Rental harmony

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Harmonious rent is a kind of fair share issue in which items that can not be shared and monetary costs should still be shared simultaneously. The problem of housemates and room-assignment-rent-division is the alternative name for the same problem.

Dalam pengaturan umum, ada ${\ displaystyle n}  ÂÂ$ mitra yang menyatukan ${\ displaystyle n}  ÂÂ$ -ruang kamar untuk biaya yang ditetapkan oleh pemilik rumah. Setiap teman serumah mungkin memiliki preferensi yang berbeda - seseorang mungkin lebih suka kamar yang besar, yang lain mungkin lebih suka kamar dengan pemandangan ke jalan utama, dll. Dua masalah berikut harus diselesaikan secara bersamaan:

• (a) Tetapkan ruangan untuk masing-masing pasangan,
• (b) Tentukan jumlah yang harus dibayar setiap mitra, sehingga jumlah pembayaran sama dengan biaya tetap.

There are some properties that we want the task to be fulfilled.

• Not negativity (NN) : all prices must be 0 or more: no partners to be paid to get a room.
• Envy-freeness (EF) : Given the pricing scheme (room lease assignment), we say that the partner prefers the space given if he/she believes the rental room parcel better than all other packages. EF means that each partner prefers the space it provides. Yes, no partner wants to take another room in the rent set for that room.
• Pareto-efficiency (PE) : No other partner assignments to a better room for all partners and strictly better for at least one partner (given a price vector).

Envy-freeness implies Pareto efficiency. Evidence: Suppose by contradiction that there is an alternative task, with the same price vector, which is strictly better for at least one partner. Then, in the current allocation, the couple feels jealous.

The lease-harmony problem has been studied under two different assumptions about partner preferences:

• In the ordinal utility version, each pair has a preferences relation on the bundle [space, price]. By providing a price vector, partners may only say which rooms (or rooms) prefer to be rented at that price.
• In the cardinal utility version, each pair has a monetary valuation vector. Partners must say, for each room, how much money he paid for the room. The partner is assumed to have a quasilinear utility, that is, if he values ​​the space as ${\ displaystyle v}$ and pay ${\ displaystyle p}$ , the net utility is ${\ displaystyle v-p}$ .

Cardinal assumptions imply an ordinal assumption, since a vector of judgments is always possible to establish a preferential relation. Ordinal assumptions are more common and place less mental burden on partners.

Video Rental harmony

Ordinal Version

Su: one person per room

The Protocol by Francis Su makes the following assumptions on partner preferences:

1. A good house : In each rental partition, everyone finds at least one room rental package that is acceptable.
2. No externalities : The preference relationships of each partner depend on the room and the rental price, but not on the choices made by others.
3. Miserable partners : any weak partners prefer free rooms (rooms with rent 0) above other rooms.
4. Topologically closed preference preferences A partner who prefers a room for a convergent set of prices, preferring that room at a price that limits.

Normalkan total sewa menjadi 1. Kemudian setiap skema harga adalah titik dalam ${\ displaystyle (n-1)} Â Â$ -dimensi simpleks dengan ${\ displaystyle n} Â$ simpul dalam ${\ displaystyle \ mathbb {R} ^ n}} Â Â$ . Protokol Su beroperasi pada versi dobel dari simples ini dengan cara yang mirip dengan protokol Simmons-Su untuk pemotongan kue: untuk setiap titik dari triangulasi dari simpleks ganda, yang sesuai dengan skema harga tertentu, ia meminta mitra pemiliknya "kamar mana yang Anda sukai dalam skema do it? " Hal ini menghasilkan pewarnaan Sperner dari simpleks ganda, den den dem demian ada sub-simpleks kecil yang sesuai dengan perkiraan ruang dan sewa kamar bebas iri.

The Su protocol returns a unified allocation order to an enviable free allocation. Price is always not negative. Therefore, the results meet the requirements of NN and EF.

and provides a popularized explanation of Su's Rental Harmony protocol.

and provides on-line implementation.

