Does Enforcement Matter? Testing Utility

Shocks in a Parking Violations Game

Anuraag Girdhar

December 14, 2013

Abstract

I propose and implement a test of the validity of mixed-strategy equi-

librium under unilateral payo↵ disturbances. When the payo↵ to one

player is changed in a mixed-strategy scenario, the theory counterintu-

itively predicts that the first player doesn’t change her randomization

strategy in equilibrium because the other player does. With parking vi-

olations data from Baltimore, Maryland, I empirically examine the game

where motorists randomize (collectively) in choosing to violate and the

police choose whether or not to enforce. Motoroists parking on streets of

di↵ereing a✏uence have di↵erent utility funcitons, and therefore diferent

payo↵s: I find, consistent with the theory, that a✏uent motorists pay the

meter at the smae rate because the police check parking meters more in

a✏uent neighborhoods. (JEL C70, R42)

1 Introduction

While game theory produces useful insights into fields as disparate as auctiondesign and industrial organization, some of its more fundamental predictionsare rarely empirically tested. The existence of mixed-strategy equilibria ensurethe exisistence of equilibrium in any n-player finihte game, as per John Nash’sfamous proof. But how much of that is just arcane mathematics?

To test the dubious but seldom questioned reality of mixed strategies, PierreAndre Chiappori, Steven D. Levitt, and Timothy Groseclose (2002) examined amatching-pennies style game with an intuitive mixed-strategies solution. Theytested whether soccer penalty kickers and goalies would play left, right andcenter (with respect to the goal) with equal probabilities, and their results wereconsistent with theory predictions. However, aggregated across heterogeneousgames, the uniformity of their probability distribution fails. This is an importantissue I will have to address.

The key model prediction that I confirm in my analysis is that the ticket yieldto the police, simultaneously a proxy for the motorists’ probability of payingthe meter and the utility payo↵ to the police, remains the same across streetsof di↵ering a✏uence. For some intuition regarding this equilibrium, suppose

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yield was higher on weatlhy streets, where motorists are more careless aboutbeing ticketed. Since law enforcement is more active in these areas, the wealthymotorists su↵er a higher probability of getting caught, so to compensate, theypay the meter more frequently than they otherwise would; namely, just as muchas regular drivers. However, since the police do enforce more on wealthierstreets, and since the yield is the same everywhere, I find that the citatin countincreases. Mechanically, this is due to the way I measure the variables: citationcount is roughly the product of yield and enforcement.

To test each part of this claim, I run three primary regressions. My right-hand-side variable remains the same in all three regressions: I use the proportionof luxury cars ticketed as a measure of a✏uence. The enforcement regressionuses a count for the number of days the police hit a given street as the depen-dent variable, while the ticketyield regression uses the number of tickets issuedby street per hour. The citation count regression uses the number of tickets is-sued as the left-hand-side variable. I control for various zip code characteristics(including race to test for bias), and street length.

Much of the existing economic literature on parking meters has to do withregulation of congestion and prime allocation (see e.g. Calthrop, Proost, 2006;Arnott, Rowse, 2007). My research is unique in examining law enforcement andparking meters from a game theoretic perspective.

The remainder of this paper is organized as follows: Section 2 develops thebasic model, Section 3 consists of empirical tests of the model predictions, andSection 4 conludes

2 Overview

2.1 The Model

Consider a large population of motorists who occupy street parking spaces andindividually choose to pay or not pay the parking meter. Consider a populationof o�cers who individually choose to enforce or not enforce on a given street.For each player, denote the former strategy as + and the latter as �, so thateach strategy space is given by (+,�).

