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Invisible Chess" with Information-Theoretic Advisors

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Invisible Chess Playing "Invisible Chess" with Information-Theoretic Advisors A.E. Bud, D.W. Albrecht, A.E. Nicholson and I. Zukerman {bud,dwa,annn,ingrid}@csse.monash, edu. au School of Computer Science and Software Engineering, Monash University Clayton, Victoria 3800, AUSTRALIA phone: +61 3 9905-5225 fax: +61 3 9905-5146 From: AAAI Technical Report SS-01-03. Compilation copyright © 2001, AAAI (www.aaai.org). All rights reserved. Abstract history of use in the AI communitybecause of its strategic complexity, and well-studied and understood properties. Makingdecisions under uncertainty remains one of the cen- Despite these advantages, chess has two signi cant draw- tral problemsin AI research. Unfortunately, most uncertain real-world problemsare so complexthat any progress in them backs as a general AI testbed: the rst is the success of is extremelydif cult. Gamesmodel some elements of the real computer chess players that use hard-coded, domain specif- world, and offer a morecontrolled environmentfor exploring ic rules and strategies to play; and the second is the fact that methodsfor dealing with uncertainty. Chess and chess-like standard chess is a perfect information domain. gameshave long been used as a strategically complextest- A number of researchers have tackled the rst of these draw- bed for general AI research, and we extend that tradition by backs. For example (Berliner 1974) investigated generalised introducing an imperfect information variant of chess with strategies used in chess play as a model for problem solving, someuseful properties such as the ability to scale the amount of uncertainty in the game.We discuss the complexityof this and (Pell 1993) introduced metagame. Metagame is a sys- gamewhich we call invisible chess, and present results outlin- tem for generating new games arbitrarily generated from a ing the basic values of invisible pieces in this game.We mo- set of games known as Symmetric Chess-Like games (SCL tivate and describe the implementationand application of two Games). Thus there is no point hand-training a computer information-theoreticadvisors that assist a player of invisible program to play one particular game, as each game requires chess to control the uncertainty in the game.We describe our different strategic planning and position evaluation. A good decision-theoretic approachto combiningthese information- computer metagame player has to somehow encapsulate a theoretic advisors with a basic strategic advisor. Finally we higher level of "strategic knowledge"than is possible in a discuss promisingpreliminary results that wehave obtained single gamesuch as chess. with these advisors. In this paper, we address the second drawback of chess as a general AI testbed -- that of perfect information. Wede- 1 Introduction scribe a missing (or imperfect) information variant of stan- dard chess which we call invisible chess (Bud et al. 1999).1 Making decisions under uncertainty remains one of the cen- Invisible chess involves a con gurable numberof invisible tral problems in AI research. An agent in an uncertain world pieces, i.e., pieces that a player’s opponent cannot see. In- needs to select actions from the action search space -- the visible chess is thus a representative of the general class set of all possible actions in that world. As the uncertain- of strategically complex, imperfect information, two play- ty increases, this task can becomeincreasingly dif cult. The er, zero-sum games. numberof possible actions may increase, the numberof pos- Manyresearchers have investigated games with missing in- sible situations in which those actions may be applied may formation including poker ((Findler 1977), (Korb, Nichol- increase, or both. The effects of these growing search spaces son, & Jitnah 1999), (Koller & Pfeffer 1997)), bridge ((Gam- are ampli ed as the agent tries to search further ahead. Each biick, Rayner, & Pell 1991), (Ginsberg 1999), (Smith, action on each possible world state requires more possible & Throop 1996)), and multi-user domains (Albrecht et al. world states to be evaluated for future moves. This property 1997). With the exception of Albrecht et al., whouse a large of imperfect information domains makes tackling real-world uncontrollable domain, all of these domainsare strategically problems extremely dif cult. simple given perfect information. Games and game theory model some of these properties On the other hand, invisible chess retains all of the strategic of real-world situations in a more controlled environment complexity of standard chess with the addition of a con- and thereby allow analysis and empirical testing of decision- trollable element of missing information. Invisible chess making strategies in these