Learning to Cooperate in a Continuous Tragedy of the Commons (Extended Abstract) Steven de Jong Karl Tuyls Dep. of Knowledge Engineering, Maastricht Univ. Postbus 616, 6200 MD Maastricht, Netherlands Eindhoven University of Technology Postbus 513, 5600 MB Eindhoven, Netherlands [email protected] [email protected] Categories and Subject Descriptors UHDFK GHVLUDEOH FRRSHUDWLYH VROXWLRQV WR WKH 3** LH DOO DJHQWV LQ YHVW C WKH PD[LPXP DPRXQW DOORZHG LQVWHDG RI WKH LQGLYLGXDOO\ UD WLRQDO LQYHVWPHQW RI 0 HYHQ LQ D FRQWLQXRXV VWUDWHJ\ VSDFH 7KH RQO\ UHTXLUHPHQW LV WKDW WKHUH DUH LQLWLDOO\ VRPH DJHQWV DOUHDG\ LQYHVWLQJ C 7KH PHWKRGRORJ\ LV EDVHG RQ WKH IROORZLQJ IRXU HOHPHQWV , >/HDUQLQJ@ , >'LVWULEXWHG $UWLILFLDO ,QWHOOLJHQFH@ - >6RFLDO DQG %HKDYLRUDO 6FLHQFHV@ General Terms &RQWLQXRXVDFWLRQ OHDUQLQJ DXWRPDWD &$/$ $JHQWV OHDUQ LQGLYLGXDO EHKDYLRU E\ PHDQV RI &$/$ >@ (DFK DJHQW i NHHSV WUDFN RI LWV FXUUHQW VWUDWHJ\ LQ WKH 3** μi ∈ [0, C] ZKLFK LV DGDSWHG DIWHU HDFK SDLUZLVH LQWHUDFWLRQ ZLWK D QHLJKERU j LQ WKH QHWZRUN $OJRULWKPV 'HVLJQ +XPDQ )DFWRUV Keywords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μi PD\ ZDQW WR SUHYHQW LQWHUDFWLQJ ZLWK UHODWLYH GHIHFWRUV DJDLQ DQ DJHQW i XQZLUHV IURP LWV QHLJKERU j DIWHU LQWHUDFWLQJ ZLWK LW ZLWK D SUREDELO LW\ pr = C1 (μi − μj ) ,I XQZLULQJ KDSSHQV i FRQQHFWV WR D UDQGRP QHLJKERU RI j DV LQ >@ $OORZLQJ DJHQW i WR VHOHFW D QHZ QHLJKERU ZRXOG JLYH i WKH RSSRUWXQLW\ WR DFWLYHO\ H[SORLW WKLV QHLJKERU 1. 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Budapest, pp. 1185–1186 Copyright Hungary, (c) 2009, International Foundation for Autonomous Agents and Multiagent©Systems (www.ifaamas.org). All rights reserved. Copyright 2009, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org), All rights reserved. ,PDJLQH DJHQW j ZLWK μj = 2 SOD\LQJ DJDLQVW DJHQW i ZLWK μi = 8 $JHQW j PD\ ZDQW WR WU\ μj = 3 'XH WR LQHTXLW\ DYHUVLRQ DJHQW i ZLOO SXQLVK j LQ ERWK FDVHV 7KHUHIRUH WKH HVVHQWLDO LGHD XQGHUO\LQJ SXQLVKPHQW LH D UHYHUVDO RI WKH LQYHUVH UHODWLRQ EHWZHHQ FRQWULEX WLRQ DQG UHZDUG IDLOV WR ZRUN μj = 2 JLYHV D KLJKHU UHZDUG WKDQ μj = 3 EHFDXVH ERWK VWUDWHJLHV DUH SXQLVKHG 1185 AAMAS 2009 • 8th International Conference on Autonomous Agents and Multiagent Systems • 10–15 May, 2009 • Budapest, Hungary :,7+ 5(:,5,1* 6000 5000 5000 4000 4000 Games per agent 3000 3000 2000 2000 1000 1000 10 9 9 8 8 Average converged strategy 7 6 5 4 3 90 10 0 70 80 7 6 5 4 3 2 2 1 1 90 10 0 80 70 60 50 40 30 0 90 10 0 70 80 50 60 30 40 10 20 0 20 0 10 Percentage FS agents Percentage FS agents 1 0.9 0.9 0.8 0.8 Performance at convergence 1 0.7 0.6 0.5 0.4 0.3 0.7 0.6 0.5 3. 