Selfish　routing　is　one　of　the　fundamental　models　in　the　study　of　network　traffic　systems.　While　most　literature　assumes　essentially　static　flows,　game　theoretical　models　of　dynamic　flows　began　to　draw　attention　recently　[1-5].　We　study　a　new　dynamic　flow　game　with　atomic　agents.　The　game　is　played　over　an　acyclic　directed　network,　with　two　special　vertices　called　the　origin　o　and　the　destination　d,　respectively,　such　that　each　edge　is　on　at　least　one　o-d　path.　Each　edge　of　this　network　is　associated　with　an　integer　capacity　and　a　positive　integer　free-flow　transit　cost.　Time　horizon　is　infinite　and　discretized　as　1,　2,....　At　each　time　point,　a　set　of　selfish　agents　enter　the　network　from　the　origin,　trying　to　reach　the　destination　as　quickly　as　possible.　When　an　agent　uses　an　edge,　two　costs　are　incurred　to　him:　The　fixed　transit　cost　of　the　edge　and　a　variable　waiting　cost　dependent　on　the　volume　of　the　traffic　flow　on　that　edge　and　the　edge　capacity　as　well　as　the　agent＇s　position　in　the　queue　of　agents　waiting　at　that　edge.　The　total　cost　to　each　agent,　the　latency　he　experiences　in　the　network,　is　the　sum　of　the　two　costs　on　all　edges　he　uses　(which　form　an　o-d　path　in　the　network).　The　waiting　time　of　each　agent　is　determined　in　a　way　called　deterministic　queuing.　At　each　time　step,　there　is　a　(possibly　empty)　queue　of　agents　waiting　at　the　tail　of　each　edge,　and　at　most　a　capacity　number　of　agents　who　have　the　highest　priority　(according　to　the　queuing　order)　start　to　move　along　this　edge.　After　the　fixed　period　of　transit　time,　each　of　these　agents　reaches　the　head　of　the　edge,　and　enters　the　next　edge　(waiting　at　its　tail　for　traveling　along　it)　unless　the　destination　d　has　been　reached.　The　queue　at　each　edge　is　updated　according　to　the　local　First-In-First-Out　rule　(restricted　to　each　edge)　and　pre-specified　Edge　Priorities　-　if　two　agents　enter　an　edge　simultaneously,　their　queue　priorities　are　determined　by　the　priorities　of　the　preceding　edges　they　just　come　from.　To　break　ties　when　agents　enter　an　edge　simultaneously　from　the　same　edge,　we　divide　each　edge　into　the　capacity　number　of　lanes　each　with　a　unit　capacity,　and　assign　priorities　among　these　lanes.　Initially,　the　agents　who　enter　the　network　at　the　same　time　are　associated　with　original　priorities　among　them,　which　are　temporarily　valid　when　they　enter　the　network.　This　is　a　crucial　difference　between　our　model　and　those　in　[2,　4],　where　pre-specified　priorities　among　all　agents　are　permanent,　working　globally　throughout　the　game.　The　essence　of　our　model　is　that　each　agent　makes　a　routing　decision　at　every　nonterminal　vertex　he　reaches.　This　is　the　second　crucial　(and　fundamental)　difference　of　our　model　from　almost　all　in　the　literature　[2,　4,　5],　in　which　agents　are　assumed　to　make　their　routing　decisions　only　at　the　origin　o　as　to　which　o-d　path　to　take.　In　addition,　it　is　usually　assumed　that　agents＇　decisions　upon　entering　the　system　are　independent　of　the　decisions　of　the　agents　who　are　currently　in　the　system.　In　game-Theoretic　terminology,　while　the　dynamic　traffic　problem　is　usually　modeled　as　a　game　of　normal　form　(or　a　simultaneous-move　game),　we　model　it　as　a　game　of　extensive　form,　which　allows　for　agents＇　online　adjustments　and　error　corrections.　Let　r　denote　our　extensive-form　game　of　dynamic　traffic,　for　which　we　apply　the　solution　concept　of　(pure　strategy)　Subgame　Perfect　Equilibrium　(SPE).　We　demonstrate　in　a　constructive　way　that　r　always　possesses　a　very　special　SPE.　To　the　best　of　our　knowledge,　this　is　the　first　SPE　existence　result　in　this　field.　Then　we　study　a　very　natural　subset　of　SPEs　of　r,　which　is　shown　to　possess　desirable　properties　that　are　intuitively　reasonable　but　technically　hard　to　verify.　This　subset　corresponds　to　the　simplified　setting,　denoted　as　rn,　where　agents　choose　their　one-off　o-d　paths　simultaneously　at　the　very　start　of　the　game.　The　SPE　existence　of　r　implies　that　rn　admits　at　least　one　(pure　strategy)　Nash　Equilibrium　(NE).　We　show　that　the　set　of　NEs　of　game　rn　forms　a　proper　subset　of　the　realized　path　profiles　of　the　SPEs　of　game　r.　Furthermore,　we　prove　that　in　rn　every　NE　is　also　a　strong　NE　and　hence　weakly　Pareto　optimal,　and　establish　many　other　nice　properties　of　all　NE　flows,　including　global　(valid　throughout　the　network)　First-In-First-Out,　hierarchal　independence,　and　so　on.　These　properties　remain　valid　even　if　the　network　has　multiple　origins　and　a　single　destination.　This　is　one　of　the　key　reasons　that　we　are　able　to　prove　the　existence　of　an　SPE　for　r　and　the　relationship　between　r　and　rn　(note　that　the　off-equilibrium　scenarios　in　r　are　essentially　new　problems　with　multiple　origins).　Since　there　may　be　infinitely　many　agents　in　models　like　ours,　a　natural　question　is　whether　the　queue　length　at　each　edge　at　equilibria　can　be　bounded　by　a　constant　that　depends　only　on　the　input　finite　network,　when　the　inflow　size　never　exceeds　the　minimum　capacity　of　o-d　cuts　in　the　network.　This　interesting　yet　theoretically　challenging　issue　has　been　listed　as　one　of　the　important　open　problems　in　the　literature　[1,　4].　We　prove　that　equilibrium　queue　lengths　are　indeed　bounded　in　our　games　on　two　classes　of　networks:　R　on　series-parallel　networks,　and　r　on　networks　in　which　the　in-degree　does　not　exceed　the　out-degree　at　any　nonterminal　vertex.　©　2017　ACM.