Market Design and Walrasian Equilibrium with Wolfgang Pesendorfer - - PowerPoint PPT Presentation

Market Design and Walrasian Equilibrium

with Wolfgang Pesendorfer and Mu Zhang May 12, 2020

the unit demand economy with transfers

Shapley and Shubik (1971)

◮ there are N agents and N (indivisible) goods ◮ each agent can consume at most one good ◮ each agent has plenty of the single divisible good

commodity money or c-money

the unit demand economy with transfers

Shapley and Shubik (1971)

◮ there are N agents and N (indivisible) goods ◮ each agent can consume at most one good ◮ each agent has plenty of the single divisible good

commodity money or c-money

◮ U(A, p) = maxj∈A u(j) − ∑j∈A pj

the unit demand economy with transfers

Shapley and Shubik (1971)

◮ there are N agents and N (indivisible) goods ◮ each agent can consume at most one good ◮ each agent has plenty of the single divisible good

commodity money or c-money

◮ U(A, p) = maxj∈A u(j) − ∑j∈A pj

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

the unit demand economy with transfers

Shapley and Shubik (1971)

◮ there are N agents and N (indivisible) goods ◮ each agent can consume at most one good ◮ each agent has plenty of the single divisible good

commodity money or c-money

◮ U(A, p) = maxj∈A u(j) − ∑j∈A pj

for arbitrary initial endowments of goods, what are efficient, core and Walrasian allocations? Shapley and Shubik answer all of these questions (LP)

transferable utility unit demand economy cont.

Shapley and Shubik show:

◮ efficient, core and WE allocations exist ◮ and are all the same ◮ and maximize the sum of utilities (surplus)

transferable utility unit demand economy cont.

Shapley and Shubik show:

◮ efficient, core and WE allocations exist ◮ and are all the same ◮ and maximize the sum of utilities (surplus) ◮ WE prices can be derived from the dual of the LP ◮ set of WE prices is a lattice

transferable utility unit demand economy cont.

Shapley and Shubik show:

◮ efficient, core and WE allocations exist ◮ and are all the same ◮ and maximize the sum of utilities (surplus) ◮ WE prices can be derived from the dual of the LP ◮ set of WE prices is a lattice

Leonard (1983) shows:

◮ efficient allocation with smallest WE prices is a strategy-proof

mechanism

unit demand economy without transfers

Hylland and Zeckhauser (1979)

◮ there are N agents and N goods ◮ each agent can consume at most one good ◮ there is no c-money

unit demand economy without transfers

Hylland and Zeckhauser (1979)

◮ there are N agents and N goods ◮ each agent can consume at most one good ◮ there is no c-money ◮ U(A, p) = maxj∈A u(j)

No Transfers cont’d

construct the following economy:

◮ all goods are initially owned by the “seller” ◮ seller does not value the goods ◮ agent i has bi > 0 units of fiat money

No Transfers cont’d

construct the following economy:

◮ all goods are initially owned by the “seller” ◮ seller does not value the goods ◮ agent i has bi > 0 units of fiat money

Results:

◮ efficient WE exist ◮ not all WE are efficient ◮ WE do not maximize sum of utilities

WE: randomization versus budget perturbation

a b c 1 2 3 1 1 1 ǫ ǫ 1 − ǫ WE with randomization: 3 gets b; 1 and 2 get 50-50 lottery of a and c. payoffs: 1

2, 1 2, 1

• Deterministic WE with budget perturbations: richest player gets a,

second richest gets b. If we randomize over budgets, expected payoffs are: payoffs: 1

3, 1 3, 2 3

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 → I

R+ ∪ {−∞},

• 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 as needed; Kelso and Crawford (1982); extensively studied.

environments

(1) transferable utility case: agents have as much c-money as needed; Kelso and Crawford (1982); extensively studied. (2) limited transfers case: agent i has as bi units of c-good.

environments

(1) transferable utility case: agents have as much c-money as needed; 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

environments

(1) transferable utility case: agents have as much c-money as needed; 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 (and upper) bound constraints

walrasian equilibrium: deterministic and random allocations

Deterministic Walrasian equilibrium is ω = (A1, . . . An), p = (p1, . . . , pL) such that 1 (feasibility) Ai ⊂ H; Ai ∩ Al = ∅ implies i = l 2 (aggregate feasibility) H =

i Ai

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 set

• f goods:

σ : 2H → [0, 1] such that ∑A⊂H σ(A) = 1

Walrasian equilibrium with randomization

a random consumption σ is a probability distribution over the set

• f goods:

σ : 2H → [0, 1] such that ∑A⊂H σ(A) = 1 a random consumption for all agents: τ = (σ1, . . . , σn) ∈ (∆(2H))n feasibility?