Azriely and Shmaya: roommates

Azriely and Shmaya generalize the Su solution for situations where the capacity of each room may be larger than one (eg, some partners can stay in the same room).

They prove an enviable free allocation under the following conditions:

1. A good home : Each partner likes at least one room given every price vector.
2. No externalities : All partners love free rooms.
3. Miserable Partners : Preferences continue in price.

The main tools used in the verification are:

• K-K-M-S theorem - generalization of K-k-m theorem.
• Hall wedding theorem.

Their solutions are constructive in the same sense as Su's solution - there are procedures that approach the solution for a certain precision.

General properties of ordinal protocol

A. In Su solution and Azrieli solution & amp; Shmaya, the preferences relation of each partner is allowed (but not mandatory) depends on the entire price-vector. Yes, the partner may say "if room A costs 1000, then I prefer room B to room C, but if room A costs only 700, then I prefer room C to room B".

There are several reasons such an announcement can be useful.

1. Future planning. Suppose the couple thinks that space A is the best, then B, then C. If A is expensive, the partner lives in B. But if A is cheaper, partners might buy C (the cheapest), and then save money and switch to A.
2. Incomplete information. Price vectors can give couples some indications on room quality.
3. Neighbors. Price vectors can allow couples to predict, to some extent, what kind of people will stay in neighboring rooms.
4. The effects of irrationality, e.g. framing effect. If room B and room C have the same quality and have the same price, then partners can buy A. But if room B becomes more expensive, then the partner can switch to C, think that "it is equal to B but with low price.. ".

B. Su solutions and solutions Azrieli & amp; Shmaya made the "Miserly partners" assumption - they assume that couples always prefer free rooms for non-free spaces. This assumption is strong and not always realistic. If one of the rooms is so bad, it is likely that some partners do not want to stay in that room even for free. This is easy to see in the cardinal version: if you believe that room A is 0 and room B is 100, and room A is free and room cost is B 50, then you definitely prefer room B.

Su advises to undermine this assumption in the following way: each partner never chooses the most expensive room if there is free space available. This does not require people to choose a free room. Specifically, this will apply if someone always prefers a free room for a room at least cost ${\ displaystyle 1/(n-1)}$ of the total rental. However, even this attenuated assumption may not be realistic, as in the example above.

Maps Rental harmony

Cardinal version

As explained above, the input to the cardinal version is a bid matrix: each pair must bid to each room, saying how much (in dollars) this room is worth for it.

Gagasan kunci dalam solusi kardinal adalah alokasi maxsum (alias utilitarian ). Ini adalah alokasi mitra ke kamar, yang memaksimalkan jumlah tawaran. Masalah menemukan alokasi maxsum dikenal sebagai masalah penugasan, dan itu dapat diselesaikan oleh algoritma Hungaria dalam waktu ${\ displaystyle O (n ^ 3)} Â$ (di mana ${\ displaystyle n} Â Â$ adalah jumlah mitra). Setiap alokasi EF adalah maxsum dan setiap alokasi maxsum adalah PE.

Ketidakcocokan EF dan NN

Both envy-freeness and non-negative payment terms are not always compatible. For example, the total cost is 100 and the valuation is:

Here, the only maxsum allocation is to provide room 1 to partner 1 and space 2 to partner 2. To ensure pair 2 is not jealous, partner 1 has to pay 115 and partner 2 has to pay -15.

In this example, the amount of valuation is more than the total cost. If the number of assessments equals the total cost, and there are two or three partners, then there is always an allocation of EF and NN. But if there are four or more partners, then EF and NN may not be compatible, as in the following example (see proof):

Note that this example does not occur in the ordinal version, because the ordinal protocol makes the assumption "Miserly Partners" - partners always prefer free rooms. When this assumption applies, there is always an allocation of EF NN. However, in the example above, the assumption does not apply and the allocation of EF NN does not exist. Therefore, protocols in the cardinal version must compromise between EF and NN. Each protocol makes a different compromise.

