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UCGIS, Feb 2000
Optimal Police Enforcement Allocation
Rajan BattaChristopher Rump
Shoou-Jiun Wang
This research is supported by Grant No. 98-IJ-CX-K008 awarded by the National Institute of Justice, Office of Justice Programs, U.S. Department of Justice. Points of view in this document are those of the authors and do not necessarily represent the official position or policies of the U.S. Department of Justice.
UCGIS, Feb 2000
Motivation
“Our goals are to reduce and prevent
crime,… and to direct our limited
resources where they can do the most
good.”
- U.S. Attorney General Janet Reno
- Crime Mapping Research Conference, Dec. 1998
UCGIS, Feb 2000
Consider Crimes Motivated by an Economic Incentive
Auto theftRobberyBurglaryNarcotics
UCGIS, Feb 2000
Literature Review
Cornish et al. (Criminology, 1987):
Criminals seek benefit from their criminal behavior.
Freeman et al. (J. of Urban Economics, 1996):
A neighborhood with higher expected monetary return is more attractive to criminals.
Greenwood et al. (The Criminal Investigation Process, 1977): A neighborhood with lesser arrest ability has a larger amount of crimes.
UCGIS, Feb 2000
Literature Review
Caulkins (Operations Research, 1993):
Drug dealers’ risk from crackdown enforcement is proportional to “total enforcement per dealer
raised to an appropriate power”.
Gabor (Canadian J. of Criminology, 1990):
A burglary prevention program may decrease local burglary rates, but increase neighboring rates - geographic displacement.
UCGIS, Feb 2000
PA(E,n) = 1- exp(-E/n)
= arrest ability value (Caulkins)
Under constant E, PA decreases in n (Greenwood et al.)
PA increases in E Effect of E is more
significant for small n
Arrest Rate (PA), Enforcement (E) & Crime Incidents (n)
Crime Level
UCGIS, Feb 2000
Monetary Return (R), Wealth (w) & Crime Incidents (n)
R(w,n) = c w exp(-n)c, depend on crime type
R decreases in n Physical Explanations:
Limited by the wealth of the neighborhood
Victims become aware and add security
Crime Level
UCGIS, Feb 2000
Expected Monetary Return (E[R]) & Crime Incidents (n)
E[R]= R(w,n)*(1-PA(E,n))=c w exp(-E/n-
n)(Freeman)
For small n, E[R] is small because of high arrest probability.
For large n, E[R] is small due to many incidents.
E forces the E[R] down.Crime Level
UCGIS, Feb 2000
Crime Rate & Socio-Economy
One area is relatively
crime-free (Amherst)
Another area is relatively
crime-ridden (Buffalo)
Expected return for
crime, E[R], may equally
attract offenders
UCGIS, Feb 2000
Crime Equilibrium
n*
Opportunity Cost of crime
n(1) n(2)
m
E[R]
Crime Level
At equilibrium, number of crimes is either 0 or n(2)
If n<n(1), high arrest rate; all criminals will leave
If n(1)<n<n(2), return>cost; attracts more criminals
If n>n(2), over-saturated; some criminals will leave
n*: organized crime equilibrium
UCGIS, Feb 2000
Crime Crackdown
Sufficient enforcement, E, can lower expected return curve E[R]
If E[R] curve < m, there is no incentive for criminals; crime collapses to 0
Crime Level
E[R]
Opportunity Cost of crime
E
m
UCGIS, Feb 2000
Minimizing Total Crime (2 Neighborhoods)
Objective 1: Minimize total number of crimes
Optimal Allocation Policy:
one-neighborhood crackdown policy is optimal: place as many resources as necessary into one neighborhood; if resources remain, into the other.
Generally, the neighborhood with better arrest ability tends to have higher priority to receive resources.
Under equal arrest ability: affluent neighborhood has priority only if both neighborhoods can be collapsed.
UCGIS, Feb 2000
Objective 2: Minimize the difference of crime numbers
Optimal Allocation Policy:
The difference of the crime numbers can be minimized to 0 unless the wealth disparity between them is large.
