Fair division experiments

Various experiments have been made to evaluate various procedures for fair division, the problem of dividing resources among several people. These include case studies, computerized simulations, and lab experiments.

Case studies

Allocating indivisible heirlooms

1. Flood[1]: case 4 describes a division of a gift containing 5 parcels: whiskey, prunes, eggs, suitcase, etc. The division was done using the Steinhaus-Banach-Knaster procedure. The resulting division was fair, but in retrospect it was found that coalitions could gain from manipulation.

2. When Mary Anna Lee Paine Winsor died at the age of 93, her estate included two trunks of silver, that had to be divided among her 8 grandchildren. It was divided using a decentralized, fair and efficient allocation procedure, which combined market equilibrium and a Vickrey auction. Although most participants did not fully understand the algorithm or the preference information desired, it handled the major considerations well and was regarded as equitable.[2]

Allocating unused classrooms

In California, the law says that public school classrooms should be shared fairly among all public school pupils, including those in charter schools. Schools have dichotomous preferences: each school demands a certain number of classes, it is happy if it got all of them and unhappy otherwise. A new algorithm[3] allocates classrooms to schools using a non-trivial implementation of the randomized leximin mechanism. Unfortunately it was not deployed in practice, but it was tested using computer simulations based on real school data. While the problem is computationally-hard, simulations show that the implementation scales gracefully in terms of running time: even when there are 300 charter schools, it terminates in a few minutes on average. Moreover, while theoretically the algorithm guarantees only 1/4 of the maximum number of allocated classrooms, in the simulations it satisfies on average at least 98% of the maximum number of charter schools that can possibly be satisfied, and allocates on average at least 98% of the maximum number of classrooms that can possibly be allocated.[3]

The partial collaboration with the school district lead to several practical desiderata in deploying fair division solutions in practice. First, the simplicity of the mechanism, and the intuitiveness of the properties of proportionality, envy-freeness, Pareto optimality, and strategyproofness, have made the approach more likely to be adopted. On the other hand, the use of randomization, though absolutely necessary in order to guarantee fairness in allocating indivisible goods such as classrooms, has been a somewhat harder sell: the term "lottery" raised negative connotations and legal objections.

Resolving international conflicts

The adjusted winner procedure is a protocol for simultaneously resolving several issues under conflict, such that the agreement is envy-free, equitable, and Pareto efficient. It has been commercialized through the FairOutcomes website. While there are no account of it actually being used to resolve disputes, there are several counterfactual studies checking what would have been the results of using this procedure to solve international disputes:

  • For the Camp David Accords, the authors construct approximate numeric valuation functions for Israel and Egypt, based on the relative importance of each issue for each country. They then run the AW protocol. The theoretical results are very similar to the actual agreement, which leads the authors to conclude that the agreement is as fair as it could be.[4]
  • For the Israeli-Palestinian conflict, the author constructs the valuation functions based on a survey of expert opinions, and describes the agreement that would result from running the AW protocol with these valuations.[5]
  • For the Spratly Islands dispute, the authors construct a two-phase procedure for settling the dispute, and present its (hypothetic) outcome.[6]

Allocating rooms and rent

Rental harmony is the problem of simultaneously allocating rooms in an apartment and the rent of the apartment among the housemates. It has several solutions. Some of these solutions were implemented in the Spliddit.org website[7] and tested on real users.[8]

공유협력잉여금

서로 다른 주체들이 협력하면 복지에 경제적 흑자가 발생한다. 협동 게임 이론은 선수들의 다양한 연합 옵션을 고려하여 이 잉여금을 어떻게 배분해야 하는지에 대한 문제를 연구한다. 샤플리 가치와 같은 개념에 비추어 그러한 협력의 몇 가지 사례가 연구되었다.[9]

페어 바겐팅

Flood는[1] 상품(예: 자동차)을 구입하는 가격에 대해 구매자와 판매자 사이의 몇 가지 협상 사례를 분석했다. 그는 "분할-차이" 원칙이 두 참가자가 모두 받아들일 수 있다는 것을 발견했다. 같은 협동원리가 보다 추상적인 비협조 게임에서 발견되었다. 그러나 경매에서 입찰자가 협력적 해결책을 찾지 못한 경우도 있었다.

