158 Acta všfs, 2/2008, roč. 2
Redistribution systems theory as a key to reality
decoding
Teorie redistribučních systémů jako klíč
k dekódování reality
In any theory development, we may generally distinguish three stages. Its origin is based
on an idea enabling to make first small steps. For some time, they remain unconvincing,
insufficient, and seem to be just little utilisable. The emerging theory suffers not only from
lack of supporters, but also opponents. There are namely not many of those who would
notice that something new and prospective is emerging. By entering the second stage,
smaller and larger discoveries made within the theory increase faster and faster. They accumulate
with a snow-ball effect. The theory proves prospective, consistent, and it opens
the space for research that attracts expert supporters. It offers certain style and method
adopted by higher number of experts. In the third stage, the space for further discoveries
gets gradually exhausted. Some partial elements of the theory are rather fine-tuned.
Theory results are however used plentifully in practice, and procedures of their utilisation
are continuously improved. The theory has matured, it remains tied up by its own developed
paradigm, and it waits until it is preceded by new one.
The fact that a new theory is emerging becomes apparent in the transition period from the
first to the second stage. Indications of the fact that it is a prospective theory include:
●	Increasing new knowledge and discoveries, relative quick transition from one knowledge
level to the next.
●	 Each new step in the theory development raises great amount of questions. Figuratively
speaking – upon opening one door, there are many others awaiting their
opening.
●	 Frequency of questions and possible continuations in theory development lead up
to certain scepticism, when we ask whether we are able to defend theoretical issues
of the theory in question.
●	 Various directions of theory distribution start interfering, and the new knowledge
gained this way becomes more conclusive, obvious and distinct, concerning both
the meaning and the method of use.
The above applies in both the natural sciences area (e.g. theoretic mechanics, molecular
kinetic theory of gases, elementary particles theory, etc. were born like this) and in case
of social sciences (theory of use, Keynesian theory of macroeconomics, etc.).
	��������������������������Compare e.g. Kuhn (1982).
Petr Budinský, Radim Valenčík
159Acta všfs, 2/2008, roč. 2
Six months after we published the English version of the article called Nash Equilibrium
in Redistribution Systems (calculation, meaning, and utilisation) in ACTA VŠFS
magazine
in February 2008, we acquired much more knowledge in this area in a relatively short
time, and walked through another part of the path leading to recognition of what goes
on in social systems. It seems to us that we are at the beginning of a prospective theoretic
concept.
Let’s remember that under the term “redistribution system” we understood such system,
in which certain movement of compensations as against performance of individual
players takes place, resulting in its reduced efficiency. The cause of compensation movement
as against performance of individual players is considered development of alliances
inside the system, when players in the alliance controlling the system are favoured at
the expense of players who are not part of this alliance. In other words, we can see the
social system as a system, which various lobbies win their recognition in, when those
controlling the system increase their compensations at the expense of those who are not
part of these lobbies, which at the same time results in reduced efficiency of the entire
system. The area of application of the redistribution systems theory is very broad. More
specifically – we would barely find a social system that is not more or less influenced or
at least threatened by what the redistribution systems theory tries to express using its
mathematical apparatus. In the article published in February 2008, we presented a model
of the elementary redistribution system, showed some of its features, and determined
and calculated discriminating equilibrium and Nash equilibrium. We at the same time
indicated possibilities of further model extension and practical importance of modelling
social systems using the apparatus of the redistribution systems theory.
1	 How we recognise the reality through abstract models
We can recognise the reality in various ways. In some cases, we identify, designate, and
classify what we encounter. We find the general and search for the causative or other connections.
This way, we may proceed also in case of the analysis of various social systems.
We may for example find that various alliances are formed not only inside these systems
but also between them. These alliances are often hidden and decide on how the situation
inside individual social systems would develop. When analysing procedures based
on monitoring and experience only, we can find out soon that there are too many various
connections or links as well as their intermediations. The possibility to foresee the development
in individual systems is thus limited.
Another possibility is to use an abstract model. The process of reality recognition then
looks like a process of “deciphering”or “decoding”of that area, when the key or the code is
the model itself. The experience from reality recognition says that the condition of success
consists in revealing what is very simple, i.e. really elementary. By expanding, enlarging,
supplementing, generalising (extending) the preconditions, incl. hidden ones, the model
then gets an image when it enables to capture the richness of the reality, and it is capable
to describe it more comprehensively. It concurrently shows that it is necessary to define
	���������������������������Budinský – Valenčík (2008)
	 Nash equilibrium was interpreted mainly according to Carmichael (2005).
160 Acta všfs, 2/2008, roč. 2
(capture) root definitions again and again, and more precisely. Their precise approach is
just the key to revealing hidden preconditions and thus also to model development.
