How to extend the use of value-based strategy models to situations with large quasi-rents shared among multiple actors, such as ecosystems? How to consider how players understand competition in value-based models? How to overcome some limitations of these models such as lack of uniqueness of solutions? In this paper, we extend the reach of value-based strategy by revisiting the celebrated biform games model to answer these questions. 

Operationally, we make players evaluate their payoff from the cooperative stage of the game according to a generalized expectation over their value capture. Our solution has several advantages: (i) It subsumes the original biform framework and seamlessly integrates recent works providing bounds to value capture (ii) It allows solving issues such as the possible non-uniqueness of solutions and invariance to the competitive environment structure while maintaining the role of competition in determining value capture (iii) It remains axiomatically justified on behavioral grounds (iv) It permits richer preferences representations that, for example, can include subjective distortions of objective chances of value capture (v) It further leads the way to the use of generalized preference representations in the value-based framework.