Robust protection of fisheries with COmPASS

William B. Haskell, Debarun Kar, Fei Fang and Milind TambeUniversity of Southern California, Los Angeles, CA, 90089

Sam Cheung and Lt. Elizabeth DenicolaU.S. Coast Guard Research Development Center and Atlantic Area

Abstract

Fish stocks around the world are in danger from ille-gal fishing. In collaboration with the U.S. Coast Guard(USCG), we work to defend fisheries from illegal fish-erman (henceforth called Lanchas) in the U.S. Gulf ofMexico. We have developed the COmPASS (Conserva-tive Online Patrol ASSistant) system to design USCGpatrols against the Lanchas. In this application, we facea population of Lanchas with heterogeneous behaviorwho fish frequently. We have some data about theseLanchas, but not enough to fit a statistical model. Previ-ous security patrol assistants have focused on counter-terrorism in one-shot games where adversaries are as-sumed to be perfectly rational, and much less data abouttheir behavior is available. COmPASS is novel because:(i) it emphasizes environmental crime; (ii) it is based ona repeated Stackelberg game; (iii) it allows for boundedrationality of the Lanchas and it offers a robust approachagainst the heterogeneity of the Lancha population; and(iv) it can learn from sparse Lancha data. We report theeffectiveness of COmPASS in the Gulf in our numeri-cal experiments based on real fish data. The COmPASSsystem is to be tested by USCG.

IntroductionFish stocks are at risk of imminent collapse (see (Gillig,Griffin, and Ozuna Jr 2001; Baum and Myers 2004)). Ille-gal fishing greatly exacerbates the decline of fish stocks; itaccounts for as much as 30% of the total catch in some ma-jor fisheries, and it is estimated that 11 - 26 million tons offish were caught illegally worldwide each year (Petrossian2012). As a specific example, the Gulf of Mexico is heav-ily exploited by illegal fisherman from Mexico, hence-forthcalled Lanchas. The U.S. Coast Guard (USCG) wants toprotect fisheries within the U.S. Exclusive Economic Zonesfrom overfishing. The system reported in this paper is devel-oped in collaboration with USCG with this aim.

Illegal fishing in the Gulf poses unique challenges toUSCG. First, USCG can neither police nor observe the en-tire Gulf simultaneously. Second, it is assumed by USCGthat Lanchas can observe their assets moving to and fromtheir bases, and on the open water. Third, USCG does not

Copyright c© 2014, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

have perfect information about Lanchas making it difficultto forecast their future behavior. USCG detects many butnot every Lancha incursion. Since we have limited data, itis hard to reliably fit a statistical model to observed Lanchabehavior.

To respond to these challenges, we have developed theCOmPASS (Conservative Online Patrol ASSistant) system.COmPASS is a decision aid for designing USCG patrols ofthe Gulf. This system is novel in four ways. First, COmPASSis a pioneering application of security game theory and ro-bust optimization to environmental protection, while mostresearch on security games emphasizes counter-terrorism(Tambe 2011; Paruchuri et al. 2008; Shieh et al. 2012;Conitzer 2012; Letchford and Vorobeychik 2011). Second,COmPASS models the interaction between Lanchas andUSCG as a repeated Stackelberg game, while the literatureon security games focuses on one-shot counter-terrorismgames. Third, in contrast to earlier applications on secu-rity games which emphasize perfectly rational attackers,we adopt a bounded rationality model for Lancha behav-ior. However, we face an entire heterogeneous population ofLanchas and each could have a different behavioral model.COmPASS is novel because it is robust against this hetero-geneity; the resulting patrols are conservative and protectUSCG against uncertainty in Lancha type. Fourth, COm-PASS combines robust optimization with learning to makeuse of available data to update its recommendations. In thisway, it can still adapt to changes in Lancha behavior. COm-PASS will be tested by USCG in the Gulf.

