Royalty bargaining in Public–Private Partnership projects: Insights from a theoretic three-stage game auction model

•It is a transformed first-price sealed-bid auction with independent private values.•We consider interactions between bidding and bargaining behaviors in a franchise.•The true values bidders assign to a project influence royalties and subsidies.•The payoff of a government from outside options affects royalties and subsidies.•The payoff of a government from outside options is vulnerable to collusion.