The Efficient Allocation of Individuals to Positions by Aanund Hylland and Richard Zeckhauser Presented by Debreu Team: Justina Adamanti, Liz Malm, Yuqing Hu, Krish Ray Hylland and Zeckhauser consider the problem of efficiently allocating individuals to positions with capacity constraints (maxima, minima, or strict equalities) when the preferences of individuals are unknown and there is not an external medium of exchange that can facilitate the allocation. The trick here is developing a social choice mechanism using an implicit market procedure that induces individuals to express their true preferences (rather than behaving strategically by misrepresenting their preferences). Ideally, position assignments will be Pareto optimal (this is the criteria for efficiency). Further, the mechanism must be able to accommodate situations when all individuals are treated equally and also when some are systematically favored (this difference in weighting is achieved by giving some individuals a larger endowment in the pseudomarket). Conditions for an equilibrium allocation are (must occur simultaneously): 1. Elicits honest indication of preferences from the individuals being assigned. 2. Allocates individuals efficiently to positions given these preferences (an efficient allocation is one that is ex ante Pareto optimal, which guarantees it also to be ex post Pareto optimal). 3. Meets given distributional objectives. In general, such a solution will exist if the number of individuals to be assigned is large compared to the position capacities. Variables & Notation i I j J M j u ij u i U w ij w i W p ij p i P q j q B i indicator of individuals total number of individuals indicator of jobs total number of jobs number of individuals assigned to job j true utility for individual i if he/she gets job j true utility vector for individual i true utility set submitted utility for individual i if he/she gets job j submitted utility vector for individual i submitted utility set the share of probability of individual i getting job j probability vector of job assignment for individual i set of probabilities of job assignment market clearing price of job j set of market clearing price budget of individual i 1
SECTION I: Formulation of the Problem Assumptions More specific mathematical assumptions can be found in the appendices.. 1. Individuals are indivisible (one person cannot be divided up and assigned to different positions). 2. Individual preferences are unknown (and are only discovered by asking or by allowing individuals to respond to the assignment procedure, but individuals are not required to respond truthfully when asked). 3. Individuals know the probability of obtaining each job (probability of assignment to each job). 4. Individuals are von Neumann-Mortenstern maximizers of expected utility. 5. Each person s preferences are assumed to depend only on his own assignment (he does not care about others assignments). 6. Jobs do not have preferences for individuals (makes this differ from the Gale-Shapley problem). 7. Values external to individual preferences play no role. Problem Statement I individuals must be assigned to J jobs. M j individuals must be assigned to job j for all jobs j = 1, 2,..., j. Note that M 1 +M 2 +...+M j = I. Individual i gets utility u ij from being assigned to job j. This is an expected utility from the lottery of jobs and is the weighted average of utilities from all jobs j weighted by their probabilities of being assigned. An individual engages in strategic behavior to maximize expected utility (i.e., wants to increase the probability of getting his most favored job this would increase his expected utility and may be why a person would behave strategically and not reveal their true preferences about assignments). We are given I, J, M 1,..., M j, where j n=1 M j = 1. U = (u ij ) is an i x j matrix with i = 1,..., I and j = 1,..., J. Note that the matrix U=(u 1.,...,u i.,...u I. ) has rows u i. where each represents a 1 x j row vector where each individual i lists their utility from each job j such that u i. = (u i1,..., u ij ). The U is matrx is u 11... u 1i... u 1I... u i1... u ii... u ii... u I1... u Ii... u II Each individual i submits a preference statement W i =(w i1,...,w ij ) that tells his preferences over all J of the jobs. This may or may not be identical to his true preferences, u i (it could be truthful or strategic). Thus, the assignment procedure receives a total submission W=(w 1.,...,w i.,...w I. ) and then generates allocations that satisfy the constraints given below. The objective is to find an allocation procedure that (in general) elicits W and processes it to generate a Pareto efficient outcome; to induce people to give preference rankings that create efficient allocations and constraints and not behave strategically. Note that evaluations of efficiency depend on U (true preferences), not W (state preferences). The procedure allows individual s preferences to be weighted different ways (it does not 2
