MA5 - Théorie des jeux / Game theory 1

We propose a joint pricing and capacity investment game model for coopetitive port authorities. The problem is formulated as an equilibrium problem with equilibrium constraints and solved using reformulation, linearization, and a diagonalization method. Numerical experiments are conducted to reveal model characteristics and generate managerial insights.

Keywords: Port pricing and investment; multi-leader-follower game; co-opetition

This study models optimal R&D intensity, investment timing, and capacity size under stochastically evolving quality. We start from a monopoly and extend to a duopoly situation. Minimum technology standards mitigate preemptive investment. Crucially, while moderate standards stimulate innovation, excessively strict standards prompt firms to reduce research to offset prolonged mandatory waiting costs.

Bilevel optimization models hierarchical decisions in which a leader anticipates a follower’s optimal response. Optimistic and pessimistic approaches assume
that, when multiple optimal responses exist, the follower behaves either fully cooperatively or fully adversarially. In practice, however, tie-breaking behavior is not restricted to these extremes, especially when data on past follower responses or beliefs about follower behavior suggest more nuanced follower behavior. We investigate the intermediate bilevel optimization program (I-BO), in which the follower’s selected optimal response is a decision-dependent random event, with a probability measure influenced by the leader’s decision. We formally introduce a class of such endogenous measures that capture distinct follower behaviors, including the special case of strong-weak decision-dependent cooperation I-BO (SWDI-BO). We reformulate the I-BO as a Transformed I-BO (T-I-BO) with exogenous uncertainty by defining inverse and Markov-chain transformations, which represent the follower’s response as a function of the leader’s decision and exogenous randomness. We handle the T-I-BO’s uncertainty via sample-average approximation (SAA), and we propose tailored approaches for its SAA program according to the chosen transformation. Our results show that accurately modeling follower tie-breaking behavior is crucial: distinct follower behaviors can strategically influence the leader, often leading to substantially different leader and follower optimal values. Misspecifying the follower’s behavior also produces leader solutions with significantly worse leader objective values. The proposed method also outperforms directly solving the deterministic equivalent (DE) of the SWDI-BO, which admits a closed-form expectation.

Generalized Nash Equilibrium Problems (GNEPs) provide a framework for modeling strategic interactions among multiple decision-makers with coupled constraints. Solving GNEPs is challenging due to the lack of convexity of each player’s objective and constraint functions with respect to other players’ strategies, the difficulty of ensuring equilibrium existence, and the potential non-emptiness failure of players’ strategy sets. This paper proposes a practical, relatively simple search-based method for solving GNEPs. We call this method Adaptive Greedy Local Search (AGLS), which is a stochastic method where players sample candidate actions within a dynamically adjusted trust-region radius, and only the player with the most promising improvement updates its strategy. AGLS can solve the academic GNEP test examples, which include four jointly convex, six general (non-shared constraints), and one parameterized GNEP, as well as two real-world GNEPs, namely autonomous driving and internet switching.