## Abstract

In a series of laboratory experiments, we explore the impact of different market features (the level of information, search costs, and the level of commitment) on agents’ behavior and on the outcome of decentralized matching markets. In our experiments, subjects on each side of the market actively search for a partner, make proposals, and are free to accept or reject any proposal received at any time throughout the game. Our results suggest that a low information level does not affect the stability or the efficiency of the final outcome, although it boosts market activity, unless coupled with search costs. Search costs have a significant negative impact on stability and on market activity. Finally, commitment harms stability slightly but acts as a disciplinary device to market activity and is associated with higher efficiency levels of the final outcome.

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## Notes

In fully centralized two-sided matching markets, the matchmaker produces a matching of the two sides of the market using lists of preferences regarding the other side of the market that each agent submits. Examples of fully centralized markets are the medical residency match and school allocation in the U.S. Other markets are characterized by a decentralized phase preceding the centralized procedure or are not fully centralized, i.e. not all matches are achieved through the matchmaker. Good examples of these are college admissions and the market for junior economists in the U.S.

Given that the scarce experimental literature on decentralized matching markets does not agree on the design—and because of the absence of theoretical models to test—we implemented this intuitive market with real-time interaction, primarily inspired by the search game designed by Eriksson and Strimling (2009).

We also test markets that differ in complexity, as captured by the number of stable matchings and the number of rounds required for the deferred-acceptance algorithm—a protocol that is used by matchmakers in various centralized markets to produce a matching—to converge under truth-telling. Given that we do not find significant or meaningful patterns along complexity, we do not emphasize it or consider it as a treatment variable.

The feature that proposals are held and not immediately accepted from one step of the algorithm to the other explains why this algorithm is sometimes referred to as the deferred-acceptance algorithm.

The latter class was later expanded in Hoffman et al. (2013).

See also Boudreau (2008).

The difference in utility between being matched to a subject’s

*k*th and*k*+ 1th choice is either 20¢ or 70¢. Our design is comparable to the 20¢-treatment in Echenique and Yariv (2013), given that we normalized the difference between any given partner and the next best to 10 EMU (10/35 \(\approx\) 0.29) in all our treatments.The per-proposal cost was 4 Experimental Monetary Units (EMU) with an initial budget of 16 EMU. The part of the budget that was not spent on sending proposals was added to the subjects’ final payoff in each round. Given our objective to analyze the effect of market frictions on market outcome, we chose the amount of the per-proposal cost and the initial budget as to introduce significant search costs in the corresponding treatments, without them being too restrictive. In order to illustrate the impact of search costs, we have simulated 10 thousand markets for each of our six preferences profiles. Simulated interaction happened through random encounters between myopically-rational agents who would never repeat partners who had rejected or abandoned them before. We find that the shortest paths towards a stable matching is of 6 to 13 interactions long in small markets, and it is of 18 to 53 interactions in large markets.

In low information environments, each subject knows her own payoff table and knows that the others’ “are similar.” We do not specify any probability distribution or upper and lower limits for the others’ valuations.

Small-market treatments involved 180 subjects and took place in May 2010. Large-market treatments involved 240 subjects and were conducted in December 2012.

The addition of this flexible ending rule in the large-market and longer treatments aimed at shortening idle time. We do not believe that it had any significant impact on individual behavior or the final outcome. Time limits are unavoidable design elements of any laboratory experiment. Echenique and Yariv (2013) do not set an exogenous limit for the duration of a round, instead they force each market to end after 30 s of inactivity. And, when receiving a proposal, subjects have at most 10 s to respond. Comola and Fafchamps (2018) opt for a design in which subjects take turns to make proposals. In this case, even though there are no explicit time limits, other constraints are in place. Namely, each subject has a maximum number of turns to propose (8), in each turn each subject can make a limited number of proposals (5), and each subject has at most 15 s to make a decision (either to propose or to react to a received proposal). Similar restrictions were introduced by Echenique et al. (2016) who consider a dynamic version of the deferred-acceptance algorithm. In their setup, subjects are simply not allowed to repeat proposals; additionally, proposers have 30 s to make a proposal and respondents have 25 s to react.

