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A family of ordinal solutions to bargaining problems A family of ordinal solutions to bargaining problems with many players Z. Safra, D. Samet

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Shapleys ordinal solution ShapleyShapley proposed a solution to three-player bargaining problems which is Ordinal Efficient Symmetric Individually rational ShapleyShapley proposed a solution to three-player bargaining problems which is Ordinal Efficient Symmetric Individually rational In Safra, Samet (2004) we extended Shapleys solution to any number of players 3.Safra, Samet (2004) Here, we construct Shapleys solution in a way that lends itself to the construction of a continuum of solutions for [0,1], (where 1 is Shapleys solution), with these properties, for any number of players 3. In Safra, Samet (2004) we extended Shapleys solution to any number of players 3.Safra, Samet (2004) Here, we construct Shapleys solution in a way that lends itself to the construction of a continuum of solutions for [0,1], (where 1 is Shapleys solution), with these properties, for any number of players 3. A solution is ordinal if it is covariant with respect to monotone transformations of each players utility. By the way, you can click underlined words and see the reference.

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. Shapleys ordinal solution a Consider a three player bargaining problem (a,S ) with a disagreement point a and a bargaining set with Pareto surface S. S We consider S as the graph of a function π 3 (x) defined on the plane where 3s utility is fixed at a 3.. x. π 3 (x)

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Shapleys ordinal solution a.... Consider π 3 equi-valued lines on S … S

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Shapleys ordinal solution a … and their projections on the plane x 3 = a 3 S Consider π 3 equi-valued lines on S …

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Shapleys ordinal solution a 1 2 a.... These projections form a family of Pareto surfaces for 1 and 2 parameterized by the value of π 3, that is, players 3 utility. For each surface in the family choose the ideal point. The path crosses the surface S at exactly one point 3 These ideal points form a monotonic path p 3 parameterized by the utility of player 3 p3p3 S 3

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Shapleys ordinal solution a 1 2 a. S 3. The solution (a,S) 3 (a,S) is ordinal, and symmetric with respect to 2 and 3. Using 3 we define now an ordinal symmetric solution.

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Shapleys ordinal solution a 1 2 a.... x3x3 The point 3 is attained for utility level x 3 of player 3. Increasing 3s utility to x 3 results in a point. 3 S It is easy to see that the projections of on the plains x 1 = a 1 and x 2 = a 2 are 1 and 2. The solution (a,S) (a,S) is ordinal and symmetric, but alas, it does not lie on S. This is fixed now...

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a 0 = a Starting with the problem (a, S) we generate the sequence a 1 = (a 0, S) a 2 = (a 1, S) a 3 = (a 2, S) a 4 = (a 3, S) a 5 = (a 4, S) a 6 = (a 5, S) a 7 = (a 6, S) a 8 = (a 7, S)... The sequence (a k ) converges to a point x on S. The solution defined by 1 (a,S) = x is an ordinal, efficient, symmetric and individually rational solution. The sequence (a k ) converges to a point x on S. The solution defined by 1 (a,S) = x is an ordinal, efficient, symmetric and individually rational solution. Shapleys ordinal solution

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Other constructions of 3 To construct Shapleys solution we generated a family of Pareto surfaces for all players but 3 … … construct the ideal points of each surface … … which form a path p 3. The point 3 is the intersection of p 3 with S. We now generalize the idea of this construction. 1 2 a.... p3p3 3

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Other constructions of 3 Starting with the same family of surfaces, we introduce two paths p 3,1 and p 3,2 which we call guidelines. We construct the ideal points of each surface with respect to the guidelines… … this form a path p 3 The point 3 is the intersection of p 3 with S. The guidelines play the role of the axes in the construction of Shapleys solution 1 2 a.. 3 p3p3 p 3,1 p 3,2... The question is how to construct the guidelines ordinally.

