The assignment game: core, competitive equilibria and multiple - PowerPoint PPT Presentation

The assignment game: core, competitive equilibria and multiple partnership Marina N´ u˜ nez University of Barcelona Summer School on Matching Problems, Markets and Mechanisms; Budapest, June 2013

Outline 1 Coalitional games 2 The assignment game The core Lattice structure Competitive equilibria Some properties of the core Markets with the same core 3 Multiple-partners assignment market 1 Pairwise-stability Optimal pairwise-stable outcomes Competitive equilibria The core 4 Multiple-partners assignment market 2 Dual solutions and the core Differences with the assignment game

Outline 1 Coalitional games 2 The assignment game The core Lattice structure Competitive equilibria Some properties of the core Markets with the same core 3 Multiple-partners assignment market 1 Pairwise-stability Optimal pairwise-stable outcomes Competitive equilibria The core 4 Multiple-partners assignment market 2 Dual solutions and the core Differences with the assignment game

Outline 1 Coalitional games 2 The assignment game The core Lattice structure Competitive equilibria Some properties of the core Markets with the same core 3 Multiple-partners assignment market 1 Pairwise-stability Optimal pairwise-stable outcomes Competitive equilibria The core 4 Multiple-partners assignment market 2 Dual solutions and the core Differences with the assignment game

Outline 1 Coalitional games 2 The assignment game The core Lattice structure Competitive equilibria Some properties of the core Markets with the same core 3 Multiple-partners assignment market 1 Pairwise-stability Optimal pairwise-stable outcomes Competitive equilibria The core 4 Multiple-partners assignment market 2 Dual solutions and the core Differences with the assignment game

Coalitional TU games A coalitional game with transferable utility is ( N , v ), where N = { 1 , 2 , . . . , n } is the set of players and v : 2 N − → R v ( S ) is the characteristic function. S �→ An imputation is a payoff vector x = ( x 1 , x 2 , . . . , x n ) ∈ R N that is Efficient: � i ∈ N x i = v ( N ) Individually rational: x i ≥ v ( i ) for all i ∈ N . Let I ( v ) be the set of imputations of ( N , v ) and I ∗ ( v ) be the set of preimputations (efficient payoff vectors).

The core Let it be ( N , v ) and x , y ∈ I ∗ ( v ): y dominates x via coalition S � = ∅ ( y dom v S x ) ⇔ x i < y i for all i ∈ S and � i ∈ S y i ≤ v ( S ). y dominates x ( y dom v x ) if y dom v S x for some S ⊆ N . Definition (Gillies, 1959) The core C ( v ) of ( N , v ) is the set of preimputations undominated by another preimputation. If C ( v ) � = ∅ , then it coincides with the set of imputations undominated by another imputation. Equivalently, C ( v ) = { x ∈ I ( v ) | � i ∈ S x i ≥ v ( S ) , for all S ⊆ N } .

The assignment game (Shapley and Shubik, 1972) The assignment game is a cooperative model for a two-sided market (Shapley and Shubik, 1972). A good is traded in indivisible units. Side payments are allowed and utility is identified with money. Each buyer in M = { 1 , 2 , . . . , m } demands one unit and each seller in M ′ = { 1 , 2 , . . . , m ′ } supplies one unit. Each seller j ∈ M ′ has a reservation value c j ≥ 0 for his object. Each buyer i ∈ M valuates differently, h ij ≥ 0, the object of each seller j . Buyer i and seller j , whenever they trade, make a join profit of ( h ij − p ) + ( p − c j ). Hence, a ij = max { 0 , h ij − c j } . All these data is summarized in the assignment matrix A :   a 11 a 12 a 1 m ′ . . . a 21 a 22 a 2 m ′ . . .     · · · · · · · · · · · ·   · · · a m 1 a m 2 a mm ′

The assignment game Cooperation means we look at this market as a centralized market where a matching of buyers to sellers and a distribution of the profit of this matching is proposed: ( u , v ) ∈ R M × R M ′ . � A matching µ is a subset of M × M ′ where each agent appears in at most one pair. Let M ( M , M ′ ) be the set of matchings. � A matching µ is optimal iff, for any other µ ′ ∈ M ( M , M ′ ), � � a ij ≥ a ij . ( i , j ) ∈ µ ( i , j ) ∈ µ ′ Let M ∗ A ( M , M ′ ) be the set of optimal matchings. The cooperative assignment game is defined by ( M ∪ M ′ , w A ), the characteristic function w A being (for all S ⊆ M and T ⊆ M ′ ) � w A ( S ∪ T ) = max { a ij | µ ∈ M ( S , T ) } . ( i , j ) ∈ µ

The core The core:  � � i ∈ M u i + �  j ∈ M ′ v j = w A ( M ∪ M ′ ) �  ( u , v ) ∈ R M × R M ′   � C ( w A ) = u i + v j ≥ a ij for all ( i , j ) ∈ M × M ′ , � � u i ≥ 0 , ∀ i ∈ M , v j ≥ 0 , ∀ j ∈ M ′ .  � Given any optimal matching µ , if ( u , v ) ∈ C ( w A ) then u i + v j = a ij for all ( i , j ) ∈ µ and u i = 0 if i is unmatched by µ . Fact In the core of the assignment game, third-party payments are excluded

The core Theorem (Shapley and Shubik, 1972) The core of the assignment game is non-empty and coincides with the set of solutions of the dual program to the linear assignment problem. w A ( M ∪ M ′ ) = max � � j ∈ M ′ a ij x ij min � i ∈ M u i + � j ∈ M ′ v j i ∈ M � where i ∈ M x ij ≤ 1 , ∀ j ∈ M ′ , u i + v j ≥ a ij ∀ ( i , j ) ∈ M × M ′ , � j ∈ M ′ x ij ≤ 1 , ∀ i ∈ M , u i ≥ 0 , v j ≥ 0 . x ij ≥ 0 , ∀ ( i , j ) ∈ M × M ′ .

