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, MonashUniversity
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
Makingdecisions under uncertainty remains one of the central problemsin AI research. Unfortunately, most uncertain
real-world problemsare so complexthat any progress in them
is extremelydif cult. Gamesmodelsomeelementsof the real
world, and offer a morecontrolled environmentfor exploring
methodsfor dealing with uncertainty. Chess and chess-like
gameshave long been used as a strategically complextestbed for general AI research, and we extendthat tradition by
introducing an imperfect information variant of chess with
someuseful properties such as the ability to scale the amount
of uncertainty in the game.Wediscuss the complexityof this
gamewhichwecall invisible chess, and present results outlining the basic values of invisible pieces in this game.Wemotivate and describe the implementation
and application of two
information-theoretic
advisorsthat assist a player of invisible
chess to control the uncertainty in the game.Wedescribe our
decision-theoretic approachto combiningthese informationtheoretic advisors with a basic strategic advisor. Finally we
discuss promisingpreliminary results that wehave obtained
with these advisors.
1 Introduction
Makingdecisions under uncertainty remains one of the central problems in AI research. An agent in an uncertain world
needs to select actions from the action search space -- the
set of all possible actions in that world. As the uncertainty increases, this task can becomeincreasingly dif cult. The
numberof possible actions may increase, the numberof possible situations in which those actions may be applied may
increase, or both. The effects of these growingsearch spaces
are ampli ed as the agent tries to search further ahead. Each
action on each possible world state requires more possible
world states to be evaluated for future moves. This property
of imperfect information domains makes tackling real-world
problems extremely dif cult.
Games and game theory model some of these properties
of real-world situations in a more controlled environment
and thereby allow analysis and empirical testing of decisionmakingstrategies in these domains. In this capacity, games
have long been used as a testbed for general arti cial intelligence research and ideas. In particular, chess has a long
Copyright(~) 2001, AmericanAssociation for Arti cial Intelligence(www.aaai.org).All rights reserved.
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history of use in the AI communitybecause of its strategic
complexity, and well-studied and understood properties.
Despite these advantages, chess has two signi cant drawbacks as a general AI testbed: the rst is the success of
computer chess players that use hard-coded, domain specific rules and strategies to play; and the secondis the fact that
standard chess is a perfect information domain.
A number of researchers have tackled the rst of these drawbacks. For example (Berliner 1974) investigated generalised
strategies used in chess play as a model for problemsolving,
and (Pell 1993) introduced metagame. Metagameis a system for generating new games arbitrarily generated from a
set of games known as Symmetric Chess-Like games (SCL
Games). Thus there is no point hand-training a computer
program to play one particular game, as each game requires
different strategic planning and position evaluation. A good
computer metagame player has to somehow encapsulate a
higher level of "strategic knowledge"than is possible in a
single gamesuch as chess.
In this paper, we address the second drawback of chess as
a general AI testbed -- that of perfect information. Wedescribe a missing (or imperfect) information variant of standard chess which we call invisible chess (Bud et al. 11999).
Invisible chess involves a con gurable numberof invisible
pieces, i.e., pieces that a player’s opponent cannot see. Invisible chess is thus a representative of the general class
of strategically complex, imperfect information, two player, zero-sum games.
Manyresearchers have investigated games with missing information including poker ((Findler 1977), (Korb, Nicholson, & Jitnah 1999), (Koller &Pfeffer 1997)), bridge ((Gambiick, Rayner, & Pell 1991), (Ginsberg 1999), (Smith,
& Throop 1996)), and multi-user domains (Albrecht et al.
1997). With the exception of Albrecht et al., whouse a large
uncontrollable domain, all of these domainsare strategically
simple given perfect information.
Onthe other hand, invisible chess retains all of the strategic
complexity of standard chess with the addition of a controllable element of missing information. Invisible chess
1Invisible chess is in the set of Invisible SCLGames,an extension to the set of SCLGamesintroduced by Pell. Wehave chosen
invisible chess as the rst gameto explorebecauseof the availability of standard chess programs.
is related to kriegspiel ((Li 1994) and (Ciancarini, DallaLibera, &Maran1997)), a chess variant in which all the
opponent’spieces are invisible and a third party referee determines whetheror not each moveis valid.
In addition to the complexityof playing standard chess, a
player of invisible chess mustmaintainthe possible positions of the opponent’sinvisible pieces. Thesepositions may
be representedas a probability distribution over the possible
squares on the chess board. Section 4 describes our design
that enables a central moduleto maintainapproximationsto
the invisible piece probabilitydistributions for bothplayers.
Section5 presents a brief analysis of the advantagesof playing with various invisible pieces. Interestingly, we show
that the relative value of the minorinvisible pieces differs
from the standard chess piece ranking. Wereport on the
uncertainty causedby the different invisible pieces and the
effect that those pieces have on the opponent’sability to
play strategically. Section6 discusses the relationship betweenuncertainty and the information-theoretic concept of
entropy,andthe applicationof entropyto our results. It also
motivatesthe design of information-theoreticadvisors, describes these advisors and presents results that demonstrate
the ef cac y of our approach.Section 7 contains conclusions
and ideas for further work.
2 Related
ions do not tend to occur in a signi cant numberof nodes
in the gametree, 2 and therefore partition search is not likely to be very effective on any variants of chess. Morerecently (Ginsberg 1999) included MonteCarlo methods
his bridge playing programto simulate manypossible outcomes,choosingthe action with the highest expectedutility
over the simulations. Ginsbergclaims that trying to glean
or hide information from an opponentis probably not useful for bridge. In contrast, wepresent results that showthat
both informationhiding and gleaning can be useful in invisible chess.
