ExtensiveFormGames2/14/18

Alpha-BetaPruningExercise

+

− −

++ ++ ++

−

− − − − − − − − −

5 6 7 4 5 3 6 8 6 9 4 7 6 9 9 8 6 25

DiscussionQuestions

1. Howcanweorganizethesearchtomaximizetheamountofworksavedbypruning?A. Explorebestnodesfirst(estimatewithheuristic).

2. Howcanwemodifyalpha/betapruningtoworkonnon-zero-sumgames?A. Wecan’t.

3. Howcanwemodifyalpha/betapruningtoworkon3-(ormore)-playergames?A. Wecan’t.

IterativeDeepeninginMin/Max

• Wegenerallyimposeadepthlimitonalpha/betamin/max,basedonhowmuchtimewewanttoallow.

• Ifwehavetimeleftover,wecouldre-runwithalargerdepthlimit. Ifwerunoutoftime,wecanjustreturnthemovefromthelastrun.

Whatcanwemodelsofar?Withminimax,wecansolve:• two-player,zero-sum,completeinformation,sequentialmovegames• lotsofclassicboardgames:chess,checkers,connect4…• notmuchelse,andthesearchspacefortheaboveisoftentoobig

Withbackwardsinduction(sofar)wecansolve:• completeinformation,sequentialmovegames• simplemodelsofeconomiccompetition,afewothereconomicapplications

Gametheoryisn’tjustaboutplayinggames.

• We’dliketomodeldecisionmakinginallsortsofmulti-agentinteractions.

• Todoso,weneedtocomeupwithamodel(suchasagametree)thatdescribestheinteraction.

• Weneedtoidentifytheplayers,whentheymakedecisions,andhowwhattheywanttoachieve.

ExampleApplication:ResourceSharingAlsoknownascake-cutting,asin“Icut,youchoose”.• Oneagentproposesadivisionofadesirableresource,theotheracceptsorrejectsthatdivision.• Here,wemodeltheagentsasgettingutility+1foreachunitofresource,butthey’realsospitefulsotheyfeeladisutilityof0.5iftheyreceivelessthanhalf.• MANYvariationsonthismodelarepossible.

1

2 2 2 2

3,-.5 0,0 2,.5 0,0 .5,2 0,0 -.5,3 0,0

3,02,1 1,2

0,3

A R A R A R A R

Whatcan’twemodelsofar?

RandomOutcomes

SimultaneousMoves

IncompleteInformation