Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and...
Transcript of Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and...
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Market Design and Walrasian Equilibrium
with Wolfgang Pesendorfer and Mu Zhang
May 12, 2020
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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)
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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)
![Page 4: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/4.jpg)
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)
![Page 5: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/5.jpg)
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)
![Page 6: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/6.jpg)
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
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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
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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
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unit demand economy without transfers
Hylland and Zeckhauser (1979)
I there are N agents and N goods
I each agent can consume at most one good
I there is no c-money
I U(A, p) = maxj∈A u(j)
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unit demand economy without transfers
Hylland and Zeckhauser (1979)
I there are N agents and N goods
I each agent can consume at most one good
I there is no c-money
I U(A, p) = maxj∈A u(j)
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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
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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
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WE: randomization versus budget perturbation
a
b
c
1 2 3
1 1 1
ε ε 1− ε
0 0 0
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
)
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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 .
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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
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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
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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
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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
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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 =⋃
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).
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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.
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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.
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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.
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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.
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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.
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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
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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
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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
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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
the implementability problem
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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.
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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.
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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.
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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.
A B
a1 a2 c b1 b2
a1 a2 c b1 b2
a1 a2 c b1 b2
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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.
A B
a1 a2 c b1 b2
a1 a2 c b1 b2
a1 a2 c b1 b2
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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.
A B
a1 a2 c b1 b2
a1 a2 c b1 b2
a1 a2 c b1 b2
![Page 35: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/35.jpg)
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.
A B
a1 a2 c b1 b2
a1 a2 c b1 b2
a1 a2 c b1 b2
![Page 36: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/36.jpg)
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.
A B
a1 a2 c b1 b2
a1 a2 c b1 b2
a1 a2 c b1 b2
![Page 37: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/37.jpg)
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.
A B
a1 a2 c b1 b2
a1 a2 c b1 b2
a1 a2 c b1 b2
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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
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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?
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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?
![Page 53: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/53.jpg)
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?
![Page 54: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/54.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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)
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
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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)
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 thenontransferable utility economy
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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)
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
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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)
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
![Page 73: Market Design and Walrasian Equilibrium · the unit demand economy with transfers Shapley and Shubik (1971) I there are N agents and N (indivisible) goods I each agent can consume](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f754876aa7f752875426bc3/html5/thumbnails/73.jpg)
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
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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
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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
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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 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⊥.
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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 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⊥.
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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
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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
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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
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existence of WE with production
theorem 4: every nontransferable utility economy with matroidtechnology has a strong equilibrium.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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(B)
where
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
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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(B)
where
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
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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 .