# Nash Equilibria and Undecidability in Generic Physical Interactions—A Free Energy Perspective

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

Bob ${s}_{B}$ | |||

C | D | ||

Alice ${s}_{A}$ | C | $(3,3)$ | $(0,5)$ |

D | $(5,0)$ | $(1,1)$ |

- In such a setting, D is the dominant strategy and (D, D) is the Nash equilibrium. The optimal joint strategy is (C, C), in which both players cooperate, but this is clearly unstable.

## 2. Representing Generic Interactions as Games

#### 2.1. Physical Interaction Is Information Exchange

#### 2.2. Actions Require Quantum Reference Frames

#### 2.3. VFE Provides a Generic Payoff Function

#### 2.4. Normal-Form Games Are Special Cases

#### 2.5. Example: The IPD as a Prediction Game

## 3. Generic Limits on Observation

**Theorem**

**1**

**.**Let S be a finite system and Q be a QRF implemented by ${H}_{S}$. The following statements hold:

- 1.
- S cannot determine, by means of Q, either Q’s dimension $\mathrm{dim}\left(Q\right)$, Q’s associated sector dimension $\mathrm{dim}\left(\mathrm{dom}\right(Q\left)\right)$, or Q’s complete I/O function.
- 2.
- S cannot determine, by means of Q, the dimension, associated sector dimension, or I/O function of any other QRF ${Q}^{\prime}$ implemented by S.
- 3.
- S cannot determine, by means of Q, the I/O function or dimension of any QRF ${Q}^{\prime}$ implemented by any other system ${S}^{\prime}$, regardless of the relation of S to ${S}^{\prime}$, from ${S}^{\prime}=S$ to ${S}^{\prime}=E$, inclusive.
- 4.
- Let $S={S}_{i}{S}_{j}$, in which case ${E}_{i}=E{S}_{j}$. Then, ${S}_{i}$ cannot determine, by means of a QRF ${Q}_{i}$, the I/O function or dimension of any QRF ${Q}_{j}$ implemented by ${S}_{j}$.

**Proof.**

## 4. Convergence and Equilibria in Generic Interactions

#### 4.1. Convergence Driven by the FEP

- Does an equilibrium that minimizes VFE—maximizes predictive accuracy—for both S and E exist?
- Can S or E determine by finite observation that they have reached such an equilibrium?
- Can S or E determine that they are on a trajectory toward such an equilibrium?

- Nash’s theorem [3] gives a positive answer to the first of these questions. Such equilibria have also been characterized in the classical FEP literature, where it has been shown how mutual predictability induces generalized synchrony [67,68]; see, e.g., [69,70,71,72] for relevant earlier work and [73,74] for applications. A simple and limiting example of generalized synchrony is convergence to thermodynamic equilibrium, in which prediction ceases at “stasis” because there is no longer available thermodynamic free energy to support computation. Unlike the classical formulation, the quantum formulation of the FEP does not employ an embedding space to enforce separability between S and E; here, the limit of perfect mutual predictability corresponds to entanglement [29], as discussed further in Section 5 below.

#### 4.2. Example: IPDs and Generalized Imitation Games

#### 4.3. Example: The SPD and Its Limit Sets

#### 4.4. Prediction, Regulation, and Generalized Synchronization—A Circularity of Idealizations

- Good Regulator ⟹ Identical Synchronization ⟹ Winnable Imitation Game
- (or decidable Turing Test) ⟹ Good Regulator
- The source of undecidability in every case is clear: if two interacting systems are identical, self-reference and other-reference are indistinguishable, and Gödel’s theorem applies equally to either.

## 5. Quantum Games

#### 5.1. Definitions and Formalism

**Definition**

**1.**

- (a)
- The underlying Hilbert space $\mathcal{H}$ of the physical system;
- (b)
- The initial state $\nu \in \mathcal{S}\left(\mathcal{H}\right)$, where $\mathcal{S}\left(\mathcal{H}\right)$ is the associated game-state space;
- (c)
- ${S}_{A}$ and ${S}_{B}$ are sets of permissible quantum operations of players A and B, respectively;
- (d)
- ${P}_{A}$ are ${P}_{B}$ are the utility functions specifying the respective utility for each player.

