Systematic Political Science

The Primary and Secondary Precursors to Mathematical Formulations in Dynamic Games

by Dallas F. Bell, Jr.

Presented at the New Economic Windows: Complexity Hints for Economic Policy conference in September 2004 hosted by the Department of Economics and Statistics at the University of Salerno, Italy.

Abstract: To move successfully beyond the use of simple game theory matrixes toward complex mathematical tools first requires precise input. The primary and secondary precursors to such equations are indexed. They are derived from theological, epistemological, psychological, sociological, and eschatological elements and variables found in systematic political science. Their implications should make the mapping of individual and societal decision-making behavior more efficient by producing higher outcome probability. Thus, a realistic learning domain for multiagent scenarios can be created in simulations. This nonlinear effort to decrease the current gap between abstract notions and confident policy applications is of obvious benefit to analysts from all anthropocentric disciplines.

1. Introduction

Analysts are well aware of the fact that dynamic games are used to examine situations between players. Limited options are posed for those players in the attempt to forecast a likely outcome. The probable outcome is based on the player's apparent strategies for his or her best payoff. The classic game of the Prisoner's Dilemma has two players (Mary and Joe). Arresting officers tell each that if he or she confesses to the crime and the other doesn't, the confessor will go free and the other will receive 5 years of jail time. If both deny the crime, enough circumstantial evidence exists to convict both, and each will get 2 years. But if both confess, each will receive a 4-year sentence. They may choose to either cooperate (C) or to defect (D). Their payoff options have often been represented by the following matrix.

                             C           D
                 Mary   C  (4,4)       (0,5)
                          D  (5,0)       (2,2)

It is commonly predicted that both players will choose a defect (D) strategy, the rationality being, of course, that Joe and Mary would receive a higher payoff by not cooperating despite what the other player chose to do. How and why does an analyst in this variation arrive at what equates to a higher payoff?

More questions are also raised by the next three extensions of this matrix. The first case has the same two players (Mary and Joe). Mary loves Joe, and Joe is aware of her devotion. Mary may choose to accept all of the blame and let Joe go free. She would perceive herself as having a winning strategy and a higher payoff. Uncaring Joe would also consider himself a winner. In another situation we find that Joe was elated by not having experienced negative consequences in the first case, so he soon faced the same decision with another player (Dave). Joe remembered Mary's loving sacrifice and remorsefully decided that it was just for him to confess all and accept the punishment. He perceived this course of action as having the highest payoff. Dave also considered himself a winner. Finally, Dave, who has an average IQ, and Jack, who has a genius IQ, were arrested for writing a novel unsanctioned by the government. Jack confessed to having written the book. He testified that Dave was merely his landlord and knew nothing about the manuscript. Dave and Jack would both consider themselves as having the highest payoff.

In the first case, Mary's decision was affected by love. Why was this need dominate in her strategy? What caused Joe to change from uncaring to remorseful in the next situation? Did Jack and Dave's IQ scores matter in the last scenario? Why did Jack consider the consequences of defying the law as less important than having his physical freedom? Why did the government oppose Jack's freedom of speech?

The questions posed from the last three cases indicate that more input on the players and influencing agents is necessary before a mathematical formula is begun as seen in the Prisoner's Dilemma. They also demonstrate that the value judgment of the analyst concerning what a higher payoff may be is of importance to the process. Therefore, the focus will be on the precursors needed to understand the decision-making motivations and influences of all participants in games and situations.

2. Primary Precursors

Primary precursors are defined as the elements and variables that are fundamental to the input of each player, agent, and analyst in every equation. A systematic analysis of the primary precursors must begin with indexing how each participant determines good and evil. Then the category of theological options of belief should be the first element introduced.

2.1 (T) Theological Variables

God either is or is not believed to be infinite in ability and existence. The latter bifurcation point or category of belief is that god either is or is not celestial. These three options and their operating symbols are as follows.

