The Combinatorial Assignment Problem

The Combinatorial Assignment Problem

Fuhito Kojima1

April 14, 2015

The Combinatorial Assignment Problem

1 Agents have multiunit demand, 2 _{Monetary transfer is prohibited}

As usual, the social planner has potentially conflicting goals: 1 efficiency,

2 _{incentive compatibility,} 3 fairness

Example: Course Allocation at Universities 1 The objects are seats in courses

The Combinatorial Assignment Problem

Combinatorial assignment is relatively new, though there are related literatures:

No restriction on money → Combinatorial Auction Problem (Vickrey 1961, Cramton et al. 2005)

Single-Unit Demand → School Choice /House Allocation Problem (Shapley and Scarf 1974, Abdulkadiroglu and Sonmez 2003)

Two-Sided Preferences →Two-Sided Matching (Gale and Shapley 1961, Roth and Sotomayor 1990)

Draft Mechanism

Consider the following “draft” mechanism.

1 Students report rank-order lists over individual courses 2 Students are ordered in some way (deterministic for now) 3 Students are allocated 1 course per round for k rounds, based

on their reported preferences and remaining availability. This looks like a sensible generalization of serial dictatorship (priority mechanism) at least if there is no complementarity or subsitutability.

Used in many environments: 1 Course allocation.

Serial Dictatorships

Another natural generalization of the serial dictatorships, call it (once-and-for-all) serial dictatorship.

1 Students report rank-order lists over (subsets of, or individual) courses

2 Students are ordered in some way (deterministic for now) 3 The highest-priority student receives her best bundle of

courses, then the next student obtians his best bundle of courses, etc.

Serial dictatorships are

1 Strategy-proof (unlike draft mechanisms). 2 Pareto efficient.

Characterization Theorems

Roughly, the only mechanisms that are ex-post Pareto efficient and strategyproof are serial dictatorships (Papai 2001; Ehlers and Klaus, 2003; Hatfield 2007, Kojima 2013)

Theorem (Hatfield, 2009)

A mechanism is strategy-proof, Pareto efficient, nonbossy, and neutral if and only if it is a serial dictatorship.

Should we conclude that the serial dictatorship is the desirable mechanism? → unlikely (so unfair!).

What about the random serial dictatorship?

Draft versus RSD

Budish and Cantillon (2012) compare a version of the draft mechanism used at HBS and the random serial dictatorship.

Findings

1 HBS mechanism is simple to manipulate in theory, and heavily manipulated in practice.

The HBS Mechanism: Simple Manipulations

Say that a course ispopular at a given preference profile if it runs out with strictly positive probability.

Theorem (Budish and Cantillon)

Fix an agent i and preference profile of others. Form the strategy

b

Pi by taking the first k courses in her true preference Pi and

rearranging them so that cPb_ic0 whenever:

1 cP_ic0 and both are popular or both are unpopular

2 c is popular and c0 is unpopular

The strategyPbi generates weakly greater utility than truthful

reporting Pi.

The HBS Mechanism: Efficiency and Welfare

Strategic behavior has ex-post and ex-ante redistributive consequences (Example 1)

Students who value popular courses less benefit from the opportunity of ranking them higher

Students who value popular courses highly are hurt by congestion

The HBS mechanism may be ex-post inefficient due to risk-taking by students (Example 3)

There are special cases in which eqm is ex-post efficient (Theorem 5)

Data

Data on the allocation of elective courses at HBS during the 2005-2006 academic year

Reported (potentially strategic) preferences: actual submitted rank-order lists

Welfare Consequences of Strategic Play: Ex-Post

Data – ordinal over individual courses – means we can only detect a fraction of profitable trades

100 trials.

Ex-Post Pareto Improving Trades

Mean Std. Dev. # of Executed Trades per Student 1.54 (0.04) % of Allocated Course Seats Traded 15.4% (0.31%) % of Students Executing

0 Trades 16.4% (1.1%)

1 Trade 35.4% (1.7)

2 Trades 30.5% (1.6)

Welfare Consequences: Ex-Ante, Individual Level

Challenge: preference data are ordinal rankings over courses, but ex-ante welfare depends on lotteries over bundles

Look at first-order stochastic dominance Findings:

45% prefer HBS truthful 5.5% prefer HBS strategic 1% indifferent

Ex-Ante Welfare Evaluation: Methodology

To resolve indeterminates, we need to assume more structure on preferences

1 Average-rank preferences: A sufficient statistic for an outcome bundle is the average true preference rank of the courses it contains

