Andy McLennan. Workshop on Game Theory Institute for Mathematical Sciences National University of Singapore June 2018

p. 1/16 Efficient Disposal Equilibria of Pseudomarkets Andy McLennan Workshop on Game Theory Institute for Mathematical Sciences National University of Singapore June 2018

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate.

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate. Agents are endowed with artificial currency or commodity bundles.

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate. Agents are endowed with artificial currency or commodity bundles. They trade on a market.

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate. Agents are endowed with artificial currency or commodity bundles. They trade on a market. Several unusual features are natural:

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate. Agents are endowed with artificial currency or commodity bundles. They trade on a market. Several unusual features are natural: Objects may be indivisible.

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate. Agents are endowed with artificial currency or commodity bundles. They trade on a market. Several unusual features are natural: Objects may be indivisible. Free disposal can be important.

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate. Agents are endowed with artificial currency or commodity bundles. They trade on a market. Several unusual features are natural: Objects may be indivisible. Free disposal can be important. Some consumers may be sated.

p. 2/16 Introduction A pseudomarket is a mechanism for allocating a collectively owned endowment in settings where monetary transfers are impossible or inappropriate. Agents are endowed with artificial currency or commodity bundles. They trade on a market. Several unusual features are natural: Objects may be indivisible. Free disposal can be important. Some consumers may be sated. Outcomes may need to be computed.

p. 3/16 Plan of the Talk Closely related literature will be reviewed.

p. 3/16 Plan of the Talk Closely related literature will be reviewed. The main contribution is an existence-ofcompetitive-equilibrium result.

p. 3/16 Plan of the Talk Closely related literature will be reviewed. The main contribution is an existence-ofcompetitive-equilibrium result. This result encompasses existence results in related literature, as well as classical existence theorems.

p. 3/16 Plan of the Talk Closely related literature will be reviewed. The main contribution is an existence-ofcompetitive-equilibrium result. This result encompasses existence results in related literature, as well as classical existence theorems. It is more general in ways that are relevant for certain applications.

p. 3/16 Plan of the Talk Closely related literature will be reviewed. The main contribution is an existence-ofcompetitive-equilibrium result. This result encompasses existence results in related literature, as well as classical existence theorems. It is more general in ways that are relevant for certain applications. The proof has some interesting and novel features.

p. 3/16 Plan of the Talk Closely related literature will be reviewed. The main contribution is an existence-ofcompetitive-equilibrium result. This result encompasses existence results in related literature, as well as classical existence theorems. It is more general in ways that are relevant for certain applications. The proof has some interesting and novel features. Two open problems are described.

p. 4/16 Hylland and Zeckhauser (1979) Hylland and Zeckhauser (henceforth HZ) study a setting in which there are:

p. 4/16 Hylland and Zeckhauser (1979) Hylland and Zeckhauser (henceforth HZ) study a setting in which there are: A finite set of agents.

p. 4/16 Hylland and Zeckhauser (1979) Hylland and Zeckhauser (henceforth HZ) study a setting in which there are: A finite set of agents. A finite set of indivisible objects.

p. 4/16 Hylland and Zeckhauser (1979) Hylland and Zeckhauser (henceforth HZ) study a setting in which there are: A finite set of agents. A finite set of indivisible objects. Each object has an integral capacity.

p. 4/16 Hylland and Zeckhauser (1979) Hylland and Zeckhauser (henceforth HZ) study a setting in which there are: A finite set of agents. A finite set of indivisible objects. Each object has an integral capacity. Each agent has a vnm utility over the objects.

p. 4/16 Hylland and Zeckhauser (1979) Hylland and Zeckhauser (henceforth HZ) study a setting in which there are: A finite set of agents. A finite set of indivisible objects. Each object has an integral capacity. Each agent has a vnm utility over the objects. The goal is to find a probability distribution over feasible assignments of an object to each agent that is efficient and fair.

