of
Cooperation in Games Through Modeling in Terms
of
Formally Non-Cooperative Action in a Repeated
Game
Context
A
few years ago I gave a talk on the topic of the use
of
the "Prisoner's Dilemma" game, in a context of repetition and evolution
of strategies, by theoretical biologists
who
were interested in studying the natural evolution of cooperative adaptations.
And after giving the talk I thought more about the concept of studying
a game by studying it as a repeated game and through this viewpoint I got
an idea of how to eliminate all of the "verbal" complicationsthat
could become involved in the consideration of coalitions
and
coalition formation.
In
principle, coalitions, and specifically coalitions
as
considered by Von Neumann and Morgenstern in "Theory of Games and Economic
Behavior", are things that could be implemented by contracts, like contracts
in roman law. But of course a contract is quite intrinsically a "verbal"
thing because indeed it could (or should!) be written down in words.
On
the other hand, if in Nature a form of cooperation has evolved, like with
a species of insects providing fertilization for a flowering species of
plants, then the cooperation exists and is maintained, not by the enforcement
of a verbal contract but presumably by the action of "natural selection"
affecting the genetics of both species as time passes.
My
idea was that in a repeated game context that the players could be given
the right to vote for "agencies" or "agents" among themselves. Thus at
a first step a player, say player A, would have the option to accept player
B
as his agent. And the effect of this would be that the coalition (A,B)
would be formed (like a committee with B as chairman) without any verbal
processes occurring between A and B.Furthermore,
this process adapts to successive steps of coalescence since if another
step of elections is held then B, as the "agent" representing the coalition
(A,B), can vote to accept the agency of player C and then C will thus become
the agent representing the coalition (A,B,C).
And
in this manner, with generalized "agencies" being electable, the level
of a "grand coalition" can always be reached (for a game of finitely many
players), and as a consequence of that the requisites for "Pareto efficiency"
will be available.
With
regard to the actual results which have been obtained and which we can
report on now, they have been calculations of equilibria for models and
specific games where there were two or three players. The models relate
to
the representation of the coalescence (or coalition
formation)
possibilities in terms of the concept of agencies and the models also incorporate
some form of "reactivity" by which players can react affirmatively when
well treated by others (acting as agents) and negatively when badly treated
by other players. (And of course there is no bound, a priori, to the complexity
of reaction patterns. But it
is
somewhat plausible that with a "reasonable" level of refinement in the
modeling of reactive behavior that we
can
find results, in terms of calculated equilibria of
behavior,
that will be plausible in relation to appraising the bargaining/negotiation
"values" for the game and
also
compare calculated value vectors for it with other calculable value vectors
such as the Shapley value or the nucleolus. And the interest in these comparisons
is a good reason for studying games given as "CF games" (which we
will
discuss further below) so that the Shapley value, the classic nucleolus
and the "Harsanyi nucleolus" can directly be calculated for these games.
We
have seen that there is much to do further, even on the level of games
of merely three players, to refine the modeling. We have worked with parameters
called "epsilons" which enable some smoothness and good mathematical solutions
for equilibria when they are finite and, as they tend towards zero, limiting
results that have Pareto efficiency that becomes asymptotically perfect
as "the epsilons" pass to zero limits. (What is less clear at this stage
of the research, for three players, is that we can actually evaluate the
"division of the profits" as the payoffs
to
the players.)
In
the continuing text below we will go into specific topics and also into
some details of calculations and of programming developed to achieve the
calculation results.
Agencies
It
was a few years ago that I got what seemed like an inspired idea that offered
a means for studying cooperative (or "coalitional") games in a manner with
parallels to
the
ways by which theoretical biologists have studied the
topic
of "the evolution of cooperation" (Axelrod et al).
The
idea also offered an escape from all of the verbal complexities that might
otherwise naturally attach to coalition possibilities because of the unlimited
nature
of
the conceivable elaboration of verbal contracts.
In
effect the concept allows the game to be transformed into a game that is
in certain senses equivalent and which is to be considered in the repeated
gamecontext
that is directly analogous to the repeated game context studied
by
theoretical biologists studying the possibility of cooperation evolving
in the context of a repeated game of "Prisoners' Dilemma" form.
Instead
of there being unlimited means by which coalitions might actually be formed
or form, dissolve,
and
reform, we have an election procedure through which
any
player may elect to accept any other player as his agent. And in the context
of studying a repeated game we
can
afford to prescribe that this election process is such
that
the agent is entirely uncommitted and his election is irrevocable for each
specific playing of the game. (Of course election choices are expected
to vary as the game
is
repeated.)
A
set of rules can be devised so that there are election stages in each Of
which all of the players remaining independent (not represented by another
player as agent) have, each of them, the option of electing another player
as an accepted agent. It is natural for these rules to require convergence
so that no more than (n-1) stages of election will ever be needed for a
game of n players.
Election
rules need to do something to resolve the impasse of a situation where
A votes to accept B as his agent but B simultaneously also votes similarly
for A.
It
is not exactly clear which rule version handles these situations in the
most preferable fashion, we have worked with more than one variant.
When
we more recently found, in the course of the use of specific model games
for calculations, that it seemed to be desirable to allow elections to
be repeated when an election had failed to result the election of any agency
power, this finding had the effect of suggesting that election rules which
had the effect that at most one agency could be elected at any stage of
the election process would be most convenient.
Concerning
the general concept of transforming a cooperative game into a form where
all cooperation must be realized by means of the election of agencies,
we can remark that this is analogous to thinking of committees as being
such that all effective actions of a committee must take
the
form of an action by the "chairperson" of the committee.
If
one begins with a quite general CF game and then if one introduces a game
formally requiring that all coalition benefits must be attained through
the means of the action of agents who have the authority to represent all
the members of the coalition then the "agencies game" resulting from this
still has the same validly derivable characteristic function as the original
game. In essence the coalitions simply have the same potentialities as
before, but in a formal sense, for these potentialities to be exploited,
the members of a coalition would need TO CONSPIRE on a practical procedure
for electing agents successively and finally taking the effective action
of the coalition as an action of the agent finally elected to represent
all of the members of that coalition.
Remark:
The agencies concept inspired the origins of this research project on which
we are reporting. And earlier we considered various alternative specific
schemes of rules for the election of agents and agencies. Some earlier
texts which describe some of these alternative ideas are available on the
author's "web page". In particular the text of the file "agentt7c.c" gives
a good picture of these ideas with alternative varieties of election rules
being considered.
The
Currently Applied Election Rules for Agencies
It
was initially used just for the purpose of simplifying the format of model
games and simplifying the calculations needed to find the corresponding
equilibria (for the game
of
the realization of "coalescence" (and coalition benefits) through the election
of agencies). But we find that a simplifying format of election rules seems
to be adequate and convenient. If in a stage of elections more than
one
of the players has voted to give his acceptance to
another
player to become his agent (and as if with "power
of
attorney") then a random process, or "chance move", determines only one
of those votes as effective, and one
of
the players has elected one of the other players to be agent for him.