[Ontological discussions and explanations of all elements and variables have been detailed in the paper "How Theology Dictates Political Systems."]

(T1) Theology of the infinite God

An active individual choice of faith to believe in a balanced God of love and justice—to whom one can relate only by God's grace.

(T2) Theology of a finite god

An active or de facto individual choice to reject the infinite God for a faith to believe in an unbalanced god of
(l) love,
(j) justice. or
(f) force
to whom one can relate by human effort.

(T3) Theology of finite gods

An active or de facto individual choice to reject the infinite God for a faith to believe in an unbalanced god of
(o) objects,
(a) animals,
(h) other humans,
(i) societal institutions,
(s) one's self, or
(e) beings of evil
to which one can relate by human effort.

2.2 (R) Epistemological Variables

Information is taken in through the senses and processed for future use based on the corresponding T category of belief.

[Fuzzy analysis is required to compare most, many, or less compliance.]

(R1) Rational epistemology

A rational individual seeking and complying with natural laws in submission to their immutable and inerrant designer for mankind's good—the infinite God (T1).
(R1+) is perfect compliance with natural laws, which requires infinite understanding.
(R1) is finite compliance with natural laws, which requires extrahuman understanding.
(R1-) is the highest consistently achievable human level of compliance with natural laws because of finite understanding and the option to choose to disobey known natural laws.

(R2) Irrational epistemology

An irrational individual rejection of some natural laws to comply with an errant designer of good—a finite god (T2).
(R2+) is compliance with most known natural laws.
(R2) is compliance with many known natural laws.
(R2-) is less compliance with many known natural laws.

(R3) Irrational epistemology

An irrational individual rejection of natural laws to comply with situational perceptions of the source of good—a finite god (T3).
(R3+) is the rejection of many known natural laws.
(R3) is the rejection of most known natural laws.
(R3-) is the lowest humanly survivable rejection of natural laws.

2.3 (B) Human Behavioral Variables

Based on the R category from the T belief, the need levels common to all mankind are pursued in ascending numerical order.

(B1) Common individual needs with a T1 standard/authority for behavior
(B11) Survival
    Serve others, trusting God for provisions
(B12) Economic Security
    Help others achieve their needs, resting in the security of God
(B13) Love and Affection
    Love others as God loves
(B14) Status and Self-esteem
    Seek to reflect godly attributes to glorify God
(B15) Self-Actualization
    Experience the overflowing joy in knowing and obeying God

(B2) Common individual needs with a T2 standard/authority for behavior
     (B21) Survival
        Use deceit and violence if necessary
     (B22) Economic Security
        Hoard assets greedily
     (B23) Love and Affection
        Lust toward others
     (B24) Status and Self-esteem
        Use power to control the weak
     (B25) Self-actualization
        Suffer self-destruction

(B3) Common individual needs with a T3 standard/authority for behavior
(B31) Survival
    Deceive and murder if necessary
(B32) Economic Security
    Steal assets
(B33) Love and Affection
(B34) Status and Self-esteem
    Enslave others
(B35) Self-actualization
    Suffer death

2.4 (W) Societal Variables

To meet human needs, all people interact within the following descending societal institutions:

(Fa) Family
(Ch) Church
(Bu) Business
(Go) Government
Individual need levels directly correlate to ascending numerical societal levels. Those levels will be listed in section 4.2 after the secondary precursors are covered in section 3.

2.5 (E) Eschatological Variables

Individual and societal behavior provide evidence of the option chosen for the belief in the concept of eternity. This belief is based on the beginning T faith of what the future may hold. That vision provides the goals for behavior.

[Eschatological variations may be identified linguistically with latent semantic analysis.]

(E1) Eschatology for the T1 belief
Balanced God of love and justice—Eternal future of life in heaven with God for those who have been forgiven for violating God's law; eternal separation in hell for those who rejected God's grace.