Walrasian equilibrium with randomization

a random consumption σ is a probability distribution over the set

• f 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 ∑Ai∋j σ(Ai) ≤ 1 for all j necessary but not sufficient.

the implementability problem

two agents, three goods B1 = {1, 2}, B2 = {1, 3}, B3 = {2, 3}, B4 = ∅

the implementability problem

two agents, three goods B1 = {1, 2}, B2 = {1, 3}, B3 = {2, 3}, B4 = ∅

◮ σi(Bj) = 1/4 for i = 1, 2, j = 1, . . . 4: each agent chooses

each Bj with probability 1/4.

◮ each agent consumes each good with probability 1/2 ◮ adding up constraint is satisfied: ∑i ∑A∋j σi(A) = 1 for all j. ◮ 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 an existence problem H = {1, 2, 3}, N = {1, 2} u1(A) = u2(A) =

• if |A| ≤ 1

2 if |A| ≥ 2

existence

without some restriction on preferences, indivisibility creates an existence problem H = {1, 2, 3}, N = {1, 2} u1(A) = u2(A) =

• if |A| ≤ 1

2 if |A| ≥ 2

◮ p1 = p2 = p3 ◮ if p1 > 1, aggregate demand = 0 (∅) ◮ if p1 = 1, aggregate demand = 0, 2 or 4 units ◮ if p1 < 1, aggregate demand = 4 units

randomization does not help

u1(A) = u2(A) =

• if |A| ≤ 1

2 if |A| ≥ 2

◮ p > 1 implies aggregate demand = 0 (∅) ◮ p = 1 implies demand aggregate demand = 4, 2 or 0 units ◮ p < 1 implies aggregate demand = 4 units

• nly possible candidate for eq. price: p = 1

even at p = 1 demands never add up to 3

randomization does not help

u1(A) = u2(A) =

• if |A| ≤ 1

2 if |A| ≥ 2

• nly 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 the implementability problem

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

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

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.

examples of substitutes preferences: academic preferences

D ⊂ 2H is an M♯-convex set if

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

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

• r there is k ∈ B\A such that

(A\{j}) ∪ {k}, (B\{k}) ∪ {j} ∈ D.

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

• r there is k ∈ B\A such that

(A\{j}) ∪ {k}, (B\{k}) ∪ {j} ∈ D. A B a1 a2 c b1 b2

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

• r there is k ∈ B\A such that

(A\{j}) ∪ {k}, (B\{k}) ∪ {j} ∈ D. A B a1 a2 c b1 b2 a1 a2 c b1 b2

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

• r there is k ∈ B\A such that

(A\{j}) ∪ {k}, (B\{k}) ∪ {j} ∈ D. A B a1 a2 c b1 b2 a1 a2 c b1 b2 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) =

• max A⊃B∈D v(B)

if there is B ∈ D such that A ⊂ B −∞

• therwise

substitutes preserving operations

for substitutes v, w endowment: u(A) := v(A ∪ B) − v(B) restriction: u(A) := v(A ∩ B)

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)

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)

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) A B a1 a2 c b1 b2

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) A B a1 a2 c b1 b2 a1 a2 c b1 b2

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) A B a1 a2 c b1 b2 a1 a2 c b1 b2 a1 a2 c b1 b2

transferable utility: existence and properties of equilibrium

◮ WE exist (Kelso and Crawford (1982)) ◮ WE are efficient

transferable utility: existence and properties of equilibrium

◮ WE exist (Kelso and Crawford (1982)) ◮ WE are efficient ◮ WE maximize the sum of utilities (surplus)

transferable utility: existence and properties of equilibrium

◮ WE exist (Kelso and Crawford (1982)) ◮ WE are efficient ◮ WE maximize the sum of utilities (surplus) ◮ WE allocations = surplus maximizers ◮ WE has a product structure

