market design and walrasian equilibrium · the unit demand economy with transfers shapley and...

Market Design and Walrasian Equilibrium

with Wolfgang Pesendorfer and Mu Zhang

May 12, 2020

the unit demand economy with transfers

Shapley and Shubik (1971)

I there are N agents and N (indivisible) goods

I each agent can consume at most one good

I each agent has plenty of the single divisible goodcommodity money or c-money

I U(A, p) = maxj∈A u(j)−∑j∈A pj

for arbitrary initial endowments of goods, what are efficient, coreand Walrasian allocations?

Shapley and Shubik answer all of these questions (LP)

transferable utility unit demand economy cont.

Shapley and Shubik show:

I efficient, core and WE allocations exist

I and are all the same

I and maximize the sum of utilities (surplus)

I WE prices can be derived from the dual of the LP

I set of WE prices is a lattice

Leonard (1983) shows:

I efficient allocation with smallest WE prices is a strategy-proofmechanism

unit demand economy without transfers

Hylland and Zeckhauser (1979)

I there are N agents and N goods

I there is no c-money

I U(A, p) = maxj∈A u(j)

unit demand economy without transfers

Hylland and Zeckhauser (1979)

I there are N agents and N goods

I there is no c-money

I U(A, p) = maxj∈A u(j)

No Transfers cont’d

construct the following economy:

I all goods are initially owned by the “seller”

I seller does not value the goods

I agent i has bi > 0 units of fiat money

Results:

I efficient WE exist

I not all WE are efficient

I WE do not maximize sum of utilities

No Transfers cont’d

construct the following economy:

I all goods are initially owned by the “seller”

I seller does not value the goods

I agent i has bi > 0 units of fiat money

Results:

I efficient WE exist

I not all WE are efficient

I WE do not maximize sum of utilities

WE: randomization versus budget perturbation

ε ε 1− ε

WE with randomization: 3 gets b; 1 and 2 get 50-50 lottery of aand c .

payoffs:(12 , 1

2 , 1)

Deterministic WE with budget perturbations: richest player gets a,second richest gets b. If we randomize over budgets, expectedpayoffs are:

payoffs:(13 , 1

3 , 23

the multi-unit consumption setting

finite number of agents 1, . . . ,N;

finite number of goods H = 1, . . . , k

utility functions Ui (A, p) = ui (A)− p(A) where

- A ⊂ H is the set of discrete goods that i consumes

- ui : 2H → IR+ ∪ −∞,

- dom u := A | u(A) > −∞ is the consumption set

- A ⊂ B implies ui (A) ≤ u(B) (monotone)

- pj is the price of good j and p(A) = ∑j∈A pj .

environments

(1) transferable utility case: agents have as much c-money asneeded; Kelso and Crawford (1982); extensively studied.

(2) limited transfers case: agent i has as bi units of c-good.

(3) nontransferable utility case: no c-money

(4) no c-money, aggregate constraints, individual lower (andupper) bound constraints

environments

walrasian equilibrium: deterministic and random allocations

Deterministic Walrasian equilibrium is ω = (A1, . . .An),p = (p1, . . . , pL) such that

1 (feasibility) Ai ⊂ H; Ai ∩ Al 6= ∅ implies i = l

2 (aggregate feasibility) H =⋃

3 (optimality) ui (Ai )− p(Ai ) ≥ ui (B)− p(B) for all B ⊂ H or

A ∈ B(bi , p) and ui (Ai ) ≥ ui (B) for all B ∈ B(bi , p).

Walrasian equilibrium with randomization

a random consumption σ is a probability distribution over the setof goods:

σ : 2H → [0, 1]

such that ∑A⊂H σ(A) = 1

a random consumption for all agents:

τ = (σ1, . . . , σn) ∈ (∆(2H))n

feasibility?

adding up constraint: ∑i ∑Ai3j σ(Ai ) ≤ 1 for all j

necessary but not sufficient.

σ : 2H → [0, 1]

τ = (σ1, . . . , σn) ∈ (∆(2H))n

feasibility?

σ : 2H → [0, 1]

τ = (σ1, . . . , σn) ∈ (∆(2H))n

feasibility?

the implementability problem

two agents, three goods

B1 = 1, 2, B2 = 1, 3, B3 = 2, 3,B4 = ∅

I σi (B j ) = 1/4 for i = 1, 2, j = 1, . . . 4: each agent chooseseach B j with probability 1/4.

I each agent consumes each good with probability 1/2

I adding up constraint is satisfied: ∑i ∑A3j σi (A) = 1 for all j .

