First Report on a Project Studying the Analysis
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" complications that
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 game context 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 those which 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}];
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}