Let each ticket issued have a utility (net enforcement cost) u for the o�cer,in accordance with some ticekt quota. Denote the parking meter fee m andthe fine f . Let d be a marginal extra cost of enforcement in case the e↵ort isunsuccessful: it is most helpful to think of this as a “disappointment factor”.Finally, define A as the prior stock of utility for the motorist and B as theprior stock of utility for the o�cer (this is in order to keep utility numbers non-negative, so that the results of the randomization are interpretable. The payo↵matrix is then given by:

When the strategy combinations are (.,�), the o�cer is not working andobtains zero additional utility. With (+, .) or (+,�), the motorist incurs thecost of the meter when paying and the cost of the fine when not, respectively.With (.,+), the o�cer incurs the disappointment factor when the motorist pays

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Motorist

O�cerEnforce Not Enforce

Pay (A�m), (B � d) (A�m), BNot Pay (A� f), (B + u) A,B

and the ticket utility when the motorist does not.It can be shown with elementary algebra (see Appendix) that this game is

in equilibrium where the o�cers choose to work mf of the time, and motorists

choose to pay uu+d of the time. However, if I multiply f by some discounting

factor (1 � r) to represent the wealthy motorists, such that each instance of fin the game specification is replaced with f(1 � r), then the o�cers choose towork m

f(1�r) : a greater proportion of the time.Notice, however, that even though the ‘f’ term is only present in in the

motorists’ utility function, the motorist’s randomization across the decision topay does not change with the utility shock.

Moreover, an intuitive sequential equilibrium concept would seem to suggestthat if there is a place where ticket count should be higher; say, where the mo-torists are welathy and careless about paying the fine, enforcement will increase,forcing even the wealthy motorists to pay more often and driving the ticket yieldback to an unchanging, equilibrium level.

2.2 Previous Literature

To account for the heterogeneity issue, I turn to John Harsanyi’s (1973) proof ofthe Purification theorem, which treats mixed-strategy equilibrium as the limit ofpure-strategy equilibria for a Bayesian game of incomplete information. In otherwords, if players are heterogeneous due ot noise in each player’s utility functionthat is unobserved by other players, then as the size of the noise factor tends otzero, the pure strategy Nash Equilibrium of the incomplete information gameapproaches a mixed-strategy Nash equilibrium. However, to prove this, Harsanyimakes a strong simplifying assumption; namely, that the noise for each playeris independently distributed. To ensure such an independent distribution inmy empirical design, proper analysis would involve clustering across all relevantmotorist observations.

John Knowles, Nicola Persico, and Petra Todd (2001) conducted a test forconducted a test for racial bias in tra�c stop drug checks that is very similarto how I interpret my yield regression. Through their model, they posited thatin the absence of bias, the probability of being guilty should not be related torace or any other driver characteristic. According to my model, if o�cers arenot enforcing optimally, I should see lower yields in areas they over-police andhigher yields in areas they under-police. Otherwise, race should be unrelated toyield.

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3 Empircal Tests

3.1 The Data

I obtained parking violations data for the city of Baltimore from the OpenBalti-more online database provided through Socrata. The data span an observatioonperiod from roughly November of 2011 through October of 2013, along withresidual records of those individuals who had outstanding fine balances goingback about 7 years. It should be noted that I have data on every ticket writtenin the observation period, not aggregate data.

The original data included a vehicle tag, citation number, date of ticket,location of ticket, description of violation, violation code, fine amount, and anoutstanding balance. I also obtained zip codes corresponding to the location(in order to merge in other relevant variables), and these zip code level data arefrom the 2010 census. My final, filtered dataset contains 133,022 observationsover a period of 23 months. There are 31 zip codes represented therein.

3.2 Emprical Methodology

Apart from the expected e↵ects on each of my dependent variables as outlinedin the model section of this paper, I am also interested in whether police arealtering their enforcement decision based on the race of the motorist. If I ob-serve a significant negative coe�cient on the variable black, I will have evidencethat the police are over-policing an area and obtaining a suboptimal yield, justbecause of bias against black motorists.

Furthermore, I include the log of street length (where street length is mea-sured by the number of street blocks ticketed per street) as a control in all threeregressions. This is operating under the assumption that a very long street willhave higher enforcement figures, but yields will remain insignificant and closeto zero.

