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In this paper, we consider a game between an Internet Service (access) Provider (ISP) and content provider (CP) on a platform of end-user demand. A price-convex demand-response is motivated based on the delay sensitive applications that are expected to be subjected to the assumed usage-priced priority service over best-effort service. Thus, we are considering a two-sided market with multiclass demand wherein one class (that under consideration herein) is delay-sensitive. Both the Internet and proposed Information Centric Network (ICN, encompassing Content Centric Networking (CCN)) scenarios are considered. For our purposes, the ICN case is basically different in the polarity of the side-payment (from ISP to CP in an ICN) and, more importantly here, in that content caching by the ISP is incentivized. A price-convex demand-response model is extended to account for content caching. The corresponding Nash equilibria are derived and studied numerically.