Tyrant Model using Decision Process Theory

A classic game theory model for a tyrant might be a game of chicken, with a tyrant against the poor, each choosing between swerving or crashing. The difference however is that a tyrant has more power and influence than the poor. One aspect of game theory that we believe needs to be changed is that the size of the payoffs should matter.

As an example of something that plays no role in game theory, consider the concept of engagement. Multiplying the payoff matrix by the engagement does not change the max-min solution of game theory and is thus of no importance. Moreover, the payoff matrix strength is not relevant either. In physics, this is like saying that the charge of a particle does not influence its motion in a magnetic field, nor does the strength of the magnetic field. This is only true for a charge moving parallel to the field. The circulation around the field clearly depends on the charge as well as on the strength of the field.

The hard part is to disentangle the charge from the field strength. This is accomplished in physics by the field equations that say that charges and currents are the sources of the fields. Strong engagement leads to changes in the players behaviors through changes in their payoff and valuation fields. What is amazing however, is that engagement like viscosity enters the decision process theory equations in a non-linear way. Therefore, dramatically increasing or decreasing the engagement changes the qualitative behavior of the decision flow. You get an inkling of why when you realize that the engagement is itself a conserved flow and therefore contributes to the energy momentum of the system. It thus gets shared with other component parts.

Like viscosity, when the engagement is small it has almost no effect other than adjusting the time scale. As it gets larger, the non-linear nature of its behavior becomes evident. It is even possible that we might get “chaotic” or “turbulent” behaviors, though there is currently no basis for this conjecture. The following is the result of a computation (using a decision engineering notebook described in the Stationary Ownership Model Update white paper and implemented using a Mathematica notebook) in which the poor outnumber the tyrant by 10:1 and the engagement of the tyrant to the poor is 1:10.

We have the following interpretation of this picture. From a game theory perspective, the model has two Nash equilibriums. The poor see their strategy to be “swerve” and assume that the wealthy will “crash”. The wealthy see the opposite. This is one variant of the classic game of chicken. It is clear in the figure that by assuming the wealthy are much less engaged, we are closer to one of the Nash equilibrium: we see the Nash equilibrium of the poor, which is to “swerve” and for the wealthy to “crash”.

We see more however. Because of the inequality between the two players, the preference of the poor to crash is going to be smaller over time than for the wealthy to crash. So, in some sense, the wealthy crashing is not as devastating as it is for the poor. There is much less incentive for the wealthy to avoid the choice “swerve”.

The strategic flows add some interesting aspects to the story. The magnitude of the flows in this model run are all comparable. Nevertheless, they support the fact that the net behavior for the wealthy to swerve is zero and the net behavior of the wealthy to crash is small and positive. It is also clear that the net flow for the poor to swerve is much bigger than any of the other net flows.

We interpret this as follows. The poor will suffer and avoid the game of chicken. On average they will swerve. The wealthy will crash, making what appears to be the correct assumption that the poor will always swerve. These assumptions, however, rely on the small engagement of the wealthy. This appears to be the trademark of the tyrant They suffer no consequence for their action and are thus rewarded. The full decision process theory would go on to predict that the payoffs for the wealthy should not change since their source is small. The poor however might change their payoffs over time to better reflect what is happening.

Is there anything that can be done? One possibility is demand an equality of effort between the tyrant and the people, through laws, the courts or the press, as examples. An example computation is provided below, using a gravitational like field to force equality by imagining a potential well that is centered when the strategic effort of the poor matches the strategic effort of the tyrant.

The model generates a time component of the metric, which can be thought of as a “gravitational” field that pulls all strategies to this common center. We see the consequence of this in the above figure. We now expect that the tyrant will see consequences to their action, despite their initial lack of engagement. In fact it might be that that engagement will also change as a consequence.


Propaganda, Narrative and Geometry

We often hear the word propaganda in the press and in the media. I wonder what kind of force this represents. The competitive force is always the one that comes first to mind. This is because I think of economics and social behavior. Yet propaganda is the ability to set the narrative, it is the ability to change a person’s mind. How does one include such a force in a detailed theory of behavior? Is the narrative a gravitational field that draws all under its sway to move in the same direction, friend and foe alike? Does this field owe its strength to some type of concentration of mass or energy at some strategic location? Might we call it strategic capital? How do I see that such a narrative effectively describes the process? Is this narrative distinct from other cooperative forces as well as any competitive forces? How would I tell?

