Demand-Elasticity Misperception in Cournot Duopoly

• The paper reveals that quantum-induced output mixing and stochastic belief errors lead to misperceptions of demand elasticity, profoundly affecting equilibrium prices and profits.
• It utilizes advanced methodologies including quantum game theory, stochastic order analysis, and fractional memory models to capture strategic interdependence and dynamic learning.
• The study shows that misestimated elasticity results in suboptimal pricing and chaotic dynamics, highlighting critical implications for market stability and efficiency.

Demand-elasticity misperception in Cournot duopoly refers to systematic bias or error in how firms infer, estimate, or react to the responsiveness of market demand to changes in total output, often due to non-classical strategic interactions, stochastic beliefs, dynamic updating processes, or model misspecification. The phenomenon is central to both the comparative statics and stability properties of duopolistic competition, as misperception can shift equilibrium quantities, prices, profits, and even alter the existence and uniqueness of equilibrium outcomes.

1. Quantum Effects and Strategic Interdependence

Quantum game-theoretic extensions of Cournot duopoly (Rahaman et al., 2012) demonstrate that enlarging the strategic space via quantization fundamentally distorts the link between output and market price, leading to altered perceptions of demand elasticity. In the quantized Hotelling–Smithies model, the classical choice of output is replaced by quantum mechanical operators acting on entangled electromagnetic field states. The quantity outcomes for each firm are given by:

$$q_1 = x_1 \cosh \gamma + x_2 \sinh \gamma,\quad q_2 = x_2 \cosh \gamma + x_1 \sinh \gamma$$

where $\gamma$ is the entanglement (squeezing) parameter.

The mixing of output decisions (via $\cosh \gamma$ and $\sinh \gamma$) means each firm’s strategic intent is partially "leaked" into the observed output of its rival, a property without classical analog. Thus, when classical methods are used to infer the demand elasticity from observed outcomes, the result may be erroneous: the quantum-induced correlation is mistaken for altered market responsiveness. Strong entanglement ($\gamma \gg 0$) produces higher profits and output levels at equilibrium, misleading analysts to attribute changes to demand-side elasticity rather than to non-classical strategic interdependence.

2. Supply Chain Uncertainty, Stochastic Orders, and Elasticity Misperception

When demand is uncertain and the supplier must optimally determine wholesale pricing without knowing the actual market realization, demand-elasticity misperceptions arise as errors in the supplier’s belief distribution (Koki et al., 2018, Melolidakis et al., 2018). The supplier’s optimal wholesale price $r^*$ is the unique fixed point of the mean residual life (MRL) function:

$$r^* = m(r^*) = \mathbb{E}[\alpha - r^* \mid \alpha > r^*]$$

where $F$ is the supplier’s belief about the demand parameter $\alpha$.

A misperceived, less elastic demand (i.e., overstating the "stiffness" or underestimating the variability of $F$) shifts $m(r)$ upward, resulting in a higher $r^*$, lower equilibrium retail quantity, and higher retail price. Comparative statics using stochastic orders rigorously characterize how changes in $F$ (including those caused by misperceived elasticity) affect equilibrium prices and supply chain efficiency. The inefficiency due to misperception is quantified by the Price of Anarchy (PoA), which increases when $r^*$ is set suboptimally.

3. Misspecified Learning and Evolutionary Stability

Misestimation of demand elasticity enters through agents’ choice of model specification in evolutionary competition (He et al., 19 Sep 2025). Resident agents hold the correct slope $r$ in the linear demand function $P = \beta - r(a_1 + a_2) + \epsilon$, while entrant agents misperceive the slope as $\hat{r} \neq r$. Bayesian updating is used for the intercept, but the misperceived slope persists in the agent’s equilibrium beliefs and subjective best response:

$$a_{\text{entrant}} = \frac{\beta - c}{2 \hat{r} + r}$$

Entrants whose misperceived elasticity (lower $\hat{r}$) yields output close to the Stackelberg level can achieve higher fitness than residents, especially when rare. The evolutionary stability of the resident population can be undermined by such misspecifications, leading to a persistent wedge between actual and perceived demand responsiveness in the strategic environment.

