Sequential Duopolistic Newsvendor Game

• The paper introduces a sequential duopolistic newsvendor model where a leader commits to an inventory level before a follower optimizes, capturing key Stackelberg dynamics.
• It demonstrates how strategic quantity commitment and market allocation rules lead to demand cannibalization and competitive preemption.
• Analytical solutions via backward induction reveal equilibrium shifts influenced by demand uncertainty, cost asymmetries, and supply disruptions.

A sequential duopolistic newsvendor game is a game-theoretic extension of the classical newsvendor (newsvendor inventory) problem in the context of two competing firms (duopoly), under sequential decision-making. In these models, two firms facing uncertain demand participate in a supply chain or market, deciding on inventory (order) levels in a leader–follower (Stackelberg) structure: the leader chooses quantity first, the follower observes and optimizes its own quantity in response. This model captures non-cooperative, strategic interaction under demand uncertainty, limited market information, and possibly asymmetrical cost structures, channel power, or supply disruptions.

1. Mathematical Structure and Key Definitions

Let firm 1 (the leader) and firm 2 (the follower) make inventory decisions $q_1, q_2 \geq 0$ to satisfy stochastic demand $D$ in a market with uncertain single-period demand. Each firm faces its own per-unit cost $c_i$ and may set its own retail price $p_i$, but classical formulations often set $p_1 = p_2 = p$ for baseline comparison.

The demand allocation depends on the structure—commonly, in the duopolistic newsvendor game, market demand is filled by both firms’ inventories up to the realized demand $D$, with specific market sharing rules (e.g., proportional, priority-based, or random allocation). Let $Q = q_1 + q_2$ be total available inventory.

A prototypical Stackelberg game is played as:

• Stage 1: The leader chooses $q_1$.
• Stage 2: The follower observes $q_1$ and chooses $q_2$.
• Stage 3: Demand $D$ is realized, and each firm sells up to its available inventory, according to market allocation.

The payoff (expected profit) for firm $i$ typically takes the form: $$\Pi_i = p \, \mathbb{E}[\min\{q_i, D - (\text{other firm's sales})\}] - c_i q_i$$ under the assumption that unsold inventory is salvaged or written off. Firms solve for Nash (simultaneous) or Stackelberg (sequential) equilibria, depending on their order of play and information structure.

2. Analytical Solutions and Equilibrium Concepts

Sequential duopolistic newsvendor games are analytically characterized by backward induction. Given $q_1$, the follower solves: $$\max_{q_2 \geq 0} \; p \mathbb{E}[\min\{q_2, D - q_1\}] - c_2 q_2$$ The optimal $q_2^*(q_1)$ may be found using the single-firm newsvendor critical fractile: $q_2^*(q_1) = F^{-1}\left(1 - \frac{c_2}{p}\right)$ with $F$ the cumulative distribution function of residual demand $D - q_1$, subject to $q_2 \geq 0$. The leader anticipates this response and optimizes: $$\max_{q_1 \geq 0} \; p \, \mathbb{E}[\min\{q_1, D - q_2^*(q_1)\}] - c_1 q_1$$ This yields the Stackelberg equilibrium. Comparisons against simultaneous-move (Nash) equilibria delineate the value of channel power and information.

An important pattern is that the leader often benefits from first-mover advantage, but this can depend on demand distribution characteristics, cost asymmetries, and the structure of residual demand.

3. Strategic Effects: Inventory, Competition, and Market Allocation

Two primary competitive externalities arise:

• Inventory depletion: High inventory by the leader reduces the market opportunity for the follower (demand cannibalization).
• Quantity commitment: The leader may deliberately overstock to preempt the follower (“stock leadership"), forcing the follower to take a minor market share or none at all.

The form of the market allocation (proportional, priority, or random) critically affects strategic behavior. In priority-based models, the leader's inventory is exhausted first, enhancing first-mover advantage. In proportional allocation, both firms share available demand according to set rules or capacities.

Such interactions can be further modulated by supply chain contracts, buy-back policies, or decentralized versus centralized procurement.

4. Relations to Pull-to-Center Bias and Stereotype Modeling

Pull-to-center bias, as formalized in (Bavaud, 2010), is a context-dependent systematic shift in statistical estimation or group characterization toward the global mean, modulated by intra-group variability. While not directly a part of the classical newsvendor game, analogous effects manifest in sequential newsvendor competition:

• The leader's order exerts a centralizing force on the follower’s response, especially under incomplete information or uncertainty in the competitor's inventory (the Stackelberg framework constrains the follower’s optimal action toward the realized leader quantity).
• In stochastic/heterogeneous demand models, as group variability increases, follower responses tend to shift toward aggregate demand means—reflecting the mathematical structure of stereotype bias minimization via relative inertia ratios (Bavaud, 2010).

A plausible implication is that in sequential games with group-mean constraints (e.g., when both firms have correlated demand), the optimal response function will exhibit a form of pull-to-center behavior, where optimal follower orders are biased toward the estimated global market mean, particularly under incomplete demand revelation or signal noise.

5. Practical Implications and Extensions

The sequential duopolistic newsvendor framework underpins both supply-chain design and competitive assortment planning:

• Supply Chain Contracts: The performance of buy-back, revenue-sharing, or wholesale price contracts depends on the timing of quantity decisions and market information dissemination.
• Retail Competition and Channel Coordination: Sequential games motivate the design of mechanisms to counteract double marginalization, overstocking, and inefficient risk sharing, especially in decentralized supply chains.
• Information Structure: Asymmetry in demand signals or order observability can alter equilibrium outcomes, possibly reversing leader–follower advantages.
• Dynamic and Multi-Period Extensions: Extensions to dynamic games allow incorporation of demand learning, multi-period decision-making, strategic inventory carryover, and capacity investment.

6. Quantitative Effects and Computational Experiments

Quantitative results in the duopolistic sequential newsvendor game show:

• Overcorrection or systematic bias in inventory levels if firms base their demand estimation or competitive strategies on mis-specified distributions, analogous to the overestimation error ($\sim$2–6%) documented in weak-lensing mass inference when ignoring the alignment of centering proxies (Sommer et al., 18 Dec 2024, Sommer et al., 2023).
• Sensitivity of optimal policies and payoffs to the specific rules of sequentiality, information sharing, and demand allocation.
• Computational experiments on Stackelberg equilibria further delineate the impact of leader commitment and the potential inefficiency due to strategic competition, particularly when uncoordinated.

7. Broader Theoretical Connections and Research Directions

Sequential duopolistic newsvendor models provide testbeds for robust optimization, mechanism design, and empirical calibration of inventory competition under uncertainty. They are tightly linked to general Stackelberg games, preemption games, and models of market entry deterrence. Open research problems include:

• Characterization of equilibrium uniqueness and sensitivity to cost and demand parameters.
• Extension to $n$-player versions (oligopolistic newsvendor games), including coalition formation and collusion detection.
• Integration with behavioral phenomena such as the anchoring effect, risk aversion, and bounded rationality, leveraging advances in bias modeling from psychology and empirical economics (Xiong et al., 2019, Rothkegel et al., 2016).
• Investigation of endogenous demand learning feedback (market signals shaping future quantity choices), and the paper of contract-theoretic mechanisms for sequential inventory competition.

Sequential duopolistic newsvendor games thus constitute a foundational theoretical framework for understanding strategic inventory behavior under uncertainty in competitive markets, bridging operational research, economics, and the mathematical analysis of context-dependent bias.