Critical Position Identiﬁcation in Games and Its Application to

Speculative Play

Mohd Nor Akmal Khalid

, Umi Kalsom Yusof

, Hiroyuki Iida

and Taichi Ishitobi

School of Computer Sciences, Universiti Sains Malaysia, 11800 Georgetown, Pulau Pinang, Malaysia

School of Information Science, Japan Advance Institute of Science and Technology,

1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan

Keywords:

Speculative Play, Critical Position, Games.

Abstract:

Research in two-player perfect information games have been one of the focus of computer-game related studies

in the domain of artiﬁcial intelligence. However, focus on an effective search program is insufﬁcient to give

the “taste” of actual entertainment in the gaming industry. Instead of focusing on effective search algorithm,

we dedicate our study in realizing the possibility of applying speculative play. However, quantifying and

determining this possibility is the main challenge imposed in this study. For this purpose, the Conspiracy

Number Search algorithm is considered where the maximum and minimum conspiracy numbers are recorded

in the test bed of a simple Tic-Tac-Toe game application. We analyze these numbers as the measures of

critical position identiﬁer which determines the right moment for possibility of applying speculative play

through operators formally deﬁned in this article as ↑ tactic and ↓ tactic. Interesting results are obtained with

convincing evidences but further works are still needed in order to prove our hypothesis.

1 INTRODUCTION

In the domain of speculative play, understanding the

the game or mastering the game intricacy is the most

important aspect for achieving successful outcome

in the respective competitive combat (van den Herik

et al., 2005). However, the main challenge in the

domain of speculative play is to identify a (criti-

cal) position for applying a speculative play. The

opponent-model search (Iida et al., 1993; Carmel and

Markovitch, 1993; Iida et al., 1994) is such a specu-

lative play, but without the knowledge of when is the

most optimum position to apply it during a game. In

other words, determining when one should change his

strategy from minimax to any speculative way is the

puzzling issue.

The mechanics of computer-games in two-player

perfect information games or two-player games in

short, such as board games (e.g. tic-tac-toe, othello,

checkers, chess, etc.), involve using a tree searching

algorithm to evaluate and decide the possible moves

to take. However, even in the best known search al-

gorithms, the search space possesses exponential time

complexity with the growingdepth of the tree (Lorenz

et al., 1995). Innovative search algorithms, search en-

hancements, and learning ideas have been applied by

ample research efforts towards creating a computer

program that dominates the games against their oppo-

nents with better strength and performance (Schaeffer

and van den Herikb, 2002), as well as overcoming the

time complexity limitation of the tree search (Kishi-

moto et al., 2012). In order to better understand the

nature of any computer-game, studying the progress

of the games is important for improving the game’s

value as a form of entertainment.

When progressing on the board game, different

positions made by a player throughout the game’s

time horizon can effect the outcome of the endgame.

Usually, playing well throughout the game when

competing against top human players is not enough

but playing optimally during the endgame (or certain

parts of the games) is very important (Jansen, 1992;

Donkers, 2003). Focusing on a certain stage of the

game is essential in order to apply different specula-

tive play to boost its excitement. However, the out-

come of the computer-games (lose, win, or draw) is

unclear until the game ends. Predicting this outcome

during the progress of computer-games is mainly de-

pendent on the likeliness of a position to result in ei-

ther winning, losing, or draw. This situation is for-

mally deﬁned as critical position, where at a cer-

tain point of the game progress, the game outcome

Khalid M., Yusof U., Iida H. and Ishitobi T..

Critical Position Identiﬁcation in Games and Its Application to Speculative Play.

DOI: 10.5220/0005179900380045

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence (ICAART-2015), pages 38-45

ISBN: 978-989-758-074-1

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

is measurable and eventually becomes certain and in-

evitable.

Identifying critical position of a progressing com-

puter game involve comprehending the computer

players tendencyof changing its strategy during a par-

ticular moment of the game play. In other words,

knowing the right moment of based on this critical

position at a certain state of the game enables the pos-

sibility of applying speculative play which essentially

produces interesting outcome of the game (Ramon

et al., 2002). Thus, identifying this critical position

is highly dependent on the quantifying capabilities of

the indicator. Therefore, a suitable search algorithm

that acts as an indicator during the games progress is

necessary in order to identify its critical position.

A well-known search indicators studied by sev-

eral researchers in the late-80s is the Conspiracy-

Number Search (CNS). Introduced by McAllester

(McAllester, 1985; McAllester, 1988), CNS is a best-

ﬁrst search algorithm for minimax tree framework,

which determines the cardinality of the smallest set of

leaf nodes which have to “conspire” to change their

values in order to change the minimax value of the

root. This present a suitable opportunity for determin-

ing the moments for applying speculative play (e.g.

opponent model), thus acts as the motivating factor of

selecting CNS for this study. One of the ideas under-

lying CNS is that the distribution of the values over

the leaf nodes of the tree, and the shape of the tree,

should inﬂuence the selection of the next node to be

investigated (Kishimoto et al., 2012). However, fo-

cus on CNS as a search algorithm in computer-games

research was faced with discouraging results (Lorenz

et al., 1995; Schaeffer, 1989; Elkan, 1989; Schaef-

fer, 1990; van der Meulen, 1990). Later, Allis et al.

(Allis et al., 1994) derived and specialized CNS con-

cepts to the AND/OR tree framework where the study

matured into Proof-Number Search (PNS). Although

CNS is different compared to PNS, CNS’s conceptual

frameworks bear certain correlation to PNS.