To approach an answer to these questions that might be acceptable to most reasonable audiences, I anticipate that I must overcome a general fear such audiences have of numbers being used to describe human behaviors. Perhaps it is more general. In many cases we don’t like using numbers even when describing physical events. Take our conversation of weather. Are we more comfortable in reducing weather to a yes/no prediction of rain tomorrow than in understanding the complexities of airflow, humidity and pressure as a function of time? The former is like a person deciding, albeit an unknown weather god. The latter involves taking a stand on understanding. I think we are most comfortable with a yes/no prediction.

What is involved in the more complex understanding? Weather phenomena are not the acts of a capricious god but are the result of a process involving interacting parts spread out in both space and time. This process occurs in a continuum over space and time. What is happening here and now is dictated by what has happened elsewhere in the past. This is true at every level of scale, not only from a macroscopic but a microscopic perspective. A process view is based not just on a qualitative and discrete yes/no understanding but a quantitative description. The quantitative description provides the geometry; without this geometry the process is hidden.

But, you raise the valid objection that social phenomena are inherently uncertain and intrinsically about decisions which are discrete yes/no type outcomes. How can such events possibly be described as part of a continuum? My choice at this instant does not continuously flow from choices made in the past. It is a gamble and I might in fact, if given the chance, make exactly the opposite choice. Yet even in physics we have phenomena at the quantum level that from a certain point of view are uncertain and if viewed in a certain way, appear to be discontinuous. That fact has not prevented us from looking at them from another point of view as being described by a differential geometry in both time and space. Indeed, we suggest that game theory has provided us a perspective that social interactions in fact can be viewed geometrically if we focus on the mixed strategies and how they evolve in time as opposed to the pure strategies that focus on the uncertainties. We don’t focus on the actual decision, but on the mixture of decisions that are possible at any point in time.

As an example, consider that over the last decade, wealth has redistributed itself dramatically (Piketty, 2014). It has not happened discontinuously. It has evolved over time and appears to change continuously across social strata. Some members of the middle class have become wealthy whereas others have become poor. The changes reflect a process, not a capricious set of changes. The process is even more in evidence over long time scales of centuries. These then might be examples that we can focus our attention.

Suppose we analyze the flow of wealth and assume for simplicity it is distributed to two distinct populations. If the populations are valued similarly and if the interaction is zero-sum, we expect each to receive the same payoffs in the sense that what one wins, the other loses. But what if one population believes their payoff, if they win, is much higher than the other population?  We still achieve a zero sum type game if we balance the product of the population and value for each side. If it is agreed that one side is 10 times more valuable, then this works if the other side has 10 times the population. What matters here is the interplay between competition and propaganda, since there may in fact be no objective reason for the valuation other than a possibly enforced agreement. This then is an example from the data supporting the geometric geometry view, since we can see how the flows change as a function of wealth distribution.

So we return to the question of propaganda and narrative, which we view as a question of process. In the above example, the valuation of wealth is felt equally by both populations, yet it may not be factually based. From a theoretical point of view that is really fine. We are not establishing the “truth” of the valuation, but the outcome given that both sides adopt this “truth”. Whether this is a useful exercise is ultimately a question of measurement and quantitative analysis. In a differential geometry theory of decision processes, we look for confirmation in data (behaviors) that highlight the existence of the process. For example, for weather predictions, it is not enough to predict rain versus not rain; rather we must predict in addition behaviors that change continuously with space and time like air flow. Therefore, for social behaviors we must look for flows, as an example, which change continuously with strategic position and time.

We seek to gain understanding from two distinct directions. We look at data to be convinced social behaviors are in fact geometric processes and we look at results of theoretical simulations to be convinced that geometrical processes might explain such data. At some point we hope that these two approaches meet, even though we are not there yet.

Inequality in Decision Processes

It seems to me that inequities start from the following type of conversation. There is a misperception among people I talk to that I actually wish to contribute to their capital in some way; that they have some enormously interesting endeavor, which I am dying to become a part of. If I contribute in this way their capital grows. There is no reason to suppose that my capital grows as an outgrowth of this exchange. If one person is wildly successful in getting people to contribute to their capital, they will amass a large fortune at the expense of many contributors, each of which lose little perhaps, but also each gains little.