Brams and Kilgour: NN but not EF

Brams and Kilgour suggested Gap Procedures :

1. Calculate the maxsum allocation.
2. If the maximum amount is less than the total cost, then the problem can not be solved, because the partner does not want to pay the total amount required by the homeowner.
3. If the maximum amount equals the total cost, then the room is allocated and the partner pays its valuation.
4. If the maximum amount is more than the total cost, then the price is lowered by gap between this price and the next lowest valuation (see book for more details).

The idea behind the last step is that the next lowest rating represents the "competition" in the room. If there is more space desired by the next highest bidder, then it should be more expensive. This is the same as the spirit for Vickrey auction. However, while in Vickrey's auction, the payment is fully independent of the partner's offer, in the Gap procedure, the payments are only partially independent. Therefore, Gap procedures are not strategic.

Gap procedures always set a non-negative price. Since the assignment is maxsum, it is also Pareto-efficient obviouly. However, some partners may feel jealous. I.e, Gap procedure meets NN and PE but not EF.

In addition, Gap Procedures can restore non-envy-free allocations, even when EF allocations exist. Brams deals with this issue by saying that: "Price Gap takes into account the competitiveness of supply for goods, which creates a market-oriented pricing mechanism.Although jealousy is a desirable property, I prefer the market mechanism when there is a conflict between these two properties, i> must pay more when the offer is competitive, even at the sacrifice that causes jealousy ".

Haake and Raith and Su: EF but not NN

Haake et al. presents the Compensation Procedure. The problem solved is more general than the problem of harmony in certain aspects:

• The number of items that can not be shared to share ( m ) may differ from the number of partners ( n ).
• There are arbitrary constraints in item bundles, as long as they are anonymous (do not distinguish between partners based on their identity). For example, there are no constraints at all, or constraints such as "each partner must receive at least some items", or "some items must be combined together" (eg because they are land that must stay connected), etc.
• The total "cost" can also be positive, which means there is some money to share. This is characteristic of inheritance divide scenario. Similarly, "item" can have a negative utility (for example, they can represent an inseparable job).

There is a "qualification requirement" for partners: the bid amount must be at least the total cost.

The procedure works in the following steps.

1. Find the allocation of maxsum (utilitarian) - the allocation with the highest number of utilities that meets the limits on the item bundle. If there are no constraints, then the allocation that gives each item to the partner with the highest appraisal is maxsum. If there are limits (such as "at least one item per partner"), then the maxsum allocation may be harder to find.
2. Unblock from each bundle value partner allocated to it. This creates the initial set of money.
3. Pay the fee from the original set. If all partners meet the qualification requirements, then the money in the set is sufficient, and there may be leftover excess .
4. Eliminate envy by compensating an envious partner. At most ${\ displaystyle n-1}$ round compensation. The procedure is fully descriptive and explicitly says which compensation should be made, and in what order. In addition, it is quite simple to do without computer support.
5. The amount of compensation made in all rounds is the smallest amount needed to eliminate jealousy, and it never exceeds the surplus. If some surplus is left, it can be divided in any way that does not create jealousy, for example, by giving the same amount for each partner (the paper discusses other options that can be considered "fairer").

When there are many complex items and constraints, the initial step - finding the maxsum allocation - may be difficult to quantify without a computer. In this case, the Compensation Procedure may begin with an arbitrary allocation. In this case, the procedure may end with an allocation containing envy-cycles . This cycle can be removed by moving the bundle along the cycle. This really increases the total amount of utilities. Therefore, after the number of restricted iterations, the maxsum allocation will be found, and the procedure can proceed as above to create an enviable free allocation.

