# What are matching markets?

*{Epistemic Status: Just reviewing my thoughts on a book and its subject matter. I have done some independent study of the field (and have a degree in economics), but I did not try to check my impressions rigorously. Much or maybe most of what I say is extrapolation by me, rather than being in the book itself.}*

I have recently finished the book Who Gets What and Why, by Alvin Roth and found it quite intellectually stimulating. The book itself was not as systematic as I originally hoped, but that was almost better since it forced me to reconstruct definitions and patterns in ways that made sense to me. Below I have included my thoughts on the first topic of the book, matching markets. The second topic, mechanism design, will appear in a later post.

**What Are Matching Markets**
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The first model of a market that economics students learn is the perfectly competitive model. This is exceptionally useful to understand, but it can also limit the scope of what people consider as markets. In particular, this model emphasizes two characteristics of a market, price and quantity, neither of which are explicitly required to model a market. Matching markets include markets where there can be both no prices and no changes in quantity (i.e. no supply or demand curves).[^x79ugg5010o] They are also more than academic curiosities since they describe everything from dating markets to job markets to school admissions.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/289df2ed33089e839536b64c223da3cdab178ba656d5d07c.png)

You can use supply and demand to get some insight into how the market for romantic relationships works, but if you start taking it too seriously then it stops making sense. For example: Is there a single market price? Are the surpluses of each side directly opposed? Do people always prefer lower ‘prices’? Is a person with a higher willingness-to-pay always going to do better?

The primary characteristic of a matching market is that both sides of an exchange need to actively choose each other in order for it to go through. How does this actually differ from a standard (commodities) market? The key distinction is that each exchange can now be dependent on the outcome of another exchange.

For example, imagine a group of tennis players that want to enter a doubles tournament. We can imagine a player Alice who wants to team up with Bob who wants to team up with Charlie. Since Bob would prefer Charlie to Alice, Alice’s ability to team up with Bob is determined by whether Charlie will choose Bob. This alone might not seem exceptional, but just as Alice’s exchange with Bob is determined by Charlie’s decision, Charlie’s decision could itself be determined by Diane’s decision, which could, in turn, be determined by Ethan’s decision, and so on. By this simple transitivity, we can potentially make every market decision interdependent and even create unresolvable cycles of dependent decisions (e.g. if Charlie preferred Alice to Bob above).

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/fa259a7871edb3d52eb0f7e1bb01b02806f119c97f8c33b0.jpg)

We can extend this chain of dependence as far out as we want.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/9b001b3a521e5381e66abc1e26126c2c3f212dba711ded47.jpg)

This set of preferences is inherently unstable. If Bob chooses Alice, then Charlie can ask Bob to change partners. If Bob chooses Charlie, then Alice can ask Charlie to change partners. If Alice chooses Charlie, then Bob can ask Alice to change partners, returning to where we started. You might think that this is because we have only 3 people (so not everyone can pair up), but if we add Diane and make her the last choice of everyone else, then the cycle remains exactly the same (and just as unstable) except that instead of a player being unpaired, they are paired with Diane.

**Matching Markets in Practice**
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Above, I described a high-level difference between matching markets and standard (commodity) markets, but what features and behaviors does that difference actually cash out into? Specifically, (I) how can you recognize a matching market and (II) what patterns should you expect it to exhibit?

### **(I) Structural Features: how to recognize a matching market**

Three very connected structural features that seem common in matching markets are: 

(1) Highly differentiated products

(2) Highly differentiated preferences

(3) Many participants looking for only one match/exchange

Taken together, these are all qualities that make the difference between an n-th rank choice and an n+1-th rank choice larger (e.g. make the 2nd choice much better than the 3rd choice). This is important for making a matching market because, when the difference between options is too small, then choices can become independent again as the cost of getting a better match exceeds its benefit.

To illustrate what these features mean, let’s contrast two markets: a market for soda and a dating market, to see how they compare on the points above.

**Market for Soda**

(1) Most products are pretty similar: sugary/sweet liquids with carbonation.

(2) Most people have pretty similar preferences: most people like Coke or Pepsi, etc. as much as most other people do.

(3) Lots of soda is bought in bulk (many sodas to one buyer) for parties etc. and soda companies sell to lots of people. There is very little exclusivity.

This is clearly a commodities market, where the choices of an individual are not heavily dependent on the choices of others.

**Dating Market**

(1) People in the dating market are very different from each other. They have vastly different interests, locations, preferences, histories, cultures, physical attributes, professions, families, etc.

(2) What people look for in a romantic partner can differ dramatically. They might want shared or different interests, cultures, religions, etc. They might prioritize financial security or emotional vulnerability, etc. They might want to live in the suburbs or stay in the city, etc.

(3) People are (generally) looking for a single partner, rather than multiple (at least at a given point in time).

This is clearly a matching market, where the choices of an individual are heavily dependent on the choices of others.

### **(II) Behaviors in Matching Markets: what should we expect in a matching market**

Most of the interesting behaviors associated with matching markets relate to different ways they can fail, which the book refers to generally as “market unraveling,” and which –unfortunately– is common without explicit mechanism design. With that in mind, let’s analyze some of the more interesting patterns observed in matching markets: (1) Preference falsification, (2) Commitment contracts, (3) Race to the bottom dynamics, and (4) Rule/Contract breaking.

**(1) Preference Falsification**

Perhaps the most immediately apparent behavioral feature of matching markets is that people constantly lie about their preferences. Let’s look at a few examples:

*   Job interviewees will tell many job interviewers that this is their top choice of job.
*   Students will tell multiple colleges that they are their top choice.
*   People trying to join clubs, organizations, and groups will say that the current group is their favorite, even when that is clearly false.