In
our latest modeling for 3-person games we introduced the concept of repeating
an election if no player had voted to give acceptance to anyone as his
agent, but this repeated election opportunity was to be given only with
probability of (1-e4) (one minus epsilonsub4). Then we found that we got
good results mathematically by considering the limiting results as e4 (epsilonsub4)
tended to the limit of zero.
In
general, if there are n players of a game, with this procedure, it would
take n-1 steps of effective election
to
achieve the final election of an agency for the "grand coalition" (which
is the coalition including all of the players of the game).And
in general, for the realization of "Pareto Efficiency", it is needful that
the grand coalition level is attained.
CF
Games
We
want to introduce a concept or terminology applying to games that are described
by a "characteristic function" of the type developed by Von Neumann and
Morgenstern.
There
has been a tendency in the many publications on
game
theory to describe a sample game simply by specifying its characteristic
function. We wish to call such a game, for which an underlying "normal
form game" or "extensive form game" has not been specified, a "CF game".
And an important consideration comes in here: The characteristic function
cannot be considered as fully enough descriptive
of
a game so that an "evaluation concept" (or a concept of determining a "value"
for the game) that depends entirely on the information given by that function
would be well based. (This is a phenomenon that appears initially in the
context of 2-person cooperative games as studied by Nash.)
And
of course both the Shapley value and the nucleolus (imputation or value
vector) are defined in relation to a characteristic function that is presumed
given.
A
"corrected" characteristic function, consistent with Nash's theory for
2-person cases was introduced by Harsanyi around 1959 and it was further
studied by Selten in 1964. And either the Shapley Value or the nucleolus
can be alternatively calculated from this sort of a "modified"
or
corrected characteristic function.
But
there is also a paradox typical examples, IF a game has been DEFINED (for
purposes of study, perhaps as an example) by specifying a characteristic
function to describe it THEN the characteristic function is correct! So
we feel that this is a useful and convenient category of example games
and wish to have a language that allows for its convenient use.
Thus
we can seek to find, by various means, an "evaluation" of a "CF game" described
by the numbers of a characteristic function of the original type of Von
Neumann and Morgenstern). And we can obtain also, for a CF game, the modified
characteristic of Harsanyi's type. (This process of going to the Harsanyi
"modified characteristic function" can be described as applying the "Harsanyi
transform" to the original characteristic function.)
Pro-Cooperative
Games
There
seems to be the possibility that theory for cooperative games, of the sort
that could prescribe evaluation numbers, like the Shapley value or the
nucleolus, may work better for games which tend to strongly reward cooperation
of the players than for those which could be regarded as tending more towards
favoring noncooperative behavior of the players. This remark is unfortunately
both vague and rather "verbal".
A
game of the sort that most simply favors cooperation is a game which approximates
to a maximally simple game of
bargaining.
If agreement of all players in a three-person game is necessary for them
to receive the main payoff total and coalitions of merely two of the players
could get
only
comparatively quite small amounts then the game intrinsically favors the
cooperation needed for the realization of the benefits of the "grand coalition".
(We
have used games of this sort as a start for examples for explicit calculations
in our project of study.)
A
contrasting type of game is the example game presented by Alvin Roth in
1980. His illustrative game led to much discussion and arguments and counter-arguments.
But Harsanyi, whose own theory was considered by Roth not
to
give an acceptable evaluation for the example, himself accepted the critical
implications of the example. The thing that can be noted about Roth's illustrative
game is that the grand coalition is totally ineffectual (as it were) and
that all of the possible payoff is realizable by a coalition
of
only two of the players. (The context is a little complicated by the presentation
of the game in an NTU form and the need for the value vector concepts to
proceed via reductions to TU form.) (Here TU and NTU are standard terms
for transferable or non-transferable utility.)
So
it seems plausible that Roth's example game can be classed, comparatively,
as a game not of "pro-cooperative" type. And what we are thinking is that
the analysis of
the
process itself of formation of coalitions, via non-cooperative underlying
motivations, can naturally lead
to
different results for games which tend to favor or disfavor the dependence
of the players on the achievement
of
cooperation at the level of the grand coalition. (We don't know, at this
time, what should be a precise definition of "pro-cooperative game" but
we feel that it
is
likely that an analysis of negotiation and bargaining,
as
processes on the roadway towards the realization of cooperation, could
lead to unique solutions and plausible payoff outcomes for certain types
of games while for other types, perhaps like Roth's example, there could
be non- uniqueness or other deficiencies.)
Cooperation
is not always intrinsically favored, in nature, or in human affairs. Sporting
events would become absurd if Sumo wrestlers were to spend ALL of their
time in polite ceremonies and respectful bows.And
Nature allows the evolution of parasitism and predation as well as the
evolution of symbiotic relationships.
A
Consistent Value for Games of Three Players
In
connection with this project I was thinking about how games might be most
practically described in the process of preparing to study them in terms
of modeling the processes of "bargaining" and/or "negotiation" and "coalescence".
And it became clear that the modeling would be much simplified if a characteristic
function description of the game
could
be used. However I also knew that the procedure of definition of the VN&M
characteristic function (as defined in "Theory of Games and Economic Behavior")
could not be viewed as entirely "correct" because of its failure to properly
analyze the "threat" potentials of the various parties in the game.
So
the question arose of whether or not the "modified" characteristic function
defined by Harsanyi (1959, 1962) could be used instead and this issue was
stimulated by conversations with Shapley at Stony Brook 2001 from which
I
learned that it was viewed as quite appropriate to apply the Shapley value
calculation with any given characteristic function and particularly that
of Harsanyi.
It
can be remarked that this use of the "HCF" (or Harsanyi characteristic
function) with the Shapley value formula or with Harsanyi's procedure in
1959 and 1962, for
a
general 2-person game of TU type (transferable utility), yields the result
of Nash in "Two-Person Cooperative Games"
for
such a 2-person game.
And
further study of the possibility of using the HCF to describe a game led
to a surprise for me when I realized that the nature of the reduction of
information in moving to that description has the effect, since the characteristic
function is of such a form that it describes effectively a
"constant-sum"
game, that for three player games a natural value concept is definable
without making any use of an axiom of linear additivity of games. In effect,
the Shapley value for the HCF-described game emerges as appropriate
without
the use of a linearity/additivity hypothesis.
And
this is confirmed by the circumstance that for constant-sum games of three
players the nucleolus and the Shapley value coincide (as vectors of three
payoffs assigned to the players or as "imputations" in the language of
VN&M). So the nucleolus, which is not dependent on the linear additivity
axiom, confirms the value concept that is derivable for a game of three
players described by an HCF version of characteristic function.
And
these considerations suggest also the idea
of
"the Harsanyi nucleolus" which is definable as the
result
obtained by first finding the HCF (or Harsanyi characteristic function)
of a game (which might have
been
originally given as a CF game) and then calculating
the
nucleolus as usual except with the HCF used as the characteristic function
for the calculation.
