Geometry, Language and Strategy

The following sections are included:

• Dedication
• Foreword
• Preface
• Contents
• List of Tables
• List of Figures

Decision making is a distinctly human process. We constantly make decisions. What time will we get up in the morning? What activities will we do during the day? What product will we make and sell? What products will we buy? What will we eat for dinner? Who will we vote for? How will we defend against terrorists? The list is endless, covering the entire range of human activities. Our rather surprising ability is that we make decisions without apparent effort. We sift through the possible choices without seeming to enumerate them. We live with the consequence of our decision and learn from our mistakes. We recognize key patterns of decision making and take appropriate action. We are firm believers that our decisions are optimal and our decision process adequate for our needs…

Our choice to avoid a probabilistic or Bayesian approach has a cost. We must separate out the constraints from the active variables but must still describe the constraint contributions to the energy (Sec. 1.6). Thus we need to know more about the dynamics than a purely probabilistic approach would require. We shall see that there are two sources of such dynamics. One source that we deal with in this chapter is from hidden dimensions. The other we deal with in Chap. 3 is from inertial behaviors. These form the geometry of change. There is an appropriate geometric space-time with an appropriate set of dimensions, metric and units of measure. In this chapter we provide the language and set of distinctions for this discussion…

In the last chapter we arrived at a kinematic formulation of decision process theory as a self-consistent consequence of decisions viewed as events in a space of strategic choices in which some of those choices are hidden and associated with players or agents. In a population of 10, 100 or 1,000,000 players, it is possible to imagine that the theory might apply even though actually carrying out the calculation might be horrendously complicated and not practical. We generally focus on fewer players or agents and treat the rest of the players or agents as part of the background, part of the constraints. The background may also have environmental and physical effects that might influence decisions. This is the situation envisioned in our discussion of applied and constrained forces, Sec. 1.6…

In Chaps. 1-3, we developed a consistent theory of decisions built from lessons learned from game theory and the physical world. It is a mathematical and engineering language for a decision process theory. Our proposed theory is Eq. (3.31), extended to include time and both active and inactive strategies…

We have identified the mathematics of decision process theory, Eq. (4.1). The expression is concise and self-consistent. Despite its merits of brevity, to most readers unfamiliar with this mathematics, it does not give great insight into the behavior or consequences. For that we have to expend significant effort to analyze these equations. We need to identify the concepts and distinctions of that theory as applied to decisions. We need examples. We turn to engineering for guidance. In electrical engineering, whose equations are in fact similar, a large effort is devoted to learning about DC circuits since the concepts can then be applied to AC circuits. The general case of transient behavior can then be understood in that context. Similarly in mechanical engineering we can start with statics, and then proceed to a description of standing waves, steady state waves and finally general dynamics. We would like to do the same for decision process theory. We want to identify a simple set of cases that are steady state and then extend them to more interesting cases…

Our starting point for decision process theory is Eq. (4.1). If we adopt the centrally co-moving hypothesis from Vol. 1, then in the (holonomic) normal-form coordinate basis, there is a commuting Killing vector whose active vector field components K^a = ϕV^a corresponding to the time isometry are proportional to the energy flow; the inactive strategy components are zero K^j = 0. This means the centrally co-moving frame is holonomic and the partial differential equations will be independent of time…

In Part 1, we provided an in-depth development of decision process theory. In Part 2, we shall test the properties of this theory by analyzing various scenarios. There are four major types of scenarios: steady state scenarios, reflection and transmission scenarios, resonant scenarios and transient scenarios. In Vol. 1 we focused on resonant scenarios. In Part 2, we focus on the other three, starting with steady state scenarios…

Our view of decision making is that it is an active process with an underlying set of steady state codes of conduct processes specified by the existence of players and agreed codes with persistent behaviors. In this chapter we describe the geometry of those steady state processes and identify that geometry with the inactive metric components γ_jk, the inactive flows V_j and the inertial accelerations q_υ. Differential geometries help articulate our assumption that there is benefit to looking at decisions in multiple ways. An analogy relevant to this discussion is looking at wave phenomena on a lake…

Our study of the prisoner's dilemma is part of an inquiry into decision making and its connection to uncertainty. It can be thought of as a pure form of the problem in which two players can either adhere or not adhere to a contract, Cf. Sec. 11.2.2. As such, this example highlights aspects of decision making, which is one of the most human acts and seems to be the most difficult area to formalize into a theory of behaviors that are causal and deterministic. In fact one might think that the very nature of decision making is one of chance and uncertainty. Game theory focuses on these uncertainty aspects of decision making and assumes that we have access primarily to the equilibrium point that is reached based on rules provided by the theory…