In doing so, it is important to find the simplest way. The simplest within the sense of intuition,
since there is no formal regulation that would express what the simplest is and how
to find it. If we then assume the simplest (the key), the reality starts opening in front of us,
and we can start reading it.
With respect to importance of the principle to use the simplest model to read the reality,
we will dedicate couple of comments to it. The most illustrative example of finding such
key to read the reality was probably discovery of the law of inertia based on abstraction of
the uniform linear motion. It is sometimes ascribed to Galileo, but it was rather discovered
by his followers. It seems that there is nothing simpler then to express motion of an object,
which is not influenced by anything, by equation s = v*t (the stroke s is directly proportional
to the velocity v and the time t). This is thus a base of the well known formulation
that the object remains standstill or in uniform linear motion, if no power applies to it.
For more then two thousand years, there was however a concept of objects motion based
on the authority of Aristotle, saying that objects need constantly a power for their motion.
A new concept based on abstraction of the uniform linear motion was emerging (gaining
grounds as against established ideas and stereotypes) with great difficulties.
This concept then became a key (code) to read (decode) the reality. It for example raises
a question – What if the object is affected by power? The second Newton’s law then results
from the answer - generalisation of the motion to uniformly accelerated one. Another
example of extension is the power generated by mutual interaction of two objects (the
law of action and reaction, or the third Newton’s law). This was just a small step away from
the molecular kinetic theory of gasses and liquids, etc.
It is important that each step leading to extension of the original model as well as the next
step extending models generated by its extension was always managed in the direction
of the simplest expression of the overlap.
Also the transition from the classical mechanics to the new ones, based on the relativity
principle, was distinguished by the fact that the overlap featured (from the intuitive point
of view) the simplest or the most economy solution.
We may imagine each step of word decoding as a limit-simple extension of the original
(also originally limit-simplest) model. It is among others testified by some contributions
published in the book My Einstein (the original was published in 2006). These are mostly
contributions of top physicists who, with a dispassionate point of view and deep understanding,
comment on what Einstein actually came up with. We can mention one of them.
It is adumbrated by its title “Albert Einstein: Scientific “reactionary”: “Einstein resolved this
discrepancy as a twenty six year old young man in 1905. He later reported that as soon as he
realised suspicious nature of Newton’s axiom of absolute time, he was able to find a method
of modifying Newton’s mechanics within six weeks, to be in accordance with Maxwell’s equations.
Since Einstein’s mechanics and its best-known definition E = mc2
have brought revolutionary
consequences, we rarely realise that Einstein’s innovation was basically very conserva-
161Acta všfs, 2/2008, roč. 2
tive. His modification of basic physical equations valid at that time was just minimal… When
Einstein introduced the basic velocity limit, i.e. light velocity, to the mechanics of objects, it was
apparent that the Newton’s gravitation theory must also be modified, since it calculated upon
unlimited velocity to overcome gravity effects. Newton’s law of gravity does not contain any
limitation of velocity; gravity effects of rock motion on the Earth should in principle become
evident anywhere in the universe. By 1917, Einstein created successfully a new gravity theory...
His gravity theory, called regularly the general relativity, is often considered a revolutionary
change in looking at the gravity, since Newton’s gravity is a power, while in Einstein’s concept
it is a space and time curvature. In spite of that, general relativity was also actually just
a conservative modification of the existing Newton’s gravity theory… Einstein’s gravity theory
enabled curvature of both, time and space, and showed that those curvatures are mutually
related. What could be more natural?... Also Einstein’s discovery of quantum mechanics was
again a conservative modification – conservative in terms of preserving classical structure of
Newton’s physics…“ (Tipler 2007, p. 82-84.)
The fact that Einstein’s modifications of the original (Newton’s) model were “reactionary”
or “conservative” means nothing else but they were the simplest (limit-wise) extension.
Tipler’s erudite interpretation enables to understand well what this simplicity consisted in.
And in terms of methodology, it is instructive for any work with a model that strives for (if
possible) general validity in that area of reality. Both, the first model and each (emphasis
is on the word “each”) extension that is to be beneficial in discovering of what is going
on must meet the criterion of intuitive (hard to define, but apparent by internal feeling)
limit simplicity. This guide was and still is important also in our case. It was at the same
time confirmed that the above process can be used not only in the area of natural but also
social sciences. And as we show you thereinafter, we can get very far using this method.
2	 Practical and theoretical importance of identification and
calculation of discriminating and Nash equilibrium in the
elementary redistribution system
The model of the elementary redistribution system is conceived to include the most important
and also the simplest we can encounter in this area. Its logic consists of three
players (A, B, C), each of them with the same influence on the system, any two players may
form an alliance, players’ performance is acknowledged by small natural numbers (e.g.
