LEARNING MATTERS: REAPPRAISING OBJECT ALLOCATION RULES WHEN AGENTS STRATEGICALLY INVESTIGATE

• Date: 01 May 2018
• Author: Vikram Manjunath,Patrick Harless
• Published date: 01 May 2018
• DOI: http://doi.org/10.1111/iere.12280

INTERNATIONAL ECONOMIC REVIEW

Vol. 59, No. 2, May 2018 DOI: 10.1111/iere.12280

LEARNING MATTERS: REAPPRAISING OBJECT ALLOCATION RULES WHEN

AGENTS STRATEGICALLY INVESTIGATE∗

BYPATRICK HARLESS AND VIKRAM MANJUNATH1

University of Glasgow, U.K.; University of Ottawa, Canada

Individuals form preferences through search, interviews, discussion, and investigation. In a stylized object

allocation model, we characterize the equilibrium learning strategies induced by different allocation rules and

trace their welfare consequences. Our analysis reveals that top trading cycles rules dominate serial priority rules

under inequality-averse measures of social welfare.

1. INTRODUCTION

Individuals go to great lengths to investigate the value of goods that they may consume: Firms

interview candidates, students visit colleges, consumers test drive cars, and so on. How do these

decisions interact with allocation rules? Toward an answer, we directly model the preference

formation stage as a strategic game.2We model a stylized object allocation problem in which

individuals begin with common expected values and each has an opportunity to learn the true

personalized value of one object. Within this framework, we evaluate two prominent families

of assignment rules: serial priority rules and top trading cycles (TTC) rules. Our model delivers

two emphatic conclusions. First, learning incentives differ markedly across allocation rules,

and equilibrium strategies depend crucially on their details. Second, all TTC rules dominate

all priority rules according to a range of inequality-averse social welfare functions and are

equivalent in the utilitarian sense.

1.1. An Example: Visiting Colleges. To motivate our investigation and illustrate our results,

we discuss an extended example. Suppose there are three colleges, each with one available seat:

an ivy league school, a state school, and a technical school. In expectation, students rank them

in that order. The actual value a student receives by attending each school is uncertain but can

be learned by visiting. Concretely, a visit reveals whether the school is a good t or a bad t for

the student. Unfortunately, time constraints limit the student to one trip, creating a decision

problem: Which campus should he visit?

Each school is a good or bad fit for each student independently with equal probability. Table 1

summarizes the relevant utility information.

First, consider a single student free to enroll at any school. If visiting a school reveals it to be

a good fit for the student, then he will choose it over each of the other schools, about which he

∗Manuscript received March 2016; revised September 2016.

1We thank Ghufran Ahmed, Samson Alva, Sophie Bade, Rahul Deb, Srihari Govindan, Jens Gudmundsson, Guil-

laume Haeringer, Sean Horan, Annika Johnson, Sangram Kadam, Morimitsu Kurino, Romans Pancs, Arunava Sen,

Yves Sprumont, William Thomson, Rodrigo Velez, and participants of the 2013 Conference on Economic Design,

2013 Asian Meetings of the Econometric Society, Spring 2015 Midwest Economic Theory Meeting, and 26th Interna-

tional Conference on Game Theory for valuable comments and discussion. Please address correspondence to: Patrick

Harless, Adam Smith Business School, University of Glasgow, West Quadrangle, Glasgow, G12 8QQ, U.K. Phone:

+44-0141-330-6467. E-mail: patrick.harless@glasgow.ac.uk.

2Our approach departs from standard practice of the literature on object allocation and moves toward standard

practice of the mechanism design literature where information acquisition has become de rigueur (e.g., Bergemann and

V¨

alim¨

aki, 2002).

557

C

(2018) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

558 HARLESS AND MANJUNATH

TABLE 1

UTILITIES FROM EACH TYPE OF SCHOOL IN EXPECTATION AND WHEN REALIZED AS A GOOD OR BAD FIT

School Good Fit Bad Fit Expectation

Ivy 14 4 9

State 12 2 7

Technical 10 0 5

knows only the expected value. If visiting a school reveals it to be a bad fit, then the student

will prefer either of the remaining schools. The student has three options: visit the ivy, state, or

technical school. If he chooses to visit the ivy school, his expected utility is

Pr(good fit)u(ivy|good) +Pr(bad fit)E[u(state)] =1

2·14 +1

2·7=10.5.

Similar calculations show that his expected utility from visiting the state school is also 10.5 and

from visiting the technical school is 9.5. Comparing these, the student optimally visits either the

ivy or state school.

Next, suppose there is a second student who must simultaneously decide where to visit.

Visitation decisions are now more complex: Each student must consider the visitation decision

of the other student as well as how the schools will resolve potential conflict. To understand

these incentives, we compare assignment rules.