(E2) Eschatology for T2 beliefs
(l) Unbalanced god of love—Eternal future eventually for everyone in heaven regardless of violations of the god's laws.
(j) Unbalanced god of justice—Eternal future for the few in heaven who have complied with the god's laws; all others are separated eternally in hell.
(f) Unbalanced god of force—Eternal existence is in a higher or lower form depending on the degree of compliance with the god's laws now.

(E3) Eschatology for T3 beliefs
(o), (a), (h), (i) Worship of objects, animals, other humans, and/or institutions, respectively—Eternal future is uncertain and requires little compliance with god's laws.
(s) Worship of one's self—Eternal future is not a concern; laws are what one decides them to be.
(e) Worship of beings of evil—Eternal future does not necessarily exist; present existence is made more heavenly by complying with the impulses of evil spirits or is made more hellish by resisting the control of evil spirits.

3. Secondary Precursors

Secondary precursors are defined as elements and variables the input of which has the potential to exert little influence on players, agents, and analysts in every equation. For example, they might not compete between the participants and cancel each other out, such as in the case of each having a similar history (H), or a category such as intelligence (I) might not be deemed relevant.

(A) Average work productivity
(A1) 75% to 100% capacity
(A2) 50% to 74% capacity
(A3) < 50% capacity

(C) Legal classes of the population
    (C1) One tier—all are equal under the law
    (C2) Two tier—elite ruling and worker classes
    (C3) Three tier—elite ruling, worker and slave/nonhuman classes

(D) Population density
    (D1) 75% rural and 25% urban
    (D2) 50% rural and 50% urban
    (D3) 25% rural and 75% urban

(F) Individual freedoms to achieve needs within natural laws
(Rational—speech, worship, own property, defend one's self and property, etc.—and the legal notion of being treated as innocent until proven guilty)
(F1) Most freedoms
(F2) Few freedoms
(F3) Slavery

(G) Geographic position
(G1) Isolated
(G2) Neighbors have the same T beliefs
(G3) Neighbors have different T beliefs

(H) History
(H1) Y1 to Y3 of T1 history
(H2) Y1 to Y3 of T2 history
(H3) Y1 to Y3 of T3 history

(I) Intelligence
(av) average—Can't invent but can use tools to manipulate the environment
(gi) gifted—Invent complex tools to manipulate the environment
(lo) lower—Can't use tools to manipulate the environment without supervision
(I1) 85% to 90% (av), 5% to 7 1/2 % (gi), 5% to 7 1/2 % (lo)
(I2) < 85% (av), > 7 1/2 % (gi), < 5% (lo)
(I3) < 85% (av), < 5% (gi), > 7 1/2 % (lo)

(L) Levels of crisis/change
(Floods, fires, famines, war/purges, diseases, volume of immigration/ migration sufficient to change the T or I ratios of the P, mass conversion to different T beliefs, etc.)
(L1) Affect perception of 75% or more
(L2) Affect perception of 50% to 74%
(L3) Affect perception of 25% to 49%

(M) Dominant motivations for warfare
(M1) To stop violations of natural laws
(M2) To project power over others
(M3) To murder and enslave

(N) Natural resources
(N1) Requires 10% or less for imports
(N2) Requires 11% to 30% imports
(N3) Requires more than 30% imports

(O) Probability of occurrence
(O1) 90% to absolute certainty
(O2) 75% to 89% certainty
(O3) 50% to 74% certainty

(P) Percentage of the population
(P1) 51% or more
(P2) 25% to 47%
(P3) 2% to 24%

(Q) Quotient of trust to comply with natural laws
(Q1) Trust
(Q2) Don't trust
(Q3) Fear

(S) Average tolerance threshold to persistent moderate stress
(For example, an unsatisfactory occupation, marriage, or economy.)
(S1) 15 to 30 years for T1
(S2) 7 1/2 to 15 years for T2
(S3) 3 1/2 to 7 1/2 years for T3

(Y) Generations in years
(One generation is calculated beginning at birth to the average age of independent economic productivity/childbearing, generally 20 years.)
(Y1) One generations/20 years
(Y2) Two to three generations/20 to 60 years
(Y3) Three generations/60 years

4. The Decision-making Arc Precursors

The primary and secondary precursors can be applied to mapping individual, societal, and global decisions. Past decisions can be plotted to form an arc of behavior toward a T track of behavior. If a T belief is known, an arc can be constructed toward the probable behavior. These arc precursors are fundamental to the forecasting process.