공정한 하중-차단

Olabambo 등은[10] 개발 도상국에서 전기 분리의 공정한 할당을 위한 경험적 알고리즘을 개발한다. 그들은 나이지리아 상황에 적응하는 텍사스의 전기 사용 데이터에 대해 알고리즘의 공정성과 복지를 시험한다.

컴퓨터 시뮬레이션

페어 케이크 커팅

월시는[11] 온라인 공정 케이크 커팅을 위한 몇 가지 알고리즘을 개발했다. 그는 컴퓨터 시뮬레이션을 통해 케이크를 무작위 세그먼트로 나누고 각 세그먼트에 랜덤 값을 할당하여 케이크의 총 가치를 정상화함으로써 각 에이전트에 대한 평가 기능이 생성되었다. 각종 알고리즘의 평등주의 복지와 공리주의 복지를 비교했다.

슈테크맨, 고넨, 세갈-할레비는[12] 뉴질랜드와 이스라엘의 실제 토지 가치 데이터에 대해 두 가지 유명한 케이크 절단 알고리즘인 이븐-파즈와 라스트 소인기를 시뮬레이션했다. 대리점의 평가는 각 지상 셀의 시장 가치를 취하여 균일한 소음과 핫스폿 소음이라는 두 가지 다른 소음 모델에 기초한 무작위 "소음"을 추가함으로써 도출되었다. 그들은 이 알고리즘이 토지를 분할하는 두 가지 대안적 과정, 즉 토지를 매각하고 수익금을 분할하는 과정과 부동산 평가인을 고용하는 과정보다 더 잘 수행한다는 것을 보여주었다.

Welfare redistribution mechanism

Cavallo[13] developed an improvement of the Vickrey–Clarke–Groves mechanism in which money is redistributed in order to increase social welfare. He tested his mechanism using simulations. He generated piecewise-constant valuation functions, whose constants were selected at random from the uniform distribution. He also tried Gaussian distributions and got similar results.

Fair item assignment

Dickerson et al[14] use simulations to check under what conditions an envy-free assignment of discrete items is likely to exist. They generate instances by sampling the value of each item to each agent from two probability distributions: uniform and correlated. In the correlated sampling, they first sample an intrinsic value for each good, and then assign a random value to each agent drawn from a truncated nonnegative normal distribution around that intrinsic value. Their simulations show that, when the number of goods is larger than the number of agents by a logarithmic factor, envy-free allocations exist with high probability.

Segal-Halevi et al[15] use simulations from similar distributions to show that, in many cases, there exist allocations that are necessarily fair based on a certain convexity assumption on the agents' preferences.

Laboratory experiments

Several experiments were conducted with people, in order to find out what is the relative importance of several desiderata in choosing an allocation.

Fairness vs. efficiency - what outcome is better?

Sometimes, there are only two possible allocations: one is fair (e.g. envy-free division) but inefficient, while the other is efficient (e.g. Pareto-optimal) but unfair. Which division do people prefer? This was tested in several lab experiments.

1. Subjects were given several possible allocations of money, and were asked which allocation they prefer. One experiment[16] found that the most important factors were Pareto-efficiency and Rawlsian motive for helping the poor (maximin principle). However, a later experiment found that these conclusions only hold for students of economics and business, who train to acknowledge the importance of efficiency. In the general population, the most important factors are selfishness and inequality aversion.[17]

2. 두 사람 사이에 불가분의 항목이 분할된 것에 관한 설문지에 답하도록 피험자에게 요청하였다. 각 (가상)인이 각 항목에 붙이는 주관적 가치를 피험자에게 보여주었다. 고려된 주요 측면은 형평성으로 각 개인의 선호도를 충족시켰다. 효율성 측면은 부차적인 것이었다. 이러한 효과는 경제학과 학생에서 약간 더 뚜렷하게 나타났으며, 법대 학생(파레토 효율적 할당을 더 자주 선택한 학생)에서는 덜 뚜렷하게 나타났다.[18]