In constructing my measure of a✏uence, the proportion of luxury cars tick-eted on a street, I considered the following as luxury makes: Alfa Romeo, AstonMartin, Audi, Bentley, Mercedes Benz, BWM, Corvette, Ferrari, Lamborghini,Land Rover, Lexus, Porsche, and Range Rover. For the dependent variabelsin all of my regressions, I aggregated observations of individual motorists upto the street level. Finally, I estimated the following equations using Poissonregression with a log link.

My first baseline regression is of the following form:

ln(enforcei) = �0 + �1Li + �2density + �3black + �4income+

+ �5 ln(stlen) + �6state+ �7make+ �8college+ ✏i(1)

Where Enforcei is the total number of days in the observation period thata street i was hit by enforcement, and Li is the proportion of luxury vehiclesobserved in the ticket data for street i. The population density, median income,race and college (percent college educated) controls are aggregated on the zip

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code level. I also include state and make fixed e↵ects. I expect to find a positivecoe�cient on Li

The second regression I run, which seeks to determine the e↵ect on motoristbehavior, has the following specification:

ln(Y ieldi) = �0 + �1Li + �2�8controls+ ✏i (2)

Where Y ieldi is the ticket yield (# tickets)/(� hours worked) for street i, andthe control variables are the same as the first regression. For reasons statedpreviously, I expect the coe�cient on Li to be indistinguishable from zero inthis regression.

Finally, my last regression takes the following form:

ln(Citationsi) = �0 + �1Li + �2�8controls+ ✏i (3)

Where Citationsi is the number of citations observed for street i over the entireobservation period. The controls from the previous two regressions reprise theirroles, and I expect to find a positive coe�cient on Li

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3.3 Baseline Findings

[]article

(1) (2) (3) (4) (5) (6) (7) (8)EQUATION VARIABLES d1 st d1 st d1 st d1 st d1 st d1 st d1 st d1 stheightd1 st luxprop 3.334*** 3.154*** 3.098*** 3.202*** 4.271*** 4.266*** 4.262*** 4.266***

(0.370) (0.390) (0.322) (0.309) (0.524) (0.525) (0.526) (0.514)popdensity 4.94e-05*** 5.02e-05*** 4.18e-05*** 3.15e-05*** 3.15e-05*** 3.15e-05*** 3.49e-05***

(1.12e-05) (9.55e-06) (9.66e-06) (6.89e-06) (6.88e-06) (6.88e-06) (1.05e-05)black -0.0745 -0.506 -0.947*** -0.946*** -0.946*** -0.994***

(0.245) (0.339) (0.271) (0.271) (0.271) (0.323)medincome -2.18e-05** -3.80e-05*** -3.80e-05*** -3.80e-05*** -3.64e-05***

(9.37e-06) (9.57e-06) (9.56e-06) (9.55e-06) (8.41e-06)stlengthlog 0.464*** 0.464*** 0.464*** 0.462***

(0.0655) (0.0654) (0.0654) (0.0650)state 0.00138*** 0.00137*** 0.00137***

(0.000357) (0.000357) (0.000352)make2 -0.000133 -0.000104

(0.000242) (0.000221)college -0.00365

(0.00705)Constant 5.545*** 4.950*** 4.992*** 5.916*** 4.904*** 4.859*** 4.864*** 4.891***

(0.0934) (0.161) (0.219) (0.455) (0.500) (0.496) (0.499) (0.531)

Observations 133,018 133,018 133,018 133,018 133,018 133,018 132,832 132,832Population Density * * * * * * *Race Dummies * * * * * *Median Income * * * * *Street Length * * * *

State FE * * *Make FE * *College *

height Robust standard errors in parentheses*** p¡0.01, ** p¡0.05, * p¡0.1

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[]article

(1) (2) (3) (4) (5) (6) (7) (8)EQUATION VARIABLES yield st yield st yield st yield st yield st yield st yield st yield stheightyield st luxprop 0.894 0.436 0.277 -0.734 -0.738 -0.737 -0.632

