An Adjusted Matching Market: Adding a Cost to Proposing

Joschka TrybaBrian Cross

Stephen Hebson

Introduction

• Gale-Shapely matching method:– (1) Two sided market– (2) Strict preferences – (3) Use of the Deferred Acceptance Algorithm– Under these assumptions, there will always exist an

optimal stable matching• But the dating market does not always yield this

optimal matching– Analysis of data from online dating websites shows a

disparity between the predicted Gale-Shapley matching and the actual matching (Hitsch et al)

Introduction

• Our Question: Can the Gale-Shapley model be adjusted to more accurately predict actual matching outcomes?

• Our Method: Adding to the model a constant cost to proposing.

• Would this change the final matching? Proposal strategies? Is there now an incentive to “settle” for a less than ideal mate?

• We will explore this cost-added model by executing an in-class demonstration as well as proposing a nuanced analysis of existing data from an online dating website

Background Literature

• "College Admissions and the Stability of Marriage" (Gale and Shapley, American Mathematical Monthly 1962)

• "Matching and Sorting in Online Dating" (Hitsch et al, American Economic Review 2010)

• "What Makes You Click" (Hitsch et al, January 2010)• "A Model of Price Adjustment" (Peter Diamond, Journal of

Economic Theory 1971)• "Interviewing in Two-Sided Matching Markets" (Lee and

Schwarz, NBER working paper 2009)

Model

• Assumptions:– Preferences for all agents are strict– All agents have cardinal preferences, with the expected

payoff for agent i to be matched with agent j being some valuation Vj.

– Every time a proposal is made, the proposer is charged some cost c

– Every proposer has some sense of where on other people’s preference list s/he falls• From this, agent i can roughly deduce the probability of his

proposal being accepted by agent j

• The expected value to agent i of participating in the market:

EVi v j jc pij 1 pik k

j 1

j1

n

Model

In-Class Demonstration

Let’s play two deferred acceptance games. At the end, you’ll be able to use your accrued points to buy brownies.

Game 1: With Proposal Costs

• Take one of the sheets that we’re passing around

• The sheet will group you as a “proposer” or “non-proposer”

• Your sheet will list your preferences and your payouts for ending the game with each participant

• Your sheet will also show you your average position on other people’s lists

Round 1: Part A

• “Proposers:” you may propose to one of the people on your preference list if you wish to do so

• If you make a proposal, you will lose 1 of your accrued points

• It is possible to to have negative points• You may also withdraw from the game

without a match, for free

Round 1: Part B

• “Non-Proposers:” you must decide whether or not to accept any proposals you have received

• You may not have more than one accepted proposals at any time

• If you receive a better proposal in a later round, you will be able to accept that proposal and reject any proposal you accept in this round

Round N: Part A

• “Proposers:” you may make another proposal as you did in Round 1, but you don’t have to

• If you make a proposal, you will lose 2 of your accrued points

Round N: Part B

• “Non-Proposers:” Accept and reject proposals as you did in Round 1

Game 2: No Proposal Costs

• Take one of the sheets that we’re passing around

• The sheet will group you as a “proposer” or “non-proposer”

• Your sheet will list your preferences and your payouts for ending the game with each participant

• Your sheet will also show you your average position on other people’s lists

Round 1: Part A

• “Proposers:” you may propose to one of the people on your preference list if you wish to do so

• You may also withdraw from the game without a match

Round 1: Part B

• “Non-Proposers:” you must decide whether or not to accept any proposals you have received

• You may not have more than one accepted proposals at any time

• If you receive a better proposal in a later round, you will be able to accept that proposal and reject any proposal you accept in this round

Round N: Part A

• “Proposers:” you may make another proposal as you did in Round 1, but you don’t have to

Round N: Part B

• “Non-Proposers:” Accept and reject proposals as you did in Round 1

Outcomes

• Did we see different strategies in the two games?

• How did game length compare?• Qualitatively, which game seemed more

realistic?• Would the matchings in Game 1 be stable

without transaction costs?

Empirical Test

Using online dating data to test our model against the standard Gale-Shapley deferred acceptance game.

The Data

• Profile and contact-history data for 3,000 men and 3,000 women on an online dating site– From Hitsch, Günter J.; Hortaçsu, Ali; Ariely, Dan

“Matching and Sorting in Online Dating,” The American Economic Review, March 2010

Step 1: Estimate Preferences

• Use number of received contacts as a proxy for attractiveness

• Using this proxy as the outcome, regress on a vector of demographic and profile data– BMI, age difference, race (with interactions), etc.

• Using these regression coefficients and standard errors, draw monte carlo samples estimating the value of each participant to each participant of the opposite gender

Step 2: Simulate Games• For each monte carlo repetition, find the man-optimal and woman-

optimal outcome of the standard Gale-Shapley model and our proposal-cost model

• In our model: Normalize all values from 0 to 1. For the man-optimal game, we will use womanj’s normalized value of mani as a proxy for the p of womanj accepting mani’s proposal– This is a simplification and assumes a great deal of knowledge, but it

makes the model testable and manageable• Repeat our model with many different costs• In the Gale-Shapley game, players will propose in descending order

of their values• In our model, players will propose using the order that maximizes

their expected value

Step 3: Compare models

• If, with any cost, our model explains significantly more variation than Gale-Shapley, this supports adding a cost to the standard Gale-Shapley model– The actual cost does not matter, and may vary

between situations

Limitations of the Experiment

• To do this simulation by brute force would require a lot of computation time. As there are similarities between our model and a hidden Markov chain, we may be able to make computation possible by using dynamic programming.

• Our simulation is somewhat vulnerable to our experimental assumptions: Using contacts as a proxy for attractiveness and using our estimate of attractiveness as a proxy for the probability of being accepted

Limitations of the Model: Costs

• Assumes cost of proposal and probability of being accepted are uncorrelated

• Doesn’t differentiate between types of costs– Cost of Searching – Cost of Proposal– Cost of Rejection

Future Applications: Design

• How does the presentation of information influence peoples perception of probabilities?

• Is more information always better?• Can people reduce their costs of searching,

proposing, and rejection via learning?• How does the fixed cost of “putting oneself

out there” factor into this?