DomainOur goal is to offer USCG a more efficient patrol strat-egy against Lanchas in the Gulf in the face of uncertaintyabout the true Lancha behavioral model. In the Gulf, thered snapper and shark populations are considered to be themost at risk from illegal fishing. Red snapper stocks are non-migratory, the location of these fisheries typically does notchange over time. Sharks are migratory and spread through-out the Gulf. Empirically, it appears that the Lanchas haveconsiderable knowledge about the distribution of these fishpopulations.

Lanchas enter U.S. waters frequently in small powerboatswith crews of two to four, which are difficult to see againstthe water. Lanchas are either spotted directly, or their gear

Figure 1: (a) Coast Guard boat used to patrol Gulf of Mexicoand (b) a captured Lancha boat

is detected. A captured Lancha and an USCG patrol assetis shown in Fig. 1. Throughout this paper, we assume thatthe Lanchas have knowledge of the distribution of USCGpatrols.

Building COmPASSWe formally present the COmPASS system in this section.First we describe our game-theoretic framework and then wedevelop the details of COmPASS’s implementation.

Game-theoretic modelWe model the interaction between USCG and Lanchas as arepeated Stackelberg game (Myerson 2013). USCG plays asthe leader and generates patrols, then the Lanchas observeand react. This game is repeated because USCG can react toLancha movement and design new patrols, which the Lan-chas can then observe and use to improve their own strategy.

We discretize the Gulf into a grid of identical cells (tar-gets) indexed by T = 1, . . . , T. We use real data on fishdensity and distance from the Mexican border to constructthe payoff of each target to the Lanchas, Ra

t . Rat is directly

proportional to the amount of fish in a grid cell but inverselyproportional to the distance of that cell from their base to ac-count for greater fuel cost. We also introduce a penalty termfor Lanchas for the cost of being intercepted by USCG. P a

tis the value of the Lancha vessel which is confiscated byUSCG at target t if it is intercepted. If USCG protects tar-get t, and the Lancha attacks target t, then USCG receivesreward Rd

t from the Lancha penalty. If the Lancha attackstarget t and it is not defended, then USCG loses P d

t - themonetary value of the stolen fish. This game is zero sum. Ifthe Lancha earns a high reward, they must have caught a lotof fish from near the shore (because most of the fish is nearthe shore). Thus, the USCG is penalized harder by missinga well known area of high fish density.

For r boat hours, USCG’s strategy can be represented bya probability distribution x = (x1, . . . , xT ) over targets inthe set

X =

x ∈ RT :

∑t∈T

xt = r, 0 ≤ xt ≤ 1,∀t ∈ T

,

where xt is the marginal probability of protecting target tand r is the total patrol boat hours. If the Lancha attacks

target t, then for a given defender strategy x, the defender’sexpected utility is

Ut (x) = xtRdt + (1− xt)P

dt .

We are interested in randomized strategies because theymake USCG more unpredictable. Each target t has a vec-tor of attributes, in our case, the attribute vector will in-clude the current USCG patrol density xt, and the rewardRa

t and penalty P at for that target. Actual patrols are gen-

erated from x taking into account time and distance con-straints on USCG assets (Korzhyk, Conitzer, and Parr 2010;Jain et al. 2010).

We base our Lancha behavioral model on the SubjectiveUtility Quantal Response (SUQR) model from (Nguyen etal. 2013). SUQR is a stochastic choice model; it includessome randomness in adversary decision making which is notpresent in perfectly rational decision makers. SUQR gives usa realistic bounded rationality model of the Lanchas, sincewe cannot assume the Lanchas are perfectly rational. Fur-ther, (Nguyen et al. 2013) shows that SUQR outperformsother models like quantal response (McKelvey and Palfrey1995) via extensive human subjects experiments in securitygames. In our case, using the data (xt, P

at , R

at ) available to

the Lancha for target t, the probability that the Lancha willattack target t is

qt (ω |x) =eω1xt+ω2P

at +ω3R

at∑

v∈T eω1xv+ω2Pa

v +ω3Rav.