4.1 Individual Decision-making Arc

When an individual T belief and the pursuit of the specific need are designated for each player, the participants can be thought of as homogeneous agents. They have identical need levels, senses, knowledge, and decision processes, but their input is calibrated by their R category. The behavioral output would be generally consistent with the B1, B2, or B3 set of needs.

From a distance time perspective the axis of decision-making from need achieved to pursued need may seem to resemble a straight line toward the T track instead of the fractal reality. The connection of decisions would look more like arcs that combine to create waves because of fluctuations between T influences. Momentum is created after each decision toward the same T influence, which, in turn, influences the direction of the next decision. Immediate negative feedback or the totality of a direction, remorse due to conversion to a higher T belief, or other T changes and stress from cognitive dissonance may alter the direction of the momentum.

The same fractal qualities can be seen in those who have efficient human relationships. From a distance, the cooperating behavior might look like a line from a harmonic 50%-50% total equality of abilities and direction. In reality, the differences in abilities and needs of the participants fluctuate between having more or less cooperation, which form waves the total of which may nearly equal 100% harmony. Extreme fluctuations caused by interactions between those with different T beliefs make cooperation difficult and have a polarizing effect witnessed by a drop from Q2 to Q3. Polarization is the natural gravitation of those with congruous T arcs toward like-mined others for efficient cooperation.

Decisions can also be made predicated on anticipation of a planned future decision. This could include a decision made with the intent to prevent the future decision from being predictable—deception strategy. The total decision direction as well as the present arc should be considered when determining realistic probabilities. Therefore, a sound analysis must entail constant inducing, behavior to T belief, and deducing, T belief to behavior.

The decision process begins by the perception of a circumstance that requires the selection of a course of action. A judgment is made to be consistent (or not) with the chosen T belief. The less time there is to consider options, the greater the impact of relevant influences, especially if they are compatible with the past decision arcs or one's core T belief. Absent those influences, the momentum direction likely will be maintained. The course of action is rationalized (R) and behavior begun to accomplish the choice. The next decision-making process is begun at this point on the arc. Unless momentum is altered, the direction will continue toward the same T belief.

There are no neutral decisions regarding the influence of T beliefs, the operative word being decisions and not absent mindedness. Some courses of action might seem to be insignificant because they require little effort to decide (e.g., the need to select the clothes to wear for an outdoor soccer match with friends). The temperature is expected to be 95 degrees Fahrenheit. The choice to wear (or not) light clothes to keep the body from overheating reflects a belief in complying (or not) with known natural laws. If a series of decisions is forcibly made toward the compliance with natural laws, it will create a momentum toward a higher T belief arc (e.g., a person lost in the wilderness must dress appropriately and avoid dangerous animals, etc., to survive).

(Barc) Individual Decision-making Arc
(Barc1) O1 of T1 arc momentum
(Barc2) O1 of T2 arc momentum
(Barc3) O1 of T3 arc momentum

4.2 Societal Decision-making Arc

Polarization of individuals with the same T beliefs occurs, and groups are formed within societies. They have P1, P2, or P3 categories of numerical influence on the society or nation-state's four institutions (Fa, Ch, Bu and Go.) The institutions are made up of individuals, which act as homogeneous agents in a multiagent scenario.