Hylland and Zeckhauser (1979) Hylland and Zeckhauser (henceforth HZ) study a setting in which there are: A finite set of agents. A finite set of indivisible objects. Each object has an integral capacity. Each agent has a vnm utility over the objects. The goal is to find a probability distribution over feasible assignments of an object to each agent that is efficient and fair. They propose equilibrium allocations of a market with currency endowments and goods that are probabilities of being assigned to each object. p. 4/16

p. 5/16 Compact Consumption Sets Bergstrom (1976), Mas-Colell (1992), and Polemarchakis and Siconolfi (1993) study existence of general competitive equilibrium with compact consumption sets.

p. 5/16 Compact Consumption Sets Bergstrom (1976), Mas-Colell (1992), and Polemarchakis and Siconolfi (1993) study existence of general competitive equilibrium with compact consumption sets. This encompasses classical general equilibrium with more-is-always-better consumers.

p. 5/16 Compact Consumption Sets Bergstrom (1976), Mas-Colell (1992), and Polemarchakis and Siconolfi (1993) study existence of general competitive equilibrium with compact consumption sets. This encompasses classical general equilibrium with more-is-always-better consumers. Mas-Colell allows redistribution of sated consumers excess income.

p. 5/16 Compact Consumption Sets Bergstrom (1976), Mas-Colell (1992), and Polemarchakis and Siconolfi (1993) study existence of general competitive equilibrium with compact consumption sets. This encompasses classical general equilibrium with more-is-always-better consumers. Mas-Colell allows redistribution of sated consumers excess income. Following Gale and Mas-Colell (1975, 1979), Mas-Colell allows quite general externalities.

p. 5/16 Compact Consumption Sets Bergstrom (1976), Mas-Colell (1992), and Polemarchakis and Siconolfi (1993) study existence of general competitive equilibrium with compact consumption sets. This encompasses classical general equilibrium with more-is-always-better consumers. Mas-Colell allows redistribution of sated consumers excess income. Following Gale and Mas-Colell (1975, 1979), Mas-Colell allows quite general externalities. Probably that could be done here, but this has not been pursued.

p. 5/16 Compact Consumption Sets Bergstrom (1976), Mas-Colell (1992), and Polemarchakis and Siconolfi (1993) study existence of general competitive equilibrium with compact consumption sets. This encompasses classical general equilibrium with more-is-always-better consumers. Mas-Colell allows redistribution of sated consumers excess income. Following Gale and Mas-Colell (1975, 1979), Mas-Colell allows quite general externalities. Probably that could be done here, but this has not been pursued. These papers do not allow free disposal.

p. 6/16 Course Allocation At some business schools students report preferences and allocations of seats in courses are computed.

p. 6/16 Course Allocation At some business schools students report preferences and allocations of seats in courses are computed. Budish and Cantillon (2012): versions of random priority used at Harvard.

p. 6/16 Course Allocation At some business schools students report preferences and allocations of seats in courses are computed. Budish and Cantillon (2012): versions of random priority used at Harvard. Budish and Kessler (2016): auction and marketlike mechanisms used at Wharton.

p. 6/16 Course Allocation At some business schools students report preferences and allocations of seats in courses are computed. Budish and Cantillon (2012): versions of random priority used at Harvard. Budish and Kessler (2016): auction and marketlike mechanisms used at Wharton. Budish, Che, Kojima, and Milgrom (2013) (henceforth BCKM) study probabilistic allocations of seats.

p. 6/16 Course Allocation At some business schools students report preferences and allocations of seats in courses are computed. Budish and Cantillon (2012): versions of random priority used at Harvard. Budish and Kessler (2016): auction and marketlike mechanisms used at Wharton. Budish, Che, Kojima, and Milgrom (2013) (henceforth BCKM) study probabilistic allocations of seats. They give conditions under which assignments of probabilities can be realized by distribution over pure assignments.