1.1.1. Priority rule. Suppose the first student has higher priority at all schools, meaning that

he is able to attend his most preferred school regardless of the other student’s preferences. This

student may proceed as if facing a one-student problem, but the second student’s problem is

more complicated. How should he respond?

The second student expects the first student to choose one of his optimal visitation strategies.

As we saw above, the two strategies of the first student have the same implications for the

second student: Between the ivy and the state schools precisely one will remain available to

him as the first student will take the other. If the second student visits either of these schools

and it is unavailable, he will have “wasted” his opportunity to visit. To see this, suppose that he

visits the ivy school. It is either available or unavailable to him with equal probability. So his

expected utility is

Pr(ivy unavailable)E[u(state)] +Pr(ivy available) (Pr(good fit)u(ivy|good)

+Pr(bad fit)E[u(technical)])=1

2·7+1

21

2·14 +1

2·5=8.25.

A similar calculation shows that his expected utility from visiting the state school is 8.75. If

instead he visits the technical school, his visit always reveals useful information: If it is a good

fit, he enrolls; if it is a bad fit, he opts for whichever of the other schools is available. His expected

utility is

Pr(good fit)u(technical|good) +Pr(bad fit) (Pr(ivy available)E[u(ivy)]

+(1 −Pr(ivy available))E[u(state)])=1

2·10 +1

21

2·9+1

2·7=9.

The second student optimally visits the technical school. He prefers to visit a school that is

certain to be available, even though it has a lower expected value. Altogether, the students’

expected utilities in equilibrium are 10.5 and 9, respectively.

LEARNING MATTERS 559

1.1.2. Endowment (TTC) rule. Suppose the first student has higher priority at the ivy school,

but the second student has higher priority at the state school. Intuitively, these priorities give

each student a right-of-refusal at the school where he has higher priority, effectively endowing

him with the seat at that school.

We suppose that the first student visits the ivy school3and focus on the second student. Given

his priority, the second student knows that the state school will always be available to him. In

contrast, the ivy school will be available only if the first student finds it to be a bad fit. This

suggests that visiting the state school, the second student’s “endowment,” is optimal. To confirm

this, first suppose he visits the ivy school. If the ivy school is available, he enrolls if he finds

it to be a good fit and otherwise chooses the state school. If the ivy school is unavailable, he

compares the state and technical schools by expected value and chooses the state school. His

expected utility is

Pr(ivy available)(Pr (good fit)u(ivy|good) +Pr(bad fit)E[u(state)])

+Pr(ivy unavailable)E[u(state)]) =1

21

2·14 +1

2·7+1

2·7=8.75.

On the other hand, if he visits the state school, he is able to enroll whenever he finds it to be

a good fit. If he finds it to be a bad fit, he chooses the remaining school. His expected utility is

then

Pr(good fit)u(state|good) +Pr(bad fit) (Pr(ivy available)E[u(ivy)]

+Pr(ivy unavailable)E[u(technical)])=1

2·12 +1

21

2·9+1

2·5=9.5.

The technical school is also guaranteed to be available, and analogous computation shows the

second student’s expected utility when visiting it is 9. Thus, the second student optimally visits

the state school and obtains an expected utility of 9.5. Reasoning similarly, the first student’s

equilibrium expected utility is 10.

Comparing the rules, the change in priority at the state school moves it from the first stu-

dent’s effective endowment to the second student’s effective endowment. As a result, the

second student changes his visitation strategy and is also better off. This illustrates our cen-

tral conclusion: Compared with priority rules, TTC rules lead to more equal distributions of

utility.4

1.2. Preview of Results. We elaborate on the features of our model, all of which appear in

the example. First, objects have common expected values. This is appropriate when individuals

have access to similar information or consult the same source.5We model values as the common

expected value plus an idiosyncratic private value. With this specification, we are able to capture

private values that follow essentially any symmetric distribution, including uniform and normal

distributions, as well as binary “good-news/bad-news” distributions as in the example. Beyond

symmetry, we assume that private values are independent and identically distributed across

individuals and objects. Thus, as in the example, whether one individual is a good or bad fit at

one school provides no information about whether he is a good or bad fit at another school or

whether a different student is a good or bad fit at the original school.

Each individual has access to a learning technology that permits him to learn the realization of

one private value. We interpret this as a time constraint. Although simple, this technology allows

3In fact, as further computation shows, this is his dominant strategy.

4That the sums of the expected utilities are equal (9 +10.5=9.5+10) also generalizes; the rules are never Pareto-

comparable but instead entail a redistribution of expected utility.

5For example, many college-bound seniors use the ranking of U.S. News and World Reports as a baseline. More

prosaically, we assume a common prior.