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Ordinal guidelines 1 2 a. We describe how to construct ordinal guidelines for a family of Pareto surfaces. Suppose the guideline has been defined up to point x. It is enough to show the direction of the path at x. x The rate of utility exchange between 1 and 2 at x on this surface is the negative of the slope of the tangent at x. The slope of the direction of the path at x is this rate. The ordinality of this construction is shown in ONeill et al. (2000) ONeill et al. (2000) π 3 = x 3 Consider the Pareto surface at x.

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Ordinal guidelines 1 2 a. x More formally, dx2dx2 dx1dx1 dx 2 dx 1 The ratio between the marginal changes in 2 and 1s utility at x, as a result of changing x 3 = The rate of exchange of 2 and 1s utility at x along the surface where π 3 is fixed at x 3 π 3 x 1 π 3 x 2 This can be described by a pair of equations, dx2dx2 = π 3 x 2 dx3dx3 dx1dx1 = [ ] π 3 x 1 dx3dx3 [ ] 1/2 These factors guarantee, that at x 3 the path reaches the right surface. π 3 = x 3

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Ordinal guidelines We can generate infinitely many ordinal guidelines by changing the relative weight of the equations. We fix [0,1] and choose the guideline p 3,2 to be the solution of (1- ) dx2dx2 = π 3 x 2 dx3dx3 dx1dx1 = [ ] π 3 x 1 dx3dx3 [ ] 1/2 1 2 a.. 3 p3p3... p 3,2 p 3,1 Similarly the guideline p 3,1 is the solution of Note that when =1, the guidelines coincide with the axes and we get Shapleys solution.

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Ordinal guidelines 1 2 a.. 3 p3p3... The pair {p 3,1, p 3,2 } is symmetric with the respect to 1 and 2, and therefore 3 is also symmetric with respect to 1 and 2. In a similar manner we can construct also 1 and 2 such that for each i, i is symmetric with respect to the other two players. p 3,2 p 3,1

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Constructing For Shapleys solution, the points 1, 2, and 3 are the projections of a single point which we denoted by a.. 3 S This does not hold for the points we have constructed using the guidelines. We choose the minimal coordinate on each axis... We define to be the point with these coordinates.

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The solution The constructing of is done in the same iterative way as in Shapleys solution. That is, (a, S) is the limit of a k = (a k-1, S). Some of the points a k may lie above the surface S. (Indeed, for Shapleys solution they oscilate). For points above S the construction carried out before is slightly different. Instead of ideal points with respect to the guidelines, which are defined by the maximal payoff to the players, we take horror points which are similarly defined by the minimal payoffs. The coordinates of the point are the maximal on each axis rather then minimal.

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Ordinal solution Z. Safra, D. Samet (2004) An Ordinal Solution to Bargaining Problems with Many Players, Games and Economic Behavior, 46, Presents an ordinal solution for any number of players greater than or equal three. This solution extends Shpaleys solution for three players. Z. Safra, D. Samet (2004) An Ordinal Solution to Bargaining Problems with Many Players, Games and Economic Behavior, 46, Presents an ordinal solution for any number of players greater than or equal three. This solution extends Shpaleys solution for three players. back

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Ordinal path-valued solutions ONeill B, Samet, Z. Wiener and E. Winter, (2000) Bargaining with an Agenda, forthcoming GEB. Axiomatizes a path-valued solution for gradual bargaining. This solution is ordinal. ONeill B, Samet, Z. Wiener and E. Winter, (2000) Bargaining with an Agenda, forthcoming GEB. Axiomatizes a path-valued solution for gradual bargaining. This solution is ordinal. back

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Shapleys solution Shubik, M., (1982), Game Theory in the Social Sciences: Concepts and Solutions, MIT, Press, Cambridge. Thomson, W., (1994), Cooperative models of bargaining, In Handbook of Game Theory 2, Aumann R. J., and Hart S.Eds, North Holland, p Shubik, M., (1982), Game Theory in the Social Sciences: Concepts and Solutions, MIT, Press, Cambridge. Thomson, W., (1994), Cooperative models of bargaining, In Handbook of Game Theory 2, Aumann R. J., and Hart S.Eds, North Holland, p back

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