Home Microeconomics • Axiomatic Theory of Bargaining with a Variable Number of by William Thomson

Axiomatic Theory of Bargaining with a Variable Number of by William Thomson

By William Thomson

This publication extrapolates at the Nash (1950) therapy of the bargaining challenge to contemplate the placement the place the variety of bargainers may perhaps range. The authors formulate axioms to specify how options may still reply to such alterations, and supply new characterizations of all of the significant recommendations in addition to the generalizations of those recommendations.

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Similarly, applying MON to each agent / e Q\l to compare what she gets in T to what she gets in S' = TP we obtain Ft{T) g F/(5') = b. 34 3 The Kalai-Smorodinskv solution p 1 < .. 5. The relationships between the members of Q. These inequalities together yield F(T)^beQ. O results. D. Remark: It should be pointed out that the general construction applies to the case | P | = 2 as well but does not coincide with the one given for that case. This first argument is simpler. 6. 5 we indicate schematically with which group each agent is associated so as to face a problem identical to that faced by agent 1 in S.

Xi=y{ and V/e P 1 , xyi(j) =yj}> The problem 5'is a replica of S in which agent / plays the role played in S by agent 1, and the agents in P 2 play the roles played in S by the agents in P 1 . The operation that we just performed for each of the members of P 1 in relation to the set P 2 is then successively carried out for each of the members of P 2 in relation to the set P 3 and for each of the members of P 3 in relation to the set P 1 . More precisely, let y2 be a one-to-one function from P 2 to P 3 , and for each / e P 2 , let S ' C I R ^ " be defined by Also, for each / e P 3 , let S ' C I R ^ ' 1 be defined by Now let x* = aeQ and finally let Tc IR2 be given by 7 = cch[S, S' for ieQ\l,x*}.

After S has been replicated (as before), the family (P* of subsets of Q different from P, all the Pi9 and any subset of these is introduced. For each P e (P*, let xp be the point of IR/5 that has all of its coordinates equal toe + (a- c)( | Q\-\P\)/( | Q\-\P\) if | P | > | P | and a + (l-a){\P\-\P\)/(\P\-l) if | P | < | P | . Note that x<2 = x* of the original construction. Then Tc IRQ is defined by T = cch{5, S! for / e Q\l, xp for Pe (P*j. The tedious verification that T satisfies all the desired properties is omitted.

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