Competition among news sources may encourage some sources to share fake news and misinformation to influence the public. While sharing misinformation may lead to a short-term gain in audience engagement, it may damage the reputation of these sources, resulting in a loss of audience. To understand the rationale behind sharing misinformation, we model the competition as a zero-sum sequential game, where each news source influences individuals based on its credibility-how trustworthy the public perceives it-and the individual's opinion and susceptibility. In this game, news sources can decide whether to share factual information to enhance their credibility or disseminate misinformation for greater immediate attention at the cost of losing credibility. We employ the quantal response equilibrium concept, which accounts for the bounded rationality of human decision-making, allowing for imperfect or probabilistic choices. Our analysis shows that the resulting equilibria for this game reproduce the credibility-bias distribution observed in real-world news sources, with hyper-partisan sources more likely to spread misinformation than centrist ones. It further illustrates that disseminating misinformation can polarize the public. Notably, our model reveals that when one player increases misinformation dissemination, the other player is likely to follow, exacerbating the spread of misinformation. We conclude by discussing potential strategies to mitigate the spread of fake news and promote a more factual and reliable information landscape.