3. 피험자는 2인 1조로 나누어 교섭하고 4개 항목 세트를 어떻게 나눌 것인지 결정하도록 하였다. 항목들의 각 조합은 미리 정해진 화폐가치를 가지고 있었는데, 이는 두 과목마다 달랐다. 각 과목은 자신의 가치관과 파트너의 가치관을 모두 알고 있었다. 분단 후, 각 주체는 화폐가치로 항목들을 상환할 수 있었다. 항목은 몇 가지 방법으로 나눌 수 있다: 일부 부문은 공평했고(예: 각 파트너에게 45의 가치를 부여), 다른 부문은 Pareto 효율적이었다(예: 한 파트너 46과 다른 파트너 75). 흥미로운 질문은 사람들이 공평한 분업을 선호하느냐 아니면 효율적인 분업을 선호하느냐 하는 것이었다. 그 결과는 사람들이 "너무 불공평하지 않은" 경우에만 더 효율적인 분리를 선호한다는 것을 보여주었다. 대부분의 과목에서는 2-3의 가치 단위의 차이가 충분히 작다고 여겨져 효율적인 배분을 선호했다. 그러나 (45:45 대 46:75 예와 같이) 20-30 단위의 차이는 너무 큰 것으로 인식되었다. 51%는 45:45 분할을 선호했다. 전체 화폐가치가 아니라 각 피험자에 대한 항목 조합의 순위만 보여주었을 때 효과는 덜 뚜렷했다. 이 실험은 또한 협상 중에 사용된 반복적인 과정을 밝혀냈다: 실험 대상자들은 먼저 가장 공평한 상품 분할을 발견한다. 그들은 그것을 기준으로 삼아 파레토 개선책을 찾으려고 한다. 개선은 그것이 야기하는 불평등이 너무 크지 않은 경우에만 시행된다. 이 과정을 CPIES: Equal Split에서 조건화된 파레토 개선이라고 한다.[19]

Intra-personal vs. inter-personal fairness - which is more important?

What is the importance of intra-personal fairness criteria (such as envy-freeness, where each person compares bundles based only on his own utility-function), vs. inter-personal fairness criteria (such as equitability, where each person views the utilities of all other agents)? Using a free-form bargaining experiment, it was found that inter-personal fairness (e.g. equitability) is more important. Intra-personal fairness (such as envy-freeness) are relevant only as a secondary criterion.[20]

Fairness vs. simplicity - what procedure is more satisfactory?

Divide and choose (DC) is a fair and very simple procedure. There are more sophisticated procedures that have better fairness guarantees. The question of which were more satisfactory was tested in several lab experiments.

1. Divide-and-choose vs Knaster-Brams-Taylor. Several pairs of players had to divide among them 3 indivisible goods (a ballpoint pen, a lighter and a mug) and some money. Three procedures were used: the simple DC, and the more complicated Adjusted Knaster (an improvement of adjusted winner) and Proportional Knaster. The authors asked the subjects to select their favorite procedure. Then, they let them play the procedure in two modes: binding (strict adherence to the protocol rules) and non-binding (possible renegotiation afterwards). They compared the procedures performance in terms of efficiency, envy-freeness, equitability and truthfulness. Their conclusions are: (a) The sophisticated mechanisms are advantageous only in the binding case; when renegotiation is possible, their performance drops to the baseline level of DC. (b) The preference for a procedure depends not only on the expected utility calculations of the negotiators, but also on their psychological profile: the more "antisocial" a person is, the more likely he is to opt for a procedure with a compensatory mechanism. The more risk-averse a person is, the more likely he is to opt for a straightforward procedure like DC. (c) The final payoff of a participant in a procedure depends a lot on the implementation. If participants cannot divide the goods under a procedure of their own choice, they are more eager to maximize their payoff. A shortened time horizon is equally detrimental.[21]

2. Structured procedures vs. Genetic algorithms. Two pairs of players had to divide between them 10 indivisible goods. A genetic algorithm was used to search for the best division candidates: out of the 1024 possible divisions, a subset of 20 divisions was shown to the players, and they were asked to grade their satisfaction about the candidate division on a scale ranging from 0 (not satisfied at all) to 1 (fully satisfied). Then, for each subject, a new population of 20 divisions was created using a genetic algorithm. This procedure continued for 15 iterations until a best surviving allocation was found. The results were compared to five provably-fair division algorithms: Sealed Bid Knaster, Adjusted Winner, Adjusted Knaster, Division by Lottery and Descending Demand. Often, the best divisions found by the genetic algorithm were rated as more mutually satisfactory than the ones derived from the algorithms. Two possible reasons for that were: (a) Temporal fluctuation of preferences - the valuations of humans change from the point they report their valuations to the point they see the final allocation. Most fair division procedures ignore this issue, but the genetic algorithm captures it naturally. (b) Non-additivity of preferences. Most division procedures assume that valuations are additive, but in reality they are not; the genetic algorithm works just as well with non-additive valuations.[22]

