Affinity of one player to another shall refer to the benefits (utility) the player gets just by forming a coalition with another player, whereas such benefits (utility) may be expressed in denominations that are used for payoffs within the original game. In case both players generate benefits (utility), it is referred to as mutual affinity; however, the extent may vary for each of the players. Positive affinity may also be referred to as sympathies of one player to another, with negative affinity being antipathy of one player to another.
Affinity may be expressed as follows: a player, who forms coalition with another player, generates specific benefits (utility) just by forming the coalition, whereas such benefits (utility) are expressed in the same denominations as their payoffs. The total payoff of a player (referred to as xij*) within a coalition with another player, under a relationship of certain affinity, shall then equal to the player's payoff in the original game plus the player's payoff corresponding to the benefits (utility) arising from the formation of the coalition (the additional payoff shall be referred to as sij):
xij* = xi + sij.
0001pt;line-height: normal">The value of si may be both positive (positive affinity – i.e. sympathies) or negative (negative affinity – i.e. antipathy). It would seem that if a one-sided or mutual affinity exists between two players, with no affinity existing between either of the players and a third player, the formation of coalition between the two players is predetermined. However, this may not be the case and if the third player is informed about the affinity of the other two, he may offset such positive affinity through a lower payoff. Let us assume that all players are fully informed about all affinities of the players. The original set of equations shall be modified as follows:
S(x12*, x21*, 0) = s12 + s21
S(x13*, 0, x31*) = s13 + s31
S(0, x23*, x32*) = s23 + s32
Right-hand side of equations shall be interpreted by saying that additional payoffs arise within the game on the basis of the relevant affinities. The following shall then apply to the payoffs within the original game: x1 = 1/2(x12* + s12 + x13* + s13), etc.
The original generalized Raiffa sequential solution does not allow the assessment of the role of affinities, because it does not contain an alternative to the formation of two-member coalitions with regard to the alternative of a three-member coalition formation, measured by opportunity costs. For the same reason, other point solutions of the Nash bargaining problem do not make it possible to assess the role of affinities. Therefore, we have created an original theoretical concept, on the basis of which we are able to identify (ascertain and assess) how external factors – in the form of affinities – affect any community or partnership of the redistribution system type.
In line with the specified objectives, the project solution is aimed at expressing (modeling, evaluating, and assessing) the impact of the following affinities and social networks interconnected by such affinities:
- Those that arise by investments in social status and are associated with a creation of social networks (affinities between players) derived from investments in social status.
- Those that arise by violations of principles generally accepted within the given system as well as its social environment and that lead to the creation of social networks relying on mutual covering, blackmail, and favoring of those entities that violate the generally accepted principles. We will mainly strive to describe the method of formation, development, operation, and anatomy of the structures based on mutual covering of violations of the generally accepted principles in terms of the potential elimination of their impact. We will distinguish the following:
- Effects arising through investments in social status.
- Effects arising as a result of activities of social networks derived from investments in social status.
- Effects arising as a result of violations of the generally accepted principles.
- Effects arising as a result of activities of social networks based on mutual covering, blackmail, and favoring of players that violate the generally accepted principles, i.e. as a result of what we call the structures based on mutual covering of players that violate the generally accepted principles.
(To be continued)