Quantum approach to Bertrand duopoly

Abstract

The aim of the paper is to study the Bertrand duopoly example in the quantum domain. We use two ways to write the game in terms of quantum theory. The first one adapts the Li-Du-Massar scheme for the Cournot duopoly. The second one is a simplified model that exploits a two qubit entangled state. In both cases, we focus on finding Nash equilibria in the resulting games. Our analysis allows us to take another look at the classic model of Bertrand.

On subgame perfect equilibria in quantum Stackelberg duopoly

“…For the convenience of the reader we repeat the relevant material from [13], [27], [28] and [29] in order to make our exposition self-contained. Let |00 be the initial state and J(γ) = e −γ(a † A a † B −a A a B ) be a unitary operator, where γ ≥ 0 and a † i (a i ) represents the creation (annihilation) operator of electromagnetic field i.…”

“…One of the generally accepted quantum duopoly scheme is due to Li et al [13]. A rich literature applies the Li-Du-Massar scheme to the Cournot dupoly problems [14], [15], [16], [17], the Stackelberg duopoly [23], [18], [19], [20] and the Bertrand duopoly examples [21], [22], [27] The existing results motivate further study rather than exhaust the subject. Our previous work [17] shows that the quantum Cournot duopoly given by a piecewise function requires a best reply analysis to determine Nash equilibria of the game.…”

“…Our previous work [17] shows that the quantum Cournot duopoly given by a piecewise function requires a best reply analysis to determine Nash equilibria of the game. This method found further application in the quantum Bertrand duopoly (with discontinuous payoff functions) [27]. Another problem worth restudying is the Stackelberg duopoly.…”

Quantum Approach to Cournot-type Competition

“…Let us recall the key elements of the Li-Du-Massar approach to duopoly examples [10] (see [29] for more details). Let |00 be the initial state and…”

“…Since we can restrict our attention to x 1 = x 2 = x, the derivatives (28) and (29) in this case can be written as…”