(Fa) Family
(FeY1-) Y1- female
(MaY1-)Y1- male
(FeY2) Y2 female
(MaY2) Y2 male
(FeY3+) Y3+ female
(MaY3+) Y3+ male

(Ch) Church
(Me) Members
(Le) Leaders
(Ed) Education
    (El) Elementary
    (Co) College
(Ho) Hospital
    (Pa) Patients
    (Do) Doctors
(Or) Orphanage
    (Op) Orphans
    (St) Staff
(ChNm) News media

(Bu) Business
(BuNm) News media
(Ae) Art and entertainment
(Em) Employees
(Ow) Owners

(Go) Government
(Ex) Executive branch
    (Mi) Military
    (Po) Police
    (Pr) President
    (Ca) Cabinet
(Lg) Legislative branch
    (Lo) Local representatives
    (Fe) Federal representatives
(Ju) Judicial branch
    (At) Attorneys
    (Co) Courts

From the subsets of the institutions, the R variable from the T beliefs that comprise the P1, P2 and P3 demographic groups can be assigned based on the common societal behavioral levels.

(W) Societal Behavioral Variables

(Continued from section 2.4.)

(1W) Common First World levels collective domestic and foreign relations behavior—Y time per level is D, G, I, and N dependent.

(1W1) Survival
P1=T1=R1-=B11—Build families and churches
P2=T2=R2+=B21—Use deceit and violence to get food and shelter
P3=T3=R3+=B31—Be jailed or executed for murder

(1W2) Economic Security
P1=T1=R1-=B12—Build education, businesses, and infrastructure
P2=T2=R2 =B22—Hoard assets greedily
P3=T3=R3=B32—Be jailed for stealing

(1W3) Love and Affection

[The paradigm shift in T beliefs of P1 demographics will cause fluctuations to intensify in frequency and arc direction extremes until depolarization occurs and the Q3 becomes Q2 for P groups. After a P1 majority is established in cultural T behavior norms, P2 and P3 will begin to adjust to the P1's T influence to become efficient (i.e., many members of the P2 and P3 groups may behave much like the P1, but they have other T beliefs). These individuals will have the most frequent and extreme fluctuations in their decision-making arc.]

P1=T2=R2=B23—Use government to tax others' earnings
P2=T1=R1-=B13—Build hospitals and orphanages and extend T1 missions into T2 and T3 regions
P3=T3=R3=B33—Be jailed for rape

(1W4) Status and Self-esteem
P1=T2=R2-=B24—Use the government to force T2 beliefs
P2=T3=R3=B34—Enslave others
P3=T1=R1-=B14—Be jailed and enslaved for R1- behavior

(1W5) Self-actualization
P1=T3=R3-=B35—Suffer death from homicide, suicide, infanticide, disease, and addiction
P2=T2=R2-=B25—Experience self-mutilation and individual and government bankruptcy
P3=T1=R1-=B15—Be murdered for T1 beliefs

(2W) Common Second World levels collective domestic and foreign relations behavior—Y time per level is D, G, I, and N dependent.

(2W1) Survival
P1=T2=R2=B21—Use government deceit and violence if necessary
P2=T3=R3=B31—Be jailed or executed for murder
P3=T1=R1-=B11—Be jailed or murdered for building churches

(2W2) Economic Security
P1=T3=R3-=B32—Use the government to enslave others
P2=T2=R2-=B22—Hoard assets
P3=T1=R1-=B12—Be murdered for educating others in T1 beliefs and have businesses confiscated

(2W3, 2W4, and 2W5 are not collectively achievable.)

(3W) Common Third World levels collective domestic and foreign relations behavior—Y time for this cycle is D, G, I, and N dependent.

(3W1) Survival
P1=T3=R3-=B31—Deceive, murder and enslave others
P2=T2=R2-=B21—Obtain food and shelter by violent means
P3=T1=R1-=B11—Be murdered and enslaved for holding T1 beliefs

(3W2, 3W3, 3W4, and 3W5 are not collectively achievable.)

