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Keeping track of pieces in Invisible Chess Monash University, Clayton 3800, Australia Abstract. Invisible Chess ([1],[2],[3]) is a variation of Chess where someof the pieces are invisible to the opponent. It is related to Kriegspiel [4]where all the pieces are invisible. However, in the variant of InvisibleChess that we investigated in this paper there are some significant dif-ferences in the rules from Kriegspiel.
2. Invisible pieces are revealed when they cause or block check, and 3. If an invisible piece is moved the opponent is informed which piece 4. A player of Invisible Chess misses a turn if they perform an impos- 5. There is no ’any’ type move like in Kriegspiel.
It has been demonstrated ([2],[3]) that the strategies of hiding a player’sinvisible pieces from their opponent, and seeking the opponents invisiblepieces, increases the likelihood of a player winning Invisible Chess. There-fore it follows that in order for a player to play well in Invisible Chessthey need to maintain in an efficient manner the probability distributionof the possible positions of the opponent’s invisible pieces.
Bayesian Networks provide a framework to maintain the probability dis-tribution of invisible pieces and to incorporate information gained in theprocess of playing the game. In this paper we will discuss the various is-sues that need to be handled in using Bayesian networks to keep track ofinvisible pieces in Invisible Chess. In particular the exponential increas-ing size of the conditional probability tables as the number of invisiblepieces increase, the modeling of both static constraints, (i.e., only onepiece on each square), and dynamic constraints, (i.e., no piece is allowedto move through another piece), and the handling of information gainedby the player during the game (i.e., when a player is able to move apiece, such as a rook, they learn that there are no invisible pieces onthe intervening squares). Also, we will show that the rules for InvisibleChess allows for the possibility of decomposing the Bayesian Networkinto small subnets corresponding to the movement of different types ofpieces. Finally, we will discuss how the relaxing certain constraints en-ables the rolling up parts of the network, and this leads to considerablesimplification of the network and the updating procedure.
1. Bud, A., Nicholson, A., Zukerman, I., Albrecht, D.: A hybrid architecture for strate- gically complex imprefect information games. In Jain, L., ed.: 1999 Third Interna-tional Conference on Knowledge-Based Intelligent Information Engineering Systems,Adelaide S.A., Australia (31 August – 1 September 1999) 42–45 2. Bud, A., Albrecht, D., Nicholson, A., Zukerman, I.: Information-theoretic advisors in invisible chess. In: AI and STATISTICS - Eighth International Workshop onArtificial Intelligence and Statistics, Florida, U.S.A., Morgan Kaufman Publishers,CA, USA (4–7 January 2001) 157–162 3. Bud, A., Albrecht, D., Nicholson, A., Zukerman, I.: Playing ’invisible chess’ with information-theoretic advisors. In: 2001 AAAI Spring Symposium on Game The-oretic and Decision Theoretic Agents, CA, USA, The American Association forArtificial Intelligence (26–28 March 2001) 6–15 4. Ciancarini, P., DallaLibera, F., Maran, F.: Decision making under uncertainity: A rational approach to kreigspiel. In van den Herik, J., Uiterwijk, J., eds.: Advancesin Computer Chess 8, Univ. of Rulimburg (1997) 277–298

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