The computer simulations mentioned earlier show that, on average, our markets take 19 proposals to settle when they are small and 75 proposals when they are large. The outcome is a stable matching in 99.8% of the cases in small markets and in 96.8% of the cases in large markets. If proposals take 5 s to be made and 5 s to be considered, overall stability is very high: it is equal to 99.8% in small markets and 83.5% in large markets given the above time limits. The proportion, however, drops dramatically to 86% in small markets and to 11.4% in large markets when the time required for each decision doubles to 10 s.

In treatments where rematches are allowed, this rule implies that the intermediate matches are worthless. It is as if they belonged to an interview phase.

A conveniently adjusted conversion rule from EMU to euro made sure that subjects earned the same amount of money on average in small-market and in large-market treatments. In small-market treatments subjects received 1€ per 35 EMU, whereas in large-market treatments subjects received 1€ per 40 EMU.

Each stable matching in a market with preference profile S-A, S-B and S-C has an aggregate payoff of 300, 340 and 410, while the largest possible aggregate payoff is 360, 380 and 410, respectively. As for large markets, each stable matching in a market with preference profile L-A, L-B and L-C corresponds to an aggregate payoff of 1200, 1360 and 1640, with a maximum of 1440, 1520 and 1640, respectively.

Note that, in order to make comparisons across treatments possible, in treatments with commitment we count the total number of blocking pairs including those involving agents who are matched and therefore are incapable of making or receiving new proposals.

The numbers in Table 2 are market averages, i.e. for each treatment, we first compute the average rate per round and per group and then compute the average for all groups and all rounds, thus obtaining an average rate for each treatment.

Round is an integer from 1 to 15 in the small-market treatments, and 1 to 9 in the large-market ones. Given that we essentially repeated the same games for several times (and that there are no relevant differences observed across the different preference profiles), round appears in the regressions to control for possible learning due to the static repetition of the games.

The average stability level in Echenique and Yariv (2013) of 76% compares with an average stability level of 69% obtained in our benchmark treatment S1. While Echenique and Yariv (2013) claim that results do not change as market size grows to 15 agents, the reported global average stability of 67% for markets where agents have one, two, or three stable matching partners is considerably less than the same global average of 76% for smaller markets. Our data reveal a larger decrease in a similar comparison: stability roughly halves as market size doubles (from 5 to 10).

The apparent discrepancies are probably due to statistically insignificant differences between some pairs of treatments. We do not find this problematic, because when looking at averages we focus on size effects and postpone the discussion on statistical significance to the regression analysis.

Table 4 in Supplementary material contains data on market activity for each treatment: average numbers of proposals, acceptances, rejections, cancellations, and reaction rates.

The average proportions of participants who went “bankrupt” are 1% in treatment S2, 4% in treatment S5, 5% in treatment L2, and 18% in treatment L5.

See Table 5 in Supplementary material, which contains the results from ordinary-least-squares regressions taking as dependent variables the average number of proposals (regression 11) and the average number of acceptances (regression 12). The independent variables are dummies indicating the level of information, the existence of commitment or search costs, and the size of the market, all cross-effects, as well as the number of rounds played. In specification 12 we additionally include the number of proposals as a regressor, since the number of acceptances is naturally bounded by the number of proposals.

The efficiency analysis ignores the direct, and obviously negative impact, of search costs on surplus in order to offer a more nuanced comparison of market outcomes across our treatments. The efficiency numbers for our search-costs treatments reflect the possible indirect impact of search costs, i.e. whether markets are less or more likely to find their way to the theoretically efficient outcome in their presence.

The

*p*-values from*t*-tests are 0.11, 0.69, 0.04 and 0.39 for the above-mentioned four sessions, respectively. Although the difference of the observed efficiency level from the expected one seems to be statistically significant at 4% for our treatment with low information, the size of the effect is not large.

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## Acknowledgements

We are thankful to the editor and two anonymous referees for comments and excellent suggestions. Joana Pais gratefully acknowledges financial support from the Fundação para a Ciência e a Tecnologia under project reference no. PTDC/IIM-ECO/4546/2014. Ágnes Pintér gratefully acknowledges financial support from the project SEJ2007-67135 and the Juan de la Cierva program of the Spanish Ministry of Science and Innovation.

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Pais, J., Pintér, Á. & Veszteg, R.F. Decentralized matching markets with(out) frictions: a laboratory experiment.
*Exp Econ* **23**, 212–239 (2020). https://doi.org/10.1007/s10683-019-09606-1

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DOI: https://doi.org/10.1007/s10683-019-09606-1