Course Allocation At some business schools students report preferences and allocations of seats in courses are computed. Budish and Cantillon (2012): versions of random priority used at Harvard. Budish and Kessler (2016): auction and marketlike mechanisms used at Wharton. Budish, Che, Kojima, and Milgrom (2013) (henceforth BCKM) study probabilistic allocations of seats. They give conditions under which assignments of probabilities can be realized by distribution over pure assignments. They also give a highly restricted existence theorem generalizing HZ. p. 6/16

p. 7/16 Other Pseudomarket Papers Eisenberg and Gale (1959) and Eisenberg (1961): equilibrium of a pari-mutuel betting system.

p. 7/16 Other Pseudomarket Papers Eisenberg and Gale (1959) and Eisenberg (1961): equilibrium of a pari-mutuel betting system. Varian (1974): trade from equal incomes leads to allocations of a commonly owned endowment that are efficient and envy free.

p. 7/16 Other Pseudomarket Papers Eisenberg and Gale (1959) and Eisenberg (1961): equilibrium of a pari-mutuel betting system. Varian (1974): trade from equal incomes leads to allocations of a commonly owned endowment that are efficient and envy free. Bogomolnaia, Moulin, Sandomirsky, and Yanovskaya (2017): division of a commonly owned endowment mixing goods and bads.

p. 7/16 Other Pseudomarket Papers Eisenberg and Gale (1959) and Eisenberg (1961): equilibrium of a pari-mutuel betting system. Varian (1974): trade from equal incomes leads to allocations of a commonly owned endowment that are efficient and envy free. Bogomolnaia, Moulin, Sandomirsky, and Yanovskaya (2017): division of a commonly owned endowment mixing goods and bads. Their existence results are implied by Mas-Colell s.

p. 7/16 Other Pseudomarket Papers Eisenberg and Gale (1959) and Eisenberg (1961): equilibrium of a pari-mutuel betting system. Varian (1974): trade from equal incomes leads to allocations of a commonly owned endowment that are efficient and envy free. Bogomolnaia, Moulin, Sandomirsky, and Yanovskaya (2017): division of a commonly owned endowment mixing goods and bads. Their existence results are implied by Mas-Colell s. Of course there is also a vast literature on matching and school choice. In such models usually (not always!) both sides of the market are strategic.

p. 8/16 The Model We work in a general equilibrium setting:

p. 8/16 The Model We work in a general equilibrium setting: There arelgoods indexed by h.

p. 8/16 The Model We work in a general equilibrium setting: There arelgoods indexed by h. There aremagents indexed by i. For each i there are:

p. 8/16 The Model We work in a general equilibrium setting: There arelgoods indexed by h. There aremagents indexed by i. For each i there are: a compact convex consumption set X i R l ;

p. 8/16 The Model We work in a general equilibrium setting: There arelgoods indexed by h. There aremagents indexed by i. For each i there are: a compact convex consumption set X i R l ; a continuous quasiconcave utility function u i : X i R;

p. 8/16 The Model We work in a general equilibrium setting: There arelgoods indexed by h. There aremagents indexed by i. For each i there are: a compact convex consumption set X i R l ; a continuous quasiconcave utility function u i : X i R; an endowmentω i R l.

p. 8/16 The Model We work in a general equilibrium setting: There arelgoods indexed by h. There aremagents indexed by i. For each i there are: a compact convex consumption set X i R l ; a continuous quasiconcave utility function u i : X i R; an endowmentω i R l. There are compact production sets Y 1,...,Y n R l that contain the origin.

p. 8/16 The Model We work in a general equilibrium setting: There arelgoods indexed by h. There aremagents indexed by i. For each i there are: a compact convex consumption set X i R l ; a continuous quasiconcave utility function u i : X i R; an endowmentω i R l. There are compact production sets Y 1,...,Y n R l that contain the origin. There is an m n matrixθ of nonnegative ownership shares such that i θ ij = 1 for all j.