3. Simple procedures vs. Strongly-fair procedures. 39 player-pairs were given 6 indivisible gift-certificates of the same value ($10) but from different vendors (e.g. Esso, Starbucks, etc.). Before the procedure, each participant was shown all the 64 possible allocations, and was asked to grade the satisfaction and fairness of each of them between 0 (bad) and 100 (good). Then, they were taught seven different procedures, with different levels of fairness guarantees: Strict Alternation and Balanced Alternation (no guarantees), Divide and Choose (only envy-freeness), Compensation Procedure and Price Procedure (envy-freeness and Pareto-efficiency), Adjusted Knaster and Adjusted Winner (envy-freeness, Pareto-efficiency and equitability). They practiced each of these against a computer. Then, they did an actual division against another human subject. After the procedure, they were asked again to grade the satisfaction and fairness of the outcome; the goal was to distinguish procedural fairness from distributional fairness. The results showed that: (a) procedural fairness had no significant impact; satisfaction was mainly determined by distributional fairness. (b) the results of simpler procedures (strict alternation, balanced alternation and DC) were considered fairer and more satisfactory. They explain this couter-intuitive result by showing that humans care about object equality - giving each agent the same number of objects (though this does not entail any mathematical fairness criterion).[23]

효율성 대 전략 - 어떤 절차가 더 효율적인가?

상품을 어떻게 나눌 것인가와 같이 거래를 흥정해야 하는 두 대리점을 생각해 보라. 진심으로 선호를 밝히면 윈-윈 딜을 성사시킬 수 있는 경우가 많다. 그러나 그들이 이익을 얻기 위해 전략적으로 그들의 선호를 잘못 전달한다면, 그들은 실제로 계약을 잃을 수도 있다. 좋은 거래를 성사시키는 데 있어서 어떤 협상 절차가 가장 효율적인가? 몇 가지 협상 절차가 연구실에서 연구되었다.

1. 밀봉입찰 경매: 간단한 원샷 협상 절차 연구실에서는 정보취약계층이 비대칭정보를 공격적으로 악용해 전략입찰을 통해 진정한 가치평가를 과감하게 잘못 전달했다. 이것은 종종 협상 구역을 줄이고 거래를 용서하며 낮은 경제적 효율을 초래했다. 한 실험에서, 모든 실험의 52%만이 거래가 이루어진 반면, 모든 실험의 77%는 긍정적인 협상 구역을 가지고 있었다.[24]

2. 보너스 절차: 거래를 하는 참가자에게 보너스를 주는 절차가 주어졌다. 이 보너스는 선수들이 자신의 진정한 선호를 밝히는 것이 가장 적합하도록 계산된다. 실험 실험은 이것이 도움이 되지 않는다는 것을 보여준다: 실험 대상들은 비록 그들에게 나쁘지만, 여전히 전략을 짠다.[25]