This
results, for games of three players, simply in the Shapley value (if we
start with a CF game), but for games of 4 or more it is something else
and a few examples suggested to me that for 4-player games it might tend
to assign more payoff to apparently favored players that the SV does. So
the "Harsanyi nucleolus" looks like an alternative value concept, for games
of 4 or more players, that perhaps should be studied in comparisons.
Other
Workers
The
areas of (1): analysis of cooperative games
via
means of non-cooperative theory, (2): value theory,
for
games, in general, and (3): the study of games by
means
of direct experimentation are relating areas
that
are attracting interest in recent times. We can
mention
representative names of persons doing research
in
these areas. Armando Gomes has studied a model in which cooperation is
achieved through steps that are taken by the players on a non-cooperative
basis. At one stage of his studies the result for 3-person games was sometimes
the nucleolus and sometimes the Shapley value, and thus it was
a
quite suggestive result.
And
Gianfranco Gambarelli has been studying alternative possibilities in the
area of "value formulae", where it is good to remember that any accepted
value concept can be
the
basis for an "arbitration scheme". Gambarelli has
been
interested in the connections with voting power
issues
similar to thosewhich inspired
the invention of
the
Banzhaf index.
And
Reinhard Selten in recent times has been a leader
in
the direct experimental study of games and how they are actually played
if experiments are done.
Here
I can remark that, while my research project
does
not involve experiments with human players at all,
it
is however as if experiments are being carried out on
the
behavior of robotic players. And the nature of the calculations is that
one does not know, a priori, after designing a model of robotically reactive
players, what to expect from the results of calculations based on the model.
So the process can be analogous to experimentation rather than to simply
trying to design a (somehow "proper")
arbitration
scheme without regard to the natural patterns
of
behavior of players (possibly human, possibly corporate) in a naturally
arising game context.
References
The
paper in final form will have references to various relevant papers "in
the literature". These are
not
included in this preliminary text. So that will be
the
Bibliography for this paper.
Appendices
We
include below, as texts in format, some files for programs that run under
MATHEMATICA to obtain solutions for model games. And also there are the
pages that describe calculation results for specific "cases" in terms of
the specific model most recently studied. Corresponding to
these
pages there will also be transparencies that can be displayed at the Princeton
seminar.
And
it can be remarked that the paper will need to explain in detail about
the structure of a model of reactive players whereas this issue can be
approached to some extent by talk or blackboard in a seminar.
##############################################################################
{Files
of programming for calcs.}
{First
part is of MATHEMATICA file "execpac.12v.k.s03".}
/////////////////////////////////////////////////////////////////////////////
rr[j_,
k_, l_, p_, q_] := rr12[j, k, l, p, q]
rr12[f_,
sx_, a_, m_Integer, n_Integer] := Module[{na, w, zx},
If[n
< 1, Return[no$$iterations$$error]]; na = 0; zx = rat[sx, m];
Label[o1];
If[na == n, Goto[o2]]; zx = r12f[f, zx, a, m]; na = na + 1;
Goto[o1];
Label[o2]; Return[zx]; ]
rat[x_,
k_Integer] := Rationalize[x, 1/10^(k + 2)]
r12f[f_,
sx_, a_, n_Integer] := r12fa[f, rat[sx, n], a, n]
r12fa[f_,
sx_, a_, n_Integer] := (AccuracyGoal -> n; PrecisionGoal -> n;
WorkingPrecision
-> n + 7; Module[{u, s, zx, o, nu, du, x1a, x2a, x3a,
x4a,
x5a, x6a, x7a, x8a, x9a, x10a, x11a, x12a,
fx1,
fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10, fx11, fx12, fe, w},
{x1a,
x2a, x3a, x4a, x5a, x6a, x7a, x8a, x9a, x10a, x11a, x12a} = sx;
s
= {x1 -> x1a, x2 -> x2a, x3 -> x3a, x4 -> x4a, x5 -> x5a, x6 -> x6a,
x7
-> x7a, x8 -> x8a, x9 -> x9a, x10 -> x10a, x11 -> x11a, x12 -> x12a};
{fx1,
fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10, fx11, fx12, fe} =
N[{D[f,
x1], D[f, x2], D[f, x3], D[f, x4],
D[f,
x5], D[f, x6], D[f, x7], D[f, x8], D[f, x9], D[f, x10],
D[f,
x11], D[f, x12], f} /. s,
n];
