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
As was
mentioned above in the first paragraph, this research was
inspired
by some study of the work of theoretical biologists working with
models
employing the "Prisoner's Dilemma" game. When I arrived at the
idea of
use of "agencies" as the means for the reduction of the potentially
quite
"verbal" general concept of a coalition to a concept well suited
for studies
based on analysis of the players' motivations from the
non-cooperative
viewpoint of separate and independent utility measures I
realized
also that there was a connection with the studies on the topic
of
"the evolution of cooperation" (Axelrod et al).
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 in the
process
we found 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 that appears with 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 non-cooperative 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.
So
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 included in studies that compare
alternative
ideas for values and arbitration schemes.
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}];
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}