3. 조정 우승자(AW): 총 효용을 극대화하기 위해 분할할 수 있는 사물을 할당하는 절차. 실험실에서 실험 대상자들은 분리할 수 있는 두 개의 물체에 대해 쌍으로 흥정을 했다. 두 개의 물체 각각에는 일반적으로 알려진 이전 분포에서 추출한 랜덤 값이 할당되었다. 각 플레이어는 자신의 가치에 대한 완전한 정보를 가지고 있었지만 공동취재자의 가치에 대한 불완전한 정보를 가지고 있었다. (1) 경쟁적 선호도(Competition Preferencompetiting Preferences: 참가자는 공동 바게인의 선호가 자신과 유사하다는 것을 알고 있다. (2) 보완적 선호는 다음과 같다. 선수들은 자신의 동료 선수 선호가 자신과 정반대라는 것을 알고 있다. (3) 알 수 없는 (랜덤) 선호는 다음과 같다. 선수들은 자신의 선호에 비해 공동선수가 가장 중요하게 여기는 것이 무엇인지 모른다. 조건 (1)에서, 양자간의 결정은 효율적인 결과로 수렴되지만, 단지 3분의 1만이 "완벽한" 것이다. 조건 (2)에서, 플레이어가 객체에 대한 진정한 평가를 극적으로 잘못 표현하는 동안, 효율성과 부러움-자유도 모두 최대 수준에 근접한다. 조건(3)에서는 전략적 입찰이 나타나지만, 결과적으로는 선망 없는 결과의 두 배, 효율성 수준(조건 1과 상대적)이 증가한다. 모든 경우에 구조화된 AW 절차는 구조화되지 않은 협상보다 약 3/2배 많은 윈윈 솔루션을 달성하는 데 상당히 성공적이었다. 성공 비결은 플레이어를 '고정 파이 신화'[26]에서 벗어나게 한다는 점이다.

4. Conflict-resolution algorithm: Hortala-Vallve and lorente-Saguer describe a simple mechanism for solving several issues simultaneously (analogous to Adjusted Winner). They observe that equilibrium play increases over time, and truthful play decreases over time - agents manipulate more often when they learn their partners' preferences. Fortunately, the deviations from equilibrium do not cause much damage to the social welfare - the final welfare is close to the theoretic optimum.[27]

5. Fair cake-cutting algorithms: Ortega, Kyropoulou and Segal-Halevi[28] tested algorithms such as Divide and choose, Last diminisher, Even–Paz and Selfridge–Conway between laboratory subjects. It is known that these procedures are not strategyproof, and indeed, they found that subjects often manipulate them. Moreover, the manipulation was often irrational - subjects often used dominated strategies. Despite the manipulations, the algorithms for envy-free cake-cutting produced outcomes with less envy, and were considered fairer.

How does sharing behavior develop in children?

In the lab, children were paired to "rich" and "poor" and were asked to share objects. There were differences in the perception of "initial belongings" vs. "things that have to be shared": young children (up to 7) did not distinguish them while older children (above 11) did.[29]

See also

  • The ultimatum game - a very simple game where a subject has to choose between accepting an unfair share and not getting anything. Many variants of this game were tested in lab.[30][31]
  • The Moral Machine experiment - an experiment that collected millions of decisions on moral issues related to autonomous vehicles (e.g., if a vehicle must kill someone, who should it be?).[32]
  • What is fair?[33]