w = {fx1, fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10, fx11,
fx12,
fe}; w = Rationalize[w, 1/10^(n + 2)];
{fx1,
fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10, fx11, fx12, fe} = w;
nu
= -(a*fe); du = fx1^2 + fx2^2 + fx3^2 +
fx4^2
+ fx5^2 + fx6^2 + fx7^2 + fx8^2 + fx9^2 + fx10^2 + fx11^2 +
fx12^2;
u = nu/du;
zx
= N[sx + u*{fx1, fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10,
fx11,
fx12}, n]; Return[zx]; Null; ])
rrs[z1_,z2_,z3_]
:= rrb[f,z1,9/5,z2,z3]
rra[v1_,
v2_, v3_, v4_] := rrb[v1,v2,9/5,v3,v4]
rrb[f_,
sx_, a_, m_Integer, n_Integer] := Module[{na, w, zx},
If[n
< 1, Return[no$$iterations$$error]]; na = 0; zx = rat[sx, m];
Label[o1];
If[na == n, Goto[o2]]; zx = rfb[f, zx, a, m]; na = na + 1;
Goto[o1];
Label[o2]; Return[zx]; ]
rfb[f_,
sx_, a_, n_Integer] := rf1b[f, rat[sx, n], a, n]
rf1b[f_,
sx_, a_, n_Integer] := (AccuracyGoal -> n; PrecisionGoal -> n;
WorkingPrecision
-> n + 7; Module[{u, s, zx, du, b, fx1, fx2, fx3, fx4,
fx5,
fx6, fx7, fx8, fx9, fx10, fx11, fx12, fe, fe2, w, st13},
b
= a; st13 = N[sbx[{D[f, x1], D[f, x2], D[f, x3], D[f, x4], D[f, x5],
D[f,
x6], D[f, x7], D[f, x8], D[f, x9], D[f, x10], D[f, x11],
D[f,
x12], f}, sx], n]; st13 = Rationalize[st13, 1/10^(n + 2)];
{fx1,
fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10, fx11, fx12, fe} =
st13;
du = fx1^2 + fx2^2 + fx3^2 + fx4^2 + fx5^2 + fx6^2 + fx7^2 +
fx8^2
+ fx9^2 + fx10^2 + fx11^2 + fx12^2; u = -(fe/du); Goto[o2];
Label[o1];
b = (2*b)/3; Label[o2];
zx
= N[sx + b*u*{fx1, fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10,
fx11,
fx12}, n]; fe2 = N[sbx[f, zx], n]; If[fe2 < fe, Goto[o3]];
If[b
< 6^(-n), Return[gonetoosmall]]; Goto[o1]; Label[o3]; Return[zx];
Null;
])
sbx[phi_,
sxa_] := Module[{xaa, r}, xaa = sb[sxa]; r = phi /. xaa; Return[r];
Null;
]
sb[wq_]
:= s12b[wq]
s12b[w_]
:= Module[{j1, j2, j3, a, j4, j5, j6, j7, j8, j9, j10, j11, j12},
{j1,
j2, j3, j4, j5, j6, j7, j8, j9, j10, j11, j12} = w;
a
= {x1 -> j1, x2 -> j2, x3 -> j3, x4 -> j4, x5 -> j5, x6 -> j6,
x7
-> j7, x8 -> j8, x9 -> j9, x10 -> j10, x11 -> j11, x12 -> j12};
Return[a];
]
FRTB[w1_,
w2_, w3_] := Module[{zx, o}, o = FRTA[w1, w2, w3]; zx = xx12 /. o;
Return[zx];
]
FRTA[ll_,
sx_, n_Integer] := Module[{rx, o, x1a, x2a, x3a, x4a, x5a, x6a,
x7a,
x8a, x9a, x10a, x11a, x12a},
rx
= Rationalize[sx, 1/10^(n + 2)]; {x1a, x2a, x3a, x4a, x5a, x6a, x7a,
x8a,
x9a, x10a, x11a, x12a} = rx;
o
= FindRoot[ll == zz12, {x1, x1a}, {x2, x2a}, {x3, x3a}, {x4, x4a},
{x5,
x5a}, {x6, x6a}, {x7, x7a}, {x8, x8a}, {x9, x9a}, {x10, x10a},
{x11,
x11a}, {x12, x12a}, {AccuracyGoal -> n, WorkingPrecision -> n + 7}];
o
= N[o, n]; Return[o]; ]
zz12
= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
xx12
= {x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12}
FRT[w1_,
w2_, w3_] := Module[{za, zb, zc}, za = FRTA[w1, w2, w3];
zb
= FRTB[w1, w2, w3]; zc = {zb, za}; Return[zc]; ]
SFRB[w1_,
w2_, w3_, v_] := Module[{zx, o}, o = SFRA[w1, w2, w3, v];
zx
= xx12 /. o; Return[zx]; ]
SFRA[ll_,
sx_, n_Integer, kk_Integer] :=
Module[{rx,
o, k, x1a, x2a, x3a, x4a, x5a, x6a, x7a, x8a, x9a, x10a,
x11a,
x12a}, k = kk/(1 + kk);
rx
= Rationalize[sx, 1/10^(n + 2)]; {x1a, x2a, x3a, x4a, x5a, x6a, x7a,
x8a,
x9a, x10a, x11a, x12a} = rx;
o
= FindRoot[ll == zz12, {x1, k*x1a, x1a/k}, {x2, k*x2a, x2a/k},
{x3,
k*x3a, x3a/k}, {x4, k*x4a, x4a/k}, {x5, k*x5a, x5a/k},
{x6,
k*x6a, x6a/k}, {x7, k*x7a, x7a/k}, {x8, k*x8a, x8a/k},
{x9,
k*x9a, x9a/k}, {x10, k*x10a, x10a/k}, {x11, k*x11a, x11a/k},
{x12,
k*x12a, x12a/k}, {AccuracyGoal -> n, WorkingPrecision -> n + 7}];
o
= N[o, n]; Return[o]; Null; ]
SFR[w1_,
w2_, w3_, v_] := Module[{za, zb, zc}, za = SFRA[w1, w2, w3, v];
zb
= SFRB[w1, w2, w3, v]; zc = {zb, za}; Return[zc]; ]
fsb[z_]
:= sbx[f, z]
ffsb[v_]
:= sbx[ff, v]
ff0
= s12g
////////////////////////////////////////////////////////////////////////
{Remark:
The vector ensemble of 12 quantities appearing just below represents
the
12 equations for the equilibrium, with each expression set equal to zero,
and
these
quantities are derived from the three payoff functions as differentiated
by
various
of the strategic variables, etc. The actual payoff functions, much simpler,
enter
into the vector of three components given as pay3 below (in terms of the
rather
unilluminating "anonymous" xsubi type names of the strategic variables.}
////////////////////////////////////////////////////////////////////////
s12g
