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Currently, most independent system operators in the U.S. run auctions by minimizing the total bid cost ["bid cost minimization" (BCM)], and then calculate payments based on market clearing prices. Under this setup, the payment cost could be significantly higher than the minimized bid cost. Recently, an alternative auction mechanism that minimizes the consumer payment cost ["payment cost minimization" (PCM)] has been discussed. Literature has shown that for the same set of bids, PCM leads to reduced consumer payments. However, market participants may bid differently under the two auctions, and therefore, the payment reduction may not be realized. This necessitates the study of strategic behaviors of participants. In this paper, suppliers' bidding strategies in a day-ahead energy market are investigated for both auctions by using a game theoretic framework with Nash equilibrium as the solution concept. To simplify the solution process, the originally continuous strategies are discretized to form matrix games. Discretization may cause the loss of equilibria and the creation of artificial solutions. To reduce these side effects, "approximate Nash equilibria" are introduced to recover lost equilibria, and additional strategy samples are evaluated to eliminate artificially created solutions. Games are then solved by examining supplier payoffs obtained from running auctions. Characteristics of auctions are exploited, leading to improved computational efficiency. Numerical testing results show that the PCM leads to significant payment reductions and relatively small increases of production costs as compared to BCM. Also, the ??hockey-stick?? bidding is more likely to occur under BCM. Finally, long-term impacts of PCM are discussed, and whether it would lower costs to consumers in the long run, including capacity payments, remains to be investigated.