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 superadditive.
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.