I there is no distribution α ∈ ∆[(2H)2] such that its marginals(α1, α2) = (σ1, σ2)

this is the implementability problem.

two agents, three goods

B1 = 1, 2, B2 = 1, 3, B3 = 2, 3,B4 = ∅

I σi (B j ) = 1/4 for i = 1, 2, j = 1, . . . 4: each agent chooseseach B j with probability 1/4.

I each agent consumes each good with probability 1/2

I adding up constraint is satisfied: ∑i ∑A3j σi (A) = 1 for all j .

I there is no distribution α ∈ ∆[(2H)2] such that its marginals(α1, α2) = (σ1, σ2)

this is the implementability problem.

existence

without some restriction on preferences, indivisibility creates anexistence problem

H = 1, 2, 3, N = 1, 2

u1(A) = u2(A) =

0 if |A| ≤ 1

2 if |A| ≥ 2

I p1 = p2 = p3

I if p1 > 1, aggregate demand = 0 (∅)

I if p1 = 1, aggregate demand = 0, 2 or 4 units

I if p1 < 1, aggregate demand = 4 units

existence

without some restriction on preferences, indivisibility creates anexistence problem

H = 1, 2, 3, N = 1, 2

u1(A) = u2(A) =

0 if |A| ≤ 1

2 if |A| ≥ 2

I p1 = p2 = p3

I if p1 > 1, aggregate demand = 0 (∅)

I if p1 = 1, aggregate demand = 0, 2 or 4 units

I if p1 < 1, aggregate demand = 4 units

randomization does not help

u1(A) = u2(A) =

0 if |A| ≤ 1

2 if |A| ≥ 2

I p > 1 implies aggregate demand = 0 (∅)

I p = 1 implies demand aggregate demand = 4, 2 or 0 units

I p < 1 implies aggregate demand = 4 units

only possible candidate for eq. price: p = 1even at p = 1 demands never add up to 3

randomization does not help

u1(A) = u2(A) =

0 if |A| ≤ 1

2 if |A| ≥ 2

only possible candidate for eq. price: p = 1

at price p = 1 both agents want either two units or zero units.

but if one agent gets 2 units, the other gets 1 unit

transferable utility and (gross) substitutes

a condition on the u’s that will ensure the existence of WE:

transferable utility demand:

Du(p) = A ⊂ H | u(A)− p(A) ≥ u(B)− p(B) for all B ⊂ H

u satisfies substitutes if

A ∈ Du(p)

qj ≥ pj for all j and C = j | qj = pj implies

there is B ∈ Du(q) such that A∩ C ⊂ B.

A ∈ Du(p)

examples of substitutes preferences: academic preferences

D ⊂ 2H is an M ]-convex set if

A,B ∈ D and j ∈ A\B implies either

A\j,B ∪ j ∈ D

or there is k ∈ B\A such that(A\j) ∪ k, (B\k) ∪ j ∈ D.

a1 a2 c b1 b2

A\j,B ∪ j ∈ D

a1 a2 c b1 b2

A\j,B ∪ j ∈ D

a1 a2 c b1 b2

A\j,B ∪ j ∈ D

a1 a2 c b1 b2

A\j,B ∪ j ∈ D

a1 a2 c b1 b2

A\j,B ∪ j ∈ D

a1 a2 c b1 b2

academic preferences

u is an academic preference if there is a M ]-convex D

an additive utility function over sets v is such that

u(A) =

maxA⊃B∈D v(B) if there is B ∈ D such that A ⊂ B

−∞ otherwise

substitutes preserving operations

for substitutes v ,w

endowment: u(A) := v(A∪ B)− v(B)

restriction: u(A) := v(A∩ B)

convolution: u(A) = maxB⊂A v(B) + w(A\B)

satiation: u(A) := maxB⊂A:|B |≤k v(B) for k ≥ 0.

lower bound: u(A) := maxB⊂A:|B |≥k v(B) for k ≥ 0 and:= −∞ if |A| < k.

an alternative characterization of substitutes preferences

M ]-concavity

A,B ∈ dom u, j ∈ A\B implies there is D ⊂ B\A

such that |D | ≤ 1 and

u((A\j) ∪D) + u((B\D) ∪ j) ≥ u(A) + u(B)

a1 a2 c b1 b2

M ]-concavity

u((A\j) ∪D) + u((B\D) ∪ j) ≥ u(A) + u(B)

a1 a2 c b1 b2

M ]-concavity

u((A\j) ∪D) + u((B\D) ∪ j) ≥ u(A) + u(B)

a1 a2 c b1 b2

M ]-concavity

u((A\j) ∪D) + u((B\D) ∪ j) ≥ u(A) + u(B)

a1 a2 c b1 b2

transferable utility: existence and properties of equilibrium

I WE exist (Kelso and Crawford (1982))