= {x7/E^(x1/e3) - x8/E^(x2/e3), x10/E^(x4/e3) - x9/E^(x3/e3),
x11/E^(x5/e3)
- x12/E^(x6/e3),
((-12*x11)/(1
+ (1 - x1 - x3 - x6)^4/e1^4) -
(48*x11*(1
- x1 - x3 - x6)^3*(-1 + x1 + x6))/
(e1^4*(1
+ (1 - x1 - x3 - x6)^4/e1^4)^2) -
6*x12*(2/(1
+ (1 - x1 - x3 - x5)^4/e1^4) +
(4*(1
- x1 - x3 - x5)^3*(b1 + 2*(-1 + x1 + x6)))/
(e1^4*(1
+ (1 - x1 - x3 - x5)^4/e1^4)^2)) -
(12*x7)/(1
+ (1 - x2 - x4 - x6)^4/e1^4) -
(6*(-b3
+ (b3 + 2*(-1 + x1 + x6))/(1 + (1 - x2 - x4 - x6)^4/e1^4))*
x7)/e3
- (12*x8)/(1 + (1 - x1 - x4 - x6)^4/e1^4) -
(48*(1
- x1 - x4 - x6)^3*(-1 + x1 + x6)*x8)/
(e1^4*(1
+ (1 - x1 - x4 - x6)^4/e1^4)^2))/
(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)) -
(x7*(-6*x10*(-b1
+ (b1 - 2*x4)/(1 + (1 - x2 - x3 - x5)^4/e1^4)) -
(12*x11*(-1
+ x1 + x6))/(1 + (1 - x1 - x3 - x6)^4/e1^4) -
6*x12*(-b1
+ (b1 + 2*(-1 + x1 + x6))/(1 + (1 - x1 - x3 - x5)^4/
e1^4))
- 6*(-b3 + (b3 + 2*(-1 + x1 + x6))/
(1
+ (1 - x2 - x4 - x6)^4/e1^4))*x7 - (12*(-1 + x1 + x6)*x8)/
(1
+ (1 - x1 - x4 - x6)^4/e1^4) -
6*(-b3
+ (b3 - 2*x3)/(1 + (1 - x2 - x4 - x5)^4/e1^4))*x9))/
(12*e3*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2),
-(x8*(-6*x10*(-b1
+ (b1 + 2*(-1 + x2 + x4))/(1 + (1 - x2 - x3 - x5)^4/e1^
4))
- 6*x11*(-b2 + (b2 - 2*x5)/(1 + (1 - x1 - x3 - x6)^4/e1^
4))
- 6*x12*(-b1 + (b1 - 2*x6)/(1 + (1 - x1 - x3 - x5)^4/e1^
4))
- (12*(-1 + x2 + x4)*x7)/(1 + (1 - x2 - x4 - x6)^4/
e1^4)
- 6*(-b2 + (b2 + 2*(-1 + x2 + x4))/
(1
+ (1 - x1 - x4 - x6)^4/e1^4))*x8 - (12*(-1 + x2 + x4)*x9)/
(1
+ (1 - x2 - x4 - x5)^4/e1^4)))/
(12*e3*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2) +
(-6*x10*(2/(1
+ (1 - x2 - x3 - x5)^4/e1^4) +
(4*(b1
+ 2*(-1 + x2 + x4))*(1 - x2 - x3 - x5)^3)/
(e1^4*(1
+ (1 - x2 - x3 - x5)^4/e1^4)^2)) -
(12*x7)/(1
+ (1 - x2 - x4 - x6)^4/e1^4) -
(48*(-1
+ x2 + x4)*(1 - x2 - x4 - x6)^3*x7)/
(e1^4*(1
+ (1 - x2 - x4 - x6)^4/e1^4)^2) -
(6*(-b2
+ (b2 + 2*(-1 + x2 + x4))/(1 + (1 - x1 - x4 - x6)^4/e1^4))*
x8)/e3
- (12*x8)/(1 + (1 - x1 - x4 - x6)^4/e1^4) -
(12*x9)/(1
+ (1 - x2 - x4 - x5)^4/e1^4) -
(48*(-1
+ x2 + x4)*(1 - x2 - x4 - x5)^3*x9)/
(e1^4*(1
+ (1 - x2 - x4 - x5)^4/e1^4)^2))/
(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)),
-(x9*((-12*x12*(-1
+ x3 + x5))/(1 + (1 - x1 - x3 - x5)^4/e1^4) -
(12*x10*(-1
+ x3 + x5))/(1 + (1 - x2 - x3 - x5)^4/e1^4) -
6*x11*(-b2
+ (b2 + 2*(-1 + x3 + x5))/(1 + (1 - x1 - x3 - x6)^4/e1^
4))
- 6*(-b3 + (b3 - 2*x1)/(1 + (1 - x2 - x4 - x6)^4/e1^4))*
x7
- 6*(-b2 + (b2 - 2*x2)/(1 + (1 - x1 - x4 - x6)^4/e1^4))*x8 -
6*(-b3
+ (b3 + 2*(-1 + x3 + x5))/(1 + (1 - x2 - x4 - x5)^4/e1^4))*
x9))/(12*e3*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2) +
((-12*x12)/(1
+ (1 - x1 - x3 - x5)^4/e1^4) -
(12*x10)/(1
+ (1 - x2 - x3 - x5)^4/e1^4) -
(48*x12*(1
- x1 - x3 - x5)^3*(-1 + x3 + x5))/
(e1^4*(1
+ (1 - x1 - x3 - x5)^4/e1^4)^2) -
(48*x10*(1
- x2 - x3 - x5)^3*(-1 + x3 + x5))/
(e1^4*(1
+ (1 - x2 - x3 - x5)^4/e1^4)^2) -
6*x11*(2/(1
+ (1 - x1 - x3 - x6)^4/e1^4) +
(4*(b2
+ 2*(-1 + x3 + x5))*(1 - x1 - x3 - x6)^3)/
(e1^4*(1
+ (1 - x1 - x3 - x6)^4/e1^4)^2)) -
(12*x9)/(1
+ (1 - x2 - x4 - x5)^4/e1^4) -
(6*(-b3
+ (b3 + 2*(-1 + x3 + x5))/(1 + (1 - x2 - x4 - x5)^4/e1^4))*
x9)/e3)/(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)),
-(x10*(-6*x10*(-b1
+ (b1 + 2*(-1 + x2 + x4))/(1 + (1 - x2 - x3 - x5)^
4/e1^4))
- 6*x11*(-b2 + (b2 - 2*x5)/(1 + (1 - x1 - x3 - x6)^
4/e1^4))
- 6*x12*(-b1 + (b1 - 2*x6)/(1 + (1 - x1 - x3 - x5)^
4/e1^4))
- (12*(-1 + x2 + x4)*x7)/(1 + (1 - x2 - x4 - x6)^4/
e1^4)
- 6*(-b2 + (b2 + 2*(-1 + x2 + x4))/
(1
+ (1 - x1 - x4 - x6)^4/e1^4))*x8 - (12*(-1 + x2 + x4)*x9)/
(1
+ (1 - x2 - x4 - x5)^4/e1^4)))/
(12*e3*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2) +
((-6*x10*(-b1
+ (b1 + 2*(-1 + x2 + x4))/(1 + (1 - x2 - x3 - x5)^4/
e1^4)))/e3
- (12*x10)/(1 + (1 - x2 - x3 - x5)^4/e1^4) -
(12*x7)/(1
+ (1 - x2 - x4 - x6)^4/e1^4) -
(48*(-1
+ x2 + x4)*(1 - x2 - x4 - x6)^3*x7)/
(e1^4*(1
+ (1 - x2 - x4 - x6)^4/e1^4)^2) -
6*(2/(1
+ (1 - x1 - x4 - x6)^4/e1^4) + (4*(b2 + 2*(-1 + x2 + x4))*
(1
- x1 - x4 - x6)^3)/(e1^4*(1 + (1 - x1 - x4 - x6)^4/e1^4)^2))*
x8
- (12*x9)/(1 + (1 - x2 - x4 - x5)^4/e1^4) -
(48*(-1
+ x2 + x4)*(1 - x2 - x4 - x5)^3*x9)/
(e1^4*(1
+ (1 - x2 - x4 - x5)^4/e1^4)^2))/
(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)),
-(x11*((-12*x12*(-1
+ x3 + x5))/(1 + (1 - x1 - x3 - x5)^4/e1^4) -
(12*x10*(-1
+ x3 + x5))/(1 + (1 - x2 - x3 - x5)^4/e1^4) -
6*x11*(-b2
+ (b2 + 2*(-1 + x3 + x5))/(1 + (1 - x1 - x3 - x6)^4/e1^
4))
