This note extends the results in the manufacturer-dominated game model of the paper by Li et al. to the case where the manufacturer's marginal profit is not large enough. In such situations, the profit of the entire supply chain under the co-op advertising mode is higher than the one under the Stackelberg game, which is consistent with the results of the original paper. However, the advertising expenditures of the manufacturer and the retailer under the co-op advertising model are not always larger than those under the Stackelberg game, which is different from the results of the original paper. Furthermore, the results are also compared with the simultaneous move game of the paper by Huang and Li. The manufacturer always prefers the leader–follower structure rather than the simultaneous move structure, which is consistent with the results of the original paper. However, the retailer always prefers the simultaneous move structure rather than the leader–follower structure, which differs from the results of the original paper.