This phenomenon may be related to the ideas in a new book on capital that speaks about the growing inequality of capital (Piketty, 2014), though capital is now used in its proper sense. From a theoretical viewpoint, I think this is related to decision process theory. In that theory, substances tend to accrete into big piles. When these piles become sufficiently large, they may create stable structures. I have not shown that to be the case, but such behaviors occur in physical theories that are similar. If it did occur, we would have a dynamically created code of conduct.

The idea of capitalism, as expressed by some, is that all boats rise together so there is an advantage to rewarding the entrepreneur. The argument against, by others, is that vast inequalities may arise in which one class does not rise at all. Such class members get submerged, to continue the metaphor. A theory should be able to demonstrate both behaviors and give the necessary and sufficient conditions for each behavior to be stable. In the theoretical view, for the first argument, each individual’s payoffs or desires cease to be important; only the payoffs of the dominant player are important. In this way their desires are not satisfied. At some point they must complain. As individuals this may not be sufficient to change the new status quo. Hence they must unite in order to make an impact.

This outcome need not be the only one since a modest redistribution of effort might honor the payoffs of most people while cutting back only slightly the capital going to the dominant player. Alternatively, there may be a world in which there is no dominant player, only a dominant strategy, one that everyone individually desires. I think that that strategy can’t be becoming wealthy (in terms of money) since that means most people will become poor, in terms of money. Of course the theory does not require that outcomes be measured in terms of money, only in terms of things that are valued by the player.

Nevertheless, suppose we buy into the concept that we should act based on our self-interest. This is supported perhaps by the theoretical view that we act based on our internal view of the payoffs. I say perhaps because it is not a given that our internal view of payoffs totally reflects our self-interest. To act purely in our self-interest against others who also purely act in their self-interest suggests a great deal of effort needed on our part to defend ourselves from attack. If we could agree with our fellow antagonists on some ground rules for such fights, we might be able to reduce the time we spend on defense and increase the time we spend gaining those goals we really strive for. Such an agreement on ground rules changes the nature of the interactions by adding a code of conduct. We need to be regulated. The best regulation is one that is minimal; it is best from the standpoint of where we start from, which is as antagonists.

Of course we might start from another position of being communal, in which case we might have somewhat more regulations. However there would be I think a need to limit the regulations so as not to totally smother the members of the community.

This line of thought suggests a zero sum or constant sum code of conduct. If we give something of value we should get something in return of comparable value. This “regulation” prevents outright theft and banditry. If nobody feels they are being taken advantage of, they need not spend their time defending their possessions. I suggest this might be the basis for monetary systems. I note that such regulations necessarily reflect conserved quantities in the theory. Perhaps there are reasons why such conserved quantities come into existence and form relatively stable structures. For the moment, let us just say that we have provided plausible reasons why they might exist.

So how do we proceed? I think we have to imagine what the code of conduct would be in a variety of stable configurations. Each configuration might have several components. To start, I suggest making the distinction of fair exchange between two players j,k: this occurs when their distance scales {{r}_{j}}-{{r}_{k}} is inactive. A related distinction is when the strategies {{s}_{{{i}_{1}}}},\cdots ,{{s}_{{{i}_{n}}}} form a coalition: this occurs when their sum is inactive. I think this can be generalized in a nice way to mean that these strategies form a surface. A special case is the constant sum game: the coalition consists of all players and their strategies. Although rather simple, I think these two distinctions revise both chapters 6 and the beginning of chapter 8. There is I think a new taxonomy to be considered. It includes the cases currently done but does suggest new possibilities, such as four strategies all part of a single player.

In suggesting revisions, I have in mind keeping the distinction of the internal (hidden) nature of the players. Their nature is hidden in the sense of hidden variables and constraints that I don’t wish to explicitly take into account. These hidden variables are private and inactive. They result from processes that are not publicly shared and take no direct part in the strategic decision processes. As with constraints in physical processes, we don’t need the details of these processes, just their effect in the energy considerations.

