Abstract
We introduce two novel matching mechanisms, Reverse Top Trading Cycles (RTTC) and Reverse Deferred Acceptance (RDA), with the purpose of challenging the idea that the theoretical property of strategy-proofness induces high rates of truth-telling in economic experiments. RTTC and RDA are identical to the celebrated Top Trading Cycles (TTC) and Deferred Acceptance (DA) mechanisms, respectively, in all their theoretical properties except that their dominant-strategy equilibrium is to report one’s preferences in the order opposite to the way they were induced. With the focal truth-telling strategy being out of equilibrium, we are able to perform a clear measurement of how much of the truth-telling reported for strategy-proof mechanisms is compatible with rational behaviour and how much of it is caused by confused decision-makers following a default, focal strategy without understanding the structure of the game. In a school-allocation setting, we find that roughly half of the observed truth-telling under TTC and DA is the result of naïve (non-strategic) behaviour. Only 14–31% of the participants choose actions in RTTC and RDA that are compatible with rational behaviour. Furthermore, by looking at the responses of those seemingly rational participants in control tasks, it becomes clear that most lack a basic understanding of the incentives of the game. We argue that the use of a default option, confusion and other behavioural biases account for the vast majority of truthful play in both TTC and DA in laboratory experiments.
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Notes
Roth (2015) offers a concise summary of experimental research related to market design.
For a much less selective and concise review of experimental findings on matching markets, refer to Hakimov and Kübler (2019).
See the experimental instructions in “Appendix C of ESM” for the standard quiz used in this and other studies.
For instance, “an application to the highest ranked school” was replaced by “an application to the lowest ranked school”. Refer to the experimental instructions in “Appendix C of ESM” for details.
Our design also reduces the bias that uncontrolled other-regarding preferences could create by observing human behaviour in interaction with automated opponents. As we aim at testing the empirical relevance of the theoretical concept of strategy-proofness, our approach relies on, and tests, theoretical models that typically do not assume social preferences, and this is reflected in our experimental design. Refer to Sect. 2 and to the experimental instructions in “Appendix C of ESM” for details.
Waseda University is one of the top private universities in Japan. Admission is highly selective and depends both on high-school marks and results from an entry exam administered by the university. Also, note that Japanese high schools are of an extremely high quality. They ranked #5 in the world in the last PISA international mathematics comparisons, where South Korea ranked #6 and the USA ranked #40. In conclusion, the participants in our sample are much better trained in mathematics than the ones in most previous studies.
Around the time of our experiments ¥1000 were equivalent to around $9, and would be enough to buy two lunch boxes on campus.
All our participants are volunteers who signed up for the experiment by responding to an online advertisement. They are Waseda-University students of various majors. Our average participant is 21 years old, and 60% of our participants are male.
In July 2019, Scopus listed 99 citations for our main reference (Chen and Sönmez 2006) whose experimental design follows the above-described pattern. 40% of the citing documents report new experimental results and almost 10% of them rely on a very similar and often nearly identical design. As for the instructions, we adapted the text as published by Chen et al. (2016) to our algorithms. All English instructions are to be found in “Appendix C of ESM”. The Japanese versions are available from the authors upon request.
Participants had 10 minutes to solve the quiz and were paid ¥100 for a correct solution. Given that the quiz aims at measuring participants’ understanding of the analysed strategic interaction, participants could only submit one solution and were not informed whether that was correct or not.
Participants had up to 20 minutes to complete the school-allocation task. Just like in Chen and Sönmez (2006) and the relevant related literature, the allocation task was not repeated. This departure from the tradition of using repeated games in experimental settings is typically justified by arguing that real-life participants in school-choice problems face matching mechanisms only once or twice in a lifetime and, in each occasion, those are naturally presented as one-shot games. Only research specifically targeting learning use repeated matching games (e.g., Ding and Schotter 2019).
The chosen design features not only make belief elicitation about the other decision-makers’ rationality and behaviour unnecessary, but they also strengthen the induced-value method in implementing the desired situation. Note that other-regarding considerations are unlikely to influence behaviour when the others are simply computer codes.
The only notable change in the instructions, apart from the language, is that names for schools in the explanatory example were replaced by neutral symbols that are often used in generic lists in Japanese.
As for the terms used to refer to ranking in the experiment, the submitted ranking in the example (in the instructions) talked about “first choice”, “second choice”, “third choice”, and “last choice”. When participants were requested to submit their rankings, they had to type the name (a letter) of the schools in the input cells displayed on the computer screen. Those cells were labelled as “first school”, “second school”, “third school”, and “fourth school”. The participants were also were explained (by the instructions) how the algorithm proceeds, i.e., whether it starts at the highest- or lowest-ranked school and how it moves onward to the next highest or next lowest choice.