(Frank &Basin 1998) and (Frank &Basin 2000) have
formeda detailed investigation of search in imperfectinformation games. They have concentrated almost exclusively on bridge. Frank &Basin focus on a numberof search
techniques including attening the gametree which they
have shownto be an NP-Completeproblem, and MonteCarlo methods. Their investigation suggests that MonteCarlo methodsare not appropriate for imperfect information
games.This problemis exacerbatedin invisible chess given
the relative lack of symmetry
and the size of the gametree.
In the closest workto that presentedhere, (Ciancarini, DallaLibera, & Maran 1997) considered king and pawn end
gamesin the gameof kriegspiel. Kriegspiel is an existing
chess variant where neither player can see the opponent’s
pieces or moves. They used a game-theoretic approach involving substantive rationality and have shownpromising
results in this trivial versionof kriegspiel. Thusfar they have
not applied their approachto a completegameof kriegspiel.
Kriegspiel differs frominvisible chess in a numberof important areas. All the opponent’spieces are invisible to a player
of kriegspiel. Thusthere is no wayto reduce the uncertainty
in the domainwithout reducing the strategic complexity.By
contrast, in invisible chess, the numberand types of invisible pieces is con gurable. As all pieces are invisible, every
moveinvolvessubstantial increases in uncertainty. In invisible chess, a player maychoose whetherto movea visible
or invisible piece. Finally, in kriegspiel, a player attempts
movesuntil a legal moveis performed.A third party arbitrator indicates whetheror not each moveis legal. In invisible
chess, the player is given specie information whena move
is impossibleor illegal, and maymiss a turn as a result of
attempting an impossiblemove(see Section 3).
Thesedifferences makekriegspiel a substantially morecomplex and less controllable domainthan invisible chess.
Speci cally, exploring the relationship betweenuncertainty
and strategic play wouldbe moredif cult in kriegspiel. Ultimately, the pathological case of invisible chess whereall
pieces are invisible approachesthe complexityof kriegspiel.
Thusinvisible chess provides a convenientstepping stone to
a muchmore difcult problem.
To our knowledgethere exist no computerkriegspiel players
that are able to play a full gameof kriegspiel with whichto
compareour system.
Work
Since the 1950s, when Shannonand Turing designed the
rst chess playing programs (Russell & Norvig 1995),
computershave becomebetter at playing certain gamessuch
as chess using large amountsof hand-codeddomainspecific information. Continuingthe tradition of using gamesas
a testbed for general AI investigation, (Berliner 1974)proposeda tactical analyser for chess whichused strategies and
tactics, but did not play as well as the existing hard-coded
systems.
In answer to the success of hard-coded algorithms,
(Pell 1993) introduceda class of gamesknownas Symmetric
Chess Like (SCL) Games,and a system called Metagamer
that plays gamesarbitrarily generatedfromthis class using a
set of advisorsrepresentingstrategies in the class of games.
Peli did not consider imperfect information games.However, his class of SCLGamesis easily extendedto the class
of Invisible SCLGames,where one or more of a player’s
pieces is hiddenfromtheir opponent.
(Koller &Pfeffer 1995)investigated simple imperfectinformationgameswith an initial goal of solving them. However, their approachdoes not scale up to morecomplexgames
such as invisible chess.
(Smith, Nan, &Throop1996) wrote a bridge playing program using a modi ed form of gametree with enumerated
strategies rather than actions, effecting a formof forward
pruning of the gametree. Smithet al. stated that forward
pruningworkswell for bridge, but not for chess. (Ginsberg
1996)introducedpartition search to reducethe effective size
of the gametree. He showedthat this approach workswell
for bridge and other gameswith a high degree of symmetry.
Partly becausein chess pawnsonly moveforward, and partly
becauseof the strategic nature of the game,the sameposit-
2In fact if the samepositionoccursthree times in a gameof
chess,the gameis declareda draw.
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quently the invisible black bishop on c4 is revealed, and
white must supply another move.
4: If a player is not in check, and attempts to moveinto
check, the invisible piece causing the checkis revealed,
and the player’s tumis forfeited. In standard chess, the
movewouldbe disallowed and the player wouldprovide
an alternate move.In Figure 4, white attempts to move
their king from dl to d2. However,
black’s invisible bishop on a5 is threatening d2; consequentlythe invisible
black bishop on a5 is revealed, and white forfeits their
turn.
5: Ifa player attemptsto castle throughcheckfroman invisible piece or throughan invisible piece itself, the invisible
pieceis revealedandthe turn is forfeited.
3 Domain: Invisible
Chess
Invisible chess is basedon standardchess with the following
difference. In invisible chess we differentiate betweenvisible and invisible pieces and de ne themas follows: a visible
piece is one that both players can see; an invisible piece is
one that a player’s opponentcannot see. Thus, every time a
visible piece is moved,the boardis updatedas per standard
chess. However,whena player movesan invisible piece,
their opponentis informedwhichinvisible piece has moved,
but not where it has movedfrom or to. This information
enables each player to maintaina probability distribution of
their opponent’sinvisible pieces across the squares on the
board. In this paper, we frame invisible chess pieces ~ to
distinguish themfromvisible chess pieces (~).
