A family of ordinal solutionsA family of ordinal solutions to bargaining problemsto bargaining problems

with many players

Z. Safra, D. Samet

A family of ordinal solutionsA family of ordinal solutions to bargaining problemsto bargaining problems

with many players

Z. Safra, D. Samet

www.tau.ac.il/~sametwww.tau.ac.il/~samet

Shapley’s ordinal solution

Shapley proposed a solution to three-player bargaining problems which is

• Ordinal

• Efficient

• Symmetric

• Individually rational

Shapley proposed a solution to three-player bargaining problems which is

• Ordinal

• Efficient

• Symmetric

• Individually rational

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 1 is Shapley’s solution), with these properties, for any number of players 3.

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 1 is Shapley’s 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 player’s utility.

By the way, you can click underlined words and see the reference.

.

Shapley’s ordinal solution

3

2

1

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 π3(x) defined on the plane where 3’s utility is fixed at a3 .

.

x

.π3(x)

Shapley’s ordinal solution

3

2

1

a

.

.

.

.

Consider π3 equi-valued lines on S …

S

Shapley’s ordinal solution

3

2

1

a

… and their projections on the plane x3 = a3S

Consider π3 equi-valued lines on S …

Shapley’s ordinal solution

3

2

1

a

1

2

a

..

..

These projections form a family of Pareto surfaces for 1 and 2 parameterized by the value of π3 , that is, player’s 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 p3

parameterized by the utility of player 3

These ideal points form a monotonic path p3

parameterized by the utility of player 3

p3

S

3

Shapley’s ordinal solution

3

2

1

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 .

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 .

Shapley’s ordinal solution

3

2

1

a

1

2

a

..

..x3

The point 3 is attained for utility level x3 of player 3.

Increasing 3’s utility to x3 results in a point .

3

S

3

1

2

..

It is easy to see that the projections of on the plains x1= a1 and x2= a2 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...

The solution (a,S) (a,S) is ordinal and symmetric, but alas, it does not lie on S. This is fixed now...

a0 = a

Starting with the problem (a, S) we generate the sequence

a1 = (a0, S)a2 = (a1, S)

a3 = (a2, S)a4 = (a3, S)

a5 = (a4, S)

a6 = (a5, S)

a7 = (a6, S)

a8 = (a7, S)...

The sequence (ak) 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 (ak) converges to a point x on S.

The solution defined by

1(a,S) = x

is an ordinal, efficient, symmetric and individually rational solution.

Shapley’s ordinal solution

Other constructions of 3

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 path p3.

The point 3 is the intersection of p3 with S.

We now generalize the idea of this construction.

1

2

a

..

..

p3

3

Other constructions of 3

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 path p3

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 1

2

a

..

3

p3

p3,1

p3,2

...

The question is how to construct the guidelines

ordinally.

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 O’Neill et al. (2000)

π3 = x3

Consider the Pareto surface at x.

Ordinal guidelines

1

2

a

.x

More formally,dx2

dx1

dx2

dx1

The ratio between the marginal changes in 2 and 1’s utility at x, as a result of changing x3

=

The rate of exchange of 2 and 1’s utility at x along the surface where π3 is fixed at x3

π3

x1

π3

x2

This can be described by a pair of equations,

dx2 =π3

x2

dx3

dx1 =

[ ]π3

x1

dx3[ ]

-1

-11/2

1/2

These factors guarantee, that at

x3 the path reaches the right

surface.

π3 = x3

Ordinal guidelinesWe can generate infinitely many ordinal guidelines by changing the relative weight of the equations.

We fix [0,1] and choose the guideline p3,2

to be the solution of

(1- )

dx2 =π3

x2

dx3

dx1 =

[ ]π3

x1

dx3[ ]

-1

-11/2

1/2

1

2

a

..

3

p3

...

p3,2

p3,1

Similarly the guideline p3,1

is the solution of

Note that when =1, the guidelines coincide with the axes and we get Shapley’s solution.

Ordinal guidelines

1

2

a

..

3

p3

...

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.

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.

p3,2

p3,1

Constructing For Shapley’s solution, the points 1 , 2, and 3 are the projections of a single point which we denoted by .

.

3

2

1

a

.

.

3

S

1

2

..

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.

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 surface S. (Indeed, for Shapley’s 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.