2 Lexicographic preferences: A sufficient statistic for a lottery over outcomes is the probability-weighted rank-allocation vector

Assumption on Preferences Average-Rank

Responsive FOSD Means Lexicographic

Ex-Ante Preference

HBS Truthful 45% 56% 73% 90%

HBS Strategic 5.5% 13% 26% 9%

Indifferent 1% 1% 1% 1%

HBS vs. RSD: Ex-Post versus Ex-Ante

Since strategic behavior in the HBS mechanism harms welfare, it is natural to consider a strategyproof alternative

Same exercise, but comparing HBS average rank distribution to that from anequilibrium counterfactual in which the mechanism is RSD and students are truthful

HBS vs. RSD: Ex-Ante, Individual Level

Same exercise, but comparing HBS average rank distribution to that from anequilibrium counterfactual in which the mechanism is RSD and students are truthful

Assumption on Preferences Average-Rank

Responsive SOSD Means Lexicographic

Ex-Ante Preference

RSD Truthful 0% 0% 19% 25%

HBS Strategic 0% 81% 81% 75%

Indifferent 0% 0% 0% 0%

HBS vs. RSD: Ex-Ante Utility

% Who Get % Who Get

E(Avg Rank) #1 Choice All Top 10

HBS - Truthful 7.76 83% 0.8%

HBS - Strategic 8.35 60% 1.4%

RSD - Truthful 8.94 47% 29%

Explanation: “Callousness”

In RSD, lucky students with good random draws make their last choices independently of whether these courses would be some unlucky student’s first choices

Benefit to lucky is small; harm to unlucky is large Ex-post, RSD is Pareto efficient

Ex-ante, this unavoidable callousness harms utility

What do we Learn from the HBS Mechanism?

1 Ex-post efficiency may not even be a good proxy for ex-ante efficiency

What about the probabilistic serial mechanism?

Bogomolnaia and Moulin (2001) define PS based on an “eating algorithm” for single-unit demand cases:

1 _{Imagine each good is a divisible good of “probability shares.”} 2 _{Imagine there is a time interval[0, 1].}

PS Mechanism Example

O ={a,b}, with one copy each, N ={1, 2, 3, 4}, 1 and 2 like a,b, ø (in this order),

3 and 4 like b,a, ø.

Compute the PS assignment:

1 _t =_{0: Agents 1 and 2 start eatinga, and agents 3 and 4 start} eatingb.

2 _t =₁/_{2: goodsaandbare eaten away. No (real) goods} remain.

The resulting assignment is

Good a Good b Good ø

Agents 1 and 2 1/2 0 1/2

Agents 3 and 4 0 1/2 1/2

PS can be highly manipulable (Kojima, 2009)

Two agents{1, 2}, four objects{a,b,c,d}, 2 for each agent.

1 likes a,b,c,d (in this order), 2 likes b,c,a,d.

The PS assignment is

Gooda Goodb Goodc Goodd

Agent 1 1 0 1/2 1/2

Agent 2 0 1 1/2 1/2

If agent 1 reports that she likes b,a,c,d,

Gooda Goodb Goodc Goodd

Agent 1 1 1/2 0 1/2

Agent 2 0 1/2 1 1/2

In general,

Proposition (Kojima, 2009)

If there are at least two agents with multi-unit demand, then there exists no mechanism that satisfies ordinal efficiency, envy-freeness and weak strategy-proofness.

Walrasian Mechanism

Budish, Che, Kojima and Milgrom (2013) develop a

course-allocation procedure that is ex-ante efficient (in terms of expected utility), generalizing Hylland and Zeckhauser (1979).

Goods are divisible, by considering probability share as unit (like in PS mechanism!)

Agents submit their preferences

Agents are given a unit of mock money (equal across agents) A “Walrasian mechanism” (or competitive mechanism): price-allocation pair such that (i) markets clear, and (ii) given prices, each agent is buying his or her optimal probability shares within budget.

The mechanism is

Ex ante efficient, Ex ante fair (envy-free),

Not strategy-proof, but appears to be “difficult to manipulate”

A-CEEI

Budish (2011) proposes a new mechanism satisfying a number of efficiency, fairness, and incentive desiderata.

1 Criteria of outcome fairness: maximin-share guarantee (will not talk today), and envy bounded by a single good

2 A specific mechanism: Approximate Competitive Equilibrium

from Equal Incomes

Thm 1: Approximate CEEI is approximately ex-post efficient Thm 2, 3: Approximate CEEI satisfies the proposed fairness criteria

Environment

Set ofM courses with integral capacitiesq= (q1, ...,qM). No

other goods in the economy. Set of N students.