Wede ne three terms for referring to movesin invisible
chess: A possible moveis any legal chess movegiven complete information about the board. That is, no invisible
pieces are in the wayof the move,and the piece can moveto
the desired position. An impossiblemoveis an attempted
movethat violates the laws of chess because of an incorrect assumptionas to the whereaboutsof an invisible piece.
For example, a player attempts to movea piece "through"
an invisible piece, or attempts to use a pawnto capture an
invisible piece that is not diagonallyin front of it. See Figure 1. An illegal moveis an impossible movethat would
not allow the gameto continue. For example,a player is not
allowedto movetheir king into checkby an invisible piece.
See Figure 3.
Therules of invisible chess are basedon the rules of chess.
Theonly modications pertain directly to invisible pieces
and their impact on the game.In general if a moveis possible then it is accepted;if a moveis impossiblethen it is
rejected and the player’s turn is forfeited; and if a moveis
illegal, someinformationregarding the reason the moveis
illegal is revealedto the player attemptingthe move,and the
player must supply another move.Note that we do not allow
invisible kings in this basic invisible chess, as a version of
invisible chess with invisible kings woulddrastically modify
the goal of the original game.
(Budet al. 2000) gives details of all scenarios wherenew
rules comeinto effect, but in summary,
onlythe rules for the
followingscenarios differ fromthose of standard chess.
1: Impossiblemovesare disallowed,the position of the rst
invisible piece that caused the moveto be impossible
is revealed, and the player’s turn is forfeited. Figure 1
showsan exampleof an impossible move.
2: If a player’s kingis threatened,the invisible piece causing the checkis revealed, and the threatenedplayer moves
normally. Additionally, the player can infer that there
are no invisible pieces betweenthe threatening piece
and their king (unless the king is being threatened by
a knight). Figure 2 showsan exampleof this scenario.
Blackis told that the invisible whitebishopis on g5.
3: If a player is in check, and attempts to moveinto check
again or does not moveout of check, any invisible pieces
causing the check are revealed, and the player must supply another move.In Figure 3, white is in checkand attemptsto capture the invisible black knight on fl. Conse-
3.1
The complexityof standard chess is well understood. Chess
has an average branching factor of around 35. A typical
gamelasts approximately 50 movesper player so the entire gametree has approximately351°° nodes ((Russell
Norvig1995), p 123).
In the trivial case wherethe opponenthas no invisible pieces,
the gametree for invisible chessis exactlyas large as that for
standardchess. Onceinvisible pieces are introducedinto the
game,each node of the gametree must be expandedfor each
possible movefor each possible combinationof positions of
the opponent’sinvisible pieces. In a gameof invisible chess
the player and opponent each have minvisible pieces. If
each invisible piece has a positive probability of occupying
n~ squares, then the branchingfactor is approximatelythe
branchingfactor of chess multiplied by the combinationof
invisible squaresthat could be occupiedas shownby the following formula:
Branching Factor = 35 x nl x n2 x ...
x nra (1)
If each player has two invisible pieces, and each of those
pieces has an average of only four squares with a positive probability of occupation, then the average approximate branching factor of that gameof invisible chess is
35 x 16 = 560. For three invisible pieces each, the branching
factor is around35 × 64 = 2240. Assumingthat the invisible pieces are on the board and movingfor approximately
half the game, then the complete expandedgametree for
three invisible pieces each is in the order of 22405°whichis
around 1015° nodes.
To makethe domaineven morecomplex,chess has virtually
no axes of symmetrythat allow the size of the gametree
to be reduced as in highly symmetricalgamessuch as tictac-toe, and Smith and Nan (Smith &Nan 1993) claim that
forward pruningto reduce the branchingfactor of chess has
beenshownto be relatively ineffective.
In addition to the combinatorially expansivenature of the
invisible chess gametree, in order to play invisible chess, a
player mustmaintain beliefs about the positions of the opponent’sinvisible pieces.
A player using a simplistic belief updating schemesuch as
assumingthe most likely destination will ignore manyprobable squares. Further, a player that uses any belief updating
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Domain Complexity
L......
Y/_//~
....~ .,,.’//////,~ ......
3
2
1
a
Figure 1: An impossible move. The black
bishop on b2 is invisible. Whiteattempts
an impossible moveby trying to movethe
bishopon cl to a3.
schemethat does not take into account every possible destination square whenan invisible piece is moved,will lose
important information about the o w of the game. Suppose
a player incorrectly assumesthat the opponent’sinvisible
piece is on a certain square; if the opponentthen movesa
visible piece to that square, the player nowhas no Wayof
backtracking, and has no information about the location of
the invisible piece.
Assuming
no strategic knowledge(i.e., each square that an
invisible piece can moveto is as likely to be visited as any
other), and only one invisible piece, a player can easily
maintainthe precise distribution for that piece (see (Budet
al. 2000)). In addition, wheneverany piece (other than
knight)moves,the probability distribution for each invisible
piece maybe updated to re ect the fact that other squares
in the movingpiece’s path are vacant. Similarly, whenever
a player’s king is threatened by any piece (except a knight),
visible or invisible, the probabilitydistributionsfor all invisible pieces maybe updated to re ect the knownvacancies
betweenthe piece causing check, and the threatened king.
However,for multiple invisible pieces, the positions of invisible pieces are not conditionallyindependentwith respect
to each other, so the probability of one invisible piece occupyingany particular squareaffects the probability of another invisible piece occupyingthat square. Maintainingthe
probabilitydistributions of multipleinvisible pieces involves
combinatorialcalculations in the numberof invisible pieces
or the storage of combinatoriallylarge amountsof data in
order to maintainthe completejoint distributions of all invisible piecesover the entire chess board.