P∗ = WE prices (lattice), Ω∗ = WE allocations: WE: P∗ × Ω∗

transferable utility: existence and properties of equilibrium

◮ WE exist (Kelso and Crawford (1982)) ◮ WE are efficient ◮ WE maximize the sum of utilities (surplus) ◮ WE allocations = surplus maximizers ◮ WE has a product structure

P∗ = WE prices (lattice), Ω∗ = WE allocations: WE: P∗ × Ω∗

◮ substitutes preferences are a maximal class for which WE

existence can be guaranteed

transferable utility: existence and properties of equilibrium

◮ WE exist (Kelso and Crawford (1982)) ◮ WE are efficient ◮ WE maximize the sum of utilities (surplus) ◮ WE allocations = surplus maximizers ◮ WE has a product structure

P∗ = WE prices (lattice), Ω∗ = WE allocations: WE: P∗ × Ω∗

◮ substitutes preferences are a maximal class for which WE

existence can be guaranteed

◮ randomized WE allocations are mixtures of WE allocations

the limited transfers case

agents have limited endowments of the c-money (bi) utility maximization problem is: max

A⊂H ui(A) − p(A) subject to p(A) ≤ bi

(the constraint p(A) ≤ bi is new)

the limited transfers case

agents have limited endowments of the c-money (bi) utility maximization problem is: max

A⊂H ui(A) − p(A) subject to p(A) ≤ bi

(the constraint p(A) ≤ bi is new) can we ensure existence of WE?

the limited transfers case

agents have limited endowments of the c-money (bi) utility maximization problem is: max

A⊂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?

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

2

existence of Walrasian equilibrium

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

existence of Walrasian equilibrium

E = {(ui, Bi, bi)i∈N} is a limited transfers economy if ui is satisfies substitutes, bi > 0 and Bi ∈ dom ui for all i theorem 1: every limited transfers economy has a Walrasian equilibrium.

existence of Walrasian equilibrium

E = {(ui, Bi, bi)i∈N} is a limited transfers economy if ui is satisfies substitutes, bi > 0 and Bi ∈ dom ui for all i theorem 1: every limited transfers economy has a Walrasian equilibrium. with substitutes preferences, implementability problem is resolved/bypassed

existence of Walrasian equilibrium

E = {(ui, Bi, bi)i∈N} is a limited transfers economy if ui is satisfies substitutes, bi > 0 and Bi ∈ dom ui for all i theorem 1: every limited transfers economy has a Walrasian equilibrium. with substitutes preferences, implementability problem is resolved/bypassed equilibria are Pareto efficient

how the proof works

for each i choose λi ∈ [0, 1] and replace every Ui with ˆ Ui(Ai, p) = λiui(A) − p(Ai) ignore constraints, find equilibrium for the transferable utility economy such that

how the proof works

for each i choose λi ∈ [0, 1] and replace every Ui with ˆ Ui(Ai, p) = λiui(A) − p(Ai) ignore constraints, find equilibrium for the transferable utility economy such that every agent spends at most bi

how the proof works

for each i choose λi ∈ [0, 1] and replace every Ui with ˆ Ui(Ai, p) = λiui(A) − p(Ai) ignore constraints, find equilibrium for the transferable utility economy such that every agent spends at most bi if λi < 1, agent i spends exactly bi

how the proof works

for each i choose λi ∈ [0, 1] and replace every Ui with ˆ Ui(Ai, p) = λiui(A) − p(Ai) ignore constraints, find equilibrium for the transferable utility economy such that every agent spends at most bi if λi < 1, agent i spends exactly bi equilibria for the transferable utility economy are implementable

how the proof works

for each i choose λi ∈ [0, 1] and replace every Ui with ˆ Ui(Ai, p) = λiui(A) − p(Ai) ignore constraints, find equilibrium for the transferable utility economy 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

nontransferable utility economies

no c-money. assign fiat money to all agents normalize the price of fiat money (i.e., = 1)

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 least expensive consumption among all her optimal consumptions fact: every strong equilibrium is Pareto efficient; other WE may be inefficient

