本研究主要聚焦於在有著消費限制效用的線上Fisher市場中，利用線上 proportional response求解市場均衡，透過數學證明進行驗證，並將此結果與[Yang, 2024]進行比較。在傳統的Fisher市場中，買家的效用是線性的，而這會造成買家將自己的預算花在單一種商品上，且也無法知道買家對商品感到滿足的條件，故我們在這裡進一步將線性效用推廣為對商品有指定的消費限制的效用。
在線上Fisher市場商品會定期進行補充，而每位買家的目標是在各自的預算限制下最大化自己的效用。傳統的方法需要有買家的效用函數與預算的完整資訊才能求解，但在現實生活中，買家通常不會希望透露太多有關自己的私人訊息，故我們採用了一種叫Proportional Response的分佈式演算法，買家僅需基於上一時間點得到的商品分配來決定出價，而市場也只需知道買家對商品的出價即可進行分配，讓買家不用透露過多的私人訊息。

This study focuses on solving market equilibrium in an online Fisher market with spending constraint utility using online proportional response, supported by mathematical proofs for validation, and compares the results with those of [Yang, 2024]. In traditional Fisher markets, buyers have linear utility, which often leads them to spend their entire budget on a single good, without accounting for the conditions under which they are satisfied with their purchases. Therefore, we extend the linear utility model to one with specified spending constraints on goods, providing a more refined approach.
In online Fisher markets, items are regularly replenished, and each buyer aims to maximize their utility within their budget constraints. Traditional methods require complete information about buyer utility functions and budgets, which may not be practical due to buyers' privacy concerns. Thus, we employ a distributed algorithm called Proportional Response, where buyers determine bids only based on the previous allocation, and the market allocates items based on these bids, minimizing the disclosure of private information.

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