Weaddress this problemby utilising a central moduleto obtain an approximationof the distribution that can be calculated quickly enoughto use in real time (Section 4). The
GCDMM
calculates the distributions for both players, and
has access to the true positionsof all invisible piecesat all
times. Weuse the true positions of other invisible pieces
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b
c
d
e
f
g
h
Figure 2: Aninvisible piece is revealed to
warnof the check. Black is in check becauseof the invisible white bishop on g5.
as an approximation
to the joint distribution of all invisible
pieces across all squares. Thus, the calculation of each invisible piece’s moveis reducedto the calculation to be performedwhenthere is only one invisible piece. Becausethe
resulting approximationis moreaccurate than the distribution that could be calculated by a real player of invisible
chess, our results slightly underestimatethe advantagesof
playingchess with invisible pieces. For the purposesof this
research, using this methodto maintainprobability distributions allowsus to focus our efforts on the effects of manipulating the amountof uncertainty inherent in these distributions.
4 Basic Design
In Section 3.1 we estimate the branchingfactor of the game
tree for invisible chess with 2 invisible pieces to be around
560, and for 3 invisible pieces to be in the order of 2000,
comparedto the branchingfactor of standard chess whichis
about 35 ((Russell &Norvig 1995), p. 123). To cope
this combinatorial explosion, and the strategic complexity
of invisible chess, we employa "divide and conquer" approach. Wesplit the problemof choosingthe next moveinto
a numberof simpler sub-problems,and then use utility theory (Raiffa 1968) to recombinethe calculations performed
for these sub-problems into a move.Weuse informationtheoretic ideas (R6nyi1984)to deal with the uncertainty
the domain, and standard chess reasoning to deal with the
strategic elements.
This modular, hybrid approach is implementedwith advisors or experts connected and controlled by a GameController and Distribution MaintenanceModule(GCDMM).
the current implementation,we include three advisors: (1)
the strategic advisor; (2) the movehide advisor; and (3)
the move seek advisor. The GCDMM
controls the game
state, has knowledge
of the positions of all invisible pieces
and maintainsdistributions of invisible pieces on behalf of
N®N
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386
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-24 81.50
-24 87.75
61 107.50
64 107,25
21 189.25
29 198.25
24 384 376 54 209.50
30,395 374
55 213.50
b263
Ialbl , 26 395 376 58 213.75
2
glh3 I 33 397 374 52 214.00
e2e4 t 24 387 376 54 210.25
62 216.25
dld3[ 31~395377
h2h3 31 395 376 63 216.25
e2a6l 241 397 376 54 212.75
glf3 34 394 376 64 217.00
e2d3I 32 395 377 63 216.75
1
b
c
d
e
f
g
g
h
Hide Seek Total
IStrat (W1:1) - [[~]{Pos
Move b4 a3 c5 d6
EU W2:5 Ws:5
N N
°N
a
f
Figure 4: Attempting to moveinto check.
Whiteattempts to movethe king from dl to
d2. This is impossiblebecauseit results in
the king being in check fromthe invisible
black bishop on a5.
Figure 3: An illegal move. White is in
check fromthe black bishop on a5, and attempts to capture the invisible black knight
on fl whereit wouldbe in check from the
invisible black bishop on c4.
8x
b
h
Figure 5: Aninvisible bishop each. White
believesthe invisible black bishopon b4 is
on one of a3, b4, c5 or d6. Black believes
the invisible white bishop on e2 has possible squarese2, d3, c4, b5 and a6.
0.77
0
0
0
0.77
-0.22
0
~
0.77
0
0.77
0 213.35
O, 213.50
0 213.75
0~ 214,00
0 214,10
O! 215,15
0 216.25
0l 216.60
0 217.00
0 ] 220.60
Figure 6: Possible Movessorted by their
relative total scores.
the two players. The GCDMM
is responsible for deciding
whethera moveis legal, impossibleor illegal and ensuring
that the gameprogresses correctly. Whenit is a player’s
turn to move,the GCDMM
requests the next movefrom the
Maximiserfor that player. The Maximiser,responsible for
choosingthe best movesuggestedby the available advisors,
generatesall possible boardsand all possible moves,and requests utility values fromeach of the advisorsfor each move.
Eachadvisor evaluates the possible moves, across as many
boards as possible in the available time, accordingto an internal evaluation function. The strategic advisor, a modied version of GNU
Chess whichreturns all possible moves
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0 0.56
84,30
0 0.56~ 90.55
0
0 107150
0 0.77 111.10
0.77
0 193,10
0
0 198,25
and their utilities (evaluatedto a speci ed depth), gives the
ExpectedUtility (EU) of each movefrom a strategic perspective. 3 The EUfor each moveor action (A) is calculated by multiplyingits utility value by the probability of the
gamestate (X) for whichthe utility value was calculated,
3Notethat GNU
Chesshas no knowledge
of invisible pieces.
Themodications to GNU
Chessare to performx ed depthsearching, andto returnall moves
andutilities rather thanonlythe best
move.In standardchess, a poormovemaybe discountedearly in
the search so that the searchon the gametree canfocus on more
promising
moves.However,
in invisible chess, a movethat is poor
in oneboardpositionmaybe goodin other positions.
and summing
this across all possible gamestates (G) as per
Equation2.
