# Is the essence of a quantum game captured completely in the original classical game?

## TL;DR

This paper demonstrates that certain equilibrium solutions in quantum Hawk-Dove games are unique to quantum strategies and cannot be replicated classically, highlighting the distinct nature of quantum game solutions.

## Contribution

It provides an analytical solution to the quantum 2x2 Hawk-Dove game using random strategies, showing their Pareto optimality and uniqueness from classical strategies.

## Key findings

- Quantum strategies can produce unique equilibrium solutions not achievable classically.
- Random strategies in the quantum Hawk-Dove game are Pareto optimal.
- Correlated strategies in quantum games yield classical Nash equilibria.

## Abstract

S. J. van Enk and R. Pike in PRA 66, 024306 (2002) argue that the equilibrium solution to a quantum game isn't unique but is already present in the classical game itself. In this work, we contest this assertion by showing that a random strategy in a particular quantum (Hawk-Dove) game is unique to the quantum game. In other words, one cannot obtain the equilibrium solution of the quantum Hawk-Dove game in the classical Hawk-Dove game. Moreover, we provide an analytical solution to the quantum $2\times2$ strategic form Hawk-Dove game using randomly mixed strategies. The random strategy which we describe is Pareto optimal with their payoff classically unobtainable. We compare quantum strategies to correlated strategies and find that correlated strategies in the quantum Hawk-Dove game or quantum Prisoner's dilemma yield the Nash equilibrium solution.