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 least expensive consumption among all her optimal consumptions fact: every strong equilibrium is Pareto efficient; other WE may be inefficient theorem 2: every nontransferable utility economy has a strong equilibrium.

how the proof works

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

how the proof works

define the transferable utility economy En = {(nui, bi)i∈N} all ui’s have been multiplied by n pretend agents value fiat money and find WE for the limited transfers economy (previous theorem)

how the proof works

define the transferable utility economy En = {(nui, bi)i∈N} all ui’s have been multiplied by n pretend agents value fiat money and find WE for the limited transfers economy (previous theorem) let (αn, pn) be a WE for the economy with En

how the proof works

define the transferable utility economy En = {(nui, bi)i∈N} all ui’s have been multiplied by n pretend agents value fiat money and find WE for the limited transfers economy (previous theorem) let (αn, pn) be a WE for the economy with En find a convergent subsequence of (αn, pn)

how the proof works

define the transferable utility economy En = {(nui, bi)i∈N} all ui’s have been multiplied by n pretend agents value fiat money and find WE for the limited transfers economy (previous theorem) let (αn, pn) be a WE for the economy with En find a convergent subsequence of (αn, pn) the limit of that subsequence is a strong equilibrium for the nontransferable utility economy

matroids and production

I ⊂ 2H is a matroid if (i) ∅ ∈ I

matroids and production

I ⊂ 2H is a matroid if (i) ∅ ∈ I (ii) A ∈ I, B ⊂ A implies B ∈ I

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 that B ∪ {j} ∈ I

matroids and production

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 only if they have maximal cardinality.

matroids and production

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 only if they have maximal cardinality. fact: if B is the ppf of some matroid I, then I = {A ⊂ B | for some B ∈ B} and B⊥ = {A ⊂ B | for some Bc ∈ B} is the ppf of I⊥ = {A ⊂ B | for some B ∈ B⊥}.

matroid technology

H is the set of all possible goods (outputs)

matroid technology

H is the set of all possible goods (outputs) I ⊂ 2H is the set of feasible output combinations

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 with matroid technology if ∅ ∈ dom ui, bi > 0 and I is a matroid

existence of WE with production

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

how the proof works

H =

A∈I A

B is the set of maximal elements of I

how the proof works

H =

A∈I A

B is the set of maximal elements of I B⊥ = {Bc | B ∈ B}

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

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) = max

B∈I⊥ |A ∩ B|

this u satisfies substitutes

how the proof works cont.

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

how the proof works cont.

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

how the proof works cont.

replace production with agent 0 who has utility u to get an n + 1 person exchange economy with aggregate endowment H give agent 0 a lot of money WE of n + 1 person exchange economy and aggregate endowment H 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, every student must take at least 2 science courses

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, every student must take at least 2 science courses (g) at least 50% of the slots in a school have to go to students who live in that district

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, every student must take at least 2 science courses (g) at least 50% of the slots in a school have to go to students who 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 resources limit the total enrollment two courses ≤ 90

group constraints

maximal enrollment in phy 311 is m; n < m physics majors have priority, they must take the course. group constraint (A, n) for group I ⊂ {1, . . . , N} means agents in I can collectively consume at most n units from the set A, where A is a collection of perfect substitutes (for all agents).

group constraints

maximal enrollment in phy 311 is m; n < m physics majors have priority, they must take the course. group constraint (A, n) for group I ⊂ {1, . . . , N} means agents in I can collectively consume at most n units from the set A, where A is 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 a given set.

individual constraints

simplest individual constraints: bounds on the number of goods an agent may consume from a given set. student is required to take 4 classes each semester, is barred from enrolling in more than 6. u6

4(A) = max

B⊂A |B|≤6

u4(B) where u4(A) =

• u(A)

if |A| ≥ 4 −∞ if |A| < 4 We can impose multiple constraints even hierarchies of constraints provided constraints and preferences line-up nicely

theorem 5: every nontransferable utility economy, {(ui(ci, ·), 1)N

i=1, I} with matroid technology and modular

constraints has a strong equilibrium if there are I-feasible A1, . . . , AN+1 ∈ dom ui(ci, ·) for all i.