Notice that∑

t∈T qt (ω |x) = 1. We see that the vectorω = (ω1, ω2, ω3) encodes all information about Lanchabehavior, and each component of ω determines how muchweight the Lancha gives the corresponding attribute in hisdecision making. In (Nguyen et al. 2013), ω is set to a fixedvalue to represent an attacker; there is no heterogeneity inattackers. Instead, we face an entire population of heteroge-neous Lanchas, so we introduce a set Ω ⊂ R3 to representthe range of all possible ω that represent real Lanchas. Weallow a separate value of ω for each Lancha incursion. Wewill refer to ω as a specific Lancha type.

Given that we have heterogeneous Lanchas, rather thanjust one, we are dealing with a population of ω. A naturalidea is to assume that there is a prior distribution F over Ω.Then, the stochastic optimization problem

maxx∈X

ˆΩ

[∑t∈T

Ut (x) qt (ω |x)

]F (dω) (1)

can be solved. Problem (1) maximizes the expected utilityfor USCG, where expectation is taken over Lancha types.Furthermore, if Lancha data is available, then Bayesian up-dates can be performed on F .

However, there are several problems with the stochasticoptimization approach in Problem (1). First, we cannot jus-tify the general assumption that there is a probability distri-bution F over Lancha types. There is just not enough data tomake a reasonable hypothesis about the distribution of ω forthe Lancha population. Second, when observations of ω areavailable we can perform Bayesian updating in Problem (1)

to improve our estimate of the true probability distributionon Ω. However, this type of Bayesian updating requires a lotof data even if a prior distribution F is available. Even whena lot of data is available, it may take time to gather these dataand Bayesian updating would not be effective while that datagathering takes place. Runtime presents itself as a third is-sue, since solving Problem (1) is computationally expensive.

Robust optimizationRobust optimization offers remedies for the shortcomings inProblem (1). It does not require a distribution F over Lanchapopulation parameters. Also, robust optimization will workwith very small data sets - even a single observation. Wecontinue to use the SUQR model from (Nguyen et al. 2013),since we do not rely on the assumption of perfectly rationalattackers as in earlier security games. We now enhance theSUQR model by hedging against the heterogeneity of theLancha population with robust optimization.

We start with an uncertainty set Ω ⊂ Ω to capture the pos-sible range of reasonable Lancha behavior parameters. Ω canbe initialized based on domain expertise and human behav-ior experiments. As new observations come in, COmPASSupdates Ω as described in the next section.

Given an uncertainty set Ω, we solve the initial robust op-timization problem

maxx∈X

minω∈Ω

∑t∈T

Ut (x) qt (ω |x) , (2)

to get a patrol for USCG. We continue to assume that in-dividual Lanchas follow an SUQR model. However, we donot know the distribution of the SUQR parameters ω overthis population. Problem (2) is distinct from Problem (1).No probability distribution is assumed and no Bayesian up-dating is performed in Problem (2). Additionally, the objec-tive of Problem (1) is the USCG expected utility over SUQRparameters with respect to F . In contrast, the objective ofProblem (2) is the worst-case expected utility over all allow-able SUQR models for the Lancha. Since we do not knowthe distribution of ω, it is safest for us to hedge against allpossible Lancha types.

Problem (2) is a nonlinear, nonconvex, nondifferentiableoptimization problem. The function∑

t∈T Ut (x) qt (ω |x) is continuous and differentiable in xfor any fixed ω, but it is not convex in x. For easier imple-mentation, we transform Problem (2) into the constrainedproblem

maxx∈X, s

s : s ≤

∑t∈T

Ut (x) qt (ω |x) , ∀ω ∈ Ω

, (3)

by introducing a dummy variable s ∈ R. This transforma-tion replaces the nonsmooth objective in the original Prob-lem (2) with a set of smooth constraints, so that we eliminatethe nondifferentiability to make numerical algorithms workbetter. Specifically, we solve Problem (2) in MATLAB withthe function fmincon.

