Energy markets in the United States and Europe are getting more liberalized. The question of whether the liberalization of the gas industry in both markets has led to stable prices and less concentrated markets has appealed great interest among the scientific community. This study aims to measure the power and efficiency of an information structure contained in the gas prices time series. This assessment is useful to the oversight duty of regulators in such markets in the post liberalized era. First, econometric and mathematical methods based on game theory, records theory, and Shannon entropy are used to measure the following indicators: level of competition, price stability, and price uncertainty respectively—for both markets. Second, the level of information generated by these indicators is quantified using the information theory. The results of this innovative two-step approach show that the functioning of the European market requires the regulator’s intervention. This intervention is done by applying additional rules to enhance the competitive aspect of the market. This is not that case for the U.S. market. Also, the value of the information contained in both markets’ wholesale gas prices, although in asymmetric terms, is significant, and therefore proves to be an important instrument for the regulators.