I also keep the notion that there are active variables, which all players agree are part of the decision making process. It may happen that some of these variables or collections of these variables are part of a code of conduct, which could mean there is a coalition. A coalition acts just like an inactive variable in all respects, except that these inactive variables are public; all players are aware of these degrees of freedom and so they form a necessary part of the solution to any decision problem. To make the distinction clear, when there is a doubt, I now call such variables code-inactive. I include the fair exchange and possibly other distinctions as they become known in this set as well. I will call the hidden inactive variables private-inactive.

Another possibility for a code-inactive variable is time. Time is interesting since we normally don’t think of time as a strategy. Yet I have considered time as the hedge player that allows a symmetrized version of the game. It thus has structural significance. As a stretch distinction, it is the learning strategy which brings us into the future. From this formulation, it is clear that time and space strategies are being treated on the same footing, with the exception of the metrical difference with time. Thus for time to be a code-inactive variable, I suggest that learning is occurring at a steady (conserved) rate. The code of conduct thus may consist of different types of code-inactive variables: the learning strategy; coalitions; and fair exchange.

For every inactive variable, there will be a payoff matrix. This is true whether or not the variable is private-inactive or code-inactive. For any solution of the active variable problem, there will be a reduction to an equivalent problem that includes code-inactive variables. This problem will have payoffs for the expanded set of players that includes the code of conduct. Once solved, one can go back to the original problem and compute the payoff matrices for each of the private-inactive players. These are the native players that provide the definition of the private-inactive variables. This has been the approach of the numerical computations carried out so far.

What provides new insight is that these different types of codes of conduct, in different combinations, lead to different solutions; we have expanded the taxonomy of possible behaviors. We have to understand in more detail what decision process we have in mind. For example, the notion of a game is centered on the idea of their being payoffs. Whenever we have payoffs we essentially have a game. For any game we should look for the code of conduct which defines the game; it is the strategy that is orthogonal to all the strategies that are active in the exchange.

We say more by characterizing the types of forces that must be associated with a code of conduct. We take this from the theory. First and foremost is the exchange or payoffs that reflect a competitive push-pull force made famous in game theory. It has analogs in physical theories: Coriolis forces in mechanics, magnetic forces in electromagnetism. The second force is less well known in economics, but we believe is also important: a gradient force whose behavior is set by surfaces of constant potential and whose strength and direction are set by the gradient normal to that surface. When the learning strategy is part of the code of conduct, we expect and see in our computations a gradient force that encourages an accumulation of value, of wealth. Other codes of conduct produce forces that discourage such accumulation. In physical theories, gravity and centripetal forces provide examples respectively. We suggest looking for such forces for each code of conduct.

You might consider it natural that the overall decision is characterized by a coalition of the whole: it is a constant sum game. From this we expect payoffs or exchanges. Indeed it is also natural to consider that economic behaviors in general are characterized by the exchange of money and value. Moreover, economic behaviors are also considered to involve fair exchange. We see a strong reason for agreeing with (Von Neumann & Morgenstern, 1944) that such behaviors should be considered as constant sum and fair exchange games with a learning strategy. This however does not mean every application of decision process theory should be thought of as such; only a subset of applications with this particular set of codes of conduct.

For example, we can have a Robinson Crusoe environment in which the total scale is not constant. This works not only for a single strategy controlled by one person, but for multiple strategies controlled by a single person. Clearly this can be generalized to multiple players where the overall sum is not a constant. These are not constant sum games. We might envision that of such possibilities, there will be those with a fair exchange between every pair of players, which thus identifies a different code of conduct. This is not the game theory of (Von Neumann & Morgenstern, 1944). Yet it is an interesting model that we can study in detail. A particular example to revisit might be the prisoner’s dilemma.

We are now ready to return to the question of inequalities in economic behaviors. Why might one group get rich and another group get poor? In our formulation of game theory, we consider not only games that are constant sum, but games in which there is fair exchange between each of the players including the learning strategy. Thus each player scale  is part of the code of conduct. If we relax the idea of fair exchanges and even perhaps the idea that the learning strategy are inactive, then our results may generate inequalities and thus shed light on how inequalities would occur as a consequence of the theoretical assumptions. We might learn whether such behaviors are stable or not. To date, we have not focused on behaviors associated with inequality in our numerical computations. This is something that should be done.