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This paper analyzes the competing pricing mechanisms of uniform and pay-as-bid pricing in an electricity market. Game theory and auction theory are adopted to analyze the strategic behavior of a big player and a small player in a short-term auction game. Contrary to what would be expected from the conclusion of the "revenue equivalence theorem," we prove that for a two-player static game the Nash Equilibrium (NE) under pay-as-bid pricing will yield less total revenue in expectation than under uniform pricing when demand is inelastic. To confirm this theoretical result we simulated the model using a mixed-strategy NE solver. We extended the model to an elastic demand case and showed that pay-as-bid pricing also led to a larger expected total demand being served when demand is elastic.