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Information-Driven Search Strategies in the Board Game of Clue 609 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009 607 Information-Driven Search Strategies in the Board Game of CLUE Silvia Ferrari, Senior Member, IEEE, and Chenghui Cai, Member, IEEE Abstract—This paper presents an information-driven sensor room in the mansion. Therefore, a game strategy must plan management problem, referred to as treasure hunt, which is rele- the suggestions’ sequence and enabling pawn motions based vant to mobile-sensor applications such as mine hunting, monitor- on the evidence that becomes available over time. By viewing ing, and surveillance. The objective is to infer a hidden variable or treasure by selecting a sequence of measurements associated the suggestions as measurements and the rooms as targets, an with multiple fixed targets distributed in the sensor workspace. approach is developed for computing optimal strategies from an The workspace is represented by a connectivity graph, where each influence diagram (ID) representation of the game. node represents a possible sensor deployment, and the arcs repre- As shown in [1] and [3], the treasure hunt is a basic sent possible sensor movements. An additive conditional entropy information-driven sensor management problem that is relevant reduction function is presented to efficiently compute the expected benefit of a measurement sequence over time. Then, the optimal to several mobile-sensor applications, such as robotic mine treasure hunt strategy is determined by a novel label-correcting hunting [4], cleaning [5], and monitoring of urban environments algorithm operating on the connectivity graph. The methodol- [3], manufacturing plants [6], and endangered species [7]. ogy is illustrated through the board game of CLUE , which is Recently, several authors have demonstrated that information- shown to be a benchmark example of the treasure hunt problem. driven sensor management is a general and effective framework The game results show that a computer player implementing the for planning a measurement sequence based on a proba- strategies developed in this paper outperforms players implement- bility mass function (PMF) that is referred to as sensor ing Bayesian networks, Q-learning, or constraint satisfaction, as well as human players. model [8]–[13]. Relative entropy was used in [8] to solve a multisensor–multitarget assignment problem and also in [9] and Index Terms—Bayesian networks (BNs), computer game play- [13] to manage agile sensors with Gaussian models for target ing, influence diagrams (IDs), label-correcting algorithms, mine hunting, path planning, search theory, sensor planning, value of detection and classification. Entropy and the Mahalanobis dis- information. tance measure were used in [10] for sensor selection in ad hoc sensor networks. The theory of optimal experiments was im- I. INTRODUCTION plemented in [11] to develop optimal adaptive-search strategies for the detection and classification of buried targets. In [12], N THIS paper, the basic treasure hunt problem and al- the Rényi information divergence applied to the prior and gorithms are developed and illustrated through the board I posterior joint multitarget probability density functions was game of CLUE . The same algorithms are applied in [1] to used to manage the trajectories of multiple aerial vehicles. manage a robotic sensor in a mine-hunting application. As The objective of information-driven sensor management is to was recently pointed out in [2], games are ideal benchmarks decrease the uncertainty or entropy of one or more hypothesis for testing computational intelligence theories and algorithms variables that are not observable or hidden. A basic difficulty because they provide challenging dynamic environments with associated with the use of entropy-based functions is that they rules and objectives that are easily understood. CLUE con- are typically nonadditive and myopic, in that they do not stitutes an excellent benchmark for the treasure hunt problem consider the effects of prior measurements on those that are because the information obtained during the game depends on performed subsequently [8]–[10], [13]–[15]. In this paper, the the position of the pawn. The objective is to infer hidden cards effect of prior measurements is taken into account by defining from suggestions made upon entering the rooms of the CLUE the measurement benefit as the expected entropy reduction mansion. However, in order to make a suggestion involving a (EER) conditioned on prior measurements, which is shown to particular room card, the pawn must occupy the corresponding be additive over time in Section IV. The sensor workspace is represented by a novel connectivity graph with a subset of nodes that are referred to as obser- Manuscript received February 18, 2008; revised June 16, 2008, August 28, vation cells. Each observation cell represents