have to be equally). Individuals cannot be assured their first choice job unless some individual s preferences get a much higher weigh than others or preference orderings among individuals are very different. Based on W i, individual i will be assigned to jobs 1 through j with probabilities p i = (p i1,..., p ij,..., p ij ) and can be summarized for all individuals in the matrix P=(p 1.,...,p I. ) and represents the lottery presented to the all individuals to be assigned. Each person s individual probabilities of being assigned to each of the J jobs must sum to one: j n=1 p ij = 1 for all i (CONSTRAINT 1 each person must be assigned to one and only one position with certainty). Each job s individual probability for every person i must sum to that job s capacity: i n=1 p ij = M j for all j (CONSTRAINT 2 position capacities must be exactly filled). These two constraints tell us that the rows of P must sum to 1 and the columns must sum to M j, respectively.. All p ij 0 (CONSTRAINT 2b nonnegativity of probabilities). Interpretations of Pareto Optimality Recall the definition of Pareto Optimality: an outcome is Pareto optimal when one agent cannot become better off without making another agent worse off. There are two possible interpretations of Pareto Optimality: 1. Ex Post Pareto Optimality. An allocation is EPPO if the final assignment of positions to individuals produced by performing the lotteries (the allocation procedure) is Pareto optimal relative to their preferences among positions. Preferences over positions are measured by direct utility (where the job giving the highest utility is the most preferred). 2. Ex Ante Pareto Optimality. An allocation is EAPO when the procedure s assignment of lotteries to individuals is Pareto optimal relative to their preferences among lotteries. Preferences over lotteries are measured by expected utility (where the job giving the highest expected utility is the most preferred). If a procedure is Pareto optimal ex ante then it is also Pareto optimal ex post, but the converse need not be true. Consider the following three individuals (1, 2, 3) and positions (A, B, C): A B C 1 100 10 0 2 100 10 0 3 100 80 0 Any allocation procedure that selects A, B, or C randomly and offers him his first choice and then selects on of the remaining two at random and offers him his first choice and then gives the final individual the only remaining position will result in each individual getting the same lottery on positions ( 1 3, 1 3, 1 3 ) (these are the probabilities of being assigned to jobs 1, 2, or 3, respectively). If we apply these probabilities to the given utilities, we can calculate each individual s expected utility as E[U] 1 = 36 2 3 ; E[U] 2 = 36 2 3 ; and E[U] 3 = 60. No individual has an incentive to misrepresent their preferences because there is no way for them to increase the probability of getting their most preferred job by behaving strategically, so this procedure elicits honest statements of preferences and w i = u i. Note that the individuals rank the positions in the following way: A B C 1 Best Second Worst 2 Best Second Worst 3 Best Second Worst 3
All individuals have the same ordinal ranking of jobs (each person chooses position A is their favorite and C for their least favorite). Any allocation based on this procedure will be ex post Pareto optimal because there would be no way to make someone better off without making someone else worse off. With this procedure, all individuals obtain their most-favored position out of the ones that are available to them. This outcome is not ex ante Pareto efficient, however, because this allocation procedure does not maximize all individuals expected utilities subject to the other individual s utilities. If the allocation procedure were instead: give individual 3 job C and then randomly choose between individuals 1 and 2 for job A. Then individual 3 would face the lottery (100, 0, 0, ) and individuals 1 and 2 would face the lottery (0, 1 2, 1 2 ). Their new expected utilities given this new allocation procedure are E[U] 1 = 50; E[U] 2 = 50; and E[U] 3 = 80. All three individuals are better off (have higher expected utility). There is no way to make one better off without making another worse off (when comparing to these new expected utilities), so this allocation is ex ante Pareto optimal. This is also true when looking at direct utilities, so this allocation is ex post Pareto optimal, as well. For example, if we gave individual 3 job A instead of job B, he would go from having utility of 80 to utility of 100 he is better off. But either individual 1 or 2 it does not matter which would have to accept a job that gave him utiliity less than 100, clearly making him worse off. An efficient allocation is one that is ex ante and thus also ex post Pareto optimal. The main idea of ex ante Parteo optimality is whether or not any individuals can be made better off in terms of their expected utility. SECTION II: Pseudomarket for Probability Shares To achieve an efficient allocation (as defined above), a pseudomarket for probability shares is created. What better way to find an efficient allocation than through the use of the traditional market mechanism? Imagine that individuals are able to trade probabilities of being assigned to a certain job. Imagine the example in the previous section what if