The Compensation Procedure may charge a number of negative payment partners (ie, give partners a certain amount of positive money). This means that the Compensation Procedure is EF (hence also PE) but not NN. The authors say:

"we do not rule out the possibility that someone may end up being paid by another to take a bunch of goods In the context of fair distribution we do not find this problem at all, indeed, if a group does not want to exclude one of its members, then there is no reason why the group should not subsidize members to receive unwanted bundles. In addition, qualification requirements ensure that subsidies are never a consequence of inadequate judgment of players, the full set of objects to be distributed ".

However, other authors claim that, in the usual housemaid scenario:

"a housemate who was given a room with a negative room rent was subsidized by some housemates.In such situations, some housemates might prefer to leave the room with an unused negative room rent and exclude their roommate. who are assigned the room, because they may receive a larger discount.To avoid such situations, negative room rent should be avoided.

Abdulkadiroglu and Sonmez and Unver: EF and NN if possible

Abdulkadiro? Lu et al. suggest a market-based approach. This is a combination of auction up and auction down. It's simplest to describe as a sustainable price auction:

1. Initialize the price of each room for ${\ displaystyle 1/n}$ of total home cost.
2. Calculate sets of requests from each partner: the room or set of rooms that he likes best with the current prices.
3. Calculate the amount of excess space (rooms requested by more partners than the number of rooms; see paper for exact definition).
4. Increase the price of all billed rooms in the same rate;
5. Simultaneously, reduce the prices of all other rooms in the same rate, so the total price of all rooms is always equal to the total cost.
6. At any time, update the request of each partner and the set of rooms that are required to be redundant.
7. When the over-charged room set is empty, stop and apply Hall's wedding theorem to allocate to each partner a room on their set of requests.

In practice, there is no need to change prices constantly, because the attractive price is only the price at which the demand set of one or more partners changes. You can calculate a series of attractive prices first, and convert a sustainable price auction to a separate pricing auction. This separate price-auction stops after a limited number of steps.

Allocation returned is always envious. The price may be negative, as in the procedure of Haake et al. However, unlike the procedure, the price is not negative if there is an allocation of EF with a non-negative price.

Sung and Vlach: EF and NN if possible

Sung and Vlach prove that every free envision allocation is maxsum, and every free envy maxsum allocation for multiple price vector options. Based on this evidence, they propose the following algorithm:

1. Find the maxsum allocation.
2. Find the minsum price vector (vector where the price amount is minimized), subject to envy constraints. Such price vectors are the solution of the linear programming problem, and that can be found by the Bellman-Ford algorithm.
3. If the minimum amount equals the total cost, apply the infall allocation at minute and done.
4. If the minimum amount is smaller than the total cost, then raise all prices in a constant rate until the sum equals the total cost (i.e. add to each price: ${\ displaystyle (cost-minsum)/n}$ ). Changing all prices by the same amount ensures that the task is still free of envy.
5. If the minimum amount is more than the total cost, then there is no solution that satisfies NN and EF. There are several possible ways to proceed:
   • Lower all prices at a constant level until the sum equals the total cost (ie, subtract from each price: ${\ displaystyle (minsum-cost)/n}$ ). Some prices will always be negative, as in the solution of Haake et al.
   • Lower only a positive price at a constant rate, until the sum equals the total cost. Here, the price does not change with the same amount, so some partners will certainly feel jealous, as in Brams and Kilgour solutions. However, in this solution, envious partners get their rooms for free .

Kompleksitas runtime dari keduanya menemukan alokasi maxsum dan menemukan harga minsum adalah ${\ displaystyle O (n ^ 3)} Â Â$ .

The Sung and Vlach solutions seem to have all the desired properties of the previous protocol, namely: PE and EF and NN (if possible) and polynomial operating times, and in addition, this ensures that each partner is envious of a free space. provide implementation of similar solutions, also based on linear-programming problem solving but cite different papers.