- 6*(-b3 + (b3 - 2*x1)/(1 + (1 - x2 - x4 - x6)^4/e1^4))*
x7
- 6*(-b2 + (b2 - 2*x2)/(1 + (1 - x1 - x4 - x6)^4/e1^4))*x8 -
6*(-b3
+ (b3 + 2*(-1 + x3 + x5))/(1 + (1 - x2 - x4 - x5)^4/e1^4))*
x9))/(12*e3*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2) +
((-12*x12)/(1
+ (1 - x1 - x3 - x5)^4/e1^4) -
(12*x10)/(1
+ (1 - x2 - x3 - x5)^4/e1^4) -
(48*x12*(1
- x1 - x3 - x5)^3*(-1 + x3 + x5))/
(e1^4*(1
+ (1 - x1 - x3 - x5)^4/e1^4)^2) -
(48*x10*(1
- x2 - x3 - x5)^3*(-1 + x3 + x5))/
(e1^4*(1
+ (1 - x2 - x3 - x5)^4/e1^4)^2) -
(6*x11*(-b2
+ (b2 + 2*(-1 + x3 + x5))/(1 + (1 - x1 - x3 - x6)^4/
e1^4)))/e3
- (12*x11)/(1 + (1 - x1 - x3 - x6)^4/e1^4) -
6*(2/(1
+ (1 - x2 - x4 - x5)^4/e1^4) + (4*(1 - x2 - x4 - x5)^3*
(b3
+ 2*(-1 + x3 + x5)))/(e1^4*(1 + (1 - x2 - x4 - x5)^4/e1^4)^
2))*x9)/(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)),
((-12*x12)/(1
+ (1 - x1 - x3 - x5)^4/e1^4) -
(12*x11)/(1
+ (1 - x1 - x3 - x6)^4/e1^4) -
(48*x11*(1
- x1 - x3 - x6)^3*(-1 + x1 + x6))/
(e1^4*(1
+ (1 - x1 - x3 - x6)^4/e1^4)^2) -
(6*x12*(-b1
+ (b1 + 2*(-1 + x1 + x6))/(1 + (1 - x1 - x3 - x5)^4/
e1^4)))/e3
- 6*(2/(1 + (1 - x2 - x4 - x6)^4/e1^4) +
(4*(1
- x2 - x4 - x6)^3*(b3 + 2*(-1 + x1 + x6)))/
(e1^4*(1
+ (1 - x2 - x4 - x6)^4/e1^4)^2))*x7 -
(12*x8)/(1
+ (1 - x1 - x4 - x6)^4/e1^4) -
(48*(1
- x1 - x4 - x6)^3*(-1 + x1 + x6)*x8)/
(e1^4*(1
+ (1 - x1 - x4 - x6)^4/e1^4)^2))/
(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)) -
(x12*(-6*x10*(-b1
+ (b1 - 2*x4)/(1 + (1 - x2 - x3 - x5)^4/e1^4)) -
(12*x11*(-1
+ x1 + x6))/(1 + (1 - x1 - x3 - x6)^4/e1^4) -
6*x12*(-b1
+ (b1 + 2*(-1 + x1 + x6))/(1 + (1 - x1 - x3 - x5)^4/
e1^4))
- 6*(-b3 + (b3 + 2*(-1 + x1 + x6))/
(1
+ (1 - x2 - x4 - x6)^4/e1^4))*x7 - (12*(-1 + x1 + x6)*x8)/
(1
+ (1 - x1 - x4 - x6)^4/e1^4) -
6*(-b3
+ (b3 - 2*x3)/(1 + (1 - x2 - x4 - x5)^4/e1^4))*x9))/
(12*e3*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2),
((-6*E^(x2/e3)*(-b2
+ (b2 - 2*x2)/(1 + (1 - x1 - x4 - x6)^4/e1^4)) -
6*E^(x1/e3)*(-b3
+ (b3 - 2*x1)/(1 + (1 - x2 - x4 - x6)^4/e1^4)))*
(1
+ x10 + x11 + x12 + x7 + x8 + x9) - (E^(x1/e3) + E^(x2/e3))*
((-12*x12*(-1
+ x3 + x5))/(1 + (1 - x1 - x3 - x5)^4/e1^4) -
(12*x10*(-1
+ x3 + x5))/(1 + (1 - x2 - x3 - x5)^4/e1^4) -
6*x11*(-b2
+ (b2 + 2*(-1 + x3 + x5))/(1 + (1 - x1 - x3 - x6)^4/
e1^4))
- 6*(-b3 + (b3 - 2*x1)/(1 + (1 - x2 - x4 - x6)^4/e1^4))*
x7
- 6*(-b2 + (b2 - 2*x2)/(1 + (1 - x1 - x4 - x6)^4/e1^4))*x8 -
6*(-b3
+ (b3 + 2*(-1 + x3 + x5))/(1 + (1 - x2 - x4 - x5)^4/e1^4))*
x9))/(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2),
((-6*E^(x4/e3)*(-b1
+ (b1 - 2*x4)/(1 + (1 - x2 - x3 - x5)^4/e1^4)) -
6*E^(x3/e3)*(-b3
+ (b3 - 2*x3)/(1 + (1 - x2 - x4 - x5)^4/e1^4)))*
(1
+ x10 + x11 + x12 + x7 + x8 + x9) - (E^(x3/e3) + E^(x4/e3))*
(-6*x10*(-b1
+ (b1 - 2*x4)/(1 + (1 - x2 - x3 - x5)^4/e1^4)) -
(12*x11*(-1
+ x1 + x6))/(1 + (1 - x1 - x3 - x6)^4/e1^4) -
6*x12*(-b1
+ (b1 + 2*(-1 + x1 + x6))/(1 + (1 - x1 - x3 - x5)^4/
e1^4))
- 6*(-b3 + (b3 + 2*(-1 + x1 + x6))/
(1
+ (1 - x2 - x4 - x6)^4/e1^4))*x7 - (12*(-1 + x1 + x6)*x8)/
(1
+ (1 - x1 - x4 - x6)^4/e1^4) -
6*(-b3
+ (b3 - 2*x3)/(1 + (1 - x2 - x4 - x5)^4/e1^4))*x9))/
(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2),
((-6*E^(x5/e3)*(-b2
+ (b2 - 2*x5)/(1 + (1 - x1 - x3 - x6)^4/e1^4)) -
6*E^(x6/e3)*(-b1
+ (b1 - 2*x6)/(1 + (1 - x1 - x3 - x5)^4/e1^4)))*
(1
+ x10 + x11 + x12 + x7 + x8 + x9) - (E^(x5/e3) + E^(x6/e3))*
(-6*x10*(-b1
+ (b1 + 2*(-1 + x2 + x4))/(1 + (1 - x2 - x3 - x5)^4/
e1^4))
- 6*x11*(-b2 + (b2 - 2*x5)/(1 + (1 - x1 - x3 - x6)^4/
e1^4))
- 6*x12*(-b1 + (b1 - 2*x6)/(1 + (1 - x1 - x3 - x5)^4/
e1^4))
- (12*(-1 + x2 + x4)*x7)/(1 + (1 - x2 - x4 - x6)^4/
e1^4)
- 6*(-b2 + (b2 + 2*(-1 + x2 + x4))/
(1
+ (1 - x1 - x4 - x6)^4/e1^4))*x8 - (12*(-1 + x2 + x4)*x9)/
(1
+ (1 - x2 - x4 - x5)^4/e1^4)))/
(12*(1
+ x10 + x11 + x12 + x7 + x8 + x9)^2)}
sqsum[kk_]
:= Module[{r, f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12},
{f1,
f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12} = kk;
r
= f1*f1 + f2^2 + f3*f3 + f4^2 + f5*f5 + f6^2 + f7^2 + f8*f8 + f9*f9 +
f10*f10
+ f11^2 + f12^2; Return[r]; ]
s3sum[kk_]
:= Module[{r, f1, f2, f3}, {f1, f2, f3} = kk; r = f1 + f2 + f3;
Return[r];
]
posqq
= {1 - x2 - x4 - x6, 1 - x1 - x4 - x6, 1 - x2 - x4 - x5,
1
- x2 - x3 - x5, 1 - x1 - x3 - x6, 1 - x1 - x3 - x5}
pay3
= {((-2*x12*(-1 + x3 + x5))/(1 + (1 - x1 - x3 - x5)^4/e1^4) -
(2*x10*(-1
+ x3 + x5))/(1 + (1 - x2 - x3 - x5)^4/e1^4) -
x11*(-b2
+ (b2 + 2*(-1 + x3 + x5))/(1 + (1 - x1 - x3 - x6)^4/e1^4)) -
