Routers are called upon to insert their address into Record Route, Strict Source and Record Route, Loose Source and Record Route, or Timestamp Options. When a router inserts its address into such an option, it MUST use the IP address of the logical interface on which the packet is being sent. Where this rule cannot be obeyed because the output interface has no IP address (i.e., is an unnumbered interface), the router MUST instead insert its router-id. The router's router-id is one of the router's IP addresses. The Router ID may be specified on a system basis or on a per-link basis. Which of the router's addresses is used as the router-id MUST NOT change (even across reboots) unless changed by the network manager. Relevant management changes include reconfiguration of the router such that the IP address used as the router-id ceases to be one of the router's IP addresses. Routers with multiple unnumbered interfaces MAY have multiple router-id's. Each unnumbered interface MUST be associated with a particular router-id. This association MUST NOT change (even across reboots) without reconfiguration of the router.
This specification does not allow for routers that do not have at least one IP address. We do not view this as a serious limitation, since a router needs an IP address to meet the manageability requirements of Chapter  even if the router is connected only to point-to-point links.
One possible method of choosing the router-id that fulfills this requirement is to use the numerically smallest (or greatest) IP address (treating the address as a 32-bit integer) that is assigned to the router.