individuals 1, 2, and 3 could trade the three items (probabilities) in their probability basket such that if someone valued a particular job more, they could trade or pay for a higher probability of obtaining that job. Individuals would then be able to construct their own lotteries based on how much they valued certain jobs (as reflected by their direct utilities). Individuals would be able to buy probability shares for each job from a central marketer that would announce prices and shift them in response to excess demand. The same result will occur if instead of directly selling probability shares, individuals are asked to provide the utility levels they receive from each of the possible jobs. When the purchased commodity is a lottery...and the preferences for outcomes are expressed as von Neumann- Morgenstern utilities, all that is required is that each individual provide a vector of J numbers that gives his utility value for each of the J positions. Let this be called the individual s von Neumann-Morgenstern utility vector (VNM utility vector). This saves time and effort search costs that would have been expended engaging in the market (they no longer need to search for their optimal lottery that maximizes their expected utility over various price iterations). 4
SECTION III: Detailed Procedure First assume that all individuals submitted their honest VNM utility vector when asked (we will deal later with whether or not this is a valid assumption) and that each individual has his own budget constraint, B i (CONSTRAINT 3). The objective is to efficiently allocate lotteries to individuals subject to the three constraints listed at the beginning of the paper. A market is simulated in the following way: The commodity to be bought and sold on the pseudomarket are probability shares and these shares are infinitely divisible. Imagine that the good to be bought and sold on the psdeudomarket is a j a single probability share (this is thinking of this in the discrete case, when in truth we are dealing with a continuous case). The more a j you accumulate, the higher chance you have of obtaining job j. Thus, p ij is your basket of shares and the proportion of your budget B i you put towards trying to obtain jobj, which you would do if you value this job more. It is important to note that the shares, regardless of their magnitude, must add to one. Essentially, individual i determines, through purchase, his overall probability of obtaining job j (p ij ), based on his preferences, his budget constraint, and the given price of shares. These probability shares are summarized in matrix P, where the rows denote individuals and the columns denote jobs. Each entry p ij is the probability that individual i will obtain job j. Individual purchases preferred lottery given the prices of various probability shares offered on the market and his budget constraint. The sum of his probabilities must equal one since he is to be assigned to exactly one position (recall constraint 2). If more than one person could be placed in each position, this number would add to more than one (these are the sums of the columns of P). Price of probability shares is given by the vector q = (q 1,..., q j,..., q J ) for q i > 0. This gives the price of probability shares in posotion j (a 0.75 chance at position j would cost 0.75q j ). The price a person would pay for probability share p ij is p ij q j. Here, p ij is the percentage of their total budget B i they will devote to purchasing a probability share for job j. Their individual p ij is determined by how much utility job j gives person i it is determined by their utility function. Note that the menu of q j s is the same for everyone. Further, there will always be on price that is equal to zero (this is to ensure that if one individual s budget is equal to zero, they can still obtain a position). Individuals chose p i1,..., p ij to maximize the following objective equation. (That is, they choose how many shares of a job to buy based on how much utility that job would give them, if they were assigned to it.) J EU i = p ij u ij subject to 0 p ij 1 j and j=1 J p ij q j B i j=1 and I p ij = 1 i=1 5
If more than one lottery gives an individual the same expected utility, the individual will choose the lottery that he can obtain at the lowest cost (i.e., he will minimize cost while simultaneously maximizing expected utility). This results in no weak preferences and no indifference between lotteries). The goal of the mechanism, after each individual has chosen p i1,..., p ij, is for the total demand for probability shares for jobs to equal M 1,..., M j,..., M J (satisfy constraint 2). Prices will then adjust since in general, the above will not happen and there will be excess demand and supply. Prices will adjust until excess demand is equal to zero. There will always exist a price vector q such that there exist numbers P chose to maximize individuals expected utilities in accordance with above and satisfying constraint 2. A price vector can be chosen such that at least one price is zero. This assignment procedure will always guarantee an ex ante (and thus ex post) final assignment. Conducting the Lottery