understood that pie is a surrogate for what is to be divided.
Everyone is familiar with the two person case, in which one person cuts the pie into two pieces and the other chooses one. This scheme is fair and envy free since both get the biggest piece. One because she cut the pie exactly into halves and the other because he chose the biggest. Last Chooser: Start with 6 participants, who figuratively are put around a table with the pie on the table. The order in which the participants sit is determined in some way so that we have a first and a second etc. 1) The first person cuts what he thinks is 1/6" of the pie. 2) The second person decides if It is less than or equal to 1/6th in which case she passes It is more than 1/6 th in which case she reduces it to 1/6th. 3) The succeeding persons duplicate what the second person has done, i.e. either pass or reduce to 1/6 th. 4) The last person may choose to take the piece of pie or passes. If she passes, then the last person who modified the piece of pie takes it. If no one modified it, the first person gets the piece. Now the person who took the piece made the piece 1/6 th so is satisfied. Everyone else passed on this piece or on a piece that was bigger. Hence they think that what remains is at least 5/6 of the pie. The person who got the piece is now removed and we repeat the above with only 5 people so in the above scheme 6" is replaced by 5". One person gets ( 1/5)(5/6) = 1/6. Then we go to four people, then 3 and finally the two person divide and choose. The other method starts at the other end with two people and then adds the third. Lone Chooser. This scheme begins with the two person scheme. (I use to indicate the
individual persons value system) Step 1. Two people do the divide and choose scheme so each thinks that have at least ½ of the pie. Step 2. Enter participant 3 who now does the following with each of the first two participants; a) the ones with the pieces of pie divide them into 3 equal pieces. b) participant 3 chooses one of them. If we stop here, this is fair. For the first two retain 2/3" of ½ which is 1/3". Meanwhile participant 3 by picking the biggest got at least 1/3" of two parts which add up to 1 so gets at least 1/3". Step 4 Enter participant 4 who does the following with each of the first three. a) each of the first three divides their chunk of pie into 4 equal parts and b) participant 4 chooses one of them. Etc. The final method we will discuss is the Adjusted Winner method due to Alan Taylor of Union College and Steven Brams of New York University. This is for two people and allows for sentimental value considerations. There are several items to be shared. Each participant assigns numbers to the items. These numbers should add up to 100 and represent relative values of the items. The method to be described below will: 1. Be equitable. Each participant receives the same number of points; 2. Is fair. The number of points is bigger than 50. The other person get less than 50 so this is envy free.
3. Be efficient. A technical property that means that no subsequent trade will be beneficial to both parties. The method works best when the number of items is about 8 and the items have roughly the same commercial value. Here is an example. John and Jill make the following assignments NAME NAME JOHN SHIFT JILL Y/X LABELS ITEM file cab. 8 20 bookcase 10 3 VCR 12 15 chair # 1 8 6 desk 16 18 sofa 12 10 TV 8 8 dining table 20 12 chair # 2 6 8
NAME NAME JOHN SHIFT JILL Y/X LABELS ITEM file cab. 8 20* bookcase 10* 3 VCR 12 15* chair # 1 8* 6 desk 16 18* sofa 12* 10 TV 8 8 dining table 20* 12 chair # 2 6 8*
TEMPORARY TOTALS 50 61 FIGURE 1 The first instruction is to circle (here starred) the biggest value in each row. And if there is a tie, leave it for the moment. Total the encircled items and put the total at the bottom. If these two totals are the same, then we are done. Award the items encircled to each participant. In most instances this will not be the case so we proceed here. In the first line of the table, put a 'A' in the column for theperson that has the smallest total and a 'B' (for biggest) in the column with the largest total. This is entered in the first row of the table. See figure 2. Next assign the tied items to participant B, i.e. encircle the value in participant B's column then recompute the total as shown in figure 2. NAME NAME JOHN SHIFT JILL Y/X LABELS ITEM A B file cab. 8 20* bookcase 10* 3 VCR 12 15*