EU(A) = ~ (Utility(AlX)
x Prob(X))
(2)
VX6 G
The moveseek advisor scores movesaccording to howmuch
they reduce the player’s uncertainty. For example, a move
that fails becauseit "collides" with an invisible piece removesthe uncertaintyas to the position of that piece. Similaxly, the movehide advisor scores movesaccording to how
muchthey increase the opponent’suncertainty. For example,
movinga previouslyrevealedinvisible piece will greatly increase the opponent’s uncertainty. The movehide and seek
advisors are describedin Section6.
Eachadvisor has a weightassociated with it that re ects the
relative value of its advice. The Maximisermultiplies the
value returned from each of the advisors by its weight (W0
and sumsthese values. The movewith the highest overall
value is passed back to the GCDMM
which implementsthat
moveand requests a movefromthe other player. If there are
multiple moveswith equal highest score, one of these moves
is chosen at randomby the Maximiser.
4.1
Example
Figure 5 showsa typical position in a gameof invisible
chess with each player playing with one invisible bishop;
it is white’s turn to move.Black’sbelief aboutthe position
of white’s invisible bishopis representedby the probability
distribution:
e2 0.2, d3 0.2, c4 0.2, ]35 0.2,
a6 0.2, and white believes that black’s invisible bishop
has the probability distribution: a3 0.25, b4 0.25,
c5 0.25, d6 0.25. White knows that the invisible
black bishop cannot be on e7 as the black queen traversed
e7 to get to its current position at h4. TheGCDMM
asks the
white Maximiserfor its next move. The white Maximiser
passes each possible boardposition, i.e., the current board
with black’s invisible black bishop in each of b4, a3, c5
and d6 to the strategic advisor. Figure 6 showssomepossible movesacross the four possible boardstogether with their
utilities and the EUof eachmovefromthe strategic advisor’s
perspective. Noticethat the moveglf3 has the highest strategic EUof 217. However,movingthe knight on gl has no
effect on either player’s information.If there wereno other
advisors present, the Maximiserwouldchoosethis move.
Somemovesdo assist in discovering the location of the invisible black bishop, e.g. d4c5, a2a3 and b2a3, however, noneof these movesis strategically strong. For this reason, and becauseof the small numberof squares in the invisible black bishop’s distribution, the moveseek advisor has
little effect at this point in the game.Later in the game,once
the invisible black bishop has movedseveral times, many
squareswill have a probability of being occupiedby the invisible black bishop, and the moveseek advisor will have
moreeffect.
The movehide advisor evaluates each movebased on the
increase in the opponent’suncertainty. Whenthe invisible
white bishopmoves,its destination fromthe opponent’sperspective, maybe any square accessible fromany square currently in the invisible white bishop’sdistribution. Thusall
Y.I.
X
39.0
35.0
65.0
82.8
58.5
56.4
10.2
61.0
X
46.6
60.8
86.8
62.6
72.4
11.0
65.0
53.4
X
61.8
74.6
65.6
72.6
7.0
35.0
39.2
38.2
X
56.2
58.6
52.4
7.0
17.2
13.2
25.4
43.8
X
28.0
42.4
4.4
41.5
37.4
34.4
41.4
72.0
X
54.2
3.8
43.6
27.6
27.4
47.6
57.6
45.8
X
1.8
89.8
89.0
93.0
93.0
95.6
96.2
98.2
X
Table 1: Winpercentages for different combinationsof invisible pieces.
possible destination squares need to be addedto its distribution, and regardless of wherethe invisible white bishop
movesto (unless it causes check), the opponent’suncertainty is increased. Onthe other hand, whena visible piece
traverses a square that has a positive probability of occupation by an invisible piece, the opponent’suncertainty decreases as they can infer that the traversed squareis actually
vacant. For example,the movedld3 tells the opponentthat
the square d3 is empty.
The Maximisermultiplies each movevalue from each advisor by the advisor’s weight(Wi)and sumsover all the advisors, giving the "Total" columnin Figure 6. Notice that the
movewith the highest overall value is nowe2d3 (220.60).
The extra addeduncertainty has beenenoughin this case to
makethis movebetter than the strategic choice of glf3.
5 The Strategic
Advisor
In this section wepresent results for playinginvisible chess
with a single strategic advisor for each player. Eachplayer
played with a different con guration of invisible pieces, using only a strategic advisor to decide the next move.Each
result is obtained from500gamesrun with a particular conguration of invisible pieces that endedwith a win.4 White
has a slight advantagein standard chess, and the strategic
advisor movespieces differently dependingon whetherit is
playing white or black. To removecolour biases fromthe resuits, each set of 500gameswasbrokeninto two runs of 250
gameseach, the invisible piece con gurations were swapped
betweenwhite and black for each run, and the results were
averagedbetweenthe two runs.
Table 1 showsresults for gamesplayed with major (nonpawn)invisible pieces against each other and against no invisible pieces (N.I.). Readingacross a row, it showsthe percentage of gameswonby the combinationof invisible pieces
in the rowheadingagainst the invisible piece combinationof
a particular column.For example,one invisible bishop (1B)
won65%of the time against one invisible rook (1R), but
only 43.6%of the time against twoinvisible rooks (2R).
Theseresults showthat the values of several invisible piece
combinationsdiffer betweeninvisible chess and standard
chess. For example,in standard chess, a rook is considered
morevaluable than a bishop. However,in invisible chess,
4Drawn
gamesare largelythe result of repeatingmoves
continuously.Consistingof less than 10%of gamesplayed,drawsare
notcounted
in theseresults.
