The Mystique of Values; Can There Be
Independence of Irrelevant Alternatives?
In the Nash (or Zeuthen) theory of 2-party bargaining it naturally derives that
some portions of the Pareto boundary of the total set of possible alternative bargaining
compromises or arrangements can become "irrelevant" in relation to the determination
of the axiomatically preferred bargaining solution.
This phenomenon of the occasional irrelevancy of some alternative arrangements
(or "imputations") on the Pareto boundary carries over into the Nash theory of
"2-Person Cooperative Games" or to the "NTU" Shapley value theory for games of any
number of players.
And now my work attempting to study the games of three players that are of
a cooperative bargaining type by means of modeling these games through special types
of infinitely repeated non-cooperative games has led me to study the resulting payoff
(or value) predictions in comparison with the Shapley value and the nucleolus. And
that study naturally leads to the question about the "irrelevance" phenomenon since
the nucleolus exhibits a version of this phenomenon. The nucleolus can effectively
ignore the worth or value of coalitions that are relatively weak while appreciating
the relevance of the same coalitions when they become stronger.
This behavior of the nucleolus can be studied by studying its dependence on the data
of the characteristic function describing a cooperative game.
Neither the Shapley value nor the results of my modeling attempt give rise to such
a phenomenon of complete independence of the axiomatically calculated value or the
observed equilibrium payoffs on the strengths of relatively weaker coalition aggregates.
It is probably most illuminating to illustrate the behavior described
(of the nucleolus) through some examples.
In a simple 3-person game of cooperative bargaining type (where we assume that
a characteristic function validly describes the essential structure of the game) suppose
that all players together can realize a "payout" of +1 and that the only favored coalition
of fewer players is that of P1 and P2 with v(1,2) = b3. Then so long as b3 <= 1/3
the result is that the nucleolus imputation is {1/3,1/3,1/3}, as would seem clearly
appropriate when b3 = 0.
And then if b3 increases above the value of 1/3 and moves up to its maximal possible
value of 1 then the nucleolus vector (or imputation) varies linearly in dependence on b3
and goes to the limiting value of {1/2,1/2,0}. And here the nucleolus and the Shapley value,
as vectors, coincide again, as for b3 = 0.
I thought of checking out the theme of "irrelevance" on a four person game example
where there would be coalition strengths so structured as to seem to favor a specific
pattern of "alliances" among the players. Suppose that with 4 players a game is defined
as determined by a characteristic function of payouts and that v(1,2,3,4) = 1,
v(1,3) = v(2,4) = 1/2, and all other coalition values are the minimal non-negative
quantities such that the total characteristic function is properly super-additive. Then
this seems to favor the natural alliance of P1 and P3 and dually that also of P2 and P4.
But suppose we perturb this picture by allowing the coalition of P1 and P2 to have
a modest value, say v(1,2) = k. It is possible to calculate the result of the nucleolus
vector as k varies. (For this I have used some programming done by Sven Klauke
at Bielefeld.)
And as long as k <= 1/4 it happens that the nucleolus is the same as if k = 0
(where the only coalitions favored at all are those of P1 and P3 and of P2 and P4).
When k rises above the level of 1/4 and goes to the level of 1/2 then
the nucleolus vector varies linearly as a function of k and reaches the final value
of {1/3,1/3,1/6,1/6} with k = 1/2.
So here again the determination of the nucleolus, viewed as an "arbitration scheme",
seems to act as if the coalition of P1 and P2 were an "irrelevant alternative" as long
as k <= 1/4. But then that coalition becomes effectively quite "relevant" with k = 1/2
so that then its members get the assignment of 2/3 of the total payout available if
the nucleolus is viewed as the operative arbitration scheme.
A vintage example where the question of the possible irrelevance of some weaker
coalitions arises naturally can be found in a paper of 1980 by Alvin Roth. He was
concerned with issues relating to the NTU value of Shapley and his published remarks
led to a sort of debate via publications with Aumann.
And what I notice about the example, which was a game of three players, is that
the issue arises of whether or not the two weaker coalitions are so weak that their
conceivable advantages should be considered irrelevant in relation to an evaluation
of the game.