chair # 1 8* 6 desk 16 18* sofa 12* 10 TV 8 8* dining table 20* 12 chair # 2 6 8* TEMPORARY TOTALS 50 61 50 69 Figure 2
On this page we will fill in the last two columns. NAME NAME JOHN SHIFT JILL Y/X LABELS ITEM A x B y file cab. 8 20* 2.5 V bookcase 10* 3 VCR 12 15* 1.25 III chair # 1 8* 6 desk 16 18* 1.125 II sofa 12* 10 TV 8 8* 1 I dining table. 20* 12 chair # 2 6 8* 1.33 IV TEMPORARY TOTALS 50 61 50 69
figure 3 Column 5 now will be filled in. For each item which has temporarily assigned to participant B (ie encircled in B's value column) let y be B's value and x be A's value. Divide y by x and put the result in column 5. See figure 3. All of these numbers should be bigger than 1. The labels column is filled in this way. Write a I in the row where y/x is least. Then II in the next smallest y/x etc. If there are ties, just pick one first then the others and continue. See figure 3. The idea is that we now start to transfer some items from B to A in order to equalize the sums. This is done on the here. NAME NAME JOHN SHIFT JILL Y/X LABELS ITEM A x B y file cab. 8 20* 2.5 V bookcase 10* 3 VCR 12 15* 1.25 III chair # 1 8* 6 desk 16? no 18* 1.125 II sofa 12* 10 TV 8* y? ok 8 n 1 I dining table 20* 12
chair # 2 6 8* 1.33 IV TEMPORARY TOTALS 50 61 50 69 58 61 74 43 figure 4 We now test whether we can shift some from B to A. We start in the row marked I. So we put a? in the shift column and recompute the totals as if the item was shifted to A. This increases A's total and decreases B's. See figure 4. These are the third temporary totals 58, and 61. Since B's total is still larger we write ok in the shift column and put a y (for yes) in A's value to remind that this item now belongs to A and an n (for no) in B's column to remind us that it no longer belongs to B. Now we look at the row marked with a II and do the same thing. The shift contains a? and we recompute the totals if that item were shifted. The result is shown in figure 4 and we see that now A's total is larger so we may not shift the entire item over. So we write a "no" in the shift column. If we could have shifted all of II we do and then go on to the next one. Since we cannot shift the entire amount, this item needs to be shared by a fractional amount. We now compute this fraction. I copy the relevant lines from the above figure.
Item participant A participant B Desk 16 a 18 b temp. Total 58 c 61 d If a fraction t is shifted from B to A then the total for A becomes 58 + 16 t and for B becomes 61-18t. We want to make them equal, i.e. 58 + 16t = 61-18 t. The solution is t= (51-58)/(16+18)=3/34. After we have shifted this fraction of the desk from B to A the total for each participant is 58 + 16(3/34) = 59.41= 61-18(3/34). In general, t = (d-c)/(a+b). How do we shift a fraction of the desk? This is where we use our first method. Each participant gives a value and we award the desk to the high bidder and the payment is based on the average. If B is the high bidder, then B retains the desk and pays A (3/34) times the average. t times the average. If A is the high bidder, then A gets the desk and pays (31/34) times the average. (1-t)times the average. Example: A bids $40 and B bids $50. The average is $45 so B retains the desk and pays A 45(3/34)=39.70. If the bids are reversed, A bids $50 and B bids $40 then A gets the desk and pays B 45(31/34)=45.03. Having many ways to divide assets allows us to do various things. Here are some examples. Example 1: Use the Adjusted Winner for the furniture in the dining room Use it again for the furniture in the bedroom.
Etc. Example 2: Use adjusted winner with the items being Bedroom furn. Dining room furn. Family room furn.without the grand piano Living room furn. Etc. Then do the grand piano with the first scheme and do the kitchen with the divide and choose scheme. Acknowledgments: The data for the Bond s home run is from the San Francisco Chronicle. The entire finding of Judge McCarthy and these articles can be found by googling Bond s 73 rd home run. The Last Chooser method is due to Hugo Steinhaus and the Lone Chooser is my contribution.