6:4:2). We can then display players’ compensation in a 3D space, when each coordinate
axis corresponds with the compensation to any player. The impact of reduced system
efficiency due to compensation departure from players’ performance can be expressed
by redistribution equation, e.g.:
x + y + z = 12 - η.R(x - 6, y - 4, z - 2)
where x + y + z is a summary of compensations to individual players; 12 is the maximum
compensation that could be distributed, if the performance of the redistribution system is
maximal, which means that no redistribution would occur, and compensations would be
distributed according to the performance; η is a coefficient of performance drop; R(x - 6,
y - 4, z - 2) is a function of distance of distributing actual compensations from compensations
according to players’ performance.
162 Acta všfs, 2/2008, roč. 2
With respect to the fact that the redistribution equation links three variables (compensations
of individual players) by single dependency, all possible redistributions may be displayed
in the redistribution area in a 3D space. If we assume that the lowest compensation
for each player is equal to 1, we can then form an idea on the arrangement of individual
types of equilibrium in the redistribution area as follows:
Diagram 1: Various types of equilibrium in the elementary redistribution system:
(1;y;1) = compensation of the player B
					 (x;1;1) = compensation of the player A
(1;1;z) = compensation of the player C
Where:
●	 The triangle determines a simplified image of the redistribution area, i.e. s set of all
possible redistributions.
●	 Thin lines of the coordinate represent compensations of individual players.
●	 ○ Is the point corresponding with the distribution of compensations according to
players’ performance, in case of both, discrimination equilibrium (if on the border
line) and the distribution according to the performance of all players (if inside the
redistribution area).
●	 ● Is the point corresponding with discrimination equilibrium (if on the border line),
or Nash equilibrium (if inside the redistribution area).
●	 ● Is the point corresponding with equalitarian distribution of compensations in
case of both, discrimination equilibrium (if on the border line) and the distribution
according to the performance of all players (if inside the redistribution area).
●	 Arrows indicate possible movements of discrimination equilibrium and Nash
equilibrium in case of external influences on the redistribution system.
Graphic display of individual equilibrium types and their movements enables to form
an idea of what is going on due to various influences acting on any system (that is also
always a redistribution system). Everyone of us estimates what would happen. The ability
163Acta všfs, 2/2008, roč. 2
of our estimate is considerably individual. We have available various information, our own
experience, we include the fact that “wish is the father to the thought” in our estimates
in various ways, etc. General scheme may however significantly consolidate such our individual
ability.
In doing so, it is very important to mention the following: As soon as we encounter that
something develops differently than it should, it is a signal that there is a new influence
or factor, which we did not envisage, e.g. a network link of any game participant, or the
fact that he/she is a member of certain hidden cross alliance.
It applies in general that the role of social networks influencing the results of negotiations
inside individual and relatively independent systems (of redistribution system type) is
very strong and growing. It is appropriate to take it into account. Model confrontation as
a visual image of the method of developing various types of equilibrium in redistribution
systems with real development in particular organisations or institutions enables to anticipate
respective links, and identify and correct strategic behaviour with respect to their
existence and parameters.
3	 Examples of extension of the elementary redistribution system model
We now arrive at the most important. If our model of the elementary redistribution system
is really suitable and useful abstraction, we may then extend it in various directions, so
that it increasingly becomes the true picture of the reality. Each step of such extension
then enables to imagine better what is going on in the real social systems. And in accordance
with the aforementioned, each (methodologically efficient) extension of the
elementary model will be the simplest (most economical). We select four examples from
what we have been successful with so far. They differ by their level of achievement, or the
degree of development, and all of them are instructive in a different way:
●	 The first example represents model extension using effects of the competition
and inter-organisational migration. It is in a way the most developed and most elegant
example. It is based on a very simple model extension that has however very
strong interpretation. In addition, nice (also within the aesthetic meaning) mutual
supplementation of the arithmetic and geometric expression of this extension takes
place here.
●	 The second example represents model extension using the element of mutual corruption
of players in forming alliances and distributing compensations. It is interesting,
since the primary analysis of feedback option consideration (i.e. means gained by
players in a single round to influence results in the next round) leads us to the opinion
that the whole model becomes very complicated and that we will be forced to
define and test a great number of cases to reveal the key issue (which enables reality
“decoding”). Further examination however shows that minor extension of the model
is sufficient again to capture very economically the most important matters in this
extension that form players’ behaviour.
164 Acta všfs, 2/2008, roč. 2
●	 The third example pays attention to the possibility of players from one system type
to influence forming alliances and compensation distribution in other systems. This
example is in a stage when we are able to present basic tools of model extension and
first results of their application. The model extension stated is not in a stage when the
simplification effect takes place, i.e. when from the great volume of model specifications
we select the option, which would be the key searched for, enabling to identify
respective reality as economically as possible.
