We study the fashion game, a classical network coordination/anti-coordination game employed to model social dynamics in decision-making processes, especially in fashion choices. In this game, individuals, represented as vertices in a graph, make decisions based on their neighbors’ choices. Some individuals are positively influenced by their neighbors while others are negatively affected. Analyzing the game’s outcome aids in understanding fashion trends and flux within the population. In an instance of the fashion game, an action profile is formed when all individuals have made their choices. The utility of an individual under an action profile is defined according to the choices he and his neighbors made. A pure Nash equilibria is an action profile under which each individual has a nonnegative utility. To further study the existence of pure Nash equilibria, we investigate an associated optimization problem aimed at maximizing the minimal individual utility, referred to as the utility of a fashion game instance. The fashion game with two different but symmetric actions (choices) has been studied extensively in the literature. This paper seeks to extend the fashion game analysis to scenarios with more than two available actions, thereby enhancing comprehension of social dynamics in decision-making processes. We determine the utilities of all instances on paths, cycles and complete graphs. For instances where each individual likes to anti-coordinate, graph is planar and three actions are available, we illustrate the time complexity of determining the utility of such instances. Additionally, for instances containing both coordinating and anti-coordinating individuals, we extend the results on the time complexity of determining the utility of instances with two available actions to cases with more than two actions.

我们研究时尚游戏，这是一种经典的网络协调/反协调游戏，用于模拟决策过程中的社会动态，尤其是在时尚选择中。在这个游戏中，个体（在图形中表示为顶点）根据邻居的选择做出决定。有些人受到邻居的积极影响，而另一些人则受到负面影响。分析游戏的结果有助于了解流行趋势和人群中的波动。在时尚游戏的实例中，当所有人都做出了选择时，就会形成一个动作配置文件。个人在行动概况下的效用是根据他和他的邻居所做的选择来定义的。纯纳什均衡是一种动作曲线，在该曲线下，每个个体都具有非负效用。为了进一步研究纯纳什均衡的存在，我们研究了一个相关的优化问题，旨在最大化最小的单个效用，称为时尚游戏实例的效用。具有两种不同但对称的动作（选择）的时尚游戏已在文献中进行了广泛的研究。本文试图将时尚游戏分析扩展到具有两个以上可用动作的场景，从而增强对决策过程中社会动态的理解。我们确定 paths、cycles 和 complete graphs 上所有实例的效用。对于每个个体都喜欢反坐标、图是平面的并且有三个动作可用的实例，我们说明了确定此类实例效用的时间复杂性。 此外，对于同时包含协调和反协调个体的实例，我们将确定具有两个可用操作的实例效用的时间复杂度的结果扩展到具有两个以上操作的情况。