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1B
IN
1R
1Q
2B
2N
2R
Further examination of our results showsthat the more a
player movestheir invisible pieces, the moreimpossible
movestheir opponent attempts (see (Bud et aL 2000) for
details). This movement
of invisible pieces and the corresponding opponentuncertainty can be summarisedusing entropy to quantify each player’s uncertainty in a game.Figures 7 and 8 showthe uncertainty of each player regarding
the positions of the opponent’sinvisible pieces for games
played with one invisible bishop against two invisible bishops.6 For each game,the entropy of each player’s distribution of invisible pieces is summed
across all moves.Thusthe
graphs showthe total amountof uncertainty each player had
to deal with over the course of each game.Figure 7 shows
each player’s uncertainty in the 190gamesthat werewonby
white; the solid line showsblack’s uncertainty, sorted by entropy, whilethe dashedline showswhite’s uncertainty in the
corresponding game. Figure 8 shows each player’s uncertainty in the 60 gamesthat were wonby black; the dashed
line showswhite’s uncertainty, sorted by entropy, while the
solid line showsblack’s uncertainty in the corresponding
game. As can be seen from Figure 7, all the gameswonby
white, except for one (gamenumber86) showgreater uncertainty for black. The gameswonby black (Figure 8) are
muchcloser in uncertainty and there are manygameswhere
white had greater uncertainty than black.7 This examplecorroborates our intuition that players withless uncertaintyare
morelikely to win, and underpinsour information-theoretic
advisors.
one bishop beat one rook, and two bishops beat every other
combinationof invisible pieces consideredincluding rooks.
This apparent anomalyis due to the bishop’s early involvementin the game, while rooks tend to comeinto play later. By causing uncertainty early in the game,a player with
twoinvisible bishopshas an early advantageagainst a player
withtwoinvisible rooks. Further analysis of these results is
presented in (Budet aL 2000).
6
Building Information Theoretic
Advisors
A reasonableinference fromthe results shownaboveis that
players with more information about the gametend to win
moreoften; i.e., the closer a player’sbelief abouta boardposition is to the true boardposition, the morelikely the player
is to play well strategically. In this section, wedescribeour
use of informationtheory (R6nyi1984) to quantify a player’s uncertainty about the positions of an opponent’sinvisible pieces (Section 6.1), present two information-theoretic
advisors (Section 6.2), and discuss somepreliminaryresults
obtainedwith these advisors (Section 6.3).
6.1
Uncertainty and Entropy
Informationtheory provides a measurefor quantifying information represented by a sequence of symbols. That is,
using information theory we can determine the minimum
numberof bits required to transmit the sequenceof symbols
to someoneelse. This numberrepresents a measureof the
amountof informationintrinsic to the sequenceof symbols.
Thecalculation of this numberrequires a distribution of the
probability that any particular symbolto be transmitted will
occur. Thusany probability distribution can be said to have
an entropy or information measureassociated with it. This
entropy measureis boundedfrom belowby zero, whenthere
is only one possibility and therefore no uncertainty, and increases as the distribution spreads.
In invisible chess, given a probability distribution of each
of the opponent’sinvisible pieces, it is possible to derive a
5probabilitydistribution across all possible boardpositions.
Thus we can calculate the entropy (H) of a set of board
states togetherwiththeir associatedprobabilities as follows:
H = - E Prob(X) × log2(Prob(X))
(3)
6.2 Information-Theoretic Advisors
Followingan analysis of the results in Section 5, we have
implementedtwo information-theoretic advisors. These advisors are movehide and moveseek.
YXEG
whereX represents a single gamestate, from the set of possible gamestates (G), whichis one possible combination
positions of the opponent’s invisible pieces, and Prob(X)
represents the probability of that gamestate.
As invisible pieces move,the numberof squares they may
occupyincreases. This leads to an increase in the numberof
possibleboardstates, and thereforethe entropyof the distribution of thoseboardstates, i.e., it increasesthe opponent’s
uncertainty.
5Assuming
that the piece distributions are independent,this
canbe calculatedbycombinatorially
cyclingthroughthe invisible
piecepositions.In ourimplementation,
thesedistributionsof invisible piecesare independent
becauseof the waythey are maintained
(Budet al. 2000).
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Move Hide Advisor. Working on the premise that the
moreuncertain the opponentis, the worsethey will play,
the movehide advisor advises a player to perform moves
that hide informationfrom the opponent.That is, each move
is scored accordingto its expectedeffect on the opponent’s
perceiveduncertaintyabout the positions of the player’s invisible pieces. This effect maybe an increase, a decrease
or no change in the opponent’s uncertainty. Movesby invisible pieces that do not cause checkresult in an increase
in the opponent’s uncertainty. Movesthat cause check or
movesby visible pieces maycause a decrease in the opponent’s uncertainty by revealing vacant squares or invisible
pieces themselves. As discussed in Section 6.1, we use the
entropyof the distribution of the positions of the opponent’s
invisible pieces as a measureof the opponent’suncertainty.
In order to calculate this entropy, the movehide advisor
needsto modelthe opponent’sdistribution update strategy.
Of course a player can use the real positions of each nonmovinginvisible piece and thereby avoid the cost of storing
6Themanygameswithzero entropyfor whiteare generallydue
to the invisiblebishop(queen’s
bishop)capturinga pieceonits rst
moveand subsequentlybeingcapturedwithoutmovingagain.
