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This paper proposes a doubled-sided auction mechanism with price discrimination for social welfare (SW) maximization in the electricity and heating market. In this mechanism, energy service providers (ESPs) submit offers and load aggregators (LAs) submit bids to an energy trading center (ETC) to maximize their utility; in turn, the selfless ETC as an auctioneer leverages dis-criminatory price weights to regulate the behaviors of ESPs and LAs, which combines the individual benefits of each stakeholder with the overall social welfare to achieve the global optimum. Nash games are employed to describe the interactions between players with the same market role. Theoretically, we first prove the existence and uniqueness of the Nash equilibrium; then, considering the requirement of game players to preserve privacy, a distributed algorithm based on the alternating direction method of multipliers is developed to implement distributed bidding and analytical target cascading algorithm is applied to reach the balance of demand and supply. We validated the proposed mechanism using case studies on a city-level distribution system. The results indicated that the achieved SW improved by 4%-15% compared with other mechanisms, and also verified the effectiveness of the distributed algorithm.