This paper centers around the selection of equilibrium points in robust stochastic games. In stochastic games, stationary equilibrium is a key concept. However, it is possible to have multiple stationary equilibria in a robust stochastic game. To eliminate some undesirable stationary equilibria, and establish a better equilibrium considering trembling hands, the concept of perfect stationary equilibrium is introduced in this study as a refined version of the stationary equilibrium. However, some perfect stationary equilibria may still be undesirable because they overlook the comparisons of trembling hands within each player. To address this issue, this study introduces the notion of a proper stationary equilibrium. Despite the improvements brought by proper stationary equilibria, they still fail to consider the comparisons of trembling hands across players, which results in unreasonable equilibria. To overcome this issue, this study introduces the notion of an extended proper stationary equilibrium. Hence, this dissertation focuses on the theoretical exploration of stationary equilibria, perfect stationary equilibria, proper stationary equilibria, extended proper stationary equilibria, and their computations within robust stochastic games. Since homotopy methods offer the advantage of global convergence, the main objective of this dissertation is to devise differentiable homotopy methods for the computation of these equilibria. Moreover, implementing these methods in a multi-stage repeated reward-based crowdfunding problem involving uncertain payoffs and state transition probabilities allows this study to pinpoint desirable strategies for each participant, including initiators, backers, and crowdfunding platforms.

The main theoretical and practical contributions of this paper are outlined as follows:

(1) Existing methods are primarily focused on selecting stationary equilibria in stochastic games, but they often fall short when it comes to guaranteeing the selection of stationary equilibria in robust stochastic games. To address this limitation, this study has developed a differentiable homotopy method specifically tailored for computing stationary equilibria in robust stochastic games. By utilizing this method, a stationary equilibrium has been discovered for the reward-based crowdfunding problem, revealing the following insights: The initiator has the flexibility to decide between setting a high price or a low price, as well as choosing whether to deliver the products or not. The backers should refrain from purchasing the product. The crowdfunding platform possesses the flexibility to select between the fixed funding mechanism and the flexible funding mechanism, as well as choose between regulatory measures and non-regulated measures.

(2) Considering trembling behavior, this study introduces an innovative equilibrium concept known as perfect stationary equilibrium, which is a refinement of stationary equilibrium. To compute such equilibria in robust stochastic games, this study develops a differentiable homotopy method. By employing this method, a perfect stationary equilibrium has been discovered for the reward-based crowdfunding problem with trembling hands, revealing the following insights: The desired strategies for both the backers and the platform align with those characterized by the stationary equilibrium. However, the initiator should set a high price for the product and ensure delivery to the backers.

(3) Considering the comparisons of trembling hands within each player, this study introduces a novel equilibrium concept called proper stationary equilibrium, which is a refinement of perfect stationary equilibrium. To address the numerical precision requirements associated with computing proper stationary equilibria, this study proposes the concept of perfect $d$-proper stationary equilibrium, which provides an approximate equivalence to the concept of proper stationary equilibrium. To compute such equilibria in robust stochastic games, this study develops differentiable homotopy methods. By employing these methods, a proper stationary equilibrium has been discovered for the reward-based crowdfunding problem with trembling-hand comparisons, revealing the following insights: The desired strategies for both the initiator and the backers align with those characterized by the perfect stationary equilibrium. However, the platform should opt for regulatory measures while maintaining the flexibility to select between the fixed funding mechanism and the flexible funding mechanism.

(4) Considering the comparisons of trembling hands across players, this study introduces a novel equilibrium concept called extended proper stationary equilibrium, which is a refinement of proper stationary equilibrium. To address the numerical precision requirements associated with computing extended proper equilibria, this study proposes the concept of perfect $d$-extended proper stationary equilibrium, which provides an approximate equivalence to the concept of extended proper stationary equilibrium. To compute such equilibria in robust stochastic games, this study develops differentiable homotopy methods. By employing these methods, an extended proper stationary equilibrium has been discovered for the reward-based crowdfunding problem with across-player trembling-hand comparisons, revealing the following insights: The desired strategies for both the initiator and the backers align with those characterized by the proper stationary equilibrium. However, the platform should opt for the fixed funding mechanism and regulatory measures.