(ST) Societal Transitions

After the collapse of each system's cycle, the stochastic transition behavior contributes to beginning the next First, Second, or Third World cycle.

(STr) Rational transition to a First World system—the lowest probability resulting from understanding that change is needed to a T1 belief.
(STrr) Reformation
    L2 or higher conversion to T1
(STrw) War for independence
    Military victory by T1 over T2 and T3 opposition

(STir) Irrational transition to a Second or Third World system—the highest probability. Resulting from recognizing destructive behavior, but a T1 faith is rejected for a T2 or T3 belief.
(STird) Domination by another nation-state with T2 or T3 beliefs involuntarily imposed
(STirr) Revolution
T2 or T3 military victories over other T beliefs
(STirc) Catastrophic crises
Annihilating famine, disease, war, etc., lead to the extinction of societal structures and the mass migration of survivors to other societies.

Changes in nation-state behavior arcs can reflect a move to the next societal level (e.g., the election of a president with a decision-making arc contrary to the P1 arc). Attempts by one institution to control the subset of another institution (e.g., businesses beginning to establish hospitals for profit motivations instead of benevolent motivations by churches) may reflect the change to a less efficient societal level. An L1 crisis/change may reset the Y time in the present level or hasten its end.

(Warc) Societal Decision-making Arc
(Warc1) O1 of T1 arc momentum
(Warc2) O1 of T2 arc momentum
(Warc3) O1 of T3 arc momentum

4.3 Global Decision-making Arc

The collection of nation-states or complex adaptive systems could be viewed globally as homogeneous agents in a multiagent scenario. They are made up of individuals who form P1, P2, and P3 groups within the same four institutions that become either First, Second, or Third World nation-state behavior. Unlike solitons, nation-state collisions or warfare might not result in participants keeping their same W shapes/systems.

Recent technological advances in transportation and communications have precipitated a global polarization to determine a dominate T belief, indicating a T shift of the P1 and a drop between 1W, 2W, and 3W systems from a Q2 to Q3. Traditional geographic borders of nation-states now have less impact on alignment of cooperating strategies than T beliefs. Presently, the global P1 is T2, P2 is T3, and P3 is T1, which is forming a collective Second World decision-making arc.

It might also be helpful to see the societies of the world as fuzzy fractals. They are self-similar in that they each can be broken down into the same smaller components of societal need levels and individual need levels. They can be roughly considered to have an appropriate fractional quality as well.

(GLarc) Global Decision-making Arc
(GLarc1) O1 of T1 arc momentum
(GLarc2) O1 of T2 arc momentum
(GLarc3) O1 of T3 arc momentum

5. Conclusion

The rudimentary game theory matrixes presented in the introduction indicated a profound need for additional input. The original Prisoner's Dilemma example could have had a correct outcome only by accident. Because some criminal behavior does extend from T3 beliefs, a defect, defect strategy is one of the possibilities. The analyst's assumption that both players had T3 beliefs and that neither was capable of beginning another decision arc has been proven inadequate. Such incorrect assumptions may also reflect the analyst's T category of belief.

Hopefully, by providing the elements and variables to answer the questions prompted by the opening matrixes, the reader can see the systematic analysis of primary and secondary precursors as useful in formulating mathematical tools. More accurate outcomes may foster the creation of realistic learning domains in simulations. This increase in ability to map individual and societal decision-making behavior is of obvious benefit to analysts in anthropocentric disciplines. The ensuing abstract notions could be confidently applied when enacting public policies that have the potential to enrich all of mankind.


Bell, Dallas F., Jr., "How Theology Dictates Political Systems: An Extract from the Deductive Unification of Anthropocentric Knowledge," © 2002. Paper presented in March 2002 at the Mid-America Baptist Theological Seminary in Memphis, Tennessee.

--ALL RIGHTS RESERVED (2004) Dallas F. Bell, Jr.--