Ifu i (x i ) = max x i X i u i (x i ), then agent i is sated at x i X i and x i is a bliss point fori. Otherwise i is unsated at x i. p. 9/16

p. 9/16 Ifu i (x i ) = max x i X i u i (x i ), then agent i is sated at x i X i and x i is a bliss point fori. Otherwise i is unsated at x i. Let X = i X i and Y = j Y j.

p. 9/16 Ifu i (x i ) = max x i X i u i (x i ), then agent i is sated at x i X i and x i is a bliss point fori. Otherwise i is unsated at x i. Let X = i X i and Y = j Y j. For each j and p R l let π j (p) = max y j Y j p,y j,m j (p) = argmax yj Y j p,y j.

p. 9/16 Ifu i (x i ) = max x i X i u i (x i ), then agent i is sated at x i X i and x i is a bliss point fori. Otherwise i is unsated at x i. Let X = i X i and Y = j Y j. For each j and p R l let π j (p) = max y j Y j p,y j,m j (p) = argmax yj Y j p,y j. For each i and p R l, i s total income is µ i (p) = p,ω i + j θ ij π j (p).

p. 10/16 Efficient Disposal Equilibrium A triple(p,x,y) R l + X Y is an efficient disposal equilibrium (EDE) if:

p. 10/16 Efficient Disposal Equilibrium A triple(p,x,y) R l + X Y is an efficient disposal equilibrium (EDE) if: (a) For each i there is no x i X i such that p,x i p,x i and u i (x i ) > u i(x i ), and there is nox i X i such that p,x i < p,x i and u i (x i ) u i(x i ).

p. 10/16 Efficient Disposal Equilibrium A triple(p,x,y) R l + X Y is an efficient disposal equilibrium (EDE) if: (a) For each i there is no x i X i such that p,x i p,x i and u i (x i ) > u i(x i ), and there is nox i X i such that p,x i < p,x i and u i (x i ) u i(x i ). (b) For each i, if i is unsated at x i, then p,x i µ i (p).

p. 10/16 Efficient Disposal Equilibrium A triple(p,x,y) R l + X Y is an efficient disposal equilibrium (EDE) if: (a) For each i there is no x i X i such that p,x i p,x i and u i (x i ) > u i(x i ), and there is nox i X i such that p,x i < p,x i and u i (x i ) u i(x i ). (b) For each i, if i is unsated at x i, then p,x i µ i (p). (c) For each j, y j M j (p).

p. 10/16 Efficient Disposal Equilibrium A triple(p,x,y) R l + X Y is an efficient disposal equilibrium (EDE) if: (a) For each i there is no x i X i such that p,x i p,x i and u i (x i ) > u i(x i ), and there is nox i X i such that p,x i < p,x i and u i (x i ) u i(x i ). (b) For each i, if i is unsated at x i, then p,x i µ i (p). (c) For each j, y j M j (p). (d) i x i ω + j y j.

p. 10/16 Efficient Disposal Equilibrium A triple(p,x,y) R l + X Y is an efficient disposal equilibrium (EDE) if: (a) For each i there is no x i X i such that p,x i p,x i and u i (x i ) > u i(x i ), and there is nox i X i such that p,x i < p,x i and u i (x i ) u i(x i ). (b) For each i, if i is unsated at x i, then p,x i µ i (p). (c) For each j, y j M j (p). (d) i x i ω + j y j. (e) For all h, if i x ih < ω h + j y jh, then p h = 0.

p. 11/16 The Main Result Let e = (1,...,1) R l,v 0 = {x R l : e,x = 0}.

p. 11/16 The Main Result Let e = (1,...,1) R l,v 0 = {x R l : e,x = 0}. Theorem: If, for each i there is an x 0 i X i such that x 0 i ω i, X i x 0 i +V 0, and x 0 i is in the interior (relative tox i +V 0 ) ofx i, then for anyα R m ++ there is an EDE (p,x,y) such that α i ( ) p,x i µ i (p) = i U α µ i (p) p,x i i i S for all i U, whereu is the set of i that are unsated at x i and S = {1,...,m}\U is the set ofithat are sated at x i.

