Nash Equilibrium Owen Allocation and Gain Sharing in Stochastic Biform Games for Linear Production and Sequencing

In the world of business and manufacturing, companies often face a fundamental tension between competition and cooperation. On one hand, firms compete fiercely for limited resources, trying to secure the best materials or the fastest production slots to maximize their own profits. On the other hand, once these resources are secured, the same firms often discover that working together yields far greater rewards than acting alone. Imagine a group of factories that each hold a different piece of a puzzle; individually, they can only produce a fraction of what they could if they pooled their pieces. The challenge lies in the transition: how do you decide who gets what resource in the competitive phase, and then, once the resources are gathered, how do you fairly split the massive profits that come from working together? This is the core puzzle of what researchers call "biform" games, a hybrid approach that blends the strategic maneuvering of non-cooperative competition with the value-sharing principles of cooperative teamwork.

For decades, economists and operations researchers have studied these two phases separately. They have built complex models to predict how companies compete for supplies and separate models to determine how to split the resulting gains. However, real-world scenarios are rarely so clean. Resources are often uncertain; a supplier might fail, a shipment might be delayed, or market prices might fluctuate wildly. Furthermore, the decisions made in the competitive phase directly shape the opportunities available in the cooperative phase. If a company chooses a risky supplier to save money, it might end up with a smaller resource bundle, which changes the entire calculation for how the final profits should be shared. Until now, there has been no comprehensive framework that could handle this uncertainty while simultaneously modeling both the competitive scramble for resources and the subsequent cooperative profit-sharing.

A researcher, Mohammad Reza Omidi at the University of Bonab, has developed a new mathematical framework to solve this specific problem. They focused on two critical industrial processes: linear production, where companies combine raw materials to make products, and sequencing, where jobs must be ordered on a machine to minimize delays and costs. The researcher created a model that simulates a two-stage game. In the first stage, companies compete to choose which supplier to buy from. This choice is strategic and competitive, but it is also fraught with uncertainty, as suppliers might fail to deliver the full amount of materials promised. In the second stage, once the resources are secured (or partially secured due to disruptions), the companies form a coalition to produce goods or process jobs as efficiently as possible. The key innovation is how the researcher handles the uncertainty. Instead of guessing a single outcome, they run thousands of simulations of different possible futures—some where suppliers fail, some where prices spike, and some where everything goes smoothly. They then calculate the best strategy for the companies by looking at the average result across all these possible worlds.

The researcher found that by using this "scenario-wise" approach, companies can make better decisions than if they simply ignored the risk of disruption. In their simulations, which involved five companies competing for resources from three different suppliers, the new framework increased the expected total profit by more than seven percent compared to older, deterministic models that assume the future is certain. More importantly, the new model significantly improved the worst-case scenario. When disruptions occurred, the companies using the new framework still managed to retain a much larger portion of their potential profits than those using traditional methods. This suggests that accounting for uncertainty does not just add complexity; it provides a safety net that protects value when things go wrong.

A crucial part of their discovery involves how the companies share the rewards. In the first stage, the companies compete to secure resources. In the second stage, they must agree on how to split the profits generated by their combined efforts. The researcher introduced a specific method for this sharing, called the Owen allocation, which is derived directly from the mathematical properties of the production process itself. This method ensures that the person who brings the most valuable resource to the table gets a fair share of the reward, without needing arbitrary rules or negotiations. They also applied a similar logic to the sequencing problem, where companies decide the order in which jobs are processed. In this context, they introduced an endogenous gain-sharing rule, a sophisticated mechanism where the division of cost savings is not fixed but depends dynamically on each company's bargaining power and investment levels. This means that firms that invest more in reliability or possess stronger negotiating positions automatically receive a larger share of the savings, creating a direct incentive for companies to improve their efficiency and reliability. They found that when companies can strategically influence the order in which they are processed, a unique set of stable outcomes emerges. In these stable outcomes, the companies settle into an arrangement where no single company can improve its position by unilaterally changing its request for a different spot in line.

The study also revealed that the structure of the competition matters deeply. If companies are allowed to split resources unevenly or if there are more than two suppliers to choose from, the stability of the system can break down. In some complex configurations, the researcher found that no stable agreement exists where everyone is satisfied with their choice. This highlights a delicate balance: while having more options sounds beneficial, it can sometimes lead to a chaotic situation where companies are constantly shifting strategies, unable to settle on a plan that works for everyone. The researcher demonstrated that their proposed algorithm can handle these complexities efficiently, solving problems with up to twenty companies in just a few minutes, which makes the approach practical for real-world industrial applications.

The implications for managers are clear. The research suggests that companies should not rely on a single supplier, even if it seems cheaper or more convenient. By diversifying their supplier portfolio, they can hedge against the risk of disruption and secure a more stable path to cooperation. Furthermore, the study shows that investing in the reliability of suppliers pays off. Because the profit-sharing mechanism rewards those who contribute more reliable resources, companies have a built-in incentive to ensure their supply chains are robust. The framework also addresses the issue of bargaining power. Companies with stronger negotiating positions or those that invest more in the relationship receive a larger share of the cost savings, which encourages them to continue investing in efficiency and reliability.

Ultimately, this work bridges a gap between the theoretical world of game theory and the messy reality of industrial operations. It provides a tool for understanding how uncertainty reshapes the rules of competition and cooperation. By proving that stable, profitable outcomes exist even when the future is uncertain, the researcher offers a way for businesses to navigate the volatility of modern supply chains. They show that by thinking ahead about the worst-case scenarios and planning how to share the rewards fairly, companies can turn a chaotic environment into a structured opportunity for growth. The findings confirm that in a world of uncertainty, the most successful strategy is not just to compete harder, but to compete smarter, with a clear plan for how to cooperate when the dust settles.