●	 The forth example points out the problem area of cross alliance effects (i.e. alliances
of players from various redistribution systems) on the process of negotiation inside
individual redistribution systems. This area of model extension is currently the less
developed as compared to the other three. With respect to further development of
the redistribution systems theory and its application, it is however the area of significant
importance.
3.1	 Example one: Competition and inter-organisational migration effects
In the article Budinský-Valenčík (2008), we have shown that in the systems not exposed to
the pressure of competitive environment alliances of average and less performing players
will be prevailing, while the most performing players will be discriminated. What would
happen, if the system were exposed to the competition?
It becomes apparent that we must examine competition effects always in certain connections
– e.g. with systems development and the possibility of inter-organisational migration
(i.e. in connection with the fact that anyone may leave or join the social system).
Inter-organisational migration has a decisive influence on acquisition and maintanance
the human capital of companies and other types of organisations.
To create a respective model, it is important to find appropriate simplification that would
serve as a key to precise description of the phenomenon. It became apparent that the
most appropriate simplification is to put the possibility of inter-organisational migration
in direct link with the lowest compensation, which must be paid to the most performing
player. Players then negotiate how they can distribute the compensation available by
forming various alliances. Such point of view is also nearest to the real situation. It can be
expressed mathematically as follows:
If the most performing player had to receive compensation higher then 1, e.g. a, while the
parameter 0 < a < xmax (the highest value that the player A may receive), we then get the
following equations:
a + y + z = 12 - η.R(a; y - 4; z - 2)
x + 1 + z = 12 - η.R(x - 6; 3; z - 2)
x + y + 1 = 12 - η.R(x - 6; y - 4; 1)
Very interesting and at the same time fully in accordance with what we have said on the
importance of Diagram 1 is also the graphic expression of inter-organisational migration
possibility. Let’s show how we can get to it. The simplified model of equilibration in
165Acta všfs, 2/2008, roč. 2
negotiation in pandering form can be displayed similarly as in Diagram 2 in the article
Budinský- Valenčík (2008):
Diagram 2: Equilibration in negotiation in pandering form in the simplified model:
(1;y;1)
(x;1;1)
(1;1;z)
Instead of calculation, we may then use a visual image of where individual types of equilibration
will move.
Diagram 3: Movement of discrimination equilibrations and Nash equilibration in the simplified
model:
(a;y;1)
(x;1;1)
(a;1;x)
166 Acta všfs, 2/2008, roč. 2
Thin dashed line indicates the original situation, the thick line indicates borders of the
redistribution area moved at the parameter and with axes of coordinates (x;1;1), (a;y;1),
and (a;1;z). It displays graphically the condition of discrimination equilibrations in negotiation
with pandering, and arrows show movements of discrimination equilibrations and
Nash equilibration.
The experience gained by extending the module in given direction can be summarised
as follows:
●	 Extension base is a very minor modification of equations leading to discriminating
equilibration discovery (single parameter value will change only).
● 	 This equations change corresponds with a geometrical model that is very descriptive
and intuitively understandable.
● 	 We were able to obtain the result by searching for the answer how to express effects
of the competition and inter-organisational migration as economically and simply as
possible, or using the opposite method, i.e. to made certain formal modification of
equations and try to interpret it.
● 	 In our case, we were searching for model expression of what we have identified in
the reality. The opposite procedure seems to be very tempting. The path from formal
extension of the apparatus, which describes the reality mathematically, to interpretation
of what this extension means is however not as simple as it might seem, if we
know the result.
● 	 In this case it consists in the fact that the model describes not only the competition or
not only inter-organisational migration, but concurrence of both effects. This among
others shows that good model must proceed from careful perception of the reality.
For example, trying to determine competition or inter-organisational migration effects,
etc. in “clear” form at any price would not lead to any success in this case. It is
always interaction between the model extending by “clear” mathematic generalisation
and the effort to use the model to capture the essentials that we identify in the
reality itself.
3.2	 Example two: Possibility of mutual corruption of players (investing
into a position)
The real voting power or players’ influence on game results may usually vary. Even in
the case where all players have formally the same (equal) position. Various methods are
for example used for promoting effects of assets, income level or disposable funds on
the possibility to influence forming alliances and distribution of compensations. In the
event that the income may influence game results (which happens in practice almost any
time), the feedback actually functions between the result of the previous game and the
parameters of the next game. We know that feedback existence has always a significant
impact on the behaviour of any system. We may anticipate that this will also apply to
redistribution systems. The point is to find an appropriate model that would describe this
phenomenon; i.e. that would capture consequences of investing into the player’s position
in that system.