The Main Result Let e = (1,...,1) R l,v 0 = {x R l : e,x = 0}. Theorem: If, for each i there is an x 0 i X i such that x 0 i ω i, X i x 0 i +V 0, and x 0 i is in the interior (relative tox i +V 0 ) ofx i, then for anyα R m ++ there is an EDE (p,x,y) such that α i ( ) p,x i µ i (p) = i U α µ i (p) p,x i i i S for all i U, whereu is the set of i that are unsated at x i and S = {1,...,m}\U is the set ofithat are sated at x i. This generalizes all prior existence results. p. 11/16

p. 12/16 Approaches to the Proof The set up:

p. 12/16 Approaches to the Proof The set up: In HZ the consumption set is the simplex over the objects, so it is contained in a translate ofv 0.

p. 12/16 Approaches to the Proof The set up: In HZ the consumption set is the simplex over the objects, so it is contained in a translate ofv 0. In Mas-Colell s setting we introduce an artificial worthless good to make consumption sets parallel tov 0 and production sets contained in V 0.

p. 12/16 Approaches to the Proof The set up: In HZ the consumption set is the simplex over the objects, so it is contained in a translate ofv 0. In Mas-Colell s setting we introduce an artificial worthless good to make consumption sets parallel tov 0 and production sets contained in V 0. Prices and excess demand:

p. 12/16 Approaches to the Proof The set up: In HZ the consumption set is the simplex over the objects, so it is contained in a translate ofv 0. In Mas-Colell s setting we introduce an artificial worthless good to make consumption sets parallel tov 0 and production sets contained in V 0. Prices and excess demand: The natural space of prices isv 0, which has two problems:

p. 12/16 Approaches to the Proof The set up: In HZ the consumption set is the simplex over the objects, so it is contained in a translate ofv 0. In Mas-Colell s setting we introduce an artificial worthless good to make consumption sets parallel tov 0 and production sets contained in V 0. Prices and excess demand: The natural space of prices isv 0, which has two problems: Budget sets are not lower semicontinuous at 0.

Approaches to the Proof The set up: In HZ the consumption set is the simplex over the objects, so it is contained in a translate ofv 0. In Mas-Colell s setting we introduce an artificial worthless good to make consumption sets parallel tov 0 and production sets contained in V 0. Prices and excess demand: The natural space of prices isv 0, which has two problems: Budget sets are not lower semicontinuous at 0. Aggregate demand may be less valuable than aggregate supply because of satiation. p. 12/16

p. 13/16 Our Methods There are two main innovations:

p. 13/16 Our Methods There are two main innovations: Each consumer trades in the hyperplane parallel tov 0 that contains her endowment, then free disposes to a point in her consumption set.

p. 13/16 Our Methods There are two main innovations: Each consumer trades in the hyperplane parallel tov 0 that contains her endowment, then free disposes to a point in her consumption set. We introduce a small endowment of an artificial good0that is always desired.

p. 13/16 Our Methods There are two main innovations: Each consumer trades in the hyperplane parallel tov 0 that contains her endowment, then free disposes to a point in her consumption set. We introduce a small endowment of an artificial good0that is always desired. Let V = R V 0.

p. 13/16 Our Methods There are two main innovations: Each consumer trades in the hyperplane parallel tov 0 that contains her endowment, then free disposes to a point in her consumption set. We introduce a small endowment of an artificial good0that is always desired. Let V = R V 0. For each i let X i = [ 1,τ i ] X i for some sufficiently largeτ i.

p. 13/16 Our Methods There are two main innovations: Each consumer trades in the hyperplane parallel tov 0 that contains her endowment, then free disposes to a point in her consumption set. We introduce a small endowment of an artificial good0that is always desired. Let V = R V 0. For each i let X i = [ 1,τ i ] X i for some sufficiently largeτ i. Let ũ i ( x i0,x i ) = x i0 +u i (x i ).

p. 13/16 Our Methods There are two main innovations: Each consumer trades in the hyperplane parallel tov 0 that contains her endowment, then free disposes to a point in her consumption set. We introduce a small endowment of an artificial good0that is always desired. Let V = R V 0. For each i let X i = [ 1,τ i ] X i for some sufficiently largeτ i. Let ũ i ( x i0,x i ) = x i0 +u i (x i ). For each j let Ỹj = {0} Y j.

p. 14/16 For a small ε > 0 let S ε = { p = ( p 0,p) V : p = 1 and p 0 ε}.

