Proposition 2: more quantum investment despite higher scaling cost
ProvedQuantumEcon.Duopoly.proposition_2cournoteconomicsgame-theoryp2o-batch-pfp2ap2o-gran-per-chapterp2o-plan-paperp2o-v1quantum-economic-advantage
There are parameters satisfying the paper’s standing and Lemma 1 assumptions with for which a subgame-perfect equilibrium exists and every such equilibrium has . Thus the quantum firm can invest more in market creation even when its quadratic investment cost coefficient is larger.
This statement is a source-indexed target for the two-stage model.
Preamble
import Definitions.Def_QuantumEcon_Duopoly_Setting set_option autoImplicit false
Formal statement
namespace QuantumEcon.Duopoly
/-- Bova, Goldfarb, Melko, Quantum Economic Advantage, NBER Working Paper No. 29724
(February 2022), PDF p. 14, Proposition 2; proof PDF pp. 27–28.
Formalization Note: investments and quantities are nonnegative. The exhibited instance
is `a = 20`, `c = 2`, `γ = 1`, `B₁ = 10`, `B₂ = 8`. Firm 1 is quantum. -/
theorem proposition_2 :
∃ P : Params, Standing P ∧ Lemma1Cond P ∧ P.B₂ < P.B₁ ∧
(∃ x₁ x₂ : ℝ, ∃ σ : ℝ → ℝ → ℝ × ℝ, IsSPE P x₁ x₂ σ) ∧
∀ x₁ x₂ : ℝ, ∀ σ : ℝ → ℝ → ℝ × ℝ,
IsSPE P x₁ x₂ σ → x₂ < x₁ := by sorry
end QuantumEcon.Duopoly
Source
Bova, Goldfarb, Melko, Quantum Economic Advantage, NBER Working Paper No. 29724 (February 2022), PDF p. 14, Proposition 2; PDF pp. 27–28, proof
Human review
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.