a deployment 2008, and August 29, 2008. This work was supported by the National Science Foundation under CAREER Award ECS 0448906. This paper was recom- or configuration that enables one sensor measurement. The mended by Associate Editor R. Lynch. sensor can move between adjacent nodes by paying a penalty S. Ferrari is with the Department of Mechanical Engineering and Mate- or cost that is attached to the arc between them. The treasure rials Science, Duke University, Durham, NC 27708 USA (e-mail: sferrari@ duke.edu). hunt can be viewed as an extension of the satisficing-search C. Cai is with the Department of Electrical and Computer Engineering, Duke problem of digging buried treasure chests [16]–[18] in which University, Durham, NC 27708 USA (e-mail: [email protected]). the connectivity graph specifies the ordering constraints to be Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. satisfied by the search. As in [16]–[18], the measurements’ Digital Object Identifier 10.1109/TSMCB.2008.2007629 outcomes are unknown aprioriand cannot be factored into the 1083-4419/$25.00 © 2009 IEEE Authorized licensed use limited to: IEEE Xplore. Downloaded on February 12, 2009 at 12:08 from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 608 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009 from the deck, and hidden in an envelope for the remainder of the game. The remaining cards are dealt to the players who subsequently eliminate their own cards from their list of possible hidden items. The players move their pawns about the mansion by rolling the die, and when they enter a particular room, they can make a suggestion that must include that room, and any item from the suspect and weapon categories. The following are the three ways for a pawn to enter a room: 1) a door illustrated on the game board (Fig. 1); 2) a secret passage connecting opposite corners (Fig. 1); or 3) being suggested as the murder suspect by one of the other players. After a player makes a suggestion involving the potential room, suspect, and murder weapon, the next player must refute it by showing one of these cards from his/her own deck, if possible. Otherwise, the player must state that he/she has none of the suggested cards, Fig. 1. CLUE mansion and game pieces. CLUE & 2006 Hasbro, Inc. Used with permission. and other players must attempt to refute the suggestion. When none of the players is able to refute it, it can be inferred that decision problem. However, unlike satisficing searches, in the the suggested cards are the hidden ones or that they belong treasure hunt, measurements are not necessarily binary and are to the player who made the suggestion. By entering rooms used to infer a hidden hypothesis variable. Existing dynamic and making suggestions, players gather evidence about the programming algorithms based on label-correcting [19] and adversaries’ cards and infer the values of the hidden cards. Viterbi searches [20] are not directly applicable to the connec- The problem of making decisions on the pawn movements tivity graph. Thus, a novel label-correcting pruning algorithm and suggestions that optimize the value of the information is presented in Section VI-A to reduce the number of feasible gathered during the game can be viewed as an example of the solutions based on distance while retaining the strategy with treasure hunt problem formulated in the next section. maximum measurement profit. This label-correcting algorithm generates a pruned connectivity tree that is folded into an ID to obtain a compact representation of the treasure hunt problem III. TREASURE HUNT PROBLEM (Section VI-B). The objective of the treasure hunt problem is to infer a hidden The board game of CLUE , described in Section II, is used variable or treasure y, referred to as hypothesis variable, from to illustrate the treasure hunt problem formulated in Section III. the outcomes of multiple measurements. In this paper, y is a The EER measurement-benefit function and its properties are discrete random variable with a finite range Y = {y1,...,yp}, presented in Section IV. The ID representation of the treasure where y represents the th possible value of y.Ameasurement hunt problem is obtained in Section VI and is demonstrated m is a discrete random variable that represents a clue or feature i through the board game of CLUE in Section VII. The that is observable from a corresponding target. Every mea- M { 1 N } game results presented in Section VIII show that a computer surement variable mi has a finite range i = mi ,...,mi , player implementing the strategies obtained from the ID out- where mi denotes the th possible value of mi. Although y performs computer players implementing Bayesian networks is hidden, it can be inferred from one or more measurements (BNs) [21], Q-learning [22], or constraint satisfaction (CS), in an available set M = {m1,...,mr} by means of a known as well as human players.
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