Note: from discrete solution which expect full symmetry of possibilities of players in the creation of two-person coalition exists also another NM sets having infinitive many elements. They play also important role, but we do not concerned with them. If we define S = S(x1, x2, x3), it means as the function of payoffs of players, then d = (d1, d2, d3) is given as the solution of following systems of equations:
S(x1, x2, 0) = 0
S(x1, 0, x3) = 0
S(0, x2, x3) = 0.
Here it is valid that pay-off of every player in coalition with each other player (e.g. pay-off of first player with second player or with third player) is same. This fact causes the condition that points (d1, d2, 0), (d1, 0, d3), (0, d2, d3) create discrete three-points NM set.
The generalized Raiffa´s solution and by us established NM-modified Raiffa´s solution are very similar by their logic of construction. However, they have some important differences, especially from the point of view of interpretation. NM-modified Raiffa´s solution in a certain way connects two situations: In the first case the players (each of them) decide to create only a two-person fully discriminated coalition, i.e. two players who form a coalition, can give to the third player the smallest possible pay-off. In our case this pay-off equals 0. But the smallest possible pay-off can have also different value (including negative) what is important for some interpretations and with them connected application, The simplest example is simple majority game described in Neumann and Morgenstern (1957) and following game with coalition of different power (Neumann-Morgenstern 1957, (1957, § 22). The same is valid in our case of the game with non-zero sum. In the second case the players form a great coalition, i.e. three-player coalition. The connection between both the cases can be interpreted as follows: Pay-offs of each player in the formation of fully discriminated coalitions can be seen from his perspective as an opportunity cost to the possibility of creating a great coalition. If players create a great coalition, for obvious reasons they will require pay-off higher or at least equal to the one they would have required in a two-person coalition. The problem is how to evaluate player's pay-offs for the creation of fully discriminated two-person coalitions. Here we use (introduced by us) the term average expected pay-off, which is a multiple of its pay-off in a situation where the player is the member of the winning coalition, and the probability of this coalition i.e. 2/3di, where i = 1, 2 or 3. We simplify as we do not distinguish between pay-off and utility from pay-off. If the utility function of a player has degressive character, the risk aversion would play its role. The players would in such situation prefers two-person coalition even if the value of pay-offs is lower than2/3di. The value depends on the degressivity of utility function. But the example is not important for our future ideas. "Bridge" by which we're connecting both the cases (formation of two-player coalitions and the great coalitions above), i.e. application of the principle of opportunity cost and the introduction of the concept of expected average pay-off, implicitly contains input "step-by-step" process, which results in a single point solution in the case of great coalition.
- We have a group of people that operate within a certain system. They perform some role and, based on the performance of such role, they are attributed specific funds that are subsequently redistributed among them in a certain manner.
- Coalitions may be formed within the aforementioned group of people, with a view to provide privileges to those who take part in the coalition at the costs of those who do not.
- Such privileges are in the form of the funds the players may divide among them. Two questions arise in this connection: 1. What defines (how to describe) the amount of funds that the players would be able to divide among them; 2. How (based on what rules or regularities) will they divide such funds.
- In case social networks operate within the given system, we will understand them as one-sided or mutual affinity of certain players within the given system, whereas one and the same network may operate within a number of systems of this type. Generally speaking, a system will be referred to as a redistribution system if funds are divided and redistributed within the system as a result of certain external factors: Formation of coalitions within the given system; formation of social networks within the system; reflection of roles of such networks between different redistribution systems into individual redistribution systems. It is necessary to emphasize the fact that the aforementioned factors characterize, and not define, a redistribution system. The characteristics are used to give us an idea about the types of objects, to which it is possible to apply the tools developed by us.
A game, in which we do not consider any impact of external factors, shall be referred to as the original game for the sake of explicitness. External factors shall refer to anything that may be expressed by a change in the parameters of the original game, that affects the conduct of players, and that concurrently exists as an independent parameter, the creation/development of which is not directly controlled by any of the players. The expression of the external factors through the change of the original game parameters shall be referred to as the original game extension.
(To be continued)