Svojstva Nashove ravnoteže policajca i kriminalca iz perspektive poticaja i efikasnosti sankcija na kriminal

Kroz povijest kazne za zločine su bile učestale. Iskustvo nas uči da u mnogim zemljama kroz povijest rezultati nametanja sankcija nisu urodili plodom. Kroz igru dva igrača u teoriji igara pokazali smo teorijsko objašnjenje navedenih pojava te smo zaobišli nedostatke sociološkog i ekonomskog pristupa kriminalu. Kroz igru kriminalca i policajca koju smo definirali matricom isplate Tablica 2.1 pretpostavili smo sljedeće: kriminalci vole kršiti zakon ako policija isti ne provodi, te kriminalci će se pridržavati zakona ukoliko ih policija provodi. Ostale dvije pretpostavke se odnose na to da policija neće provoditi zakon ukoliko ga kriminalci ne krše dok preferiraju provođenje zakona i uhićenja ako kriminalci krše zakon. Pod ovim pretpostavkama ne možemo izračunati čistu Nashovu ravnotežu stoga je jedina ravnoteža igre pod ovakvim pretpostavkama miješana Nashova ravnoteža. Pokazali smo da povećanje očekivane kazne za kriminalce ne utječe na intenzitet kršenja zakona. Također smo pokazali da povećanje očekivane kazne smanjuje intenzitet provođenja zakona. Pristupili smo problemu pronalaska miješane Nashove ravnoteže kroz pet različitih modela gdje smo u prvom modelu imali racionalni pristup sa savršenim informacijama igrača, u drugom modelu smo otpustili pretpostavku da svaki igrač ima samo dvije strategije te smo njegov izbor sveli na beskonačan (neprekidan) strategijski prostor. U trećem modelu, evolucijskim pristupom igrače smo podijelili u dvije podklase i pretpostavili smo kratkovidno podešavanje oba igrača, u četvrtom modelu koji je kombinacija drugog i trećeg modela pretpostavili smo da je jedan igrač potpuno racionalan a drugi kratkovidan, peti model odražava racionalnost oba igrača ali otpušta pretpostavku savršenih informacija što znači da će igrač znati koje su njegove isplate za određene poteze ali neće imati potpuno saznanje o isplatama drugog igrača. Mješovita ravnoteža svakoga od ovih modela vodi ka istom jedinstvenom rješenju što smo zapravo i htjeli prikazati u ovom diplomskom radu ,točnije da sankcije nemaju utjecaj i nisu efikasne na svojstva Nashove ravnoteže u igri policajca i kriminalca s pretpostavkama od 1 do 4. Također istu teoriju smo primijenili u primjeru 2 odnosno igri nametanja sankcija jedne zemlje drugoj zemlji.Through history awards for crime were frequent. Experience teach us that through history in many countries results of sanction impositions did not bear fruit. Through two people game in game theory we showed theoretical explanation of mentioned terms and overcame disadvantages of sociological and economic access to crime. Through the game of criminals and police which we defined with payoff matrix Table 2.1 we assumed that criminals love to break the law if police do not enforce it and that criminals will abide the law if police enforce it. The other two assumptions refer to the fact that the police will not enforce the law if the criminals do not break it but they will prefer law and arrest enforcement if the criminals break it. Under these assumptions we cannot calculate pure Nash equilibrium and therefore the only game equilibrium under this assumptions is the mixed Nash equilibrium. We showed that expected penalty enlargement for criminals does not affect the law breaking intensity. Also we showed that expected penalty enlargement reduces law enforcement intensity. We approached the mixed Nash equilibrium finding problem through five different models where in first model we had rational approach with perfect information of the players, in second model we rejected the assumption that each player has only two strategies and reduced his choice to infinite (constant) strategic area, in third model we divided players into two subclasses using evolutional approach and we assumed short-sighted adjustment of both players, in fourth model, which is combination of second and third model, we assumed that one player is completely rational and the other one is short-sighted, fifth model reflects the rationality of both players but it rejects the perfect information assumption which means that player will know what are his payments for certain moves but he will not have completely cognition of other player payments. Mixed equilibrium each of these models leads to the same unique solution which we actually wanted to show in this master thesis, apropos that sanctions do not have influence and that they are not efficient on Nash equation properties in the game of the police and the criminals with assumptions from 1 to 4. Also the same theory was applied in Example 2, apropos in the sanction imposition game of one nation to another