References

  1. ^ a b Flood, Merrill M. (1958-10-01). "Some Experimental Games". Management Science. 5 (1): 5–26. doi:10.1287/mnsc.5.1.5. ISSN 0025-1909.
  2. ^ Pratt, John Winsor; Zeckhauser, Richard Jay (1990). "The Fair and Efficient Division of the Winsor Family Silver". Management Science. 36 (11): 1293–1301. doi:10.1287/mnsc.36.11.1293. ISSN 0025-1909.
  3. ^ a b Kurokawa, David; Procaccia, Ariel D.; Shah, Nisarg (2015-06-15). Leximin Allocations in the Real World. ACM. pp. 345–362. doi:10.1145/2764468.2764490. ISBN 9781450334105. S2CID 1060279.
  4. ^ Brams, Steven J.; Togman, Jeffrey M. (1996). "Camp David: Was The Agreement Fair?". Conflict Management and Peace Science. 15 (1): 99–112. doi:10.1177/073889429601500105. ISSN 0738-8942. S2CID 154854128.
  5. ^ Massoud, Tansa George (2000-06-01). "Fair Division, Adjusted Winner Procedure (AW), and the Israeli-Palestinian Conflict". Journal of Conflict Resolution. 44 (3): 333–358. doi:10.1177/0022002700044003003. ISSN 0022-0027. S2CID 154593488.
  6. ^ Denoon, D. B. H.; Brams, S. J. (1997-02-01). "Fair Division: A New Approach to the Spratly Islands Controversy". International Negotiation. 2 (2): 303–329. doi:10.1163/15718069720847997. ISSN 1571-8069.
  7. ^ Goldman, Jonathan; Procaccia, Ariel D. (2015-01-28). "Spliddit: unleashing fair division algorithms". ACM SIGecom Exchanges. 13 (2): 41–46. doi:10.1145/2728732.2728738. S2CID 14135593.
  8. ^ Gal, Ya'akov (Kobi); Mash, Moshe; Procaccia, Ariel D.; Zick, Yair (2016-07-21). Which Is the Fairest (Rent Division) of Them All?. ACM. pp. 67–84. doi:10.1145/2940716.2940724. ISBN 9781450339360. S2CID 53223944.
  9. ^ Tijs, Stef H.; Branzei, Rodica (2004). "Cases in Cooperation and Cutting the Cake". SSRN Working Paper Series. doi:10.2139/ssrn.627424. ISSN 1556-5068. S2CID 154934030. SSRN 627424.
  10. ^ Oluwasuji, Olabambo I.; Malik, Obaid; Zhang, Jie; Ramchurn, Sarvapali D. (2018). "Algorithms to Manage Load Shedding Events in Developing Countries". Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems. AAMAS '18. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 2034–2036.
  11. ^ Walsh, Toby (2011), "Online Cake Cutting", Algorithmic Decision Theory, Lecture Notes in Computer Science, 6992, Springer Berlin Heidelberg, pp. 292–305, doi:10.1007/978-3-642-24873-3_22, ISBN 9783642248726, S2CID 501890
  12. ^ Shtechman, Itay; Gonen, Rica; Segal-Halevi, Erel (2020-05-05), "Fair Cake-Cutting Algorithms with Real Land-Value Data", Proceedings of the 19th International Conference on Autonomous Agents and MultiAgent Systems, Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems, pp. 2005–2007, ISBN 978-1-4503-7518-4, retrieved 2021-02-04
  13. ^ Cavallo, Ruggiero (2012-07-22). "Fairness and welfare through redistribution when utility is transferable". Proceedings of AAAI 2012. Aaai'12: 1306–1312.
  14. ^ Dickerson, John P.; Goldman, Jonathan; Karp, Jeremy; Procaccia, Ariel D.; Sandholm, Tuomas (2014-07-27). "The computational rise and fall of fairness". Proceedings of AAAI 2014. Aaai'14: 1405–1411.
  15. ^ Segal-Halevi, Erel; Aziz, Haris; Hassidim, Avinatan (2017-08-19). Fair allocation based on diminishing differences. Proceedings of IJCAI 2017. AAAI Press. pp. 1254–1261. doi:10.24963/ijcai.2017/174. ISBN 9780999241103. S2CID 2504915.
  16. ^ Engelmann, Dirk; Strobel, Martin (2004-08-01). "Inequality Aversion, Efficiency, and Maximin Preferences in Simple Distribution Experiments". American Economic Review. 94 (4): 857–869. CiteSeerX 10.1.1.310.3267. doi:10.1257/0002828042002741. ISSN 0002-8282.
  17. ^ Fehr, Ernst; Naef, Michael; Schmidt, Klaus M (2006-11-01). "Inequality Aversion, Efficiency, and Maximin Preferences in Simple Distribution Experiments: Comment". American Economic Review. 96 (5): 1912–1917. CiteSeerX 10.1.1.163.1204. doi:10.1257/aer.96.5.1912. ISSN 0002-8282.