167Acta všfs, 2/2008, roč. 2
The solution is always simple, and it also applies to this case. We may understand income
effects on game results (distribution of compensation) in the next round (in forming alliances
and achieving certain compensation within the alliance) so that the player will use
part of his/her funds, which he/she disposes of and which he/she gained in the previous
round, to pay to another player for the fact that this other player applies the strategy
required by the first player. A very simple principle. It is however not simple to put it in
operation, also with respect to necessary assumption of simplicity.
One of the problems we have to cope with is for example the question – how to use the
model to capture the problem of agreement compliance or violation, i.e. that the player
will apply the strategy required in return for compensation.
The following rules have been proposed:
1.	 An initial state was determined (e.g. distribution of compensation in the ratio
3:2:1).
2.	 Anybody may try to bribe anyone else from the funds available to him/her:
2.1.	Directly from the amount he/she actually disposes of.
2.2.	 And also from the amount gained from negotiation.
2.3.	 By promising compensation after round completion.
3.	 Any agreement related to compensation distribution may be proposed and any
agreement of this type may be violated at any time.
4.	 The player not violating the agreement is expected not to violate it, until he/she
violates some agreement; the player violating the agreement is expected to violate
agreements.
5.	 After completion of each round, every player will receive his/her compensation
6.	 Each player spends certain amount in each round (e.g. 1).
7.	 All players are informed of all steps (negotiation results, agreements conclusion, their
violation...) of other players.
8.	 Each player strives:
8.1.	 Not to get into the state of exigency.
8.2.	 To maximise his/her compensation in a long-term horizon.
8.3.	 To maximise the total of compensations in individual rounds.
Let’s now show an example of a game that was played in public at the regular Theoretical
seminar of PCE in VŠFS on 28.01.2008:
	 Theoretical seminar of PCE (Productive Consumption Economy) is held every week from October 2003 during
teaching. Information on this event (topics, details) can be found at www.vsfs.cz  Science and Research
 Theoretical seminar of PCE.
168 Acta všfs, 2/2008, roč. 2
A	 B	 C
3	 2	 1	 (initial state)
3(-2)	 2	 1(+2)	 A offered 2 to C, B did not respond, C accepted
1(+3)	 2(+1)	3(+2) 	 compensations in this round
4(-1)	3(-1)	 5(-1)	 consumption
3	 2	 4	 initial stat in the next round
C offers 1 to B
A offers 1 to C
B did not accept, requires 2 from C
C did not accept, offers 1 to A
A is willing to accept
B did not respond
3(+1)	 2	4(-1)
4(+2)	 2(+1)	3(+3)	 compensations in this round
6(-1)	3(-1)	 6(-1)	 consumption
5	 2	 5	 initial stat in the next round
			 A offers 2 from the result to C
			 B offers 1 to C
			 C accepts the offer of 2 from the result from A
5(+3)	 2(+1)	 5(+2)	 compensations in this round
8(-1)	3(-1)	 7(-1)	 consumption
7	 2	 6	 initial stat in the next round
7(-2)	 2	 6(+2)	 A complied with the agreement
5	 2	8
			
In this game, the necessity to particularize determination of the negotiation process has
proven among others.
If we consider the influence of redistribution to reduced performance of the organisation
and the possibility to negotiate with pandering, we will find that the system would
behave differently. We have to however assume that all players are perfectly reasonable,
perfectly informed of the results of mutual negotiations, and the negotiation process is
not limited whatsoever. Using the following rationale, we can show you how the system
will behave under different circumstances. We will search for respective equilibriums, their
definitions, interpretation and calculation. We logically also assume that some discrimination
equilibrium was formed within the system, e.g. based on the alliance B and C against
the player A. (Should another alliance be formed, the deliberation method and the results
achieved will not change.)
Case one: Players B and C will tend to pander by bribing the player A, and they try to
gain higher compensation then between each other within the existing alliance. Forming
such alliance would also mean an improvement for the player A. He/she has no reason to
decline the offer, which will bring him/her the compensation higher then 1 from any of
the players B or C in the next round. But how much can he/she ask for? Exactly as much as
defined by calculation of the discrimination equilibrium in case that he/she is in the winning
alliance. The system thus turns into the state of any discrimination equilibrium.
169Acta všfs, 2/2008, roč. 2
Case two: Players B and C may offer a certain amount to the player A from their own funds
before getting compensation in the next round, to assure getting into the alliance with
him/her and that the player A will improve his/her position in this alliance. The player B has
seemingly an advantage, since he/she disposes of greater funds and can assure the result
by “overpaying” the player A. But also in this case there is noting new. The thing is that all
players think reasonably and are perfectly informed of mutual negotiations. The player
B must aim to improve his/her compensation as compared with the previous round. He/
she may overpay the player A, giving him/her more then the player C who would have to
pander. In such case, his/her compensation would however be lower then if he continues
with the alliance with the player C, and thus such situation will not arise. In the event that
we do not consider the difference of values in time (i.e. the amount obtained in advance
from the compensation in the first round or from the compensation in the second round
with the same nominal value has also the same real value), the total of compensation in
advance and after must be exactly at the discrimination equilibrium level.
