\relax \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined \global\let\oldcontentsline\contentsline \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} \global\let\oldnewlabel\newlabel \gdef\newlabel#1#2{\newlabelxx{#1}#2} \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} \AtEndDocument{\ifx\hyper@anchor\@undefined \let\contentsline\oldcontentsline \let\newlabel\oldnewlabel \fi} \fi} \global\let\hyper@last\relax \gdef\HyperFirstAtBeginDocument#1{#1} \providecommand\HyField@AuxAddToFields[1]{} \bibstyle{ws-m3as} \citation{Trivers:1971} \citation{Axelrod:1984} \citation{Fehr:2002} \citation{Fehr:2003} \citation{Trivers:1971,Axelrod:1984,Fehr:2002,Fehr:2003} \citation{Boyd:1992} \citation{Fehr:2004} \citation{Boyd:1992,Fehr:2004} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}} \citation{Becker:1968} \citation{Nowak:2006} \citation{Becker:1968,Nowak:2006} \citation{gameTheoryEcon} \citation{gameTheoryEcon} \citation{Helbing2} \citation{Helbing2} \citation{Gordon:2009} \citation{Gordon:2009} \citation{Heckathorn:1988} \citation{Heckathorn:1988} \citation{Arenasa:2011} \citation{DOrsogna:2010} \citation{Arenasa:2011,DOrsogna:2010} \citation{Tabarrok:2007} \citation{Zimring:2001} \citation{Tabarrok:2007,Zimring:2001} \citation{USBJ2010} \citation{USBJ2010} \citation{USBJ} \citation{USBJ} \citation{Nagin:1998} \citation{Nagin:1998} \citation{Nagin:2009} \citation{Cullen:2011} \citation{Nagin:2009,Cullen:2011} \citation{Colvin:2002} \citation{Donohue:1998} \citation{Colvin:2002,Donohue:1998} \citation{Nagin:2009} \citation{Nagin:2009} \citation{Cullen:2002} \citation{MacKenzie:2002} \citation{Cullen:2002,MacKenzie:2002} \citation{Maruna:2004} \citation{Maruna:2004} \citation{Nagin:2009} \citation{Nagin:2009} \citation{Nagin:2009} \citation{Nagin:2009} \citation{Hallevy:2013} \citation{Hallevy:2013} \@writefile{toc}{\contentsline {section}{\numberline {2}Sociological background}{3}{section.2}} \newlabel{sec:sociological}{{2}{3}{Sociological background\relax }{section.2}{}} \newlabel{sec:Model}{{3}{4}{The model\relax }{section.3}{}} \@writefile{toc}{\contentsline {section}{\numberline {3}The model}{4}{section.3}} \newlabel{cons}{{3.1}{5}{The model\relax }{equation.3.1}{}} \citation{Maruna:2004} \citation{Maruna:2004} \citation{Surette:2002} \citation{Surette:2002} \citation{Maruna:2004} \citation{Maruna:2004} \newlabel{eq:pcrime}{{3.2}{6}{The model\relax }{equation.3.2}{}} \citation{Surette:2002} \citation{Surette:2002} \newlabel{ai}{{3.5}{7}{The model\relax }{equation.3.5}{}} \citation{USBJarrest} \citation{USBJarrest} \citation{Farrington:1985} \citation{Pratt:2000} \citation{Farrington:1985,Pratt:2000} \citation{Kalos:2009} \citation{Kalos:2009} \newlabel{preform}{{3.7}{8}{The model\relax }{equation.3.7}{}} \newlabel{sec:Methods}{{4}{8}{Methods\relax }{section.4}{}} \@writefile{toc}{\contentsline {section}{\numberline {4}Methods}{8}{section.4}} \citation{Kalos:2009} \citation{Kalos:2009} \citation{Nagin:2009} \citation{Nagin:2009} \@writefile{toc}{\contentsline {section}{\numberline {5}Results}{9}{section.5}} \newlabel{sec:Results}{{5}{9}{Results\relax }{section.5}{}} \newlabel{sec:pop_dyn}{{5.1}{9}{Population Dynamics\relax }{subsection.5.