# Perfect Prediction in Minkowski Spacetime: Perfectly Transparent Equilibrium for Dynamic Games with Imperfect Information

## TL;DR

This paper extends the concept of perfect prediction equilibria to dynamic games with imperfect information, using a Minkowski spacetime framework, and explores its implications for quantum physics and strategic decision-making.

## Contribution

It generalizes the Perfectly Transparent Equilibrium to extensive form games with imperfect information and introduces a Minkowski spacetime interpretation of decision structures.

## Key findings

- The generalized equilibrium is at most unique and Pareto-optimal when no ties exist.
- Strategic games correspond to spacelike-separated decisions, dynamic games to timelike-separated decisions.
- Provides a mathematical framework linking game theory with relativistic spacetime concepts.

## Abstract

The assumptions of necessary rationality and necessary knowledge of strategies, also known as perfect prediction, lead to at most one surviving outcome, immune to the knowledge that the players have of them. Solutions concepts implementing this approach have been defined on both dynamic games with perfect information and no ties, the Perfect Prediction Equilibrium, and strategic games with no ties, the Perfectly Transparent Equilibrium. In this paper, we generalize the Perfectly Transparent Equilibrium to games in extensive form with imperfect information and no ties. Both the Perfect Prediction Equilibrium and the Perfectly Transparent Equilibrium for strategic games become special cases of this generalized equilibrium concept. The generalized equilibrium, if there are no ties in the payoffs, is at most unique, and is Pareto-optimal. We also contribute a special-relativistic interpretation of a subclass of the games in extensive form with imperfect information as a directed acyclic graph of decisions made by any number of agents, each decision being located at a specific position in Minkowski spacetime, and the information sets and game structure being derived from the causal structure. Strategic games correspond to a setup with only spacelike-separated decisions, and dynamic games to one with only timelike-separated decisions. The generalized Perfectly Transparent Equilibrium thus characterizes the outcome and payoffs reached in a general setup where decisions can be located in any generic positions in Minkowski spacetime, under necessary rationality and necessary knowledge of strategies. We also argue that this provides a directly usable mathematical framework for the design of extension theories of quantum physics with a weakened free choice assumption.