Each studenti has a set of permissible schedules

Competitive Equilibrium from Equal Incomes

Competitive Equilibrium from Equal Incomes (Foley, 1967; Varian, 1974):

1 _{Agents report preferences over bundles}

2 Agents are given equal budgets b∗ of an artificial currency 3 We find an item price vector p∗ such that, when each agent is

allocated his favorite affordable bundle, the market clears 4 We allocate each agent their demand atp∗

Approximate CEEI

An allocationx∗, budget vectorb∗ and price vector p∗ constitute an(α,β)-approximate competitive equilibrium from equal incomes(Approximate CEEI) if:

1 Each agent i is allocated her most-preferred bundle in her budget set {x ∈Ψ :p∗·x ≤b∗_i}

2 Euclidean distance of market-clearing error at p∗ is≤ _α market-clearing errorj = demandj - supplyj if pj >0

market-clearing errorj = max(demandj - supplyj, 0) ifpj =0

3 The ratio of the max to the min budget in b∗ is ≤1+_β

Theorem 1

Existence of Approximate CE from Approximate EI

If we seek exact market clearing (α=0) may require arbitrarily large budget inequality (Dictatorship β) If we seek exactly equal budgets (β=0) may require arbitrarily large market clearing error (Identical prefs α)

Theorem (Budish)

Let k be the maximum number of courses in any permissible schedule. Defineσ=min(2k,M)

1 For any _β>0,there exists a ( √

σM

2 ,β)−Approximate CEEI 2 Moreover, for any budget vector b0 with inequality ratio

≤1+β, and any e>0, there exists an (

√

σM

2 ,β)−Approximate CEEI with budgetsb∗ that are

Approximate Efficiency:

√

σM

2 is small in two senses √

σM

2 does not grow withN (number of students) orq (number of copies of each good). AsN,q→_∞, we get exact market clearing (error goes to zero as a fraction of the endowment)

√

σM

2 is a small number in practical problems (especially for a worst case bound)

In a semester at HBS,k =5 andM=50, and so

√

σM 2 ≈11 Contrast with 4500 course seats allocated per semester

Approximate CEEI keeps Envy Bounded by a Single Good

envy-freeness, because all students have the same choice set.

Theorem

ifβ< _k−1₁ thenx∗ satisfies envy bounded by a single good

Intuition: suppose si enviessj. Then

1≤b_i∗ <p∗·x_j∗ ≤b_j∗ ≤ k

k−1

Since x_j∗ contains at most k goods, one of them must cost at least _k−1₁. si can afford the bundle formed by removing this

good from x_j∗

By revealed preference, si must weakly prefer her own bundle

to the bundle formed by removing this single good fromx_j∗, so her envy is bounded.

Strategyproof in the Large

CEEI procedure is not strategyproof

But, any student who regards prices as exogenous should report their preferences truthfully

Formally, the mechanism is ”Strategyproof in the Large” (Theorem 4)

i.e., SP in an appropriately-defined continuum economy. Note: can’t execute Pareto-improving trades ex-post without undermining incentives

By contrast, consider the HBS mechanism, or the Boston mechanism for school choice

Even if we hold fixed the analog of prices in these mechanisms, agents still should not report their preferences truthfully. All course-allocation mechanisms currently found in practice are manipulable even in continuum markets.

Properties of the Approximate CEEI Mechanism

Efficiency

- Ex-post efficient with respect to the allocated goods.

Fairness - Symmetric

- N+1 Maximin Share Guaranteed

-Envy Bounded by a Single Good

Incentives

Ex-Ante Welfare Performance of Approximate CEEI

The Approximate CEEI Mechanism has an element of randomness: the budgets.

Efficiency ideally should be assessed ex-ante, not ex-post Budish, Che, Kojima and Milgrom develop a course-allocation procedure that is ex-ante efficient when students’ vNM preferences for courses can be described by assignment messages (Milgrom, 2009). Some tradeoffs:

no longer satisfy the outcome fairness criteria; less flexibility in permissible schedule sets; need to assume risk neutrality

Conclusion: Next Steps for Theory and Practice

Theory

Are there classes of preferences with a better bound?

What can we achieve with respect to fairness and incentives if we seek exact ex-post efficiency? ex-ante efficiency?

Is the mechanism likely to be manipulable in large finite markets? Bayesian approach to IC?

Sequential problem: what if there are multiple sets of objects that cannot be allocated simultaneously?

Practice

How should agents report their preferences?