(-b3
+ (b3 - 2*x1)/(1 + (1 - x2 - x4 - x6)^4/e1^4))*x7 -
(-b2
+ (b2 - 2*x2)/(1 + (1 - x1 - x4 - x6)^4/e1^4))*x8 -
(-b3
+ (b3 + 2*(-1 + x3 + x5))/(1 + (1 - x2 - x4 - x5)^4/e1^4))*x9)/
(2*(1
+ x10 + x11 + x12 + x7 + x8 + x9)),
(-(x10*(-b1
+ (b1 - 2*x4)/(1 + (1 - x2 - x3 - x5)^4/e1^4))) -
(2*x11*(-1
+ x1 + x6))/(1 + (1 - x1 - x3 - x6)^4/e1^4) -
x12*(-b1
+ (b1 + 2*(-1 + x1 + x6))/(1 + (1 - x1 - x3 - x5)^4/e1^4)) -
(-b3
+ (b3 + 2*(-1 + x1 + x6))/(1 + (1 - x2 - x4 - x6)^4/e1^4))*x7 -
(2*(-1
+ x1 + x6)*x8)/(1 + (1 - x1 - x4 - x6)^4/e1^4) -
(-b3
+ (b3 - 2*x3)/(1 + (1 - x2 - x4 - x5)^4/e1^4))*x9)/
(2*(1
+ x10 + x11 + x12 + x7 + x8 + x9)),
(-(x10*(-b1
+ (b1 + 2*(-1 + x2 + x4))/(1 + (1 - x2 - x3 - x5)^4/
e1^4)))
- x11*(-b2 + (b2 - 2*x5)/(1 + (1 - x1 - x3 - x6)^4/
e1^4))
- x12*(-b1 + (b1 - 2*x6)/(1 + (1 - x1 - x3 - x5)^4/
e1^4))
- (2*(-1 + x2 + x4)*x7)/(1 + (1 - x2 - x4 - x6)^4/e1^4) -
(-b2
+ (b2 + 2*(-1 + x2 + x4))/(1 + (1 - x1 - x4 - x6)^4/e1^4))*x8 -
(2*(-1
+ x2 + x4)*x9)/(1 + (1 - x2 - x4 - x5)^4/e1^4))/
(2*(1
+ x10 + x11 + x12 + x7 + x8 + x9))}
shapval
= {1/3 + (-2*b1 + b2 + b3)/6, 1/3 + (b1 - 2*b2 + b3)/6,
1/3
+ (b1 + b2 - 2*b3)/6}
nucleo
= {1/3, 1/3, 1/3}
memo
= {nucleolus,value,vector,is,correct,ONLY,when,b1b2b3,quantities,
are,small,enough}
subcos
= {-(x10*(-1 + (1 + (-1 + x2 + x3 + x5)^4/e1^4)^(-1))*
(-3*(-2
+ x11 + x12) + (-3 + 2*x11 + 2*x12)*x7 +
(-3
+ 2*x11 + 2*x12)*x8) +
x12*(-1
+ (1 + (-1 + x1 + x3 + x5)^4/e1^4)^(-1))*
(-3*(-2
+ x10 + x9) + x7*(-3 + 2*x10 + 2*x9) +
x8*(-3
+ 2*x10 + 2*x9)))/(6*(1 - (-1 + e4)*(-1 + x11 + x12)*
(-1
+ x7 + x8)*(-1 + x10 + x9))),
-((-1
+ (1 + (-1 + x1 + x4 + x6)^4/e1^4)^(-1))*x8*(-3*(-2 + x11 + x12) +
x10*(-3
+ 2*x11 + 2*x12) + (-3 + 2*x11 + 2*x12)*x9) +
x11*(-1
+ (1 + (-1 + x1 + x3 + x6)^4/e1^4)^(-1))*
(-3*(-2
+ x7 + x8) + x10*(-3 + 2*x7 + 2*x8) + (-3 + 2*x7 + 2*x8)*
x9))/(6*(1
- (-1 + e4)*(-1 + x11 + x12)*(-1 + x7 + x8)*
(-1
+ x10 + x9))), -((-1 + (1 + (-1 + x2 + x4 + x5)^4/e1^4)^(-1))*
(-3*(-2
+ x7 + x8) + x11*(-3 + 2*x7 + 2*x8) +
x12*(-3
+ 2*x7 + 2*x8))*x9 + (-1 + (1 + (-1 + x2 + x4 + x6)^4/e1^4)^
(-1))*x7*(-3*(-2
+ x10 + x9) + x11*(-3 + 2*x10 + 2*x9) +
x12*(-3
+ 2*x10 + 2*x9)))/(6*(1 - (-1 + e4)*(-1 + x11 + x12)*
(-1
+ x7 + x8)*(-1 + x10 + x9)))}
nnn
= (1 - x11 - x12)*(1 - x7 - x8)*(1 - x10 - x9)
end
= lastline
///////////////////////////////////////////////////////////////////////////////////
{Second
part is of MATHEMATICA file "e1seq2.s19".}
///////////////////////////////////////////////////////////////////////////////////
e1seq2[tsa_,
e10_, ie1inc_, st_Integer, st2_Integer, sx0_, acc_Integer] :=
Module[{ffav,
ct, ct2, s, sxva, e1v, ie1v, r}, ffav = ff0 /. tsa; ct = 0;
ct2
= 0; sxva = sx0; e1v = e10; r = {e10, sx0}; ie1v = 1/e1v;
Label[o1];
ie1v = ie1v + ie1inc; e1v = 1/ie1v;
sxva
= sxstep[sxva, e1v, ffav, acc]; ct = ct + 1; ct2 = ct2 + 1;
If[ct2
< st2, Goto[o1]]; ct2 = 0; r = r + u^ct*{e1v, sxva};
If[ct
< st, Goto[o1]]; Return[r]; ]
sxstep[insx_,
e1a_, ffa_, acc_] := Module[{sxv}, ff = ffa /. e1 -> e1a;
f
= sqsum[ff]; sxv = rrs[insx, acc - 3, 2]; sxv = N[sxv, acc - 5];
sxv
= FRTB[ff, sxv, acc]; sxv = N[sxv, acc - 3]; Return[sxv]; ]
//////////////////////////////////////////////////////////////////////////////
{Third
part is of MATHEMATICA file "eeseq2.d.419".}
////////////////////////////////////////////////////////////////////////////
eeseq2[tsb_,
e10_, e30_, k1_, k2_, j_, st_Integer, st2_Integer, sx0_,
acc_Integer]
:= Module[{ffav, ct, ct2, sxva, e1v, e3v, r},
ffav
= ff0 /. tsb; ct = 0; ct2 = 0; sxva = sx0; e1v = e10; e3v = e30;
r
= {e10, e30, sx0}; Label[o1]; {e1v, e3v} = eestep2[e1v, e3v, k1, k2, j];
sxva
= sxstep[sxva, e1v, e3v, ffav, acc]; ct = ct + 1; ct2 = ct2 + 1;
If[ct2
< st2, Goto[o1]]; ct2 = 0; r = r + u^ct*{e1v, e3v, sxva};
If[ct
< st, Goto[o1]]; Return[r]; ]
eestep2[a1_,
a2_, k1_, k2_, j_] := Module[{b1, b2, s1, s2, r},
b1
= 1/a1; b2 = 1/a2; s1 = k1*(b1^j); s1 = IntegerPart[s1];
s2
= k2*(b2^j); s2 = IntegerPart[s2]; b1 = b1 + s1; b2 = b2 + s2;
r
= {1/b1, 1/b2}; Return[r]; ]
sxstep[insx_,
e1a_, e3a_, ffa_, acc_] := Module[{sxv},
ff
= ffa /. {e1 -> e1a, e3 -> e3a}; f = sqsum[ff];
sxv
= rrs[insx, acc - 3, 2]; sxv = N[sxv, acc - 5];
sxv
= FRTB[ff, sxv, acc]; sxv = N[sxv, acc - 3]; Return[sxv]; ]
///////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////
Five
cases of results of calculations:
Case
1:
totsubs={e1
-> 1/2192, e3 -> 1/166,
b1