7Notethat the samerelationship betweenentropyand wincan
be seen for gamesbetweenmoreevenlymatchedinvisible pieces
(Budet al. 2000).
aO
Figure8: Uncertainty(as entropy) in gameswith one invisible black bishop against twoinvisible white bishops
that were wonby black.
Figure7: Uncertainty(as entropy)in gameswith one invisible black bishop against two invisible white bishops
that were wonby white.
or calculating the completejoint distribution of the positions of all invisible pieces. This methodprovidesa modelof
the best distribution the opponentcould havewithout taking
strategic informationinto account. To incorporate strategic
informationinto the model,a player could use the combined
expectedutility for each possible destination squarethat an
invisible piece could moveto.
For a proposedmove,the movehide advisor uses Equation3
to calculate the entropy of the opponentmodelof the distribution of the player’s invisible pieces prior to and following
the move.The exponential of the perceived change in entropy is returnedas the movehide utility. Theexponentialis
taken in order to allow a comparisonbetweenthe log based
utilities returned by the movehide and moveseek advisors,
and the utility returnedby the strategic advisor whichscores
moveson a linear scale.
MoveSeek Advisor. The moveseek advisor suggests that
a player performmovesthat are morelikely to discover information about the positions of the opponent’s invisible
pieces. That is, each moveis scoredaccordingto the expected decreasein the entropy of the opponent’sinvisible pieces
following the move.This expected decrease in entropy must
be greater than or equal to zero as no movecan makea player less certain aboutthe position of the opponent’sinvisible
pieces.
A movethat covers a large numberof squares with a positive
probability of occupationby the opponent’sinvisible pieces
will yield a certain amountof informationwhetherit is successful or not. Clearly this type of movewill yield more
informationif it is successful, as the player nowknowsthat
all of those squaresare vacant. Onfailure, a player can only
concludethat at least one of those squares is occupied. On
the other hand, a movethat traverses only one square with a
positive probability of occupationby the opponent’sinvisible pieces will yield moreinformationif unsuccessful. That
is, the player nowknowsthat an invisible piece is on that
square. Thus, the moveseek advisor multiplies the projected decrease in entropy for each outcomeby the probability
of that outcometo get an expected utility. Theexponential
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]Weight]
0.5
5.0
50.0
Seek
Hide
]
J
B vsB Qvs Q BvsB
QvsQ
57.2
77.2
47.2
51.0
71.6
43.6
54.6
45.0
13.6
30.3
47.6
67.0
Table2: Theeffect of the movehide and moveseek advisors.
of this expected changein entropy is returned as the move
seekutility.
6.3
Advisor Results
This section presents and discusses somepreliminary results that showthe individual effects of the movehide and
moveseek advisors with varying weights. Eachresult was
obtained by playing 500 gamesseparated into runs of 250
gamesas before, with one player using the strategic advisor only, against an opponentusing one of the informationtheoretic advisors and the strategic advisor. The invisible piece con gurations in this section were chosen because their results are typical of those obtained using our
information-theoreticadvisors.
The move hide column of Table 2 shows the results of
adding the movehide advisor to an invisible queeneach (Q
vs Q) and an invisible white bishopversus an invisible black
bishop (B vs B). The moveseek columnshowsthe results
of adding the moveseek advisor to an invisible queeneach
and and one invisible white bishop versus one invisible black
bishop (B vs B). The rst columnheaded "Weight" shows
the weightof the information-theoretic advisor relative to
the strategic advisor.Thus, with a weight of 0.5, in games
played with an invisible queeneach, the player playingwith
the movehide advisor won57.2%of the games.
MoveHide Results. With a weight of 0.5, the movehide
advisor signi cantly assists, increasingthe winrate to 57.2%
and 72.2%for queen versus queen and bishop versus bishop respectively. However,as the movehide advisor weight
increases past 0.5, the winrate decreasesquite rapidly. This
behaviouris typical of all observedmovehide runs, and results fromthe playertaking less notice of the strategic advisor’s advice. Althoughthe opponentmaybe slightly more
uncertain whenthe movehide advisor is weighted highly,
the player is makingenoughstrategically poor movesto
counter that advantage. Nonetheless, the movehide advisor is de nitely helpful with small weight, and the results
shownfor a weightof 0.5 are statistically signi cantly different from the base case of 50%.
Figure 9 showsthe difference in entropy betweenwhite and
black, each playing with an invisible queen, averagedover
the entire gamefor each of 250 games.To makethe trends
clearer, the data points are sorted by entropy. The middie curve represents gamesplayed betweeninvisible queens
with no information-theoretic advisors present, and shows
that the entropydifference is fairly even. In approximately
half the gamesit is negative, andin the other half it is positive, and it ranges betweenaround -1 and +1. The bottom
curve represents gameswhere white played with the move
hide advisor. This represents an entropy advantageto white.
The difference betweenwhite and black’s uncertainty ranges
from under -2 to around +1. The rest of the curve is also
shifted downwards
indicating the relative increase in black’s
uncertainty. This increase in black’s uncertainty leads to
morewins for white while using the movehide advisor.
Whenan invisible piece moves,the uncertainty increases in
the same mannerregardless of the nal destination of the
invisible piece. Thusthe movehide advisor is mostvaluable
whenany strategically advantageousmoveby an invisible
piece is possible.
