{"id":"https://openalex.org/W1507028355","doi":"https://doi.org/10.1137/080734042","title":"Parallel Repetition in Projection Games and a Concentration Bound","display_name":"Parallel Repetition in Projection Games and a Concentration Bound","publication_year":2008,"publication_date":"2008-05-17","ids":{"openalex":"https://openalex.org/W1507028355","doi":"https://doi.org/10.1137/080734042","mag":"1507028355"},"language":"en","primary_location":{"id":"doi:10.1137/080734042","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080734042","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5035012558","display_name":"Anup Rao","orcid":"https://orcid.org/0000-0002-6449-9547"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Anup Rao","raw_affiliation_strings":["anuprao@cs.washington.edu#TAB#"],"affiliations":[{"raw_affiliation_string":"anuprao@cs.washington.edu#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5035012558"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.411,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.64593237,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"40","issue":"6","first_page":"1","last_page":"10"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6589605808258057},{"id":"https://openalex.org/keywords/alphabet","display_name":"Alphabet","score":0.6133028268814087},{"id":"https://openalex.org/keywords/repetition","display_name":"Repetition (rhetorical device)","score":0.5961769223213196},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5625792741775513},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5403552055358887},{"id":"https://openalex.org/keywords/repeated-game","display_name":"Repeated game","score":0.5360617637634277},{"id":"https://openalex.org/keywords/combinatorial-game-theory","display_name":"Combinatorial game theory","score":0.48902425169944763},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.477290540933609},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4706902801990509},{"id":"https://openalex.org/keywords/strategy","display_name":"Strategy","score":0.4386581480503082},{"id":"https://openalex.org/keywords/fraction","display_name":"Fraction (chemistry)","score":0.4284183382987976},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.418073832988739},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.4047507643699646},{"id":"https://openalex.org/keywords/game-theory","display_name":"Game theory","score":0.35605716705322266},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3046623766422272},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06634733080863953}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6589605808258057},{"id":"https://openalex.org/C112876837","wikidata":"https://www.wikidata.org/wiki/Q837518","display_name":"Alphabet","level":2,"score":0.6133028268814087},{"id":"https://openalex.org/C2776141515","wikidata":"https://www.wikidata.org/wiki/Q1274479","display_name":"Repetition (rhetorical device)","level":2,"score":0.5961769223213196},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5625792741775513},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5403552055358887},{"id":"https://openalex.org/C202556891","wikidata":"https://www.wikidata.org/wiki/Q1584646","display_name":"Repeated game","level":3,"score":0.5360617637634277},{"id":"https://openalex.org/C102234262","wikidata":"https://www.wikidata.org/wiki/Q1320931","display_name":"Combinatorial game theory","level":4,"score":0.48902425169944763},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.477290540933609},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4706902801990509},{"id":"https://openalex.org/C88959737","wikidata":"https://www.wikidata.org/wiki/Q1546627","display_name":"Strategy","level":3,"score":0.4386581480503082},{"id":"https://openalex.org/C149629883","wikidata":"https://www.wikidata.org/wiki/Q660926","display_name":"Fraction (chemistry)","level":2,"score":0.4284183382987976},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.418073832988739},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4047507643699646},{"id":"https://openalex.org/C177142836","wikidata":"https://www.wikidata.org/wiki/Q44455","display_name":"Game theory","level":2,"score":0.35605716705322266},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3046623766422272},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06634733080863953},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/080734042","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080734042","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},{"id":"doi:10.1145/1374376.1374378","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1374376.1374378","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the fortieth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.112.6895","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.112.6895","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ias.edu/~arao/pubs/parrep.