(0.928) (1.043) (0.960) (1.133) (1.134) (1.136) (1.147)popdensity 2.03e-05 2.45e-05 3.46e-05* 3.24e-05* 3.24e-05* 3.24e-05* 3.69e-05*

(1.70e-05) (2.13e-05) (1.97e-05) (1.93e-05) (1.93e-05) (1.93e-05) (2.01e-05)black -0.217 0.167 0.0199 0.0205 0.0208 -0.0324

(0.389) (0.352) (0.356) (0.356) (0.356) (0.331)medincome 2.01e-05*** 1.87e-05*** 1.87e-05*** 1.87e-05*** 2.22e-05**

(6.12e-06) (5.90e-06) (5.90e-06) (5.90e-06) (8.94e-06)stlengthlog 0.154*** 0.154*** 0.154*** 0.151***

(0.0253) (0.0253) (0.0253) (0.0272)state 0.000709* 0.000703* 0.000687*

(0.000392) (0.000389) (0.000387)make2 0.000432** 0.000466**

(0.000211) (0.000219)college -0.00485

(0.00520)Constant 0.725*** 0.857*** -0.0181 -0.298 -0.321 -0.335 -0.351

(0.251) (0.164) (0.371) (0.365) (0.360) (0.358) (0.372)SINGLE luxprop 3.882

(4.018)(0.347)

Constant 2.486***

Observations 132,417 132,417 132,417 132,417 132,417 132,417 132,235 132,235R-squared 0.009

Population Density NO * * * * * * *Race Dummies * * * * * *Median Income * * * * *Street Length * * * *

State FE * * *Make FE * *College *

height Robust standard errors in parentheses*** p¡0.01, ** p¡0.05, * p¡0.1

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[]article

(1) (2) (3) (4) (5) (6) (7) (8) (9)EQUATION VARIABLES cit count cit count cit count cit count cit count cit count cit count cit count cit countheightcit count luxprop 3.550*** 3.546*** 3.558*** 3.791*** 3.690*** 3.790*** 3.701*** 7.247*** 6.860***

(0.409) (0.409) (0.414) (0.327) (0.330) (0.324) (0.334) (0.730) (0.986)black 0.614 1.165 0.626 1.047 -0.125 -0.172

(0.560) (0.725) (0.794) (0.906) (0.652) (0.554)state 0.00273*** 0.00269*** 0.00274*** 0.00287*** 0.00275*** 0.00278*** 0.00188*** 0.00164***

(0.000695) (0.000696) (0.000681) (0.000639) (0.000571) (0.000555) (0.000500) (0.000453)make2 0.00151* 0.00177** 0.00177**

(0.000814) (0.000841) (0.000836)college 0.0207** 0.0218** 0.0243** 0.0110

(0.00950) (0.00902) (0.0103) (0.0106)medincome 5.83e-07 -6.24e-06 -4.28e-05*** -2.32e-05*

(1.50e-05) (1.31e-05) (1.40e-05) (1.35e-05)stlengthlog 1.036*** 1.051***

(0.129) (0.126)popdensity 5.40e-05**

(2.58e-05)Constant 7.804*** 7.714*** 7.667*** 7.253*** 6.528*** 7.230*** 6.743*** 4.003*** 3.108***

(0.209) (0.219) (0.204) (0.344) (0.510) (0.770) (0.865) (0.971) (0.836)

Observations 133,018 133,018 132,832 132,832 133,018 132,832 133,018 133,018 133,018State FE NO * * * * * * * *Make FE NO NO * * * * * * *Race * * * * *

Median Income * * * *College * * * *

Street Length * *Population Density *Race Dummies *

height Robust standard errors in parentheses*** p¡0.01, ** p¡0.05, * p¡0.1

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In Table 1, notice that the variable LUX PROP is robust in its significanceto the addition of extra control variables. The positive coe�cient in Column 5indicates that for every 10 percentage point increase in the proportion of luxuryvehicles, there is a 42.71% rise in enforcement costs.