domains. In this capacity, games have long been used as a testbed for general arti cial intel- ligence research and ideas. In particular, chess has a long 1Invisible chess is in the set of Invisible SCLGames, an exten- sion to the set of SCLGames introduced by Pell. Wehave chosen Copyright(~) 2001, AmericanAssociation for Arti cial Intelli- invisible chess as the rst gameto explore becauseof the availabil- gence (www.aaai.org).All rights reserved. ity of standard chess programs. Nolicense: PDFproduced by PStin (c) F, Siegert - http://www.this.net/~frank/pstill.html is related to kriegspiel ((Li 1994) and (Ciancarini, Dalla- ions do not tend to occur in a signi cant numberof nodes Libera, & Maran1997)), a chess variant in which all the in the gametree, 2 and therefore partition search is not like- opponent’spieces are invisible and a third party referee de- ly to be very effective on any variants of chess. Morere- termines whetheror not each moveis valid. cently (Ginsberg 1999) included MonteCarlo methods In addition to the complexityof playing standard chess, a his bridge playing programto simulate manypossible out- player of invisible chess mustmaintain the possible posit- comes,choosing the action with the highest expectedutility ions of the opponent’sinvisible pieces. These positions may over the simulations. Ginsbergclaims that trying to glean be representedas a probability distribution over the possible or hide information from an opponentis probably not use- squares on the chess board. Section 4 describes our design ful for bridge. In contrast, wepresent results that showthat that enables a central moduleto maintain approximationsto both informationhiding and gleaning can be useful in invis- the invisible piece probability distributions for both players. ible chess. Section 5 presents a brief analysis of the advantagesof play- (Frank & Basin 1998) and (Frank & Basin 2000) have ing with various invisible pieces. Interestingly, we show formeda detailed investigation of search in imperfectinfor- that the relative value of the minorinvisible pieces differs mation games. They have concentrated almost exclusive- from the standard chess piece ranking. Wereport on the ly on bridge. Frank & Basin focus on a numberof search uncertainty causedby the different invisible pieces and the techniques including attening the gametree which they effect that those pieces have on the opponent’sability to have shownto be an NP-Completeproblem, and MonteCar- play strategically. Section 6 discusses the relationship be- lo methods. Their investigation suggests that MonteCar- tween uncertainty and the information-theoretic concept of lo methodsare not appropriate for imperfect information entropy,and the applicationof entropyto our results. It also games.This problemis exacerbated in invisible chess given motivates the design of information-theoreticadvisors, de- the relative lack of symmetryand the size of the gametree. scribes these advisors and presents results that demonstrate In the closest workto that presentedhere, (Ciancarini, Dalla- the ef cac y of our approach.Section 7 contains conclusions Libera, & Maran 1997) considered king and pawn end and ideas for further work. gamesin the gameof kriegspiel. Kriegspiel is an existing chess variant where neither player can see the opponent’s 2 Related Work pieces or moves. They used a game-theoretic approach in- volving substantive rationality and have shownpromising Since the 1950s, when Shannonand Turing designed the results in this trivial versionof kriegspiel. Thusfar they have rst chess playing programs (Russell & Norvig 1995), not applied their approachto a completegame of kriegspiel. computershave becomebetter at playing certain gamessuch Kriegspiel differs frominvisible chess in a numberof impor- as chess using large amountsof hand-codeddomain specif- tant areas. All the opponent’spieces are invisible to a player ic information. Continuingthe tradition of using gamesas of kriegspiel. Thusthere is no wayto reduce the uncertainty a testbed for general AI investigation, (Berliner 1974)pro- in the domainwithout reducing the strategic complexity.By poseda tactical analyser for chess whichused strategies and contrast, in invisible chess, the numberand types of invisi- tactics, but did not play as well as the existing hard-coded ble pieces is con gurable. As all pieces are invisible, every systems. moveinvolves substantial increases in uncertainty. In invis- In answer to the success of hard-coded algorithms, ible chess, a player maychoose whetherto movea visible (Pell 1993) introduced a class of gamesknown as Symmetric or invisible piece. Finally, in kriegspiel, a player attempts Chess Like (SCL) Games,and a system called Metagamer movesuntil a legal moveis performed.A third party arbitra- that plays gamesarbitrarily generatedfrom this class
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