MoveSeek Results. Table 2 (column 5) shows the move
seek advisor’s effectiveness against an opponent’sinvisible
bishop. As the weightincreases to around5.0, the percentage of winsincreases up to 71.6%.Thisis largely a result of
the moveseek advisor assisting the player to nd and capture the opponent’sinvisible bishop early in the game.The
player then has an informationadvantageequivalentto an invisible bishopversus no invisible piecesand is verylikely to
win. Figure 10 showsthe entropy advantageof playing with
the moveseek advisor. Thedata points are sorted by entropy
to clarify the trends. Thetop curveshowsthe entropydifference with no moveseek advisor, and the bottomcurve shows
that the entropy is signi cantly lower whenplaying with a
moveseek advisor. In this example, the moveseek advisor assists white to reduceuncertaintyoverthe courseof the
gameand therefore play strategically better than black.
Table 2 (column4) showsthe moveseek advisor as applied
to a gamebetweentwo invisible queens. In this situation,
the moveseek advisor is muchless effective and wouldappear to probablybe detrimentalin somecases. It seemslikely that the large numberof squares an invisible queenmay
have a positive probability of occupying,and the frequency of queen movement,makeit dif cult for the opponent
to nd an invisible queen. The top curve in Figure 9 represents gameswherewhite played with the moveseek advisor.
Anexaminationof the entropy difference slightly favours
black comparedto the base case. This is almost certainly
because white is spending movestrying to nd black’s in-
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visible queenrather than movingtheir owninvisible queen.
Onlymovesthat traverse squares that havea positive probability of occupationby the opponent’sinvisible pieces are
valued by the moveseek advisor. These movesmayor may
not correspond to good strategic moves. The more a player listens to the moveseek advisor, the fewergoodstrategic
movesthey are able to perform.The difference betweenthe
bishop-seeking behaviour and the queen-seekingbehaviour
dependson the difculty of nding the opponent’sinvisible
piece. Aninvisible bishopcan havea positive probability of
occupyingat most half the available squares on the board,
while an invisible queencan have a positive probability of
occupyingall the available squares. Thus,the adviceprovided by the moveseek advisor often aids in the early capture of
the invisible bishop, thereby removingthe uncertainty from
the game.In contrast, following this advice whenthe invisible piece is an invisible queenleads to wastedmoves.
Thusthe moveseek advisor assists whenthe uncertainty in
the gameis low, but maybe useless or detrimental whenthe
uncertaintyis high.
7 Conclusion
and Future Work
Theresults presentedin this paper are preliminaryand further exploration of the domainis required. Thereare a number of areas that require morein depth investigation. Specifically, moreaccurate prediction of the likely positioning of
the opponent’sinvisible pieces is needed. This prediction
could take the formof using strategic informationabout the
likely destination of a movinginvisible piece, or involve
evaluating the completesearch tree for more ply. Improving this distribution wouldreduce its entropy and therefore
improvestrategic performance.Aside effect of this type of
distribution improvement
is the possibility of incorporating
blufng into invisible chess. That is, movingan invisible
piece to an unlikely position in order to confuse an opponent.
As indicated above, one wayto improvethe prediction of
the positions of the opponent’sinvisible pieces wouldbe to
modelthe uncertainty regarding a player’s pieces from the
opponent’sperspective for moreply. However,the problem
of the combinatorial expansionin the search required as a
result of this prediction needsto be resolved. The mostobvious wayto managethis explosion is to nd effective ways
to prunethe gametree.
As our systemcurrently stands, a non-integrated player of
invisible chess (whether humanor machine)could not have
the bene t of our GCDMM
which maintains the approximationto the distribution of the positions of the opponent’s
invisible pieces by using the true positions of all other invisible pieces whenupdatingthe distribution. Sucha player
wouldneed to nd other waysof approximatingthat distribution.
In summary,we have presented and discussed a complex,
but controlled domainfor exploring automatedreasoning in
an uncertain environmentwith a high degree of strategic
complexity. Wehave motivated and introduced the use of
information-theoretic advisors in the strategically complex
imperfect information domainof invisible chess. Wehave
shownthat our distributed-advisor approachusing a combi-
Invisible ~e*n vs Invlsible Oug~. Diff*r*ncs in Entropy blt~en Black and Whir*
2
¯ [n~ropy Diff*’rsn=s with No ~dvi=orl m
~ntropy Difference with ~Thits MovaSeeks"
1.5
®
1 Znv Black Bishop vl 1 inv whirs Bishop, Dlff In Entropy beb~sn Black & Whirs
2
g
Entropy
Dlf f@renc’e
wl~h ~ito
s~k
m
~tropy Difference with No Advienrn ....
1.5
O.5
....... gn~rop¥~i~Dlff*r*n¢~
with~cnlts HoveHidg.........
0
~-0.~
~
.,"""" .... .... ..... .r"
o
.......,.........-,.’.....
~ -0.5
i -1.5
i -1.5
-a.5
-2.5
I
lOO
!
15o
|
2oo
25o
Figure 10: The difference in entropy between white and
black for a base case, and with moveseek, sorted by
entropy.
Figure 9: Entropy difference between white and black
for a base case, move seek (weight 1.0) and move hide
(weight 0.5), sorted by entropy.
nation of information-theoretic and strategic aspects of the
domain lead to performance advantages compared to using
strategic expertise alone. Given the simplicity and generality of this approach, our results point towards the potential
applicability to a range of strategically complex imperfect
information domains.
Acknowledgements
The authors would like to thank Chris Wallace for his insight, advice and patience.
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