LearningThis section explains how COmPASS uses data. Learningis a core part of COmPASS, and any available data can im-prove our online patrol generation. The main intuition is thatwe will use data to construct the uncertainty set Ω for Lan-cha types that was taken as input in Problem (2). In thisway, we keep the conservatism of robust optimization butinclude the optimism that we can adapt to new observations.This idea is based on (Bandi and Bertsimas 2012), whereit is used to construct uncertainty sets for stochastic anal-ysis. This blend of robustness and learning has not yet ap-peared in the security game literature, and it is a promisingapproach to patrol scheduling in security games where somedata is available. Unlike other estimation-based approaches,our approach does not require strong statistical assumptionson the underlying data (like normality).

We will assume that each observation comes from an in-dependently distributed Lancha attack, with a possibly dif-ferent SUQR parameter ω from our population of parame-ters. We do not have the ability to know which the repeatcriminals are, so we have to assume that all incursions areindependent and each corresponds to a different SUQR pa-rameter. Let N (k) be the total number of crimes committedin round k, and let J (k) =

∑ki=1 N

(i) be the running totalnumber of crimes by the end of round k. Suppose target tis attacked during round k while the USCG patrol densityis xk, then the maximum likelihood equation (MLE) for thetype of the Lancha, i.e. the parameter ω, who committed thiscrime is

maxω∈Ω

log (qt (ω |xk)) . (4)

To explain, equation (4) finds the Lancha type ω which max-imizes the likelihood of an attack on the observed target t.Note that equation (4) only assumes the SUQR model, it isnot based on any other hidden assumptions on the data. Bysolving the MLE equation (4) for each observed incursion,we build a collection

Ξ(k) =ωj : j = 1, . . . , J (k)

of estimated Lancha SUQR parameters available at the endof round k. The set Ξ(k) is effectively a sample of estimatedLancha SUQR parameters, we will use Ξ(k) to construct Ω.

For clarity, Ξ(k) is not known until the end of round k.At the beginning of round k, USCG only knows Ξ(k−1). Wedenote the resulting data-driven uncertainty set for round kas

Ω(

Ξ(k−1))

to emphasize the dependence on the data Ξ(k−1). At the be-ginning of round k, we then want to solve the resulting ro-bust optimization problem

maxx∈X

minω∈Ω(Ξ(k−1))

∑t∈T

Ut (x) qt (ω |x) , (5)

which can be solved using the same algorithm as for Prob-lem (2). Compare Problems (2) and (5). In Problem (2), theuncertainty set Ω is initialized and no new data has yet been

collected. In Problem (5), the uncertainty set Ω(Ξ(k−1)

)in-

cludes new Lancha observations over the last k − 1 rounds.Different data sets give rise to different instances of the ro-bust Problem (5). The only remaining issue is to constructthe data-driven uncertainty set Ω

(Ξ(k−1)

).

We define our uncertainty sets in the following way: Inround k, we consider

Ω(

Ξ(k−1))

= vertexωj : j = J (k−1) −m(k), . . . , J (k−1)

.

where vertex (·) returns the vertices of the convex hull ofa set and m(k) is the number of past observations that wekeep. If m(k) = J (k−1) − 1, then we use all Lancha obser-vations. We can easily make Ω

(Ξ(k−1)

)less conservative by

having a shorter memory. In this fashion, we do not becomeprogressively more conservative but track recent Lancha ac-tivity.

Unlike Problem (2), where Ω is initialized by the user, per-haps based on historical data, the uncertainty set Ω

(Ξ(k−1)

)is data-driven and is updated online each time a new Lanchaobservation is made. In fact there are many ways to constructΩ(Ξ(k−1)

).