After individuals have stated their preferences and the pseudomarket has been run to assign individuals lotteries, the final step is to place individuals in positions in a manner that offers each individual precisely the lottery on positions that he has purchased. The example on p. 306 will be used to explain the algorithm using the following P matrix: A B C D 1 0.7 0.3 0 0 2 0 0 0.7 0.3 3 0.3 0 0.3 0.4 4 0 0.7 0 0.3 1. Eliminate from P any rows containing only one positive number (due to constraint one this would only be the case if that number is 1, since all rows must add to one). That position is assigned to the corresponding person and the algorithm proceeds among the remaining individuals (i.e., if this occurs, rewrite a new submatrix of the remaining individuals and jobs before proceeding to the next step). 2. Find a cycle of positive numbers in the remaining reduced matrix by starting at any positive (nonzero) entry and move horizontally until another positive entry is reached (it easiest to start in the row one, column one position as long as this is not zero). Then move vertically to another positive entry. Continue to alternate between horizontal and vertical until the path comes back to a row or column in which it has been before. Finish the cycle in the next step by going to the previously visited entry in that row or column. It is helpful to underline each entry chosen for the cycle and to draw arrows to signify the cycle path between entries. 6
3. If any smaller circular cycles (that cannot be escaped) are encountered, follow the procedure below in this smaller cycle and ignore the rest of the matrix. If this does not occur, ignore this step. 4. Divide the entries selected for the cycle into two classes: the first, third, fifth, etc. (odd-numbered) steps (probabilities) in the cycle are placed in one class; the second, fourth, etc. (even-numbered) steps in the cycle are placed in a different class. Thus, when we trace out the cycle we alternate between entries in class one and entries in class two. It is helpful to circle the enries in class one and place a box around the entries in the other class. 5. Choose an increment for each class by finding the lowest number (probability) in that class. The increment is chosen such that at least one positive element is reduced to zero and no element is reduced below that. Make note of the increment associated with each class. 6. Of these two increments, choose the smallest of the two and denote it i L. The larger of the two will be denoted i H. Here the smallest increment is the one associated with the square class, so i L = 0.3 and i H = 0.4. 7. The class with the largest increment (i H ) will be the addition class. The other class (with the smaller increment, i L ) will be the subtraction class. Note the shape associated with each class (circle or square). The circle class (with the larger increment i H = 0.4) is the addition class. The square class (with the smaller increment i L = 0.3). 8. Add i L to each probability in the addition class. Subtract i L from each probability in the subtraction class. Make note of this in your matrix by crossing out the old probabilities and writing in the new ones. i L = 0.3 will be subtracted from each probability in the square class and i L = 0.3 will be added to each probability in the circle class. 9. Check that all rows still add to one and all columns add to M j (constraints 1 and 2). If not, a mistake has been made. Also be sure that there are no nonnegative probabilities (constraint 2b). Put new submatrix from step 8 back into larger matrix to check these conditions. 7
10. Repeat cycle as necessary until all individuals have been assigned to positions. Individual 2 is assigned to position C (note the 1 in this position). We can now eliminate row 2 and column C and proceed through the algorithm again, using the matrix below.. Proceeding through the algorithm again (round two): 1. Round two matrix, after eliminating any rows and columns containing a 1: 2. Create cycle: A B D 1 0.7 0.3 0 3 0.3 0 0.7 4 0 0.7 0.3 3. There are no circular cycles, so step 3 can be skipped. 4. Break cycle into class one (circles) and class two (squares). 5. Choose increment. 8
6. Choose smallest increment of the two and denote it i L. i L = 0.3. Therefore, i H = 0.7. The smaller increment is associated with the square class and the larger increment with the circle class. 7. Denote addition and subtraction class. The class with the larger increment (circle) is the addition class and the class with the smaller increment (square) is the subtraction class. 8. Manipulate probabilities using increment classes chosen in the previous two steps. i L = 0.3 will be added to the probabilities in the circles. i L = 0.3 will be subtracted from the probabilities in the squares. A B D 1 1 0 0 3 0 0 1 4 0 1 0 9. Check that all rows and columns add to the appropriate number. All columns and rows add to one. 10. Assign individuals to positions when their intersection is equal to one. Individual 1 is assigned to position A, individual 3 is assigned to position D, and individual 4 is assigned to position B.. Honest Elicitation of Preferences Two conditions are required for individuals to report their true preferences over positions, rather than behave strategically. 