So why do people falsify their preferences so readily? It is because, in the face of dependent choices in a matching market, this can be more successful than telling the truth. 

The basic reason to lie is that if you manage to convince a job interviewer or admissions person that their organization is your top choice, then their choice is simplified to whether you are better than the other candidates. However, if they think you prefer another organization, then they need to factor in the chance that you will be picked by the organization you like more and become unavailable to them.

Indeed, the case is even more hazardous since, if you refuse an offer, they do not just lose you from the applicant pool, but might potentially lose everyone else as well since the others could have assumed the job was taken and accepted other (potentially worse) offers.

As a result, you are incentivized to pretend every option is your favorite option in a matching market since the more they believe you, the more likely you will get an offer.

Of course, since the cost of lying/preference falsification is low, we get a signaling equilibrium where almost everyone lies, but also almost no one believes the lies. We will revisit this problem when I discuss mechanism design.

**(2) Commitment Contracts**

A key way that people deal with the problems in matching markets is through commitment contracts. Examples of these contracts would include job contracts, marriage contracts, and sports player contracts. They generally specify that a person cannot choose a different new option (job, romantic partner, sports team) without paying some (often high) cost.

Contracts in general allow participants to reduce uncertainty about future events and they serve the same purposes in matching markets, just with an emphasis on events where one of the members of a match/pair gets a new offer (an offer from someone they prefer). Contractual commitments to the current match effectively remove the contractors from the broader market so they don’t need to worry about their partner trading up.

As should be obvious, there are considerable benefits and costs to these kinds of contracts. On the benefits side, there is the potential for some people to get matches at all. For example, if someone was sufficiently desirable, then that could work against them in the absence of commitment contracts because their partners will always need to account for the chance the desirable person will get a better offer (which is higher the more desirable they are). Having a commitment contract allows the partner to avoid this risk. On the costs side, people can get locked into poor matches, especially in cases where people are risk averse and where there is asymmetric information. A concrete example of this might be someone getting married young to the best person they can find in their local area, but later discovering a much better match in the big city/college/etc. and wishing they had waited to marry.

**(3) Race to the bottom dynamics**

Once commitment contracts have been introduced to a matching market, they can develop their own dynamics, especially in cases where match quality is uncertain. This dynamic could be characterized as a ‘race to the bottom’ where commitment contracts are continually offered at earlier and earlier points in time, when less and less information is known about the quality of future matches.

Examples of this can be seen in sports, where athletes are getting signed earlier and earlier (in college, then high school, etc.), in medicine, where medical students get residency offers earlier and earlier (after medical school, at the end of school, at the beginning of medical school, before medical school, etc.), in fraternity and sorority admissions, where students rush earlier and earlier (any year, then freshmen year, then first weeks of college, then even the summer before college), and often elsewhere, such as law and marriage.

The basic dynamic works this way: options (organizations, etc) that are not top choices can get better candidates by recruiting them earlier since people are risk averse and willing to take a guarantee of a less-than-perfect option over a low or unknown probability of a better option. This is especially true if candidates are not fully informed about their own capabilities/potential and the other options that will be available to them. This condition then repeats at every time step, pushing the contracts earlier and earlier. 

The whole situation takes the form of a collective action problem and unsurprisingly creates huge inefficiencies. As the commitments move earlier and earlier, available information is reduced and other risks are increased with only zero-sum benefits accruing. For example, when you sign a high schooler instead of a college student, you get years of extra time for an injury to occur and the deal to stop making sense. When you make a medical residency offer to a pre-med student, you risk the student dropping out of medical school. On the other side, a student might find out they were much more skilled than expected, but still end up stuck with a worse match because they made a decision before they had more information. This last case benefits the worse organization, but at a cost to both the student and whoever they would have otherwise matched with.

**(4) Rule/Contract breaking**

The last note I have on behavior in matching markets is that people break the rules a lot. Matching markets typically have two kinds of rules: the commitment contracts that were discussed in part (2), and the rules that are often imposed to deal with the collective action problem discussed in part (3). Both kinds of rules are often broken either legally or illegally in matching markets.

**Breaking commitment contracts**

The fundamental point of commitment contracts is to stop an individual from leaving their current match when a better one comes along. If you knew that your offer was literally unbeatable, then you would not need a commitment contract. For better or worse, sometimes the new offer that someone gets is superior enough that it can be worth breaking the existing contract. This can be explicitly permitted through a costly legal mechanism like paying a fine or enduring a divorce proceeding, but it can also occur through exploiting legal loopholes or just ignoring the law entirely. This ends up being exceedingly common in matching markets since it is precisely the large difference between options (highly differentiated products and preferences) that structured it as a matching market in the first place.

**Breaking collective action rules**

In many matching markets, organizations attempt to impose rules to limit ‘race to the bottom’ dynamics and solve the collective action problem. This often takes the form of requiring that offers only be made on some selected date and only to people meeting some qualifications such as ‘finished medical school’. These measures are often very ineffective because the incentives to defect on the collective action are strong and enforcement is often extremely difficult. Still, as I will discuss in a later post on mechanism design, the problem is not unsolvable.

### **Conclusion**

Those were some of the thoughts I had regarding matching markets after I finished *Who Gets What and Why*. I would caution against taking any of it as authoritative or representative of the text itself since I extrapolated wildly from the ideas mentioned in the book. I am planning to continue with a second post with my notes on mechanism design. See you then!

[^x79ugg5010o]: Obviously, some quantity must exist in every market so I am really referring to abstract quantity as a model variable or outcome.