4. Dynamic and Heterogeneous Models: Fractional Memory, Stability, and Bifurcations

Long-memory dynamics, gradient adjustment, and local estimation mechanisms are found to amplify or dampen demand-elasticity misperception in dynamic Cournot systems (Xin et al., 2019, Li et al., 2022). Fractional-order difference operators, e.g.,

$$^{C}\Delta_a^{\nu}q_i(t) = \alpha_i q_i(t^+) \Phi_i(t^+)$$

enable firms to update output based on weighted historical profits, with the fractional parameter $\nu \in (0,1]$ controlling memory. When $\nu$ is sufficiently low, chaotic dynamics emerge, and firms’ effective learning of demand elasticity is impaired, giving rise to persistent misperception.

In heterogeneous duopolies where firms differ in rationality—fully rational, boundedly rational, or local monopolistic approximation (LMA)—the stability region of the equilibrium varies counterintuitively. Models with boundedly rational rivals (who use naive past-output forecasts) produce broader stability regions, while those with fully rational counterparts yield narrower zones, due to the more aggressive output responses induced by precise elasticity expectations. Period-doubling bifurcations are the primary route to instability as adjustment speeds increase, with demand elasticity misperception functioning as a bifurcation factor via systematic error in best-response curves.

5. The Role of Strategic Inventories and Contest-Based Incentives

Strategic inventories act as temporal signals that distort the perceived demand elasticity in multi-period Cournot supply chains (Hu et al., 2021). Retailers holding inventory as a strategic tool change the effective slope of the price–quantity tradeoff, captured in the Lerner index:

$$L^{D}_{\text{Duo}}(x) = \frac{4073 - 1032x - 20592x^2}{2(7849 - 1632x - 3600x^2)}$$

where $x = h/a$ is the normalized inventory cost. An inventory build-up, facilitated by dynamic contracts with low holding costs, exaggerates apparent market power and softens the responsiveness of price to output, causing retailers to misperceive the market’s true elasticity.

Contest-based incentives (as opposed to classical profit maximization) further decouple output decisions from correct elasticity perception (Amir et al., 31 Aug 2025). The unbeatable strategy drives the equilibrium to perfect competition ($Q^* = \bar{Q}$, where $P(\bar{Q}) = c$) regardless of the underlying demand elasticity. Thus, demand elasticity is often underweighted or misperceived in the contest model, resulting in zero producer profits and a transformational recession in output when transitioning back to classical competition.

6. Mathematical Conditions for Uniqueness and Stability under Uncertainty

Uniqueness of equilibrium in Cournot duopoly under uncertain demand depends critically on statistical regularity conditions of the demand intercept distribution. The decreasing mean residual demand (DMRD) and increasing generalized failure rate (IGFR) properties ensure monotonicity and log-concavity in the adjusted per-unit profit function:

$$\tilde{\pi}(x) = m(x) - \frac{c}{F(x)} - \frac{x}{n}$$

where $m(x)$ is the mean residual demand and $F(x)$ the distribution function (Leonardos et al., 2019). If these conditions hold, even when firms misperceive point elasticity, equilibrium assignments remain unique and robust, providing systemic protection against strategic miscalculations and coordination failures.

7. Summary Table: Mechanisms and Consequences of Demand-Elasticity Misperception

Mechanism	Formulation/Agent Behavior	Misperception Consequence
Quantum entanglement	Output mixing via $\cosh\gamma, \sinh\gamma$	Elasticity misread as stronger
Stochastic orders (MRL)	Supplier beliefs about demand F	Excessive/insufficient pricing
Misspecified slope ($\hat{r}$)	Entrant updates/intercepts, Stackelberg bias	Output/fitness advantage
Fractional memory ($\nu$)	History-damped output updating	Chaos, mislearned elasticity
Inventory signaling	Intertemporal price-response alteration	Aggressive/non-optimal output
Contest incentive	Unbeatable strategies focus	Elasticity neglected, zero profit
DMRD/IGFR property	Statistical regularity in F	Unique, robust equilibrium

In conclusion, demand-elasticity misperception in Cournot duopoly is a multifaceted phenomenon, driven by quantum strategic effects, supply chain uncertainty, dynamic updating, contest incentives, and stochastic environments. Its theoretical characterization requires integrating operator-theoretic, stochastic, evolutionary, and symbolic computation techniques. The consequences manifest in equilibrium output, prices, profit allocation, stability, and sometimes—under contest or quantized environments—a fundamental shift in the perceived nature of market competition itself.