0.3 0.2 0.2 0.1 0.1 Percentage FS agents 90 10 0 80 70 60 50 40 30 20 0 90 10 0 80 70 60 50 40 30 20 10 0 10 0 0 Percentage FS agents 1 0.9 0.9 0.8 0.8 0.7 0.7 Degree distribution 1 0.6 0.5 0.4 0.6 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 CONCLUSION :H SUHVHQW D PHWKRGRORJ\ DLPHG DW DOORZLQJ D SRSXODWLRQ RI OHDUQ LQJ DJHQWV WR ILQG DQG PDLQWDLQ FRRSHUDWLYH GHVLUHG VWUDWHJLHV LQ D JDPH PRGHOOLQJ D WUDJHG\ RI WKH FRPPRQV ZLWK D FRQWLQXRXV VWUDW HJ\ VSDFH LH WKH 3** :H VKRZ WKDW RXU PHWKRGRORJ\ FRPELQ LQJ LQHTXLW\ DYHUVLRQ SUREDELOLVWLF SXQLVKPHQW DQG G\QDPLF VR FLDO QHWZRUNV DOORZV LQGLYLGXDOO\ OHDUQLQJ DJHQWV WR UHDFK WKH EHVW PRVW FRRSHUDWLYH VWUDWHJ\ LQLWLDOO\ SUHVHQW $ FHUWDLQ SHUFHQWDJH RI RXU DJHQWV LQLWLDOO\ SOD\V LQ DQ LQGLYLGXDOO\ UDWLRQDO XQFRRSHUDWLYH PDQQHU :H VKRZ WKDW RXU PHWKRGRORJ\ IRUFHV WKHVH DJHQWV WR EH FRPH PRUH FRRSHUDWLYH 7KXV WKH PHWKRORJ\ PD\ DOVR EH DSSOLHG LQ RSHQ V\VWHPV ZKHUH ZH DUH QRW DEOH WR FRQWURO WKH EHKDYLRU RI DOO DJHQWV 7KH PHWKRGRORJ\ LV WKHUHIRUH XVHIXO LQ PDQ\ SUREOHPV FRP PRQO\ DGGUHVVHG E\ PXOWLDJHQW V\VWHPV HJ UHVRXUFH DOORFDWLRQ 90 10 0 80 70 60 50 20 10 90 10 0 80 70 60 50 40 30 20 10 0 Percentage FS agents 0 0 0 UHSRUW UHVXOWV IRU n = 100 UHVXOWV IRU RWKHU n DUH KLJKO\ VLPLODU )RU DQ RYHUYLHZ RI UHVXOWV VHH )LJXUH :H XVH WKH VDPH PHD VXUHPHQWV DV LQ >@ )URP WRS WR ERWWRP ZH ILUVW ORRN DW WKH QXPEHU RI SDLUZLVH JDPHV SHU DJHQW UHTXLUHG WR REWDLQ FRQYHUJHQFH ZLWK DQ LPSRVHG PD[LPXP RI 6000 $ VWDWLF QHWZRUN UHTXLUHV VOLJKWO\ PRUH JDPHV EXW VWLOO UHTXLUHV GUDVWLFDOO\ OHVV JDPHV WKDQ UHSRUWHG LQ UH ODWHG ZRUN >@ 6HFRQG ZH ORRN DW WKH DYHUDJH FRQYHUJHG VWUDWHJ\ RI WKH FROOHFWLYH DQG PHQWLRQ WZR REVHUYDWLRQV LH ZLWKRXW DQ\ )6 DJHQWV SUHVHQW WKH FROOHFWLYH FRQYHUJHV WR LQYHVWLQJ 7 ZKLFK LV WKH PRVW FRRSHUDWLYH VWUDWHJ\ SUHVHQW ZLWK DQ LQFUHDVLQJ SHUFHQWDJH RI )6 DJHQWV WKH FROOHFWLYH OHDUQV WR LQYHVW 10 ZKLFK LV UHPDUNDEO\ HDVLHU IRU DJHQWV WKDW DUH DEOH WR UHZLUH 7KLUG ZH H[DPLQH WKH SHU IRUPDQFH DW FRQYHUJHQFH ZKLFK H[SUHVVHV WKH IUDFWLRQ RI QHLJKERU LQJ DJHQWV LQ WKH QHWZRUN WKDW KDYH VLPLODU VWUDWHJLHV 5HVXOWV DUH LQ OLQH ZLWK WKH REVHUYHG DYHUDJH VWUDWHJ\ )RU VWDWLF QHWZRUNV ZH VHH WKDW WKH '6 DJHQWV KDYH GLIILFXOWLHV WR DOLJQ WKHPVHOYHV ZLWK D ORZ QXPEHU RI )6 DJHQWV 7KLV SUREOHP LV QRW SUHVHQW LQ G\QDPLF QHW ZRUNV ZKHUH WKH SHUIRUPDQFH LV QHDUO\ SHUIHFW HYHQ ZLWK RQO\ 10 )6 DJHQWV SURYLGLQJ WKH µJRRG H[DPSOH¶ )RXUWK DQG ODVW ZH UHSRUW WKH GHJUHH GLVWULEXWLRQ RI WKH QHWZRUN RI LQWHUDFWLRQ )RU UHIHUHQFH UHVXOWV IRU VWDWLF QHWZRUNV DUH DOVR LQFOXGHG :H VHH WKDW D IHZ DJHQWV FRQQHFW WR DSSUR[LPDWHO\ 20 RI WKH RWKHUV ZKLOH PRVW DJHQWV KDYH RQO\ D IHZ QHLJKERUV ,Q G\QDPLF QHWZRUNV ZH VHH WKDW WKH PD[L PXP GHJUHH LQFUHDVHV VLJQLILFDQWO\ HVSHFLDOO\ ZKHQ WKHUH DUH IHZ )6 DJHQWV WR OHDUQ WKH GHVLUHG EHKDYLRU IURP 7KLV HPHUJHQFH RI VWURQJHU KXEV LV YHU\ XVHIXO 7KH IDFW WKDW WKH WRSRORJ\ RI WKH QHWZRUN FKDQJHV TXLWH D ORW LV VXUSULVLQJ JLYHQ RXU PHDVXUHPHQWV LH RQ DYHUDJH UHZLULQJ RQO\ KDSSHQV RQFH LQ DSSUR[LPDWHO\ 1000 JDPHV 0.4 30 Average converged strategy 10 0 Performance at convergence 50 60 Percentage FS agents Percentage FS agents Degree distribution 40 30 0 10 80 90 10 0 70 60 50 40 30 20 0 10 20 0 0 40 Games per agent :,7+287 5(:,5,1* 6000 Percentage FS agents )LJXUH ,QIOXHQFH RI SHUFHQWDJH )6 DJHQWV 2. 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