Example three: The player A who is discriminated will try to improve his/her position and
declares willingness“to get bribed”with certain amount, e.g. 2. In such case, he/she would
be in the winning alliance with higher compensation, together with the player who would
first accept his/her offer. If however some negotiation with pandering takes place and
other two players could give him/her more, discrimination equilibrium would be formed
again with the same parameters as when we do not consider bribery.
Example four: The case when a player outside the alliance (in our case the player A) makes
an offer to get bribed with certain amount and must follow this offer is even more interesting.
I.e. even in the event that another player (B or C) offers more, he/she cannot withdraw
from the agreement declared. Then the player who first makes an offer to this player will
form an alliance with him/her. However, we have to think even further here as well:
●	 It subsequently raises a question – how much he/she may ask for. It is interesting that
this amount reaches up to the discrimination equilibrium level. I.e. as soon as he/she
offers any amount that is a bit lower, one of the players will form equilibrium with
him/her.
●	 Also other players may be aware of the above and make respective offer preventively.
In such case, the system will “fall” again within some discrimination equilibrium.
Example five: Agreements however do not have to be complied with. In the first approach,
we may model this situation by the fact that the amount, which one of the players
is bribed with, must be higher exactly by much the risk of non-compliance with the
agreement is higher. But if the game is played repeatedly, its development will be interesting
again. The player who would not comply with the agreement might have never
got into the winning alliance. So we can see that there is a very strong spontaneous
pressure to comply with agreements in the system. To be more specific – agreements
will be complied with in a purely model situation (which itself is an important and not
trivial result).
Model investing into the position or the feedback bring interesting results also with respect
to behaviour of real systems, and they may be summarised as follows:
170 Acta všfs, 2/2008, roč. 2
1.	Discrimination equilibriums influence very strongly the nature of agreements and
subsequent distribution of compensations even with many modifications in the
course of negotiation.
2.	Discrimination equilibriums may be modified by some exogenous effects such as
credibility of players, difference of the real and nominal value in various rounds, etc.
These effects can be included in the system very easily, and they shift discrimination
equilibrium values.
3.	 The system is very sensitive to exogenous effects that favour one of the alliances, e.g.
to network links (an alliance with players from various redistribution systems).
The experience acquired by model extension in given direction can be summarised
as follows:
●	 Even a very simplified model of bribery, or investing into the player’s position, has
many modifications, and the task is to compare it with the reality to find those that
are relevant.
●	 Although it seems in the first approach that model extension using the element of
possibility of mutual bribery of players would result in large number of variants of
their behaviour, we can find in the detail analysis that even in such extended model,
the same discrimination equilibriums and the same Nash equilibrium remain as in the
elementary model. Some parameters may be only be slightly modified.
●	 It shows at the same time that what was considered sufficiently accurately expressed
and obvious (in this case intuitive vision of the negotiation process) it requires determination
in much more details.
3.3	 Example three: Influence of players from one redistribution system
on forming alliances in another system
We have so far considered cases when in certain system the players of that system have
sovereign influence only. We however know that the reality is different. There is usually
certain influence of players from other systems on the behaviour of players in the system
in question. It is particularly due to “networking” of players from various systems, i.e. by
forming alliances“across”various systems, the members of which are players from various
systems. These alliances may be visible or hidden. The may however significantly influence
development in any particular system. We can model this situation using for example the
following method:
We have N players in a redistribution system (A1
, A2
,…AN
), while the influence power of
each of them is:
1 + ai
qi
,
where:
ai
is a coefficient of the level of influence of i-th player of the cross-alliance (in the simplest model)
on forming alliances and the amount of compensation in his/her redistribution system;
	 For negotiation problems see for example Horniaček (2006), Osborne (2002), Selten (1999a), Selten
(1999b).
171Acta všfs, 2/2008, roč. 2
qi
is the number of players from other redistribution systems, which this player forms the
cross-alliance with.
(The above shows nothing but the fact that the more players from various redistribution
systems form the network, the higher the power of their influence inside each redistribution
system.)
