Our earlier study considered a mathematical model of the spread of a false message in the population when it is countered with inoculation and debunking. Inoculation was assumed to be non-contagious; that is, an inoculated individual was assumed to be unable to extend their resistance to fake news to other members of the population. It was shown that noncontagious inoculation and debunking, even taken together, are ineffectual against the spread of fake news. Now we present a model with inoculation performed with a contagious game. This means that psychological resistance against manipulation techniques is conferred on players of the game such that thereby inoculated individuals communicate their positive experience of gaming to other individuals, thus contributing to a wider inoculation of the population. The model considers the process in which a fake news message is published once and spreads as a rumor by the spreaders, that is, by deceived individuals. Debunking begins after the publication and is carried out continuously by the mass media and by skeptics, i.e., individuals who have internalized the debunking message. The mathematical model is studied numerically. Experiments show that the number of fake news spreaders remains relatively low only if the effectiveness of inoculation is very high.