Table 1.
List of subpopulations and of relevant parameters.
Figure 1.
Dynamics of Paladins and Unreformables
for
and variable
under initial conditions
where
and
where
.
(a) For ,
, since punishment is low and no post-release resources are allocated. For
the number of unreformables is slightly higher than for
. (b) For
, higher punishment leads to a deterrence effect and
. This trend is more evident for
as explained in the text. (c) For
,
. (d) For
higher punishment leads to
. (e), (f) Dynamics under the constraint
as in panel (c).
Figure 2.
Contours of the final ratio as a function of
and
for
, and
for
.
Note that the final ratio is an increasing function of
and
. The solid curve marks the locus
.
Figure 3.
Contours of the final values of (a) the ratio, (b) the number of crimes per player, (c) the number of punishments per player and (d) the recidivism rate as a function of
for
and
.
Initial conditions are chosen so that at the onset of the game
, and
.
Figure 4.
The final ratio plotted as a function of
under the constraint
, for (a)
(b)
(c)
and (d)
.
The constant is chosen as so that three curves are shown for each each value of
. Each curve terminates at
. Panel (b) is projected from Fig. 3(a). Note that for all
values, the most efficient allocation of resources is attained for the intermediate
. In particular, for
the final
ratio is attained at
,
and
as shown in panel (b). Also note the emergence of maxima in panels (b) and (c).
Figure 5.
Curves along which at the end of the game for different values of
.
Given , the area to the right of each curve corresponds to values of
where
and the area to the left of each curve corresponds to values of
where
. The curve for
is projected from Fig. 3(a). When no rehabilitation resources are assigned (
)
does not play a role so curves intersect at the same value of
. Note that the
curve is lowest for
, implying that for given
the best way to populate society with an equal amount of paladins and unreformables is by selecting an intermediate value for
. As explained in the text, intervention programs that are too brief or too long long yield less efficient results.
Figure 6.
The number of crimes per player over the course of a game plotted as a function of under the constraint
, for (a)
(b)
(c)
and (d)
.
The constant is chosen as so that three curves are shown for each each value of
. Each curve terminates at
. Note that a minimum arises in the case of
indicating that an optimal allocation of rehabilitation and punishment resources exists to minimize crime occurrences.
Figure 7.
Contours of the final values of (a) the ratio, (b) the number of crimes per player, (c) the number of punishments per player and (d) the recidivism rate as a function of
for
and
.
Initial conditions are chosen so that at the onset of the game
,
and all players within
are assigned
and
. Note that while qualitative trends mirror the results shown in Fig. 3 for
, there are quantitative differences between the two different initial conditions.
Figure 8.
Dynamics of Paladins and Unreformables
according to the ODEs in Eqs. 10-16 for
,
and variable
under initial conditions
where
.
Note that the dynamics are qualitatively similar to the simulation results shown in Fig. 1.