Vertical Syndication-Proof Competitive Prices in Multilateral Markets

Abstract

[eng] A multi-sided Böhm-Bawerk assignment game (Tejada, to appear) is a model for a multilateral market with a finite number of perfectly complementary indivisible commodities owned by different sellers, and inflexible demand and support functions. We show that for each such market game there is a unique vector of competitive prices for the commodities that is vertical syndication-proof, in the sense that, at those prices, syndication of sellers each owning a different commodity is neither beneficial nor detrimental for the buyers. Since, moreover, the benefits obtained by the agents at those prices correspond to the nucleolus of the market game, we provide a syndication-based foundation for the nucleolus as an appropriate solution concept for market games. For different solution concepts a syndicate can be disadvantageous and there is no escape to Aumman’s paradox (Aumann, 1973). We further show that vertical syndicationproofness and horizontal syndication-proofness – in which sellers of the same commodity collude – are incompatible requirements under some mild assumptions. Our results build on a self-interesting link between multi-sided Böhm-Bawerk assignment games and bankruptcy games (O’Neill, 1982). We identify a particular subset of Böhm-Bawerk assignment games and we show that it is isomorphic to the whole class of bankruptcy games. This isomorphism enables us to show the uniqueness of the vector of vertical syndication-proof prices for the whole class of Böhm-Bawerk assignment market using well-known results of bankruptcy problems.

[cat] Un joc d’assignació multi-sided Böhm-Bawerk (Tejada, per aparèixer) és un model per a un mercat multilateral en el que hi ha un nombre finit de béns indivisibles i perfectament complementaris (cadascun en mans de diferents venedors) i en el que les funcions d’oferta i demanda són inflexibles. En aquest treball demostrem que per a cada mercat d’aquest tipus existeix un únic vector de preus competitius (per als diferents béns) que és vertical syndication-proof, en el sentit que, a aquests preus, el fet que venedors de diferents béns s’agrupin en sindicats no afecta els compradors. Curiosament, els beneficis distribuits a tots els agents a aquests preus corresponen al nucleolus del joc, de manera que, com a subproducte del nostre resultat, obtenim una justificació per a l’ús del nucleolus. Per a d’altres solucions un sindicat pot ser perjudicial i, per tant, no hi ha sortida a la paradoxa trobada per Aumann (1973). Addicionalment, provem que vertical syndication-proof i horizontal syndication-proof són, sota unes certes condicions, restriccions incompatibles entre sí. Les proves dels nostres resultats es fonamenten en un relació, interessant per sí mateixa, entre els multi-sided Böhm-Bawerk assignment games i els bankruptcy games (O’Neill, 1982). En efecte, identifiquem un subconjunt d’aquells i provem que són isomorfs a la clase sencera d’aquests. Aquest isomorfisme ens permet provar la unicitat anteriorment citada.