The above definitions are widely accepted and used in the literature. Note that efficiency is a one-sided concept given that it only takes students’ preferences into consideration. Justified envy, however, is two-sided: both students’ and schools’ preferences are considered.
Besides the desirability of a stable matching from a normative point of view, stability as a prediction or solution to the school-allocation problem sounds reasonable when decision-makers are allowed to trade and switch partners in a frictionless decentralised way after the matchmaker has announced the outcome. Then, by definition, we should not expect unstable matchings to survive, only stable ones could prevail.
The literature refers to the problem that involves strategising school as the college-admission problem.
By student-stable we refer to the stable matching preferred by students to all other stable matchings (Roth and Sotomayor 1990). Our school-allocation problem has another stable matching, \(\{(R_1,C),(R_2,A),(R_3,D),(H,B)\}\), which is preferred by schools to all other stable matchings. Given the one-sided definition for efficiency, the latter stable matching is not efficient.
“Appendix A of ESM” describes the two algorithms in detail.
For example, our primary reference, Chen and Sönmez (2006) test three well-known school-choice mechanisms (BOS, DA and TTC) and, with the help of two treatments, show that in terms of inducing truth-telling in the laboratory, DA (72%, 56%) outperforms TTC (50%, 43%), which in turn performs better than BOS (14%, 28%). In contrast, Pais and Pintér (2008) find that in zero- and full-information settings TTC induces significantly more truth-telling (96%, 87%) than DA (82%, 67%) or BOS (87%, 47%), while under low and partial information the observed differences between TTC (82%, 76%) and DA (76%, 67%) are not statistically different from each other.
We refer to the standard approach to theoretical complexity outlined by Gilboa et al. (1993).
Some might argue that mechanisms with equilibria in dominant strategies are “easier” to play than mechanisms that only have Nash equilibria, for instance. Nevertheless, there do not exist objective and well-established criteria for complexity in the mechanism-design literature. Some argue that incentive-compatibility alone is not enough, the property should be strengthened further to dominant-strategy incentive compatibility. Recently, Li (2017) has called for an even stronger obviously-dominant-strategy incentive compatibility, which is not satisfied by any of the well-known and theoretically explored mechanisms (Ashlagi and Gonczarowski 2018). In conclusion, mechanism-design theory can not tell our reverse mechanisms apart from the original strategy-proof ones. For further discussion on complexity, refer to Sect. 4.
Figure B.1 in “Appendix B of ESM” shows the distribution of answers and reveals that it has three peaks. Beside the totally clueless mass in the center, there are also many participants who are very sure of themselves. Roughly half of those are right, the other half is wrong.
The proportion of sophisticated (strategic) decision-makers by mechanisms is: 12% for DA, 4% for RDA (p-value = 0.15 for proportion comparison), 5% for TTC, 6% for RTTC (p-value = 0.96 for proportion comparison).
Refer to “Appendix B of ESM” for the description of experimental matching outcomes and a more detailed discussion on our experimental findings related to the revelation principle.
For example, the analysis of the consequences of changes in Chicago’s assignment system in 2009, those of Barcelona’s and Beijing’s adoption of the Boston mechanism, reviewed by Pathak (2017), raises important questions related to the very foundations of market design without being able to deliver precise estimates on the proportion of sophisticated decision-makers that exist in the population.
Clearly, successful mechanisms should tolerate certain behavioural faults (McFadden 2009), however with the help of the very same experimental design, we are unable to establish a new set of empirically justified assumptions. Obviously strategy-proof mechanisms (Li 2017) do not seem to help, as stable matching mechanisms have been shown not to be obviously strategy-proof (Ashlagi and Gonczarowski 2018).
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Acknowledgements
We gratefully acknowledge useful comments from Rustamdjan Hakimov, Bettina Klaus, Dmitriy Kvasov, Jordi Massó, Jörg Oechssler, and seminar participants at Universitat Autònoma de Barcelona, Universidad Autónoma de Madrid, Université de Bordeaux and the SING15 conference at the University of Turku. Financial support from the Waseda Institute of Political Economy and Grants-in-Aid for Scientific Research 17K03634 from MEXT and JSPS has made our experimental sessions possible. We have polished and improved the manuscript with the help of the constructive criticism received from two anonymous referees, the editor, Marie Claire Villeval, and Ak Akkawi, who helped us with language editing.
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Guillen, P., Veszteg, R.F. Strategy-proofness in experimental matching markets. Exp Econ 24, 650–668 (2021). https://doi.org/10.1007/s10683-020-09665-9
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DOI: https://doi.org/10.1007/s10683-020-09665-9