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 ) = max j ∈ A u ( j ) − ∑ j ∈ A p j
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 ) = max j ∈ A u ( j ) − ∑ j ∈ A p j 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 ) = max j ∈ A u ( j ) − ∑ j ∈ A p j 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 ) = max j ∈ 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 b i > 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 b i > 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 1 2 3 a 1 1 1 b 1 − ǫ ǫ ǫ c 0 0 0 WE with randomization: 3 gets b ; 1 and 2 get 50-50 lottery of a and c . � 1 2 , 1 � payoffs: 2 , 1 Deterministic WE with budget perturbations: richest player gets a , second richest gets b . If we randomize over budgets, expected payoffs are: � 1 3 , 1 3 , 2 � payoffs: 3
the multi-unit consumption setting finite number of agents { 1, . . . , N } ; finite number of goods H = { 1, . . . , k } utility functions U i ( A , p ) = u i ( A ) − p ( A ) where - A ⊂ H is the set of discrete goods that i consumes - u i : 2 H → I R + ∪ {− ∞ } , - dom u : = { A | u ( A ) > − ∞ } is the consumption set - A ⊂ B implies u i ( A ) ≤ u ( B ) (monotone) - p j is the price of good j and p ( A ) = ∑ j ∈ A p j .
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 b i 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 b i 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 b i 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 = ( A 1 , . . . A n ) , p = ( p 1 , . . . , p L ) such that 1 (feasibility) A i ⊂ H ; A i ∩ A l � = ∅ implies i = l 2 (aggregate feasibility) H = � i A i 3 (optimality) u i ( A i ) − p ( A i ) ≥ u i ( B ) − p ( B ) for all B ⊂ H or A ∈ B ( b i , p ) and u i ( A i ) ≥ u i ( B ) for all B ∈ B ( b i , p ) .
Walrasian equilibrium with randomization a random consumption σ is a probability distribution over the set of goods: σ : 2 H → [ 0, 1 ] such that ∑ A ⊂ H σ ( A ) = 1
Walrasian equilibrium with randomization a random consumption σ is a probability distribution over the set of goods: σ : 2 H → [ 0, 1 ] such that ∑ A ⊂ H σ ( A ) = 1 a random consumption for all agents: τ = ( σ 1 , . . . , σ n ) ∈ ( ∆ ( 2 H )) n feasibility?
Walrasian equilibrium with randomization a random consumption σ is a probability distribution over the set of goods: σ : 2 H → [ 0, 1 ] such that ∑ A ⊂ H σ ( A ) = 1 a random consumption for all agents: τ = ( σ 1 , . . . , σ n ) ∈ ( ∆ ( 2 H )) n feasibility? adding up constraint: ∑ i ∑ A i ∋ j σ ( A i ) ≤ 1 for all j necessary but not sufficient.
the implementability problem two agents, three goods B 1 = { 1, 2 } , B 2 = { 1, 3 } , B 3 = { 2, 3 } , B 4 = ∅
the implementability problem two agents, three goods B 1 = { 1, 2 } , B 2 = { 1, 3 } , B 3 = { 2, 3 } , B 4 = ∅ ◮ σ i ( B j ) = 1 / 4 for i = 1, 2, j = 1, . . . 4: each agent chooses each B j with probability 1 / 4. ◮ each agent consumes each good with probability 1 / 2 ◮ adding up constraint is satisfied: ∑ i ∑ A ∋ j σ i ( A ) = 1 for all j . ◮ there is no distribution α ∈ ∆ [( 2 H ) 2 ] such that its marginals ( α 1 , α 2 ) = ( σ 1 , σ 2 ) this is the implementability problem.
existence without some restriction on preferences, indivisibility creates an existence problem H = { 1, 2, 3 } , N = { 1, 2 } � 0 if | A | ≤ 1 u 1 ( A ) = u 2 ( A ) = 2 if | A | ≥ 2
existence without some restriction on preferences, indivisibility creates an existence problem H = { 1, 2, 3 } , N = { 1, 2 } � 0 if | A | ≤ 1 u 1 ( A ) = u 2 ( A ) = 2 if | A | ≥ 2 ◮ p 1 = p 2 = p 3 ◮ if p 1 > 1, aggregate demand = 0 ( ∅ ) ◮ if p 1 = 1, aggregate demand = 0, 2 or 4 units ◮ if p 1 < 1, aggregate demand = 4 units
randomization does not help � 0 if | A | ≤ 1 u 1 ( A ) = u 2 ( A ) = 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 only possible candidate for eq. price: p = 1 even at p = 1 demands never add up to 3
randomization does not help � 0 if | A | ≤ 1 u 1 ( A ) = u 2 ( A ) = 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
transferable utility and (gross) substitutes a condition on the u ’s that will ensure the existence of WE: transferable utility demand: D u ( 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: D u ( p ) = { A ⊂ H | u ( A ) − p ( A ) ≥ u ( B ) − p ( B ) for all B ⊂ H } u satisfies substitutes if A ∈ D u ( p ) q j ≥ p j for all j and C = { j | q j = p j } implies
transferable utility and (gross) substitutes a condition on the u ’s that will ensure the existence of WE: transferable utility demand: D u ( p ) = { A ⊂ H | u ( A ) − p ( A ) ≥ u ( B ) − p ( B ) for all B ⊂ H } u satisfies substitutes if A ∈ D u ( p ) q j ≥ p j for all j and C = { j | q j = p j } implies there is B ∈ D u ( q ) such that A ∩ C ⊂ B .
examples of substitutes preferences: academic preferences D ⊂ 2 H is an M ♯ -convex set if
examples of substitutes preferences: academic preferences D ⊂ 2 H 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 ⊂ 2 H 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 .
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