Experiments and Analysis

Figure 2: (a) Sample Simulation Area in Gulf of Mexico; (b)Cumulative Expected Utility for COmPASS with and with-out the distance factor being included in the payoffs

In this section, we evaluate COmPASS on real fish andUSCG Lancha data. The fish data was taken from NOAA’sCoastal Ecosystem Maps1; for security purposes we had tosanitize the Lancha data obtained from USCG to share theseresults. We focus on part of the Gulf of Mexico and dis-cretize it into a 25*25 grid with 625 cells, 3 miles wide and4.2 miles long. This area is shown in Fig. 2, and each roundcorresponds to 3 months. For these experiments, the Lanchasare uniformly penalized across targets if they are caught byUSCG. We will return to Fig. 2(b) later.

We evaluate COmPASS and compare it against other al-gorithms. Figs. 3(a), 3(b) and 3(c) depict how the cumula-tive expected utility of the defender (shown on the Y-axis)

1http://service.ncddc.noaa.gov/website/CHP/viewer.htm

increases over rounds (shown on the X-axis) for 10, 30 and50 patrol boat hours, respectively. So, we calculate the worstcase expected utility for the defender over all Lancha typesfor each round and this gets cumulated over the rounds. Wesee that COmPASS improves based on an average of about30 Lancha observations per round. This performance is con-sistent even when varying the number of resources. We notethe superiority of COmPASS over two other approaches:first, we consider a baseline Maximin that does not use abehavioral model; second, we compare against SUQR. ForSUQR, we assume that the weights ω = (ω1, ω2, ω3) are thesame for all Lanchas, we use the weights derived from hu-man subject experiments in (Nguyen et al. 2013). The SUQRmodel performs the worst because it fails to exploit the factthat different Lanchas may use different weights for differ-ent features, and because it does not update over rounds. Onthe other hand, COmPASS successfully captures a diversepopulation of Lanchas and learns from data. Even thoughthe performance of both Maximin and SUQR improve asthe number of boat hours increases, they are consistentlyoutperformed by COmPASS. Also note that COmPASS uti-lizes resources more efficiently as compared to Maximin andSUQR, which are both sensitive to the number of resources.

The heat maps in Figs. 4(a) and 4(b) reveal the cover-age probabilities generated by COmPASS for rounds 1 and7, lighter colors correspond to higher coverage probabili-ties. These figures clearly show that COmPASS’s robust ap-proach is capable of adapting to data. Note that the grid tothe bottom right corner of Figs. 4(a) and 4(b) shows theheat map for the entire 25*25 grid, while the heat map ofa zoomed in 3*3 grid is shown at the left upper corner of thefigures. The 3*3 grid corresponds to rows 5 to 8 and columns1 to 3 in the 25*25 grid. Similar notation is followed for Fig.4(c), which shows a heat map of the strategy generated byMaximin. In contrast to the heatmaps of COmPASS whichchange over rounds, Maximin generates only one fixed strat-egy for every round. We also experiment with COmPASS tostudy the impact of distance from the shore on the defenderstrategy. Fig. 2(b) reports the cumulative defender expectedutility for COmPASS for two cases: (i) when the distancefrom shore is incorporated into the rewards and (ii) when itis not. The results suggest that traveling distance is an im-portant factor to consider while constructing the payoffs.

Related WorkOur work synthesizes the literature on fishery protection, se-curity games, and robust optimization. First, the problem ofprotecting fisheries has already received attention. (Pauly etal. 2002) discusses the urgent need for protection of fish-eries to maintain sustainability. In (Gallic and Cox 2006;Stokke 2009), an economic analysis of illegal fishing isconducted and some economic remedies are proposed. Ourwork differs from these references because it offers a tacti-cal, proactive solution to illegal fishing rather than a strate-gic, policy based solution. (Lobo 2005) solves for one-dimensional ocean patrols using self-organizing maps and(Millar and Russell 2012) uses an integer programming for-mulation to design fishery patrol routes. Both (Lobo 2005)and (Millar and Russell 2012) are based on the classic trav-

(a) 10 resources (b) 30 resources (c) 50 resources

Figure 3: Cumulative Defender Expected Utilities for different number of resources over rounds

(a) Round 1 (b) Round 7 (c) Maximin

Figure 4: Heat Maps showing defender coverage probabilities for all targets generated by (a, b) COmPASS with 50 resourcesand (c) Maximin

eling salesman problem. Our approach differs from (Lobo2005) in two ways: first, we take a game-theoretic approach;second, we account for uncertainty in the behavior and pref-erences of the adversary to get robust patrols. (Yang et al.2014) studies a wildlife security game against a populationof heterogeneous poachers. The SUQR model also appearsin this work, and it is assumed that the SUQR parameters forthe poachers are normally distributed. COmPASS does notmake any assumptions on the distribution of SUQR param-eters.

