Abstract: Gray market is a distribution channel unauthorized by the manufacturer or brand owner to sell genuine-brand products. With the rapid development of global logistics network and e-commerce, the proportion of gray market trade in the global economy is increasing. In particular, a growing number of countries are increasingly positive about gray markets or gray markets in some sectors. Therefore, how to manage the supply chain effectively under the gray market structure so as to benefit the company has become a problem that must be faced in supply chain management.
It is considered in this paper that the manufacturer does not provide additional services in the low-price market but provides better product services in the high-price market, and the service is bundled with the product and no longer charges consumers extra fees. At the same time, consumers in the high-priced market who buy products through gray channels are not able to enjoy the services provided by authorized channels. Aiming at the problem that authorized retailers may participate in gray market, a two-stage Stackelberg model composed of a single manufacturer and a single retailer is constructed. Then the optimal decision and the profits of manufacturer and retailer are solved by using backward induction method. Sufficient conditions for the existence of gray market, the impact of the gray market on the profits of both parties and the change of service level are given. By analyzing the profitability of supply chain members in four different situations, we focus on the choice of manufacturers and retailers for their respective strategies and reveal how manufacturer uses service strategies to prevent retailer from participating in the gray market. In addition, we also study the influence of consumers' sensitivity to services on service strategy's response to gray market.
The results show that: first of all, regardless of whether the manufacturer provides differentiated services in the high-priced market or not, the retailer is involved in gray market to obtain extra profits under the conditions of moderate substitution coefficient and demand elasticity in the low-priced market. And both players achieve a win-win situation under certain conditions. Second, the manufacturer can provide services to narrow the space for retailer to participate in the gray market and reduce the space for manufacturer's profit loss. Thirdly, the more consumers pay attention to service, the better service strategies can cope with the gray market. Finally, service level is correlated with sensitivity coefficient and substitution coefficient. The more consumers value services, the more willing manufacturers are to improve services. With the increase of substitution coefficient, the manufacturer first reduces the service level to encourage the retailer to participate in the gray market, and then increases the service level to prevent the retailer from participating in the gray market, which represents manufacturer's attitude towards the gray market.
The gray market research is expanded from the perspective of supply chain, and the existing literature pays less attention to the service strategy under the condition of gray market, so this paper enriches this part of research.

Key words: supply chain, gray market, service strategy, Stackelberg competition