In the last chapter we developed the prisoner's dilemma scenario with a single active strategy and three codes of conduct. We showed the importance of codes of conduct in addition to choices of active strategies. The codes of conduct provide hidden degrees of freedom and allow a payoff matrix when viewed in the normal frame…

In this chapter, we sketch a general review of game theory and economic approaches to provide further basis for the arguments from previous chapters based on two simplified scenarios. Though the scenarios are simple, they raise a number of right questions. Further, the analysis was based on a detailed and complete decision process theory from which one can study any decision process…

In the last chapter we focused on the historical treatment of Game Theory. Some issues arose that relate to the distinctions we believe are important in a dynamic treatment: in particular we identified codes of conduct and the tragedy of the commons as attributes we want to incorporate in a new way…

We have noted that the idea of using physics and differential equations for modeling economic behaviors is quite old [Edgeworth, 1881]. Though this approach has not generated a following and indeed is dated based on what we now know about the physical and economic worlds, it contains some important concepts. Physical laws are often determined by observing a phenomenon in isolation, from which a behavior is extracted that is true locally…

In Part 3, Chaps. 14-19, we focus on numerical solutions to games with three active strategies. This has applications to games with more than one person, each of which can have one or more strategies. One of the significant differences between these games and those discussed earlier is that the games have non-trivial competitive forces. Such forces can generate not only oscillations of each strategy in time, but produce cyclonic shapes or vortices in space. These form natural resonances behaviors, some of which have discrete harmonic frequencies as discussed in Vol. 1…

In Sec. 14.6, we explored the steady state surface geometry attributes of decision process theory. In this section we explore the dynamic aspects and in particular, those aspects that use game theory to define values on the initial surface geometry boundary; as an example, the harmonic behavior attributes of the energy flow that are associated with the game theory concept of Nash equilibrium. To make the correspondence to game theory, we distinguish two concepts, still point and free fall. We illustrate this for decision processes with three active strategies that have a non-trivial time development; they are not static or steady state…

Based on our work in the last chapter, we believe we have sufficient grounding to specify the known behaviors. The reasonableness of this specification may not be entirely obvious until we have worked out a few examples. In thinking about what examples to pick, we note that game theory practioners are familiar with a large number of two-person zero sum games. There is a large literature around such games. Many of those games for simplicity have been reduced to ones in which each player has just two strategies. This seems to be a good place to start…

In this chapter, we continue our analysis of two-strategy two-person games. Our analysis has much in common with engineering disciplines such as electrical engineering. In those disciplines, specific models rarely capture the full capabilities of the theory: however, one can isolate effects that are locally verifiable, which the theory then puts together into a connected whole. We gain significant insight into the causal behavior of the theory by looking in detail at local properties of steady state solutions which can then combined by the theory…

In the next two chapters, we identify distinctions in decision process theory that we believe provide increased understanding of the decision making process. In this chapter we focus on structural distinctions and in the next social distinctions. Structural distinctions arise in the theory because of the non-linear nature of the differential geometry, extending the linear ideas that provided the initial foundation. Because phenomena are coupled in time and space, they provide feedback loops that form the basis of the non-linear modeling in the theory. Social distinctions arise because decisions usually occur between more than just two individuals, which generate forces that are more social in nature than individual…

In this chapter we continue our identification of distinctions in decision process theory that increase our understanding of decision making processes. We focus here on social distinctions as opposed to purely structural distinctions. We again use specific numerical models to understand these new distinctions, returning to the work-wealth-wisdom models from Sec. 16.8 as well as the attack-defense models from Sec. 16.2…

This is phase II of a project that grew out of a desire to create a new way of talking about decision processes. In phase I, Vol. 1, I provided an introduction to the ideas and qualitative reasons and simplified calculations for seeing how one might address decision processes using the language of physics and differential geometry. The idea was to use a language that was created to describe and understand processes that evolve in time according to specified dynamic mechanisms. In the first phase, no attempt was made to provide a comprehensive solution to the equations, nor was there an attempt to educate the readers to the level where they could solve the equations themselves…

The following sections are included:

• Bibliography
• Index