The condition for forming a minimum winning alliance can then be formulated as
follows:
(1 + aij
qij
) > 1/2 (1 + ai
qi
)
		
while the following applies for all k:
				
where:
the index i assumes the value 1, 2,…N;
the index j gradually assumes the value 1, 2,…Nj
(Nj
< N) and indicates only those players
from that redistribution system who are members of the minimum winning alliance;
the index k assumes some values 1, 2,…Nj
(Nj
< N) and indicates only some (any) player
from that redistribution system, who is a member of the minimum winning alliance.
Both conditions are quite apparent. The first one (that the alliance is winning, i.e. disposes
of sufficient influence) says that total influence of all players forming the minimum winning
alliance must be higher then a half of all players in that alliance. The second one
(that the alliance is minimal) says that if any member leaves this alliance, it will not have
sufficient influence anymore.
If ai
= 0 or qi
= 0, we have the original model. Either there is no influence of the crossalliance
on development inside the system, or no cross-alliances have been formed. In
compliance with the above conditions, we can modify a system of equations leading to
calculation of discrimination equilibriums.
The above system extension distinguishes itself also by significant“economy”. We assume
that forming a cross-alliance of certain player from the distribution system in question
with each player from other redistribution system increases its influence the same way, or
by the same value (ai
). If a1
= a2
= …= an
= a, it means that this influence increase would
be identical for each player from that redistribution system. In that case, it would be possible
to write down the conditions of the minimum winning alliance using even a simpler
method.
Based on the above presented formalism, we can formulate many interesting tasks. It
is apparent that reasonably behaving player will try to maximise the number of players
he/she forms the cross-alliance with. He/she may also form an alliance with more players
172 Acta všfs, 2/2008, roč. 2
from the other redistribution system (exclusivity does not have to be preserved). The same
applies reciprocally. What would be the consequences, if there were players from different
alliances in such developed cross-alliances (expulsion from the winning alliance as well as
members of the winning alliance)? How will this influence stability of winning alliances?
What role would this play in negotiation? What consequences will player’s awareness or
lack of information on the existence and structure of cross-alliances have (i.e. differences
due to existence of hidden and visible alliances)? To answer these and similar questions,
it is necessary to formulate other supplementary specifications describing players’behaviour.
These specifications (formalised rules) must bet intuitively understandable, simple,
and correspond with the reality.
The experience acquired by model extension in given direction can be summarised
as follows:
●	 As soon as we open the door to model extension by suitable formalisation, there is
usually a great number of options of how to continue. It was also similar in the previous
second example.
●	 Various options of paths and model specifications must be examined step by step.
●	 In doing so, there are questions that would not arise without model extension. These
questions are usually both, interesting and significant, with respect to practical inter-
pretations.
●	 Sooner or later we can assume that a method will appear how to describe this problem
area and express it by relatively simple way, when many considered modification
options of extended model description will prove to be misleading.
3.4	 Example four: Influence of links between redistribution systems on
the negotiation process inside each of them
One of the basic assumptions for calculation of discrimination equilibriums in the elementary
redistribution system is that the negotiation process will pass perfectly, without any
limitations. The reality is usually different. Not all players are informed of the negotiation
process, the negotiation requires certain transaction cost, players do not have unlimited
time, they might be more or less sympathetic to each other for various reasons, to trust
each other differently, etc.
With respect to the fact that all three equilibriums are equally probable under the conditions
of perfect negotiation, imperfections in the negotiating process as well as various
external effects are decisive in what equilibrium would be formed. Even very minor effects.
It is highly probable that network links (cross-alliance) of players from different redistribution
systems may influence forming alliances also by the fact that they affect players’
preferences in negotiation. If we could find suitable model expression, it would be possible
to answer the following questions:
●	 How do redistribution systems link together within the meaning that forming certain
type of alliance in one type of systems transfers to formation of the similar type of
alliance in surrounding redistribution systems?
173Acta všfs, 2/2008, roč. 2
●	 What external influences on the redistribution system do move discrimination equilibriums
and Nash equilibrium only (they can be compensated by the fact that any
player would ask for lower compensation to enable influence of the winning alliance
with his/her participation), and what external influences do eliminate formation of
some types of winning alliances (and thus cannot be compensated)?
Both questions are from the theoretical and mainly practical point of view very important.
In this sense, the above can be formulated also in reverse order: One of the important aims
of development of the redistribution systems theory is to express the negotiation process
as respective model extension, so that it opens the path to answer the above questions.
If we attempt to answer the second question (what external effects can be compensated
and what cannot), we may consider for example the following: Effects that cannot be
compensated by concessions in negotiation of any player may relate to the fact that by
creating network links between redistribution systems the respective player is excluded
from the negotiation process. Another option is that he/she has no information on how
the negotiation process goes and whether there is any at all. There are probably also other
reasons for what predetermines certain alliance types.