p. 14/16 For a small ε > 0 let S ε = { p = ( p 0,p) V : p = 1 and p 0 ε}. If excess demand Z( p) is defined naturally (and a certain additional condition holds) then:

p. 14/16 For a small ε > 0 let S ε = { p = ( p 0,p) V : p = 1 and p 0 ε}. If excess demand Z( p) is defined naturally (and a certain additional condition holds) then: Z is upper hemicontinuous.

p. 14/16 For a small ε > 0 let S ε = { p = ( p 0,p) V : p = 1 and p 0 ε}. If excess demand Z( p) is defined naturally (and a certain additional condition holds) then: Z is upper hemicontinuous. p, z = 0 if z Z( p) (all income is spent).

p. 14/16 For a small ε > 0 let S ε = { p = ( p 0,p) V : p = 1 and p 0 ε}. If excess demand Z( p) is defined naturally (and a certain additional condition holds) then: Z is upper hemicontinuous. p, z = 0 if z Z( p) (all income is spent). Of p 0 = ε, then z 0 > 0 for all z Z( p).

p. 14/16 For a small ε > 0 let S ε = { p = ( p 0,p) V : p = 1 and p 0 ε}. If excess demand Z( p) is defined naturally (and a certain additional condition holds) then: Z is upper hemicontinuous. p, z = 0 if z Z( p) (all income is spent). Of p 0 = ε, then z 0 > 0 for all z Z( p). Thus Z is an uhc vector field correspondence that is inward pointing on the boundary of S ε, so the (generalized) Poincaré-Hopf theorem gives a p S ε such that0 Z( p ).

p. 15/16 A Technical Finesse Recall that a polyhedron in R l is an intersection of finitely many closed half spaces, and a polytope is a bounded polyhedron.

p. 15/16 A Technical Finesse Recall that a polyhedron in R l is an intersection of finitely many closed half spaces, and a polytope is a bounded polyhedron. Proposition: IfP 1 and P 2 are polyhedra in R l, Q = {q R l = (P 1 +q) P 2 }, andi : Q R l is the correspondencei(q) = (P 1 +q) P 2, then I is continuous.

p. 15/16 A Technical Finesse Recall that a polyhedron in R l is an intersection of finitely many closed half spaces, and a polytope is a bounded polyhedron. Proposition: IfP 1 and P 2 are polyhedra in R l, Q = {q R l = (P 1 +q) P 2 }, andi : Q R l is the correspondencei(q) = (P 1 +q) P 2, then I is continuous. For each i let X i be the set of bliss points in X i.

p. 15/16 A Technical Finesse Recall that a polyhedron in R l is an intersection of finitely many closed half spaces, and a polytope is a bounded polyhedron. Proposition: IfP 1 and P 2 are polyhedra in R l, Q = {q R l = (P 1 +q) P 2 }, andi : Q R l is the correspondencei(q) = (P 1 +q) P 2, then I is continuous. For each i let X i be the set of bliss points in X i. We take a sequence of expanded economies given by a sequence of endowments of the artificial good that go to zero and a sequence of polyhedrax k i X i such that X k i X i and X k i X i X i.

p. 16/16 Concluding Remarks The paper s final section points to two open problems:

p. 16/16 Concluding Remarks The paper s final section points to two open problems: In the Hylland-Zeckhauser model, is the set of equilibria finite for generic utilities?

p. 16/16 Concluding Remarks The paper s final section points to two open problems: In the Hylland-Zeckhauser model, is the set of equilibria finite for generic utilities? Is the problem of computing an equilibrium of the Hylland Zeckhauser model PPAD-complete?

p. 16/16 Concluding Remarks The paper s final section points to two open problems: In the Hylland-Zeckhauser model, is the set of equilibria finite for generic utilities? Is the problem of computing an equilibrium of the Hylland Zeckhauser model PPAD-complete? The traditional concerns of general equilibrium theory are (mostly) meaningful and conceptually pertinent in relation to pseudomarkets, so one can easily produce a host of original and meaningful problems for further research.