I WE are efficient

I WE maximize the sum of utilities (surplus)

I WE allocations = surplus maximizers

I WE has a product structureP∗ = WE prices (lattice), Ω∗ = WE allocations:WE: P∗ ×Ω∗

I substitutes preferences are a maximal class for which WEexistence can be guaranteed

I randomized WE allocations are mixtures of WE allocations

I WE are efficient

the limited transfers case

agents have limited endowments of the c-money (bi )

utility maximization problem is:

maxA⊂H

ui (A)− p(A) subject to p(A) ≤ bi

(the constraint p(A) ≤ bi is new)

can we ensure existence of WE?

do we need to make additional assumptions?

maxA⊂H

the necessity of randomization

example one good: H = 1

two agents: u1(1) = u2(1) = 2, b1 = b2 = 1

without randomization there is no equilibrium:

if p1 ≤ 1 both agents demand the good

if p1 > 1 both agents demand nothing

with randomization, the equilibrium is:p1 = 2, each agent gets the good with probability 1

existence of Walrasian equilibrium

E = (ui ,Bi , bi )i∈N is a limited transfers economy if ui issatisfies substitutes, bi > 0 and Bi ∈ dom ui for all i

theorem 1: every limited transfers economy has a Walrasianequilibrium.

with substitutes preferences, implementability problem isresolved/bypassed

equilibria are Pareto efficient

how the proof works

for each i choose λi ∈ [0, 1] and replace every Ui with

Ui (Ai , p) = λiui (A)− p(Ai )

ignore constraints, find equilibrium for the transferable utilityeconomy such that

every agent spends at most bi

if λi < 1, agent i spends exactly bi

equilibria for the transferable utility economy are implementable

fixed-point argument to find the λi ’s

how the proof works

nontransferable utility economies

no c-money.

assign fiat money to all agents

normalize the price of fiat money (i.e., = 1)

nontransferable utility economy E∗ = (ui , bi )i∈N has fiat money,bi > 0, substitutes preferences, ui , such that ∅ ∈ dom ui for all i .

typical setting for many allocation problems

school choice, class selection

nontransferable utility economies

no c-money.

assign fiat money to all agents

normalize the price of fiat money (i.e., = 1)

nontransferable utility economy E∗ = (ui , bi )i∈N has fiat money,bi > 0, substitutes preferences, ui , such that ∅ ∈ dom ui for all i .

typical setting for many allocation problems

school choice, class selection

strong equilibrium

a random allocation α and prices p are a strong equilibrium if(α, p) is a WE and α delivers, with probability 1, to each i a leastexpensive consumption among all her optimal consumptions

fact: every strong equilibrium is Pareto efficient; other WE may beinefficient

theorem 2: every nontransferable utility economy has a strongequilibrium.

strong equilibrium

a random allocation α and prices p are a strong equilibrium if(α, p) is a WE and α delivers, with probability 1, to each i a leastexpensive consumption among all her optimal consumptions

fact: every strong equilibrium is Pareto efficient; other WE may beinefficient

theorem 2: every nontransferable utility economy has a strongequilibrium.

how the proof works

define the transferable utility economy En = (nui , bi )i∈Nall ui ’s have been multiplied by n

pretend agents value fiat money and find WE for the limitedtransfers economy (previous theorem)

let (αn, pn) be a WE for the economy with Enfind a convergent subsequence of (αn, pn)

the limit of that subsequence is a strong equilibrium for thenontransferable utility economy

how the proof works

define the transferable utility economy En = (nui , bi )i∈Nall ui ’s have been multiplied by npretend agents value fiat money and find WE for the limitedtransfers economy (previous theorem)

how the proof works

let (αn, pn) be a WE for the economy with En

find a convergent subsequence of (αn, pn)

how the proof works

matroids and production

I ⊂ 2H is a matroid if

(i) ∅ ∈ I

(ii) A ∈ I , B ⊂ A implies B ∈ I

(iii) A,B ∈ I , |B | < |A| implies there is j ∈ A\B such thatB ∪ j ∈ I

(i) ∅ ∈ I

for any matroid I ⊂ 2H , B is the set of maximal elements of I :

B = B ∈ I |B ⊂ A ∈ B implies A = B

B is the production possibility frontier.

fact: B = B ∈ I | |B | ≥ |A| for all A ∈ I

elements of I are maximal (in the sense of set inclusion) if and onlyif they have maximal cardinality.

