We　propose　a　game　model　for　selfish　routing　of　atomic　agents,　who　compete　for　use　of　a　network　to　travel　from　their　origins　to　a　common　destination　as　quickly　as　possible.　We　follow　a　frequently　used　rule　that　the　latency　an　agent　experiences　on　each　edge　is　a　constant　transit　time　plus　a　variable　waiting　time　in　a　queue.　A　key　feature　that　differentiates　our　model　from　related　ones　is　an　edge-based　tie-breaking　rule　for　prioritizing　agents　in　queueing　when　they　reach　an　edge　at　the　same　time.　We　study　both　nonadaptive　agents　(each　choosing　a　one-off　origin-destination　path　simultaneously　at　the　very　beginning)　and　adaptive　ones　(each　making　an　online　decision　at　every　nonterminal　vertex　they　reach　as　to　which　next　edge　to　take).　On　the　one　hand,　we　constructively　prove　that　a　(pure)　Nash　equilibrium　(NE)　always　exists　for　nonadaptive　agents　and　show　that　every　NE　is　weakly　Pareto　optimal　and　globally　first-in　first-out.　We　present　efficient　algorithms　for　finding　an　NE　and　best　responses　of　nonadaptive　agents.　On　the　other　hand,　we　are　among　the　first　to　consider　adaptive　atomic　agents,　for　which　we　show　that　a　subgame　perfect　equilibrium　(SPE)　always　exists　and　that　each　NE　outcome　for　nonadaptive　agents　is　an　SPE　outcome　for　adaptive　agents　but　not　vice　versa.