The coe�cient on BLACK is in fact negative and signficant here, indicatingthat the police are likely to be more lenient in a neighborhood that has a higherproportion of black people as compared to other races. As expected, the coe�-cient on STLENGTH LOG is positive and significant, indicating (intuitively)that longer streets are hit more frequently by the enforcement.

In Table 2, I obtain perhaps the most interesting results of the entire analy-sis. Recall how the motorists’ randomization wasn’t supposed to change undermixed-strategy equilibrium, even when their utility function was perturbed. Inthe Column 1 regression without controls, there appears to be a positive (yetinsignificant) relationship between a✏uence and ticket yield, but as controlsare added, the coe�cient tends toward zero. This a�rms the model’s rathersurprising prediction: wealthier motorists who care less about the fine pay themeter at the same rate as less a✏uent parkers.

Also consistent with mixed-strategies, I find no e↵ect of being in a blackneighborhood on yield, indicating that there is little (if any) bias. However, thesmall positive coe�cient on median income indicates that streets in high medianincome zip codes receive higher yields, indicating that police are under-enforcingin high-income zip codes. This is contradictory to my primary result, but islikely tied to the fact that MEDIAN INCOME is aggregated di↵erently thanLUX PROP . Longer streets appear to have higher yields, but this is di�cultto interpret.

Finally, referring to Table 3, in the Column 9 regression with all the controlsadded, for every 10 percentage point increase in the proportion of luxury vehicleson a street, the citation count increases by 68.60%. Since as I have constructedit, the citation count is just the product of yield and enforcement, becauseenforcement increases in these areas, the number of parking tickets should also.

However, there is a very small but significant negative coe�cient onMEDIANINCOME, meaning that higher income in a zip code slightly reduces the num-ber of citations. This is in line with what I found in the yield regression, andalso opposes the coe�cient on LUX PROP . Again, STLENGTH LOG isinexplicably positive and significant, and there is not significant coe�cient onBLACK

4 Summary and Conclusion

Through the various tests I ran, it is obvious that the parking game playedbetween motorists and law enforcement obeys properties of a mixed-strategyequilibrium. The enforcement regression indicated that the polcie were alteringtheir behavior to a utility shock (change in the utility function across di↵erentmotorists), and the constructed equilibrium criterion, ticket yield, appeared tobe stable across that shock.

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However, contained within that equilibrium is the not-immediately-apparentnotion that yield remains constant, no matter what the utility function of themotorists looks like. Therefore, the motorists are paying the same rate whetherthey are wealthy or not. Moreover, since the yield remains the same, the payo↵to the police also remains the same across streets of di↵ering a✏uence. However,the sequential equilibrium concept I described earlier provides intuition for thissomewhat surprising result.

A limitation of my approach is that my “utility shock” isn’t really a utilityshock: I just consider di↵erent individuals with presumably di↵erent utilityfuncitons. This means that to obtain accurate results, I ought to control formany more motorist characteristics than I did here, however, my data is limitingin this regard. If I can obtain data that go far back enough, I could run similarregressions on a fine change that happened in October of 2010. If my results fromthat test corroborate these, I will have made a much stronger case supportingthe empirical validity of mixed-strategies under utility perturbations.

The results of this paper have farther reaching implications than just park-ing violations. Games with intuitive mixed-strategy solutions arise when oneplayer’s loss is another player’s gain - the most widely referenced example ofsuch a game is a zero-sum game - but the losses and gains need not sum tozero. Therefore, there are many scenarios in the general “cops and criminals”framework that can be examined as mixed-strategy games. Whenever such a

scenario can be motivated as a neat mixed-strategy game, an exogenous increase

in enforcement (a unilateral utility shock) will increase the number of crimi-

nals caught only in proportion to the increase in enforcement: it will not deter

criminal behavior

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