fact: if B is the ppf of some matroid I , thenI = A ⊂ B | for some B ∈ B andB⊥ = A ⊂ B | for some Bc ∈ B is the ppf ofI⊥ = A ⊂ B | for some B ∈ B⊥.

for any matroid I ⊂ 2H , B is the set of maximal elements of I :

B = B ∈ I |B ⊂ A ∈ B implies A = B

B is the production possibility frontier.

fact: B = B ∈ I | |B | ≥ |A| for all A ∈ I

elements of I are maximal (in the sense of set inclusion) if and onlyif they have maximal cardinality.

fact: if B is the ppf of some matroid I , thenI = A ⊂ B | for some B ∈ B andB⊥ = A ⊂ B | for some Bc ∈ B is the ppf ofI⊥ = A ⊂ B | for some B ∈ B⊥.

matroid technology

H is the set of all possible goods (outputs)

I ⊂ 2H is the set of feasible output combinations

E = (ui , bi )i∈N , I is a nontransferable utility economy withmatroid technology if ∅ ∈ dom ui , bi > 0 and I is a matroid

matroid technology

existence of WE with production

theorem 4: every nontransferable utility economy with matroidtechnology has a strong equilibrium.

how the proof works

H =⋃

A∈I A

B is the set of maximal elements of I

B⊥ = Bc |B ∈ B

I⊥ = A |A ⊂ B ∈ B⊥; I⊥ is a matroid

then define u as follows

u(A) = maxB∈I⊥

|A∩ B |

this u satisfies substitutes

how the proof works

H =⋃

A∈I A

B⊥ = Bc |B ∈ B

u(A) = maxB∈I⊥

|A∩ B |

how the proof works

H =⋃

A∈I A

B⊥ = Bc |B ∈ B

u(A) = maxB∈I⊥

|A∩ B |

how the proof works

H =⋃

A∈I A

B⊥ = Bc |B ∈ B

u(A) = maxB∈I⊥

|A∩ B |

how the proof works cont.

replace production with agent 0 who has utility u to get an n+ 1person exchange economy with aggregate endowment H

give agent 0 a lot of money

WE of n+ 1 person exchange economy and aggregate endowmentH is WE of the original production economy

individual, group and aggregate constraints

in market design problems,there may be individual, group or aggregate constraints:

(i) no student can take more than 12 courses in her major, everystudent must take at least 2 science courses

(g) at least 50% of the slots in a school have to go to studentswho live in that district

(a) two versions of an introductory physics course are to be offered:phy 101 without calculus; phy 102 with calculus.phy 101, 102 can have at most 60 students each but lab resourceslimit the total enrollment two courses ≤ 90

group constraints

maximal enrollment in phy 311 is m; n < m physics majors havepriority, they must take the course.

group constraint (A, n) for group I ⊂ 1, . . . ,N means agents inI can collectively consume at most n units from the set A, where Ais a collection of perfect substitutes (for all agents).

pick any |A| − n element subset B of A.

Replace each ui for i ∈ I with u′i such that

u′i (A) = ui (A∩ Bc)

group constraints

maximal enrollment in phy 311 is m; n < m physics majors havepriority, they must take the course.

group constraint (A, n) for group I ⊂ 1, . . . ,N means agents inI can collectively consume at most n units from the set A, where Ais a collection of perfect substitutes (for all agents).

pick any |A| − n element subset B of A.

Replace each ui for i ∈ I with u′i such that

u′i (A) = ui (A∩ Bc)

individual constraints

simplest individual constraints:

bounds on the number of goods an agent may consume from agiven set.

student is required to take 4 classes each semester, is barred fromenrolling in more than 6.

u64(A) = maxB⊂A|B |≤6

u4(A) =

u(A) if |A| ≥ 4

−∞ if |A| < 4

We can impose multiple constraints even hierarchies of constraintsprovided constraints and preferences line-up nicely

individual constraints

simplest individual constraints:

bounds on the number of goods an agent may consume from agiven set.

student is required to take 4 classes each semester, is barred fromenrolling in more than 6.

u64(A) = maxB⊂A|B |≤6

u4(A) =

u(A) if |A| ≥ 4

−∞ if |A| < 4

We can impose multiple constraints even hierarchies of constraintsprovided constraints and preferences line-up nicely

theorem 5: every nontransferable utility economy,(ui (ci , ·), 1)Ni=1, I with matroid technology and modularconstraints has a strong equilibrium if there are I-feasibleA1, . . . ,AN+1 ∈ dom ui (ci , ·) for all i .

market design and walrasian equilibrium · the unit demand economy with transfers shapley and...

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