Under equal wealth, allocation of resources is inversely proportional to arrest ability.
If the wealth disparity between the two neighborhoods is large, the affluent neighborhood has priority.
Minimizing Crime Disparity(2 Neighborhoods)
UCGIS, Feb 2000
A Numerical Example
Data: Arrest ability: 1 = .35, 2 = .10
Wealth level: w1= $30,000, w2 = $25,000
= .02; c = .01; m = $15.
Calculated Values: Enforcement required to collapse crimes in NB1=320 hours
Enforcement required to collapse crimes in NB2=990 hours
Note: Every day, Buffalo Police Department patrols 300-500 hrs in each of its five districts and the number of call-for-service in each district is about 100-150.
Decision Variable: x (proportion of enforcement allocated in NB 1).
UCGIS, Feb 2000
Total Enforcement = 1000 hours
x = .01
n1 = 149; n2 = 0
Total = 149
Difference = 149
(dominated)
x = .265
n1 = 106; n2 =106
Total = 212
Difference = 0
x = .32
n1 = 0; n2 = 110
Total = 110
Difference = 110
x = .3
n1 = 94; n2 = 108
Total = 202
Difference = 15
(non-dominated)
------- Neighborhood 1; ------- Neighborhood 2
UCGIS, Feb 2000
Total Enforcement = 520 hours
x = 0
n1 = 150; n2 = 119
Total = 269
Difference = 31
(dominated)
x = .32
n1 = 127; n2 = 127
Total = 254
Difference = 0
x = 0.62
n1 = 0; n2 = 133
Total = 133
Difference = 133
x = 0.5
n1 = 107; n2 = 131
Total = 238
Difference = 23
(non-dominated)
------- Neighborhood 1; ------- Neighborhood 2
UCGIS, Feb 2000
Total Enforcement = 300 hours
x = 0
n1 = 150; n2 = 129
Total = 279
Difference = 21
(dominated)
x = 0.4
n1 = 134; n2 = 134
Total = 268
Difference = 0
x = 1
n1 = 94; n2 = 141
Total = 235
Difference = 47
x = 0.5
n1 = 130; n2 = 135
Total = 265
Difference = 5
(non-dominated)
------- Neighborhood 1; ------- Neighborhood 2
UCGIS, Feb 2000
Objective 1: Minimize total number of crimes
The neighborhoods should be either cracked down or given no resources except for one of them.
The neighborhoods with higher arrest/wealth value have higher priority.
Objective 2: Minimize the difference of crime numbers “Evenly” distribute enforcement to the wealthier
neighborhoods such that the wealthier neighborhoods have the same number of crimes.
Optimal Enforcement Allocation (Multiple Neighborhoods)
UCGIS, Feb 2000
BPD Case Study
Buffalo Police Department
~42 Square Miles
5 Command Districts
~6700 calls for service/wk
~6400 patrol hours/week
~530 police officers
30-55 patrol cars at any time w/ 2 officers/car
UCGIS, Feb 2000
Burglary Data in Buffalo
District Median Household
Income (w)
Weekly Patrol
Hours (E)
Total Burglary
Numbers (n)
Arrest Prob.
(PA)
A $21,250 896.0 187 13.37 %
B $13,750 1485.4 350 23.86 %
C $13,750 1536.0 481 13.72 %
D $21,250 1314.1 373 9.65 %
E $21,250 1183.3 304 16.78 %
Total 6414.8 1695 15.43%
UCGIS, Feb 2000
Minimizing Burglary Disparity in Buffalo
District
Arrest
Ability
()
Opportunity
Cost
(m)
Optimal
Patrol Hours
(E*)
Required Hours
to Collapse
Crime Activity
Number of
Burglaries
(n*)
A .0300 157.0 252.1 (3.9%) 2967.6 329
B .0612 79.0 1475.8 (23.0%) 2663.3 329
C .0462 79.0 1954.9 (30.5%) 3528.0 329
D .0288 138.5 1696.7 (26.5%) 4371.7 329
E .0472 138.5 1035.3 (16.1%) 2667.5 329
Total 6414.8 (100%) 16198.1 1645