**Definition**

**2.**

- (a)
- A quantum strategy is called a dominant quantum strategy of Alice if ${P}_{A}({s}_{A},{s}_{B}^{\prime})\ge {P}_{A}({s}_{A}^{\prime},{s}_{B}^{\prime})$ for all ${s}_{A}^{\prime}\in {S}_{A},{s}_{B}\in {S}_{B}$. Likewise, a dominant strategy for Bob is defined.
- (b)
- A pair $({s}_{A},{s}_{B})$ is said to be an equilibrium in dominant strategies if ${s}_{A}$ and ${s}_{B}$ are the players’ respective dominant strategies.
- (c)
- A combination of strategies $({s}_{A},{s}_{B})$ is called a Nash equilibrium if$${P}_{A}({s}_{a},{s}_{b})\ge {P}_{A}({s}_{a}^{\prime},{s}_{B})$$$${P}_{B}({s}_{A},{s}_{B})\ge {P}_{B}({s}_{A},{s}_{B}^{\prime})$$for all ${s}_{A}^{\prime}\in {S}_{A}$ and ${s}_{B}^{\prime}\in {S}_{B}$.
- (d)
- A pair of strategies $({s}_{A},{s}_{B})$ is called Pareto optimal if it is not possible to increase one player’s payoff without reducing the other’s payoff.

#### 5.2. Example: The Decoherence Game

#### 5.3. Example: The Bell/EPR Game

#### 5.4. QRFs, Contextuality, and Asymptotic Entanglement

## 6. Discussion

#### 6.1. Rationality

#### 6.2. Entropy of Quantum Games

- high entropy ⟺ low rationality in the players’ behavior.

#### 6.3. Alternative Equilibria

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CA | Cellular automaton |

CCCD | Cone–cocone diagram |

CH | Continuum hypothesis |

CHSH | Clauser–Horne–Shimony–Holt |

EPR | Einstein–Podolsky–Rosen |

FEP | Free energy principle |

GIG | Generalized imitation game |

GT | Game theory |

HP | Holographic principle |

I/O | Input/output |

IPD | Iterated prisoner’s dilemma |

KL | Kullback–Leibler |

LOCC | Local operations and classical communication |

MB | Markov blanket |

NE | Nash equilibrium |

NP | Nondeterministic polynomial |

PD | Prisoner’s dilemma |

PPAD | Polynomial parity arguments on directed graphs |

QRE | Quantal response equilibrium |

QRF | Quantum reference frame |

RNN | Recurrent neural network |

SPD | Spatialized prisoner’s dilemma |

TM | Turing machine |

TQFT | Topological quantum field theory |

VFE | Variational free energy |

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**Figure 1.**A holographic screen $\mathcal{B}$ separating systems S and E with an interaction ${H}_{SE}$ given by Equation (1) can be realized by an ancillary array of noninteracting qubits that are alternately prepared by S (E), and then, measured by E (S). Qubits are depicted as Bloch spheres [38]. There is no requirement that S and E share preparation and measurement bases, i.e., quantum reference frames, as described below. Adapted from [33] Figure 1, CC-BY license.

**Figure 2.**“Attaching” a CCCD to an intersystem boundary $\mathcal{B}$ depicted as an ancillary array of qubits. The operators ${M}_{i}^{k}$, $k=S$ or E, are single-bit components of the interaction Hamiltonian ${H}_{SE}$. The node C is both the limit and the colimit of the nodes ${\mathcal{A}}_{i}$; only leftward-going (cocone-implementing) arrows are shown for simplicity. See [29,30,31,47] for details. Adapted from [31], CC-BY license.

**Figure 3.**Cartoon representation of a system A that deploys a QRF $\mathbf{X}$ (red triangle) to measure the state of an external system X in its informational environment (i.e., a sector X of its boundary $\mathcal{B}$), and then, deploys a second QRF $\mathbf{Y}$ (green triangle) to write the outcome to a memory sector Y. This process induces one “tick” of an internal clock ${\mathcal{G}}_{ij}$ that defines an internal elapsed time ${t}_{S}$. The process is powered by a thermodynamic loop from (thermodynamic free energy in) and back to (waste heat out) the physical environment E. Adapted with permission from [37], CC-BY license.

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Fields, C.; Glazebrook, J.F.
Nash Equilibria and Undecidability in Generic Physical Interactions—A Free Energy Perspective. *Games* **2024**, *15*, 30.
https://doi.org/10.3390/g15050030

**AMA Style**

Fields C, Glazebrook JF.
Nash Equilibria and Undecidability in Generic Physical Interactions—A Free Energy Perspective. *Games*. 2024; 15(5):30.
https://doi.org/10.3390/g15050030

**Chicago/Turabian Style**

Fields, Chris, and James F. Glazebrook.
2024. "Nash Equilibria and Undecidability in Generic Physical Interactions—A Free Energy Perspective" *Games* 15, no. 5: 30.
https://doi.org/10.3390/g15050030