The Weighted Set Covering Game: A Vaccine Pricing Model for Pediatric Immunization

The United States pediatric vaccine manufacturing market is analyzed using a static Bertrand oligopoly pricing model that characterizes oligopolistic interactions between asymmetric firms in a homogeneous multiple product market. Firms satisfy demand by appropriately pricing and selling its given set of bundles, where each bundle contains one or more products. In analyzing the pediatric vaccine market, a bundle is a vaccine, where each vaccine contains one or more immunogenic antigens. Consumers seek to purchase at least one of each antigen at an overall minimum cost. Demand is captured by defining a weighted set covering optimization problem, with the weights (prices) controlled by firms engaged in Bertrand competition. A repeated game version of the model enables multiple interactions between firms, allowing examination of tacit collusion. An iterative improvement algorithm is defined that constructs a pure strategy Nash equilibrium (some in the limiting sense) for the static game. Sufficient conditions for the existence of pure strategy Nash equilibria are provided, indicating that this class of games always yields at least one pure strategy equilibrium. Practical results of the pediatric vaccine market analysis follow from the difference in the repeated game equilibrium prices between two combination vaccines, Pediarix (R) and Pentacel (R). Assuming the manufacturers of these vaccines agree to share the market equally with respect to volume, the equilibrium prices from the repeated game indicate a price difference of $0.86, whereas the difference in price between Pediarix (R) and Pentacel (R) for contract prices ending March 31, 2010 was $2.74. Interestingly, the subsequent public sector vaccine price list (contract prices ending March 31, 2011) shows a price difference of $0.95, with the price of Pentacel (R) actually reduced from the previous year-an unusual occurrence. The results presented in this paper suggest that a smaller price difference between these two important combination vaccines is appropriate, which is what occurred. In general, such results could serve to inform both manufacturers and purchasers on the appropriate pricing of combination vaccines, given the existence of a reasonable set of collusive agreements.