Game theory in security is a highly developed field, espe-cially in counter-terrorism applications (Tambe 2011). TheStackelberg game model has been emphasized in this bodyof literature. (Conitzer 2012) surveys the state of the art inalgorithmic security game theory, in particular for Stackel-berg games; though these algorithms do not apply to our re-peated setting. Furthermore, Stackelberg games in securityoften appear in conjunction with a behavioral model for theadversary. (Marecki, Tesauro, and Segal 2012) considers re-peated Stackelberg Bayesian games, where the prior distri-bution is over follower preferences and the follower is as-sumed to play a myopic best response. We depart from thisapproach because we do not assume a prior over adversarypreferences, and we work in the specific fisheries domain.

Robust optimization, a tool for decision-making under un-certainty, is particularly pertinent to us. The usual strategy isto optimize a performance measure over the worst-case ofscenarios, thus ensuring a conservative decision. See (Ben-

Tal, El Ghaoui, and Nemirovski 2009) for a detailed sur-vey of robust optimization and see (Bandi and Bertsimas2012) for the application of robustness to stochastic systemsanalysis. Robustness is especially relevant to multiple agentsettings, and robust solution concepts have been developedfor games. (Aghassi and Bertsimas 2006) develop a solutionconcept for robust Nash games, and (Kardes, Ordonez, andHall 2011) develop a solution concept for robust Markovgames. We differ from the usual robust approach because:(i) we are focused on game theory; (ii) we are solving a se-quence of robust optimization problems; and (iii) our robustformulation makes use of data.

SummaryCOmPASS is a novel application of security game theoryand robust optimization that improves USCG’s ability to in-terdict Lanchas in the Gulf of Mexico. To summarize, COm-PASS has four main novel features. First, COmPASS ex-pands the application of security game theory and robustoptimization to the critical task of protecting fisheries. Sec-ond, COmPASS contributes a model of repeated Stackelberggames to the security literature, which traditionally empha-sizes one-shot games. Third, we develop a robust variant ofSUQR that protects against uncertainty in the Lancha popu-lation. This robustness is the key to generating defensive pa-trols. Finally, COmPASS offers a way to dynamically updateour uncertainty set online as we gather new observations

so that we can adapt to changing Lancha behavior. USCGis studying our recommendations to inform their patrols ofsea and air assets. Whenever our patrols are implemented,USCG will take detailed observations and return this infor-mation to us. Their domain experts are able to effectivelycomment on our patrol feasibility and impact. We concludeby emphasizing that our method has wide applicability toother security games beyond fishery protection where thereis repeated interaction between the population of criminalsand law enforcement. The SUQR framework is a general hu-man behavioral model for making choices, it is not specificto the fisheries domain.

DisclaimerThe views expressed in this article are those of the authorand DO NOT necessarily reflect the views of the USCG,DHS, or the U.S. Federal Government or any AuthorizedRepresentative thereof. The U.S. Government does not ap-prove/ endorse the contents of this article, therefore the ar-ticle shall not be used for advertising or endorsement pur-poses. The U.S. Government assumes NO liability for itscontents or use thereof. Moreover, the information containedwithin this article is NOT OFFICIAL U.S. Government pol-icy and cannot be construed as official in any way. Also,the reference herein to any specific commercial entity, prod-ucts, process, or service by trade name, trademark, manu-facturer, or otherwise, does NOT necessarily constitute orimply its approval, endorsement, or recommendation by theU.S. Government.

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