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 of 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.