  18. ^ Herreiner, Dorothea; Puppe, Clemens (January 2007). "Distributing Indivisible Goods Fairly: Evidence from a Questionnaire Study". Loyola Marimount University - Economics Faculty Works.
  19. ^ Herreiner DK, Puppe C (November 2010). "Inequality aversion and efficiency with ordinal and cardinal social preferences—An experimental study". Journal of Economic Behavior & Organization. 76 (2): 238–253. doi:10.1016/j.jebo.2010.06.002.
  20. ^ Herreiner, Dorothea K.; Puppe, Clemens D. (2007). "Envy Freeness in Experimental Fair Division Problems". Theory and Decision. 67: 65–100. doi:10.1007/s11238-007-9069-8. hdl:10419/22905. S2CID 154799897.
  21. ^ Schneider, Gerald; Krämer, Ulrike Sabrina (2004). "The Limitations of Fair Division: An Experimental Evaluation of Three Procedures". The Journal of Conflict Resolution. 48 (4): 506–524. doi:10.1177/0022002704266148. JSTOR 4149806. S2CID 18162264.
  22. ^ Dupuis-Roy, Nicolas; Gosselin, Frédéric (2009). "An Empirical Evaluation of Fair-Division Algorithms" (PDF). Proceedings of the Annual Meeting of the Cognitive Science Society: 2681–2686.
  23. ^ Dupuis-Roy, Nicolas; Gosselin, Frédéric (2011). "The Simpler, the Better: A New Challenge for Fair-Division Theory" (PDF). Proceedings of the Annual Meeting of the Cognitive Science Society (CogSci 2011): 3229–3234.
  24. ^ T. E. Daniel, A. Seale, A. Rapoport (1998). "Strategic Play and Adaptive Learning in the Sealed Bid Bargaining Mechanism" (PDF). Journal of Mathematical Psychology. 42 (2–3): 133–166. doi:10.1006/jmps.1998.1220. PMID 9710545.CS1 maint: multiple names: authors list (link)
  25. ^ Parco, James E.; Rapoport, Amnon (2004-11-01). "Enhancing Honesty in Bargaining Under Incomplete Information: An Experimental Study of the Bonus Procedure". Group Decision and Negotiation. 13 (6): 539–562. doi:10.1007/s10726-005-3824-4. ISSN 0926-2644. S2CID 96468159.
  26. ^ Daniel, Terry E.; Parco, James E. (2005-05-01). "Fair, Efficient and Envy-Free Bargaining: An Experimental Test of the Brams-Taylor Adjusted Winner Mechanism". Group Decision and Negotiation. 14 (3): 241–264. doi:10.1007/s10726-005-1245-z. ISSN 0926-2644. S2CID 153519701.
  27. ^ Hortala-Vallve, Rafael; Llorente-Saguer, Aniol (2010-11-01). "A simple mechanism for resolving conflict". Games and Economic Behavior. 70 (2): 375–391. CiteSeerX 10.1.1.508.8773. doi:10.1016/j.geb.2010.02.005. ISSN 0899-8256. S2CID 18938516.
  28. ^ Kyropoulou, Maria; Ortega, Josué; Segal-Halevi, Erel (2019-06-17). "Fair Cake-Cutting in Practice". Proceedings of the 2019 ACM Conference on Economics and Computation. EC '19. Phoenix, AZ, USA: Association for Computing Machinery: 547–548. doi:10.1145/3328526.3329592. ISBN 978-1-4503-6792-9. S2CID 53041563.
  29. ^ Moessinger, Pierre (1975). "Developmental study of fair division and property". European Journal of Social Psychology. 5 (3): 385–394. doi:10.1002/ejsp.2420050310. ISSN 0046-2772.
  30. ^ Güth, Werner; Schmittberger, Rolf; Schwarze, Bernd (1982-12-01). "An experimental analysis of ultimatum bargaining" (PDF). Journal of Economic Behavior & Organization. 3 (4): 367–388. doi:10.1016/0167-2681(82)90011-7. ISSN 0167-2681.
  31. ^ Gabay, Anthony S.; Radua, Joaquim; Kempton, Matthew J.; Mehta, Mitul A. (2014-11-01). "The Ultimatum Game and the brain: A meta-analysis of neuroimaging studies". Neuroscience & Biobehavioral Reviews. 47: 549–558. doi:10.1016/j.neubiorev.2014.10.014. ISSN 0149-7634. PMID 25454357.
  32. ^ Awad, Edmond; Dsouza, Sohan; Kim, Richard; Schulz, Jonathan; Henrich, Joseph; Shariff, Azim; Bonnefon, Jean-François; Rahwan, Iyad (2018-10-24). "The Moral Machine experiment". Nature. 563 (7729): 59–64. Bibcode:2018Natur.563...59A. doi:10.1038/s41586-018-0637-6. hdl:10871/39187. ISSN 0028-0836. PMID 30356211. S2CID 53029241.
  33. ^ Dickinson, David L.; Tiefenthaler, Jill (2002). "What Is Fair? Experimental Evidence". Southern Economic Journal. 69 (2): 414–428. doi:10.2307/1061680. JSTOR 1061680.