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Population Dynamics}{9}{subsection.5.1}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Correlations between $p_0$ and $h$}{10}{subsection.5.2}} \newlabel{sec:hp0}{{5.2}{10}{Correlations between $p_0$ and $h$\relax }{subsection.5.2}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Correlations between $\theta $ and $h$}{11}{subsection.5.3}} \newlabel{sec:hth}{{5.3}{11}{Correlations between $\theta $ and $h$\relax }{subsection.5.3}{}} \citation{silence} \citation{silence} \citation{Farrington:1985} \citation{Pratt:2000} \citation{Farrington:1985,Pratt:2000} \@writefile{toc}{\contentsline {section}{\numberline {6}Discussion}{12}{section.6}} \newlabel{sec:Conclusions}{{6}{12}{Discussion\relax }{section.6}{}} \citation{Hardin:1968} \citation{Hardin:1968} \citation{prop36} \citation{prop36} \citation{Arenasa:2011} \citation{Helbing:2010} \citation{Helbing2} \citation{Arenasa:2011,Helbing:2010,Helbing2} \citation{USBJ} \citation{USBJ} \@writefile{toc}{\contentsline {section}{\numberline {Appendix\ A}ODEs Corresponding to the Model}{13}{appendix.A}} \newlabel{sec:appendixODE}{{Appendix\ A}{13}{ODEs Corresponding to the Model\relax }{appendix.A}{}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Evolution of the number of paladins $P$ and unreformables $U$ with respect to time for $p_0=0.1,\tau =2$ and variable $h,\theta $ starting from a population of $N=N_0=400$ neutral citizens. (a) No resources are allocated for rehabilitation purposes and punishment is low for the set of parameter choices: $h=0, \theta =0.4$. As expected $P \gg U$, where no resources are allocated. (b) No resources are allocated for rehabilitation purposes and punishment is large: $h=0, \theta =0.8$. In this case due to the high punishment level a deterrence effect arises and $P \simeq U$. (c) Resources are allocated while keeping punishment low, $h=0.8, \theta = 0.04$ yielding the total expenditure per crime $h \tau + \theta = 1.64$. In this case, the number of paladins increases compared to panel (a) and $P \simeq U$. (d) Resources are allocated while $P > U$. (e), (f) $P>U$ while $h\tau +\theta = 1.64$ as in panel (c). \relax }}{14}{figure.caption.2}} \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} \newlabel{fig:pop_dynamics}{{1}{14}{Evolution of the number of paladins $P$ and unreformables $U$ with respect to time for $p_0=0.1,\tau =2$ and variable $h,\theta $ starting from a population of $N=N_0=400$ neutral citizens. (a) No resources are allocated for rehabilitation purposes and punishment is low for the set of parameter choices: $h=0, \theta =0.4$. As expected $P \gg U$, where no resources are allocated. (b) No resources are allocated for rehabilitation purposes and punishment is large: $h=0, \theta =0.8$. In this case due to the high punishment level a deterrence effect arises and $P \simeq U$. (c) Resources are allocated while keeping punishment low, $h=0.8, \theta = 0.04$ yielding the total expenditure per crime $h \tau + \theta = 1.64$. In this case, the number of paladins increases compared to panel (a) and $P \simeq U$. (d) Resources are allocated while $P > U$. (e), (f) $P>U$ while $h\tau +\theta = 1.64$ as in panel (c). \relax \relax }{figure.caption.2}{}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Contours of the ratio $P/U$, as a function of $p_0$ and $h$ for $\theta =0.1$, and $\tau =2$. The plot is composed of a grid of 21$\times $21 points each corresponding to 400 individuals. The color scale is logarithmic. Note that $P/U$ is an increasing function of $p_0$ and $h$. The solid curve markes the locus $P=U$.\relax }}{15}{figure.caption.3}} \newlabel{fig:P_NR_h_p0}{{2}{15}{Contours of the ratio $P/U$, as a function of $p_0$ and $h$ for $\theta =0.1$, and $\tau =2$. The plot is composed of a grid of 21$\times $21 points each corresponding to 400 individuals. The color scale is logarithmic. Note that $P/U$ is an increasing function of $p_0$ and $h$. The solid curve markes the locus $P=U$.\relax \relax }{figure.caption.3}{}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Contours of the derived quantities for (a) the ratio $P/U$, (b) the crime rate, (c) the punishment rate and (d) the recidivism rate as a function of $h,\theta $ for $p_0=0.1$ and $\tau =2$.