Abstract
Game theory is now a vital method of conflict analysis and decision making that has been applied in many disciplines, including biology, economics, political science, and law. Nash equilibrium, as one of the most important concepts in game theory, has played a fundamental role in the development of game theory and its applications. However, a game can have many Nash equilibria, some of which may be inconsistent with intuitive notions about the outcome of the game. To reduce this ambiguity and eliminate some of these counterintuitive equilibria, the concept of the perfect equilibrium was introduced. A perfect equilibrium is a Nash equilibrium that accounts for the possibility of off-the-equilibrium play by assuming that the players may choose unintended strategies with negligible probability. However, a game can still have many perfect equilibria, some of which are undesirable. To exclude some of these undesirable perfect equilibria, the concept of proper equilibrium was introduced. Nevertheless, the notion of a proper equilibrium can lead to a severe numerical problem when one intends to compute such an equilibrium. To remedy this deficiency, perfect d-proper equilibrium, as a strict refinement of the concept of perfect equilibrium, was introduced. A significant feature of perfect d-proper equilibrium is that it is numerically much easier to compute than a proper equilibrium. The introduction of perfect, proper, and perfect d-proper equilibria have significantly advanced the development of game theory and its applications.Perfect, proper, and perfect d-proper equilibria are three strict refinements of the Nash equilibrium concept and always exist in a game. Computing perfect, proper, and perfect d-proper equilibria plays an important role in the application of game theory. However, these refinements of Nash equilibrium have very few methods. To derive an alternative procedure for selection of a perfect equilibrium, we extend Nash’s continuous mapping to a special perturbed game that deforms from a trivial game to the original game as an extra variable varies from one to zero. As a result of this extension and the application of a triangulation with a continuous refinement of grid size, we develop a simplicial path-following method for approximating perfect equilibria. The method starts from a totally mixed strategy profile and leads to a perfect equilibrium at a limit. The numerical results further confirm the method’s effectiveness.
To exclude some undesirable perfect equilibria, we formulate continuous equilibrium mapping for a specifically defined perturbed game that deforms with an extra variable. As a result of this formulation and application of a triangulation with a continuous refinement of grid size, we develop a simplicial path-following method for approximating a proper equilibrium. The method follows a simplicial path that starts from any given totally mixed strategy profile and leads to a proper equilibrium. The numerical results further verify the method’s effectiveness.
Furthermore, to solve the serious numerical problem of attaining a proper equilibrium, we define continuous mapping such that every Nash equilibrium of a special perturbed game is a fixed point of the mapping. With this mapping and a triangulation with a continuous refinement of grid size, we develop a simplicial path-following method to approximate a perfect d-proper equilibrium. The numerical results further demonstrate the method’s effectiveness.
| Date of Award | 13 Apr 2017 |
|---|---|
| Original language | English |
| Awarding Institution |
|
| Supervisor | Chuangyin DANG (Supervisor) |