The experience gained by extending the module in given direction can be summarised
as follows:
●	 The option to ask the questions above is subject to achieving certain level of model
development of the redistribution systems theory. As it is usual with any theory development,
asking questions is important, it is certain theoretical result, and it contributes
to better understanding of what is going on in the reality itself.
●	 Verbal (not formalised yet) description contributes to better classification of the
problem area, but its informative ability is limited.
●	 For better capturing of the problem area, formalisation and subsequent mathematic
expression prove not only desirable but also necessary.
4	 What we have achieved
There are various concepts of how the theory develops. Each expresses the reality with
certain simplification and complies more or less with that particular case. Although
none of them is perfect, conclusions and suggestions result from each good concept,
concerning the process in any theory development, or an answer to the question whether
the theory in that particular case is prospective or not.
In our contribution, we have presented certain view of development of such theory types
that use an abstract, formalised, and subsequently also mathematically expressible model
for reality decoding. Consecutively, we showed how the elementary model is extended in
case of the redistribution systems theory, how it gradually opens the path for answering
the questions formulated, but also for asking other questions, and what role the visualisation
and the possibility to confront the model with the reality play in this process.
For this, we have used four concrete examples documenting and illustrating the pro­cess
of model extension. Each of them corresponds with different state of knowledge of such
174 Acta všfs, 2/2008, roč. 2
part of the reality that we describe by extending the model, using formalisation and mathematic
expression.
We assume that this is a suitable form of getting the expert public acquainted with continuously
achieved results, since in the stage of dynamic development of the theory, each
progress within the meaning of the answer to certain question raises even more questions.
Therefore we cannot wait until we know answers to all of them. With respect to
the fact that it is in our opinion a very prospective area of recognition, methodological
reflection and process documentation play also a very significant role. Last but not least,
using this form we aim to interest, motivate, and inspire those who are interested in cooperation
in the area of redistribution systems theory. There is namely the possibility to
achieve demonstrable and original theoretical results.
Abstract
The contribution resumes the article called Nash Equilibrium in Redistribution Systems
(Calculation, significance, use), published in the magazine ACTA 1/2008. It is based on
some general methodological questions of theory development that use mathematical
apparatus, where finding an appropriate reality simplification plays an important role.
The article then applies these bases to the problem area of redistribution systems theory,
which is one of the game variants. It specifies more the expression of discrimination equilibriums
and Nash equilibrium, presents the original scheme that enables visualisation of
these equilibrium types, extends the model of the elementary redistribution system by
considering some influences on the redistribution system. Specifically, these are competition
and inter-organisational migration effects; possibility of mutual bribery of players
(investing into the position); influence of players from one redistribution system on forming
alliances in another system; influence of links between redistribution systems on the
negotiation process inside each of them.
Keywords
Theory of games, redistribution systems theory, Pareto optimum, Nash equilibrium, competition,
inter-organisational migration, coalition, negotiation
JEL Classification / JEL klasifikace
D01, D33, D74.
Souhrn
Příspěvek navazuje na článek Nashova rovnováha v redistribučních systémech (Výpočet,
význam, využití) uveřejněný v časopisu ACTA 1/2008. Vychází z některých obecných metodologických
otázek vývoje teorií, které využívají matematický aparát a v nichž důležitou roli hraje
nalezení vhodného zjednodušení reality. Tato východiska pak konkretizuje na problematiku
rozpracování teorie redistribučních systémů, která je jednou z variant teorie her. Konkretizuje
vyjádřenídiskriminačnía Nashovyrovnováhy,prezentujepůvodníschéma,kteréumožňujenázornou
představu o těchto typech rovnováhy, rozšiřuje model elementárního redistribučního
systému o uvážení některých vlivů působících na redistribuční systém. Konkrétně pak jde o vliv
	 The work flow can be monitored at www.vsfs.cz Science and Research  Theoretical seminar of PCE An
archive, where up-to-date research results are published in annexes to individual lectures.
175Acta všfs, 2/2008, roč. 2
konkurence a meziorganizační migrace; možnost vzájemného uplácení hráčů (investování do
pozice); vliv hráčů z jednoho redistribučního systému na vytváření koalic ve druhém systému;
vliv vztahu mezi redistribučními systémy na proces vyjednávání uvnitř každého z nich.
Klíčová slova
Teorie her, teorie redistribučních systémů, paretovské optimum, Nashova rovnováha,
konkurence, meziorganizační migrace, koalice, vyjednávání
Contact address / Kontaktní adresa
h. doc. RNDr. Petr Budinský, CSc.
Institute of Finance and Administration, o.p.s., Vice-Rector for Education Affairs and External
Relations (e-mail: petr.budinsky@vsfs.cz)
doc. Radim Valenčík, CSc.
Institute of Finance and Administration, o.p.s., (e-mail: radim.valencik@vsfs.cz)
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