Bribery in elections is an important problem in computational social choice
theory. However, bribery with money is often illegal in elections. Motivated by
this, we introduce the notion of frugal bribery and formulate two new pertinent
computational problems which we call Frugal-bribery and Frugal- briberytocapturebriberywithoutmoneyinelections.Intheproposedmodel,thebriberisfrugalinnatureandthisiscapturedbyherinabilitytobribevotesofacertainkind,namely,non−vulnerablevotes.IntheFrugal−briberyproblem,thegoalistomakeacertaincandidatewintheelectionbychangingonlyvulnerablevotes.IntheFrugal−dollarbriberyproblem,thevulnerablevoteshavepricesandthegoalistomakeacertaincandidatewintheelectionbychangingonlyvulnerablevotes,subjecttoabudgetconstraintofthebriber.WefurtherformulatetwonaturalvariantsoftheFrugal−dollarbriberyproblemnamelyUniform−frugal−dollarbriberyandNonuniform−frugal−dollarbriberywherethepricesofthevulnerablevotesare,respectively,allthesameordifferent.Westudythecomputationalcomplexityoftheaboveproblemsforunweightedandweightedelectionsforseveralcommonlyusedvotingrules.Weobservethat,evenifwehaveonlyasmallnumberofcandidates,theproblemsareintractableforallvotingrulesstudiedhereforweightedelections,withthesoleexceptionoftheFrugal−briberyproblemforthepluralityvotingrule.Incontrast,wehavepolynomialtimealgorithmsfortheFrugal−briberyproblemforplurality,veto,k−approval,k−veto,andpluralitywithrunoffvotingrulesforunweightedelections.However,theFrugal−dollarbriberyproblemisintractableforallthevotingrulesstudiedherebarringthepluralityandthevetovotingrulesforunweightedelections.