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

1. A family of ordinal solutionsto bargaining problemswith many playersZ. Safra, D. Samet www.tau.ac.il/~samet

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

3. Shapley’s ordinal solution 3 2 a 1 Consider a three player bargaining problem (a,S ) with a disagreement point a . . . . S ... and a bargaining set with Pareto surface S. . π3(x) . We consider S as the graph of a function π3(x) defined on the plane where 3’s utility is fixed ata3 . . x

4. Shapley’s ordinal solution Consider π3 equi-valued lines on S … 3 S . . . 2 . a 1

5. Shapley’s ordinal solution Consider π3 equi-valued lines on S … 3 S … and their projections on the planex3= a3 2 a 1

6. Shapley’s ordinal solution 2 a 1 For each surface in the family choose the ideal point. These projections form a family of Pareto surfaces for 1and2parameterized by the value ofπ3 , that is, player’s 3 utility. Theseideal points form a monotonic path p3parameterized by the utility of player3 3 S . p3 2 . 3 . . a The path crosses the surface Sat exactly one point 3 1

7. Shapley’s ordinal solution 2 a 1 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 . 3 S 2 . . 3 a 1

8. Shapley’s ordinal solution 2 . x3 a 1 The point 3 is attained for utility level x3of player 3. Increasing3’sutility tox3 results in a point. The solution (a,S)  (a,S) is ordinal and symmetric, but alas, it does not lie on S. This is fixed now... It is easy to see that the projections of  on the plains x1= a1 and x2= a2 are1 and 2. 3 S . 1 .  2 . . 3 2 . 3 a 1

9. a0 = a Shapley’s ordinal solution Starting with the problem(a,S) wegenerate the sequence The sequence (ak) converges to a point xonS. The solution defined by 1(a,S) = x is an ordinal, efficient, symmetric and individually rational solution. a1 =(a0,S) a2 =(a1,S) a3 =(a2,S) a4 =(a3,S) a5 =(a4,S) a6 =(a5,S) a7 =(a6,S) a8 =(a7,S) . . .

10. Other constructions of3 2 a 1 To construct Shapley’s solution we generated a family of Pareto surfaces for all players but 3 … … construct the ideal points of each surface … … which form a pathp3. The point3 is the intersection ofp3withS. We now generalize the idea of this construction. . p3 . 3 . .

11. Other constructions of3 2 a 1 Starting with the same family of surfaces, we introduce two paths p3,1 and p3,2 which we call guidelines. We construct the “ideal points” of each surface with respect to the guidelines… … this form a pathp3 The point 3 is the intersection of p3 with S. The guidelines play the role of the axes in the construction of Shapley’s solution p3,2 The question is how to construct the guidelines ordinally. p3 . 3 . . . p3,1 .

12. Ordinal guidelines 2 a 1 We describe how to construct ordinal guidelines for a family of Pareto surfaces. Suppose the guideline has been defined up to pointx. It is enough to show the direction of the path at x. Consider the Pareto surface at x. The rate of utility exchange between1and2atxon this surfaceis the negative of the slope of the tangent atx. . x The slope of the direction of the path at x is this rate. The ordinality of this construction is shown inO’Neill et al. (2000) π3 = x3

13. Ordinal guidelines 2 π3 π3 π3 π3 x1 x1 x2 x2 a 1 More formally, dx2 The ratio between the marginal changes in 2 and 1’s utility at x, as a result of changing x3 = dx1 These factors guarantee, that at x3 the path reaches the right surface. The rate of exchange of 2 and 1’s utility at x along the surface where π3 is fixed at x3 . This can be described by a pair of equations, x dx2 dx1 ] [ -1 dx3 dx2 1/2 = π3 = x3 ] [ -1 dx3 dx1 1/2 =

14. Ordinal guidelines 2 p3 . 3 . . π3 π3 . x2 x1 . a 1 We can generate infinitely many ordinal guidelines by changing the relative weight of the equations. Note that when =1, the guidelines coincide with the axes and we get Shapley’s solution. p3,2 We fix  [0,1]and choose the guideline p3,2 to be the solution of Similarly the guideline p3,1 is the solution of ] [ -1 p3,1 dx3  dx2 1/2 = ] [ -1 dx3 (1-) dx1 1/2 =

15. Ordinal guidelines 2 p3 . 3 . . . . a 1 The pair {p3,1, p3,2} is symmetric with the respect to 1 and 2, and therefore 3 is also symmetric with respect to 1 and 2. p3,2 In a similar manner we can construct also1 and2 such that for eachi, i is symmetric with respect to the other two players. p3,1

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

17. The solution  The constructing ofis done in the same iterative way as in Shapley’s solution. That is,  (a,S) is the limit of ak =(ak-1,S). Some of the points ak may lie above the surfaceS. (Indeed, for Shapley’s solution they oscilate). • For points aboveS the construction carried out before is slightly different. • Instead ofideal points with respect to the guidelines, which are defined by the maximal payoff to the players, we take horrorpoints which are similarly defined by the minimal payoffs. • The coordinates of the pointare the maximal on each axis rather then minimal.