Example 1 u 1 + v 3 = 4 3 4 u 1 + v 4 ≥ 1 − 2 ≤ u 2 − u 1 ≤ 2 1 4 1 u 2 + v 3 ≥ 2 0 ≤ u 1 ≤ 4 2 u 2 + v 4 = 3 0 ≤ u 2 ≤ 3 2 3 u i ≥ 0 , v j ≥ 0 . u2 (4,3) (0,0) u1 ( u , v ) and ( u , v ), optimal core points for each side. ( u , v ) = (4 , 3; 0 , 0), ( u , v ) = (0 , 0 , 4 , 3).

Lattice structure 1 Fact (Shapley and Shubik, 1972) C ( w A ) with the following partial order(s) is a complete lattice ( u , v ) ≤ M ( u ′ , v ′ ) ⇔ u i ≤ u ′ ∀ i ∈ M . i Let ( M ∪ M ′ , w A ) be an assignment market and ( u , v ), ( u ′ , v ′ ) two elements in C ( w A ). Then, � � ( u , v ) ∨ ( u ′ , v ′ ) = (max { u i , u ′ i } ) i ∈ M , (min { v j , v ′ j } ) j ∈ M ′ ∈ C ( w A ) � � ( u , v ) ∧ ( u ′ , v ′ ) = (min { u i , u ′ i } ) i ∈ M , (max { v j , v ′ j } ) j ∈ M ′ ∈ C ( w A ) . � As a consequence the existence of a buyers-optimal core allocation and a sellers-optimal core allocation is obtained. Fact (Demange, 1982; Leonard, 1983) For all i ∈ M, u i = w A ( M ∪ M ′ ) − w A ( M ∪ M ′ \ { i } ) .

The buyers-optimal core allocation The buyers optimal core allocation ( u , v ) can be obtained by solving m + 1 linear programs. But since all buyers attain their marginal contribution at the same core point, it can easily be obtained by means of only two linear programs: the one that gives an optimal matching µ and max � i ∈ M u i u i + v j ≥ a ij ∀ ( i , j ) ∈ M × M ′ , where u i + v j = a ij , ∀ ( i , j ) ∈ µ, u i ≥ 0 , v j ≥ 0 .

Competitive equilibria � In this section let us interpret M as a set of bidders and M ′ as a set of objects. � A feasible price vector is p ∈ R M ′ such that p j ≥ c j for all j ∈ M ′ . � Add a null object O with a iO = 0 for all i ∈ M and price 0. More than one bidder may be matched to O : Q = M ′ ∪ { O } . � The demand set of a bidder i at prices p is � � D i ( p ) = j ∈ Q | a ij − p j = max k ∈ Q { a ik − p k } . � The price vector p is quasi-competitive if there is a matching µ such that, for all i ∈ M , if µ ( i ) = j then j ∈ D i ( p ). Then µ is compatible with p . � ( p , µ ) is a competitive equilibrium if p is a quasi-competitive price, µ is compatible with p and p j = c j for all j �∈ µ ( M ).

Competitive equilibria Theorem (Gale, 1960) Let ( M , M ′ , A ) be an assignment market. Then, 1 ( p , µ ) competitive equilibrium ⇒ ( u , v ) ∈ C ( w A ) where u i = h ij − p j if µ ( i ) = j v j = p j − c j , j ∈ M ′ \ { O } 2 µ ∈ M ∗ A ( M , Q ) with a i µ ( i ) > 0 ∀ i ∈ M and ( u , v ) ∈ C ( w A ) ⇒ ( p , µ ) is a competitive equilibrium, where p j = v j + c j if j ∈ M ′ and p O = 0 � The buyers-optimal core allocation corresponds to the minimal competitive price vector. � The sellers-optimal core allocation corresponds to the maximal competitive price vector.

Lattice structure 2 Given a (square) assignment market ( M , M ′ , A ), denote by i ′ the i th seller and assume µ = { ( i , i ′ ) | i ∈ M } is optimal. Then, the projection of C ( w A ) to the space of the buyers’ payoffs is � � � a ij − a jj ≤ u i − u j ≤ a ii − a ji ∀ i , j ∈ { 1 , 2 , . . . , m } u ∈ R M � C u ( w A ) = � 0 ≤ u i ≤ a ii for all i ∈ { 1 , 2 , . . . , m } . � � Notice that C u ( w A ) is a 45-degree lattice. Theorem (Quint, 1991; Characterization of the core ) Given any 45-degree lattice L, there exists an assignment game ( M , M ′ , A ) such that C ( w A ) = L. � But matrix A in the above theorem may not be unique.

Example 2 1’ 2’ 3’ 5 2 1 8 � Optimal matching: µ = { (1 , 2 ′ ) , (2 , 3 ′ ) , (3 , 1 ′ ) } . 2 7 9 6 � ( u , v ) = (5 , 6 , 1; 1 , 3 , 0), ( u , v ) = (3 , 5 , 0; 2 , 5 , 1). 3 3 0 2 6 4 u2 (=6-v3) 2 2 1 u3 (=2-v1) 0 8 0 0 2 4 6 u1 (=8-v2)