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.164.3214","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.164.3214","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.emis.de/journals/JEHPS/Decembre2007/Triclot.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.208.9505","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.208.9505","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.washington.edu/homes/anuprao/pubs/parrep.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.297.7718","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.297.7718","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://homes.cs.washington.edu/~anuprao/pubs/parrep.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.304.7430","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.304.7430","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ias.edu/~arao/pubs/parrep.pdf","raw_type":"text"},{"id":"mag:1507028355","is_oa":false,"landing_page_url":"https://dblp.uni-trier.de/db/journals/eccc/eccc15.html#Rao08","pdf_url":null,"source":{"id":"https://openalex.org/S11189987","display_name":"Electronic colloquium on computational complexity","issn_l":"1433-8092","issn":["1433-8092"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"Electronic colloquium on computational complexity","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":40,"referenced_works":["https://openalex.org/W1483170929","https://openalex.org/W1511716344","https://openalex.org/W1513663661","https://openalex.org/W1524356573","https://openalex.org/W1571037334","https://openalex.org/W1590594965","https://openalex.org/W1600412016","https://openalex.org/W1970259241","https://openalex.org/W1985123706","https://openalex.org/W2004804919","https://openalex.org/W2010631958","https://openalex.org/W2014086530","https://openalex.org/W2014323766","https://openalex.org/W2016232593","https://openalex.org/W2019578639","https://openalex.org/W2028815089","https://openalex.org/W2030257803","https://openalex.org/W2063844956","https://openalex.org/W2070208264","https://openalex.org/W2079473728","https://openalex.org/W2080578129","https://openalex.org/W2095727035","https://openalex.org/W2096397972","https://openalex.org/W2099677042","https://openalex.org/W2105892311","https://openalex.org/W2108684595","https://openalex.org/W2120358419","https://openalex.org/W2125703559","https://openalex.org/W2133263321","https://openalex.org/W2135830616","https://openalex.org/W2136735959","https://openalex.org/W2137118456","https://openalex.org/W2141815340","https://openalex.org/W2148352980","https://openalex.org/W2161719898","https://openalex.org/W2165732281","https://openalex.org/W2218607027","https://openalex.org/W3179843568","https://openalex.org/W6632549967","https://openalex.org/W6674482662"],"related_works":["https://openalex.org/W2102373437","https://openalex.org/W2191458325","https://openalex.org/W1971793441","https://openalex.org/W2949459194","https://openalex.org/W2963337108","https://openalex.org/W2951128567","https://openalex.org/W2963109078","https://openalex.org/W2964106800","https://openalex.org/W3013013019","https://openalex.org/W3104362493","https://openalex.org/W3214140134","https://openalex.org/W1996023047","https://openalex.org/W2021366584","https://openalex.org/W2077569675","https://openalex.org/W3080138781","https://openalex.org/W219192326","https://openalex.org/W2329618725","https://openalex.org/W3165348785","https://openalex.org/W2163503431","https://openalex.org/W2766798257"],"abstract_inverted_index":{"A":[0,14],"two-player":[1],"game":[2,49,52,79,130,168,176,250,443],"is":[3,50,138,161,169,220,270,306,328,378,415,437],"played":[4],"by":[5],"cooperating":[6],"players":[7,18,70,74,122,208],"who":[8,118],"are":[9],"not":[10,31],"allowed":[11],"to":[12,68,236,254,314,404,407,439],"communicate.":[13],"referee":[15,56],"asks":[16],