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Strategic considerations

All protocols surveyed thus far assume that partners disclose their true valuations. They are not strategyproof - partners can earn by reporting false assessments. Indeed, strategyproofness is incompatible with envy-freeness : there is no deterministic strategyproof protocol that always returns an enviable free allocation. This is true even when there are only two partners and when the price is allowed to be negative. Evidence : Assume that the total cost is 100 and partner ratings are as below (where ${\ displaystyle x, y}$ are parameters and ${\ displaystyle 0 & lt; x & lt; y & lt; 100}$ ):

Satu-satunya alokasi maxum adalah memberikan ruang 1 ke mitra 1 dan ruang 2 ke mitra 2. Biarkan ${\ displaystyle p_ {2}} Â$ adalah harga kamar 2 (sehingga harga kamar 1 adalah ${\ displaystyle 100-p_2}} Â Â$ ). Untuk memastikan partner 1 tidak goi, kita harus memiliki ${\ displaystyle p_ {2} \ geq x/2} Â Â$ . Untuk memastikan mitra 2 tidak goi, kita harus memiliki ${\ displaystyle p_ {2} \ leq and/2} Â Â$ .

Researchers have overcome this impossibility in two ways.

Sun and Yang: Change problem

There is a variant of the problem in which, instead of assuming that the total cost of the house is fixed, we assume that there is a maximum charge for each room. In this variant, there is a strategyproof mechanism: a deterministic allocation rule that chooses min-sum costs is a strategic strategy.

These results can be generalized for greater flexibility on inseparable objects, and evidence of robust coalition strategy.

Dufton and Larson: Using randomization

Returning to the original leased harmony issue, it is possible to consider the random mechanism . Random mechanisms return probability distributions over room assignments and lease distribution. Random mechanisms are honest in expectation if no partner can increase the expected value of his utility by misreporting his judgment to the room. The randomness of random mechanisms can be measured in several ways:

1. Ex-ante Envy-Freeness means that no partner is jealous of draws from other partners. This condition is trivial to achieve in the correct mechanism: random over all possible allocations with the same probability and fill each pair ${\ displaystyle 1/n}$ of the total cost. But this condition is not interesting, because there is a high probability that in the end result, many partners will feel jealous. They may not be comforted by the fact that the lottery is fair.

3. Expected number of Envy-Free partners (ENEF) means that there are certain integers ${\ displaystyle N}$ > ${\ displaystyle N}$ . The ENEF criterion seems to be more precise than the GPEF criterion, since it measures not only the possibility of all jealousy, but also the quality of cases where allocations are not entirely free of envy. ENEF maximum mechanism is really expected at most ${\ displaystyle n-1 1/n}$ . It is possible to reach this limit for ${\ displaystyle n = 2}$ . For ${\ displaystyle n & gt; 2}$ , there is a really expected mechanism that almost reaches this limit: ENEF is ${\ displaystyle n-1}$ . The general idea is as follows. Use the VCG mechanism to calculate assignment and maxsum payments. Select one pair at random. Ignore the couple and use VCG again. Combine the results in a way that ensures that the total payout is equal to the total cost (see paper for details). It is possible to demonstrate that: (a) the mechanism is actually expected; (b) all partners except neglected partners are not envious. Therefore, ENEF is ${\ displaystyle n-1}$ . Simulations show that in about 80% of cases GPEF mechanism is also at maximum ${\ displaystyle 1/n}$ . Andersson and Ehlers and Svensson: Achieve a strategy-partial-strategy Andersson_and_Ehlers_and_Svensson: _Attaining_partial-strategyproofness "> Andersson_and_Ehlers_and_Svensson: _Attaining_partial-strategyproofness

The likelihood of relaxation from the requirements of strategyproofness is to try to minimize the "level of manipulability". This is defined by counting, for each profile, the number of agents that can manipulate the rules. The most favored fair allocation rule is a fair and minimally adjustable budget allocation rule (individual and coalition) that can be manipulated according to this new concept. Such a rule selects allocations with the maximum number of agents for whom utilities are maximized among all fair allocations and balanced budgets.

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See also

• The setting of fair items - a fair share issue in which only items can be shared, without money transfers.

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References

Source of the article : Wikipedia