\relax }}{16}{figure.caption.4}} \newlabel{fig:stat_h_th}{{3}{16}{Contours of the derived quantities for (a) the ratio $P/U$, (b) the crime rate, (c) the punishment rate and (d) the recidivism rate as a function of $h,\theta $ for $p_0=0.1$ and $\tau =2$.\relax \relax }{figure.caption.4}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The $P/U$ ratio plotted as a function of $h$ under the constraint $h\tau +\theta =c$, where $c$ is a constant, for (a) $\tau =1$ (b) $\tau =1.5$ (c) $\tau =2$ and (d) $\tau =2.5$. The costant is chosen as $c=0.4,0.6,0.8$ so that three curves are shown for each each value of $\tau $. Each curve terminates at $\theta =0$. Panel (b) is projected from Fig.\tmspace +\thinmuskip {.1667em}\ref {fig:stat_h_th}(a). \relax }}{17}{figure.caption.5}} \newlabel{fig:lin_comb_const}{{4}{17}{The $P/U$ ratio plotted as a function of $h$ under the constraint $h\tau +\theta =c$, where $c$ is a constant, for (a) $\tau =1$ (b) $\tau =1.5$ (c) $\tau =2$ and (d) $\tau =2.5$. The costant is chosen as $c=0.4,0.6,0.8$ so that three curves are shown for each each value of $\tau $. Each curve terminates at $\theta =0$. Panel (b) is projected from Fig.\,\ref {fig:stat_h_th}(a). \relax \relax }{figure.caption.5}{}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Curves along which $P/U=1$, for different values of $\tau $. For $\tau =2$, the curve is projected from Fig.\tmspace +\thinmuskip {.1667em}\ref {fig:stat_h_th}(a). The curves all intersect at the same value of $\theta $ since when $h=0$ and no resources are assigned for rehabilitation programs $\tau $ does not play a role in the game. Note that the separatrix $P/U=1$ is lowest for $\tau =2$, implying that for given $h,\theta $ the best way to populate society with an equal amount of paladins and unreformable is by selecting an intermediate value for $\tau $. As explained in the text, intervention programs that are too brief or too long long yield less efficient results.\relax }}{18}{figure.caption.6}} \newlabel{fig:P_U_1}{{5}{18}{Curves along which $P/U=1$, for different values of $\tau $. For $\tau =2$, the curve is projected from Fig.\,\ref {fig:stat_h_th}(a). The curves all intersect at the same value of $\theta $ since when $h=0$ and no resources are assigned for rehabilitation programs $\tau $ does not play a role in the game. Note that the separatrix $P/U=1$ is lowest for $\tau =2$, implying that for given $h,\theta $ the best way to populate society with an equal amount of paladins and unreformable is by selecting an intermediate value for $\tau $. As explained in the text, intervention programs that are too brief or too long long yield less efficient results.\relax \relax }{figure.caption.6}{}} \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces For $\tau =2$, and $p_0=0.1,0.2$: (a) Time rate of crime and (b) Time rate of punishment, which are normalized to the total number of rounds and the total number of individuals. The combination $h\tau +\theta =0.8$ is held fixed. (c) Recidivism Probability. The recidivism probability is normalized to the number of criminals.\relax }}{19}{figure.caption.7}} \newlabel{fig:stat_fix}{{6}{19}{For $\tau =2$, and $p_0=0.1,0.2$: (a) Time rate of crime and (b) Time rate of punishment, which are normalized to the total number of rounds and the total number of individuals. The combination $h\tau +\theta =0.8$ is held fixed. (c) Recidivism Probability. The recidivism probability is normalized to the number of criminals.\relax \relax }{figure.caption.7}{}} \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Projected contours of (a) crime rate, (b) punishment rate, and (c) recidivism rate, while keeping the value of $h\tau +\theta =0.4,0.6,0.8$, for $p_0=0.1$ and $\tau =2$.\relax }}{20}{figure.caption.8}} \@input{ODE3.aux} \bibdata{paper_ref1}