-> 1/7, b2 -> 1/6, b3 -> 1/5}
sxa
= {0.363559352305709551694991958226659`20,
0.363557921561076956001638696615708`20,
0.339974708654198381213502283077245`20,
0.339972443473879056467419163955302`20,
0.296431911373261784065389274671264`20,
0.296431119209088920021658682547455`20,
2114.6779064860315699408562820565`20,
2114.175722488961998308090839074`20,
2293.1560625771558379268627372806`20,
2292.2939522836141951609922371819`20,
2746.1508802433459446029790820789`20,
2745.7897871970582507309309568302`20}
pp
= {0.3635458240261748, 0.33996306643187385,
0.29642441745726816}
SVa
= {0.3468253968253968, 0.3349206349206349,
0.31825396825396823}
SVe
= {437/1260, 211/630, 401/1260}
ptot
= 0.9999333079153168
Probabilities
of the various coalitions
forming
in the process:
prob23coal
= co1 = 0.3521597723247086
prob13coal
= co2 = 0.3397346289481023
prob12coal
= co3 = 0.3081055987271891
/////////////////////////////////////////////
Case
2:
totsubs
= {e1 -> 1/867, e3 -> 1/77,
b1
-> 1/4, b2 -> 2/7, b3 -> 1/3}
sxa
= {0.391848779440067626885604722563224`20,
0.391838792789722496908362650482066`20,
0.348185006310385605243056160861924`20,
0.348168443075420224049219255232638`20,
0.259852896706008588928392478345275`20,
0.259847945975483797940930658085593`20,
527.7087454188208884027418298066`20,
527.3031081108992781729154784539`20,
600.5872699971815415273494861138`20,
599.8217877957068774249296739488`20,
902.8550779742111180049136140675`20,
902.5109695675787430849845453482`20}
pp
= {0.3917925682555118, 0.3481425324376593,
0.2598376745448060}
SVa
= {0.3531746031746032, 0.3353174603174603,
0.3115079365079365}
SVe
= {89/252, 169/504, 157/504}
ptot
= 0.9997727752379771
Probabilities
that various 2-coalitions form by
elections
of agencies:
prob23coal
= co1 = 0.3699609884935736
prob13coal
= co2 = 0.3521874455795275
prob12coal
= co3 = 0.2778515659268989
////////////////////////////////////////////
Case
2B:
(with
"epsilons" e1 and e3 very large)
totsubs
= {e1 -> 1/2, e3 -> 1/2,
b1
-> 1/4, b2 -> 2/7, b3 -> 1/3}
sxa
=
{0.289762698235241678903253034842157`20,
0.311748225465505655739634105353023`20,
0.131303400034321945655721838169235`20,
0.107881131440708460301066495019401`20,
0.2415810569884195428488739633087`20,
0.21354615875616631580118705762005`20,
0.050507929199458523786204438319956`20,
0.052778367074971281320283009340478`20,
0.325377119449252628055040242879522`20,
0.31048647447960870345594601800329`20,
0.174022704589430494993878577286076`20,
0.16453379344494918071376029220869`20}
pp
= {0.26436832085030676,
0.12415896628644767,
0.20930891689734232}
ptot
= 0.5978362040340968
2-person
coalition formation chances:
23coal=0.44076965035100063
13coal=0.21044792360865588
12coal=0.3487824260403435
///////////////////////////////////////////////
Case
3:
totsubs
= {e1 -> 1/6277, e3 -> 1/216,
b1
-> 1/3, b2 -> 5/11, b3 -> 3/5}
sxa
= {0.432741371214613335476238648836203`20,
0.432736446017547789566775153071029`20,
0.448243451742605418601599532791267`20,
0.44824098057847843866062441854525`20,
0.119008427696889402521026093966642`20,
0.119007419846927669033216991490659`20,
2167.4794365988075388933530908816`20,
2165.1748058124042760059764863235`20,
3302.4971739343375818023567458945`20,
3300.7348656012511820560325232082`20,
12998.7203735390182610390443178679`20,
12995.8909174067480673389179054936`20}
pp
= {0.432731729824114880311867799299822`20,
0.448233549334670829751655363012704`20,
0.119008304902781656769637703466959`20}
SVa
= {0.397979797979798, 0.3373737373737374,
0.26464646464646463}
SVe
= {197/495, 167/495, 131/495}
ptot
= 0.999973584061567366833160865779485`20
Probabilities
that various 2-coalitions form
by
elections of agencies:
prob23coal
= co1 = 0.44127826197960357
prob13coal
= co2 = 0.41060630578890184
prob12coal
= co3 = 0.14811543223149462
///////////////////////////////////////////////////
Case
1B:
totsubs
=
{e1->1/118,
e3->1/722, b1->1/7, b2->1/6, b3->1/5}
sxa
= {0.3410830223133562, 0.3410717675905068,
0.33387899114519043,
0.33385976223535485,
0.3234680628590915,
0.3234600963630246,
51.68058669482268,
51.26233653688477,
54.16719134660171,
53.420369607338536,
57.36480199592629,
57.03579763731401}
pp
= {0.34075889426109785645331479377189`20,
0.333562147199287823238751413389702`20,
0.323171741421957822798294768067045`20}
ptot
= 0.997492782882343502490360975228637`20
SVa
= {0.34682539682539682540,
0.33492063492063492063,
0.31825396825396825397}
SVe
= {437/1260, 211/630, 401/1260}