Editing User:PronouncedSilence (section)

== Econophysics ==

Classical Physics as a Limit of Differential Game Theory

Goal: I will attempt to explain that the classical physics theory of Lagrangian Mechanics is contained within Differential Game Theory.

Resources: 
https://en.m.wikipedia.org/wiki/Lagrangian_mechanics
https://en.m.wikipedia.org/wiki/Optimal_control (optimal control theory will not be discussed, but it allows us to further generalize the argument)
https://www.amazon.com/Differential-Games-Mathematical-Applications-Optimization/dp/0486406822 (differential game theory doesn't have a detailed Wikipedia page or many resources at all.  This textbook is the standard)

Setting: Agents located in space (continuous) and time (continuous).  Each agent has an initial state (position and time) and end goal state (position and time), which it is assumed to reach.  Each agent also has a Utility Function, which assigns a score to the agent's strategy (a path) once it reaches the end goal state.  Because the utility function scores a path, it is in the form of a functional expressible as an integral over time.

Necessary Assumption: The Utility Functions change smoothly with changes in the agents' strategies.

Simplifying Assumption: All agents' Utility Functions are equal in formula, and only their initial states, final states, and strategies can differ.

Example of a Utility Function:
Utility = Int [1/2 m (dx/dt)^2 - V(x,t)] dt

Argument: Consider a strategy that achieves minimum or maximum of the agent's Utility Function.  Because the Utility Function is smooth with respect to changes in the strategy, for these 2 classes of strategies, we have that the derivative of the agent's Utility Function with respect to change in strategy is 0, or "δUtilityFunction=0". (Note: this equation also describes inflection points in addition to minima and maxima) If we rename "Utility Function" to "Action", this is Lagrangian Mechanics.

Interpretation:

Axiom: All agents attempt to maximize their utility functions.

Necessary Assumption: I will only consider games for which agents know and have access to at least one strategy better than the worst.

Conclusion: Any "agent" that ever achieves a minimum of the utility function must actually be unconscious/inanimate.  Any agent that always achieves a maximum of the utility function is perfectly rational.

Other Possible Conclusion: I won't go through the math here, but it turns out that in the example Utility Function given above, we can prove that "m > 0" and "δUtilityFunction=0" imply "δ^2 UtilityFunction > 0", which means the described strategy achieved a minimal Utility Function and is therefore the object must be inanimate.  Since this Utility Function (and "m > 0) matches Newtonian Mechanics, we know that Newtonian Mechanics describes inanimate matter.

**Why is this important**: Because physics is a boundary of game theory
1. we can expand physics to include conscious agents alongside inanimate matter.
2. we can now import any specific Action/UtilityFunction of Lagrangian Mechanics into Game Theory.  This allows us to construct games where the utility functions depends on guage fields, for instance, which will allow us to rigorously formulate and analyze the Guage Theory of Economics.