Quantum Nash equilibrium in the thermodynamic limit

Abstract

The quantum Nash equilibrium in the thermodynamic limit is studied for games like quantum Prisoner's dilemma and quantum game of Chicken. A phase transition is seen in both games as function of the entanglement in the game. We observe that for maximal entanglement irrespective of the classical payoffs, majority of players choose quantum strategy over defect in the thermodynamic limit.Quantum game theory is an important extension of classical game theory to the quantum regime. The classical games might be quantized by superposing initial states, entanglement between players or superposition of strategies, for a brief account see [1]. The outcomes of a quantum game is well known for two player case however, we want to investigate the scenario when the number of players goes to infinity, i.e., the thermodynamic limit. In recent times, there have been attempts to extend the two player classical games to the thermodynamic limit by connecting it to the Ising model [2,3,4]. We do a similar analysis and connect two player quantum games to the 1D Ising model in the thermodynamic limit to figure out the strategy chosen by majority of Shubhayan Sarkar

Entanglement renders free riding redundant in the thermodynamic limit

“…In this paper, the similarity between 1D Ising model and game theory in the thermodynamic limit, as established in Refs. [2][3][4] is used to derive the equilibrium strategy for the quantum public goods game in the thermodynamic limit.…”

“…We use the same technique as used in Refs. [2][3][4], where sites are replaced by players and the spin states at each site are analogous to the player's strategic choices. Similar to Ising model, magnetization in the thermodynamic limit of a game is defined as the difference in fraction of players choosing strategy, say s 1 over s 2 .…”

“…The thermodynamic limit of the game is also defined as was done in Refs. [2,4]. Next, the public goods game is quantized using Eisert's scheme [5] and the Nash equilibrium strategy is calculated for the quantum public goods game.…”