# Boundary Conditions and Common Ground

This is an inquiry on boundary conditions for decision process theory models and how important they are. In a way, the answer is trivial. Of course boundary conditions matter for any theory based on elliptic partial differential equations. The question is whether the boundary shapes play an essential role.

Let’s take an example from electrical engineering. A conductor shape with an applied voltage produces an electric field that is truly characterized by the shape, as much if not more so than the voltage (depending on the value of the voltage). To make the same claim for decision processes, we have to argue that there is something analogous to the “shape” in the space of strategies as well as something analogous to a conductor that dictates the field on the boundary of the shape.

We move in this direction in discussing the gauge conditions for the payoff field potential. we can argue that on the boundary it might be true that the payoff is “fair” in the sense that there should be no player bias field and that any strategic payoff moving a strategy from one direction in the surface to another should be zero. In other words, there should be no interest in changing the player engagements on the boundary and no strategic advantage to change payoffs that reside totally within the surface.

In transforming a game into a symmetric game for example, it is common practice to assume that there are no self-payoffs. The strategies owned by a player have a certain internal stability as seen by that player. Let’s call this property strategy neutral.

A surface that is both fair and strategy neutral is a strategically conducting surface. This is a new distinction. We argue that strategies that occur within a region bounded by a strategically conducting surface will be determined more by the shape than by the field values on the boundary. They thus form an important subset of surfaces to consider in the general case. These are surfaces that are strategy neutral.

A less imposing name might be common ground. This brings two concepts together: one from negotiations and one from electrical engineering. To get people to agree, one needs to go to a place which is strategy neutral for each party. The idea of a ground in electrical engineering is a surface with a constant potential: a conducting surface.

This definition of strategy neutral is based on the behavior of payoffs. However payoffs are only a part of the story. Payoffs result from payoff potentials that are the part of the measure between one active and one inactive strategy of the space. More generally, we have a metric  that represents a measure between any two points. This is a tensor potential ${{g}_{ab}}$  that generalizes the vector potentials ${{A}^{k}}_{a}$ that determine the player payoffs, which are like the electric and magnetic fields in electrical engineering. In particular, if both points are active, then the tensor potential determines the frame transformation $\mathbf{\omega }={{\omega }^{a}}_{bc}d{{x}^{c}}$, whose differential is the flux flow through a 2-surface. The covariant and gauge invariant differential is $R=d\mathbf{\omega }+\mathbf{\omega }\wedge \mathbf{\omega }$ and the analogy is that this flow is zero through the surface.

This flow of flux through a surface is a flow of the curvature of space, a flow of spatial bias. In decision processes, there are three distinct types of spatial bias flow, each arising from a distinct type of surface: there is the flux flow through mixed spatial frame between active and inactive directions ${R_{{\bf{ak}}}}$; there is the purely inactive dimension flux flow ${R_{{\bf{jk}}}}$; and there is the purely active dimension flux flow ${R_{{\bf{ab}}}}$. The constraints for each would be that the flow through the dual space $*\left( {{{\bf{U}}^{\bf{\alpha }}} \wedge {{\bf{U}}^{\bf{\beta }}}} \right)$ is zero. This would imply for each case that ${R_{\alpha \beta \alpha \beta }} = 0$ with the appropriate changes in the subscripts.

We need to further investigate whether we impose all three conditions. We also need to think how these conditions would be applied using the linear recursion method. Nevertheless, the basic conclusion holds: common ground means that this degree of cooperation does not vary on the boundary. In other words, no point on the boundary is better or worse than any other from the perspective of the cooperation or competition.

We need to assess how much constraint we impose on this common ground. If we look at electromagnetic fields on a conducting surface, we find that in fact we have curvature components based on electric fields normal to the surface and magnetic fields in the surface. Based on this observation, we suggest using the idea of a weakly conducting strategic surface in which we impose only conditions in the surface. This allows a variety of effects due to the behavior along the normal to the surface.

This suggests that we impose only the Dirichlet conditions $\delta {A^j}_{\bar a} = 0$ for the payoff fields on the boundary. A similar argument suggests imposing in addition the conditions $\delta {\gamma _{jk}} = 0$ for the inactive components and $\delta {g^{\bar a\bar b}} = 0$ for the active components. As a consequence there will be some acceleration field flow through the surfaces. These are considered part of the initial “forcing functions” that may in fact yield insightful results about the decision process.

In picking a boundary in which there is common ground, we assume that the boundary is far from the action: that there are no strategic preferences. So what does it mean to be far from the boundary? In physical problems, far from the boundary has the intuitive meaning that we are far from where there are significant interactions. We accomplish this by going far enough in any direction. The scene of the action is left.

That is not how we currently think of strategies in game-theory. We think all values of strategies are possible. In Vol. 2 however, we took a different approach. We go to the co-moving frame. Here the numerical models with “locked” behaviors have the property that action occurs in a localized region. Not all parts of space are equally populated. If we go far from the populated areas, it makes little sense to talk about strategic advantage. This distinction persists in any frame. The populated areas are associated with regions in which game theory would apply. In Vol. 2 we took the stance that we should set boundary conditions in these populated areas. We now suggest a different stance and say we should go far away from such areas to where there is common ground and no strategic preference exists and set our boundary conditions there.