1. Assignment mechanism must employ an individual s preferences so that it chooses what the individual would have chosen on their own (it must maximize expected utility subject to market prices, budget constraint, and nonnegative probabilities that sum to one). The mechanism must behave like a rational individual. 2. No individual should not be able to manipulate the market by misrepresenting his preferences, creating a more favorable outcome for himself. A person is influential is he can cause a change in the price vector q by misrepresenting his preferences (this will happen if a person is thought to be indifferent between the continuum of lotteries). A person will distort the market if the potential gain from doing so outweighs the potential loss. Distortion of the market could have two possible outcomes for the individual: 1. He may receive an efficient bundle due to the prices that are established (effectively lose ). 2. He may secure a more a favorable set of prices (effectively win ). No individual can effect price if there is a large number of market participants relative to the number of positions (individuals are price-takers because a large number of individuals are interested in each position). When no individual can gain through misrepresentation of preferences, neither can a coalition of individuals (unless they have identical preferences). Thus, individuals will provide their honest preferences rather than strategically misrepresented preferences. 9
SECTION V: Equity, Endowments, and Lack of Medium of Exchange Considerations of equity are manifested through each individual s budget constraint, B i (their endowment). For each individual to be weighted equally in the process, each individual should be given an equal endowment and thus all B i would be identical. To weight certain individuals more heavily, those individuals should be given a larger endowment. Since the situations presented in this paper are ones in which money cannot facilitate an efficient outcome, the goal is to come up with an efficient outcome when a medium of exchange is not present or available. SECTION V: Extensions and Generalizations a) Under capacity (not enough positions) This is the case where the total capacity is less than the number of persons, so some people will not get a position: Solution: J M j < 1 j=1 Create some new positions N as not being assigned, so that J j=1 M j + N n=1 M n = I where N is the not being assigned new position. Then we use the same position assignment procedure as in the paper. The person who is assigned to the not being assigned position will have no real position assigned to them. b) Over capacity (too many positions) This is the case where the total job capacity is more than the total number of individuals, so some positions will have no one assigned to them (there will be empty jobs): J M j > 1 There will be M 0 vacant positions, where M 0 = J j=1 M j I Solution: j=1 Add M 0 fictitious persons, each of whom is indifferent between any positions and thus will choose the least expensive of all lotteries. Then we use the same position assignment procedure as in the paper. The positions who get the fictitious individuals will be the unassigned (empty) positions. c) Flexible capacity (other case of over capacity) The total maximum capacity of the positions exceeds the number of persons, but we are not completely free to place the vacant slots where we wish. Solution: Assume for every M j capacity there is a minimum capacity M j that must be fulfille. Then we can create fictitious persons where the total capacity of this fictitious persons is M j M j. 10
The remaining total for every capacity will be M j for every position. Then we use the same position assignment procedure as in the paper. d) Multiple assignments This is the case where there is a possibility that a person shall get more than one assignment. Let there be k assignments, where those k positions do not need to be different. The final outcome for person i with k assignment will be a k-tuple of positions. Solution: We can regard a person who will get k assignments as k persons which each of person will get one assignment. Then we use the same position assignment procedure as in the paper.. SECTION VI: Conclusion A four-step procedure for allocating individuals to scare positions has been developed. This produces an outcome that is ex ante Pareto optimal (regarding expected utilities and the lotteries) and ex post Pareto optimal (regarding direct utilities and actual positions assuming honest preferences are given). The steps in this procedure are: - 1. Individuals are given hypothetical endowments reflecting their relative strengths of claims for positions. If all individuals are to be treated equally, equal endowments are given to each individual. 2. Individual s VNM utilities for each position are elicited. 3. Lotteries of probability shares are assigned using a pseudomarket and collected in matrix. 4. The lottery is conducted using a specialized algorithm and individuals are assigned to positions. 11