"the":[17,38,41,48,51,55,65,69,73,77,86,92,121,128,139,142,145,148,155,167,173,204,207,214,217,226,244,248,273,284,291,316,331,335,380,388,408,454,468],"questions":[19,39,62,67],"sampled":[20],"from":[21,459],"some":[22],"known":[23,35],"distribution":[24],"and":[25,40,63,109,276,355,369],"decides":[26],"whether":[27],"they":[28,88],"win":[29,76,124,209],"or":[30],"based":[32],"on":[33,230,281,296,393],"a":[34,170,188,210,260,278,311,319,325,359,410,441,450,463],"predicate":[36],"of":[37,47,61,99,141,157,175,182,213,247,272,287,290,298,318,330,334,375,384,387,395,453,467],"players'":[42],"answers.":[43],"The":[44],"parallel":[45,192,218,245],"repetition":[46,193,219,246],"in":[53,91,127,144,147,180,216,324,448,461],"which":[54,435,449,462],"samples":[57],"n":[58],"independent":[59,271,329],"pairs":[60],"sends":[64],"corresponding":[66],"simultaneously.":[71],"If":[72],"cannot":[75,123],"original":[78],"with":[80,131,349,362,444],"probability":[81,132,156,205],"better":[82,133,279],"than":[83,134],"$(1-\\epsilon)$,":[84],"what's":[85],"best":[87],"can":[89,251,343,400,456,470],"do":[90],"repeated":[93,129],"game?":[94],"We":[95,152,186],"improve":[96,315],"earlier":[97],"results":[98],"[R.":[100],"Raz,":[101],"SIAM":[102],"J.":[103],"Comput.,":[104,113],"27":[105],"(1998),":[106],"pp.":[107,116,302,398],"763\u2013803]":[108],"[T.":[110],"Holenstein,":[111],"Theory":[112,394],"5":[114],"(2009),":[115],"141\u2013172],":[117],"showed":[119],"that":[120,154,198,206,243,258,327,379,423],"all":[125,159],"copies":[126,160],"$(1-\\epsilon/2)^{\\Omega(n\\epsilon^2/c)}$":[135],"(here":[136,225],"c":[137],"length":[140,275,333,351,364],"answers":[143],"game),":[146],"following":[149,409],"ways:":[150],"(i)":[151],"show":[153],"winning":[158],"$(1-\\epsilon/2)^{\\Omega(\\epsilon":[162],"n)}$":[163],"as":[164,166],"long":[165],"\u201cprojection":[171],"game,\u201d":[172],"type":[174],"most":[177,222],"commonly":[178],"used":[179,253],"hardness":[181],"approximation":[183],"results.":[184],"(ii)":[185],"prove":[187],"concentration":[189],"bound":[190,269,305,309,377],"for":[191,199,339,424,434],"(of":[194],"general":[195],"games)":[196],"showing":[197],"any":[200,345],"constant":[201,227],"$0<\\delta":[202],"<\\epsilon$,":[203],"$(1-\\epsilon+\\delta)$":[211],"fraction":[212,452,466],"games":[215,382],"at":[221],"$\\exp\\left(-\\Omega_{\\epsilon}(\\delta^3":[223],"n/c)\\right)$":[224],"may":[228],"depend":[229],"$\\epsilon$);":[231],"our":[232,376],"result":[233],"has":[234,259,277],"applications":[235],"testing":[237],"Bell":[238],"inequalities,":[239],"since":[240],"it":[241,436],"implies":[242],"CHSH":[249],"be":[252,402,405,457,471],"get":[255],"an":[256,416,430],"experiment":[257],"very":[261],"large":[262],"classical":[263],"versus":[264],"quantum":[265],"gap.":[266],"Our":[267,308],"first":[268],"answer":[274,332,350,363],"dependence":[280],"$\\epsilon$.":[282],"By":[283],"recent":[285],"work":[286],"Raz":[288],"[Proceedings":[289,386],"$49$th":[292],"Annual":[293,390],"IEEE":[294],"Symposium":[295,392],"Foundations":[297],"Computer":[299],"Science,":[300],"2008,":[301],"369\u2013373],":[303],"this":[304],"tight.":[307],"gives":[310],"generic":[312],"way":[313,326],"soundness":[317,356,370],"probabilistically":[320],"checkable":[321],"proof":[322],"(PCP),":[323],"PCP.":[336],"Using":[337],"it,":[338],"every":[340,425],"k,":[341],"one":[342,460],"convert":[344],"q":[346],"query":[347],"PCP":[348,361],"c,":[352],"size":[353,366,432,446],"$sc$,":[354],"$(1-\\epsilon)$":[357],"into":[358],"two-query":[360],"$ck$,":[365],"$O(ck":[367],"(2s)^k)$,":[368],"$(1-\\epsilon/2q)^{\\Omega(\\epsilon":[371],"k/q)}$.":[372],"Another":[373],"consequence":[374],"unique":[381,442],"conjecture":[383],"Khot":[385],"$34$th":[389],"ACM":[391],"Computing,":[396],"2002,":[397],"767\u2013775]":[399],"now":[401],"shown":[403],"equivalent":[406],"priori":[411],"weaker":[412],"conjecture:":[413],"There":[414],"unbounded":[417],"increasing":[418],"function":[419],"$f:\\mathbb{R}^+\\rightarrow":[420],"\\mathbb{R}^+$":[421],"such":[422],"$\\epsilon>":[426],"0$,":[427],"there":[428],"exists":[429],"alphabet":[431,445],"$M(\\epsilon)$":[433],"NP-hard":[438],"distinguish":[440],"M":[447],"$(1-\\epsilon^2)$":[451],"constraints":[455,469],"satisfied":[458],"$(1-\\epsilon":[464],"f(1/\\epsilon))$":[465],"satisfied.":[472]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}