This study concerns a matter of critical position

identiﬁcation, not a matter of program strength im-

provement in games (See some approaches, e.g., done

by Jonathan Schaeffer). Hence, we have chosen a

game as simple as possible in order to clearly explain

the proposed idea. Further investigation would be, as

suggested, to implement the proposed idea in more

complicated games such as chess. To our best knowl-

edge, no other research has been done with CNs to

identify critical positions instead of using as game-

tree search heuristics (stability of the root node’s min-

imax value). In this paper, any part of speculativeplay

is not described. However, the relation between CN

and strategic change at critical position (in case where

the position is going to be disadvantageous position)

should be very important in the context of speculative

play.

2 CONSPIRACY-NUMBER

In the minimax tree framework, the ﬁrst player tries

to maximize his or her advantage, while the second

player tries to minimize it (Neumann and Morgen-

stern, 1947; Shannon, 1950). CNS searches the tree

in a manner that at least c > 1 of the leaf values

have to change in order to change the decision at the

root (Lorenz et al., 1995). Intuitively, when expand-

ing a minimax tree further, the accuracy and stabil-

ity of the root value depend on how much it changes.

Major changes on the root value make it unreliable

(McAllester, 1988). Therefore, the concept of con-

spiracy is used to measure the root value’s stability

and its likelihood to change by narrowing the range

of the plausible values of the root (Schaeffer, 1989).

The likelihood of the root taking a particular value is

reﬂected in that value’s associated conspiracy num-

ber. This conspiracy number measures the size of the

“conspirators” needed to bring about a certain change

in root value; the more conspiratorsneeded for a given

change, the less likely the change (McAllester, 1988).

This is done by keeping track so the number of leaf

nodes whose value must be changed (when searched

deeper) to change the root’s node value by a certain

amount or taking on that new value. A change in the

value of a certain set of leaf nodes is called conspiracy

between those leaf nodes.

The algorithm is a probabilistic search in nature

where there is no guarantee that the correct solution

will be found when it terminates, but the most likely

one instead. The conceptual framework behind the

CNS is to grow search trees for which one has con-

ﬁdence by measuring the number of value through

the conspiracy numbers. The search is guided in a

best ﬁrst manner, where the tree searched so far is

kept in memory. An example is probably the best

way to illustrate the function of a conspiracy number.

The following is taken from (Schaeffer, 1989; Lorenz

et al., 1995): Assume that the branching factor is 2,

the range of values are from 1 to 6, the root node is

the MAX node, inside the nodes are their names and

their minimax values, and the simple tabular for stor-

ing conspiracy numbers of the root. From Fig. 1, it

can be observed that the leaves or terminal node have

to at least change their value to cause the value of the

root to become 1, 2, 4, 5, or 6. For example, only

leaf E has to change its value to 5 in order for the root

CriticalPositionIdentificationinGamesandItsApplicationtoSpeculativePlay

A=3

D=5 E=2 F=3 G=4

B=2 C=3

Root

Interior Node

Terminal Node

value cn conspirators

1 2 (D or E) and (F or G)

2 1 F or G

3 0

4 1 E or F

5 1 E

6 2 (D and E) or (F and G)

Figure 1: Illustration of a minimax tree adopting CNS algorithm.

value to become 5.

As described by Schaeffer (Schaeffer, 1989), there

is a simple method of computing the conspiracy num-

bers. At the terminal node t, the conspiracy number

associated with node t value is 0 and for all other val-

ues is 1. For the interior node, if the value x of a

MAX node is to be increased to x

′

> x, only one of

the successor nodes need to change its value to x’. It

is clear that the conspiracy number for x’ is the mini-

mum amongst all other successors. This is denoted as

↑ needed

. If the value x is to be decreased to x

′

< x,

all successors with values greater than x’ must change

their value to one lower than x’. That is the reason

why the number of conspirators for these nodes are

summed. This is denoted as ↓ needed

. The rules of

calculating conspiracy numbers are given as the fol-

lowing (v is the associated conspiracy number of the

node and m is the minimax value of the node):

CN(v) = 0, if v = m,

At MAX node:

CN(v) =

∑

all sons i

↓ needed

(v), for all v < m,

CN(v) = min

all sons i

↑ needed

(v), for all v > m.

At MIN node:

CN(v) = min

all sons i

↓ needed

(v), for all v < m,

CN(v) =

∑

all sons i

↑ needed

(v), for all v > m.

Since its introduction, variants of CNS have

been proposed by several researchers. α − β Con-

spiracy Search proposed by McAllester and Yuret

(McAllester and Yuret, 1993) establishes lower and

upper bound of the search. The MAX strategy estab-

lishes a lower bound and the MIN strategy establishes

an upper bound. Thus, the conspiracy numbers can

be used to measure the “safety” of these two strate-

gies. Lorenz et al. (Lorenz et al., 1995) proposed

a Controlled-Conspiracy Number Search where in-

stead of variable depth, an α − β quiescence search

is used and search bound is provided. The conspiracy

number vectors are compressed into 3-tuples, allow-

ing the CCNS to be independent of the granularity of

the evaluation function (e.g. positional play in chess).

3 CONSPIRACY-NUMBER

SEARCH AS CRITICAL

POSITION IDENTIFIER

Determining critical position is vitals due to the fol-

lowing reasons: First, the speciﬁc outcome of the

game can be estimated in advance (e.g. resignation)

and this estimation can be utilized for game outcome

prediction. Second, the critical positions are expected

to expose the possibilities of tactic changes, denoted

as ↑ tactics or ↓ tactics, respectively. The ↑ tactics

implies that the computer player tries to force a draw

when it’s losing, while ↓ tactics implies that the com-

puter player tries to force a draw when achieving a

win is impossible. Third, critical positions can be

used to estimate the moment for computer player to

apply speculative play and change the outcome of the

game.

However, why do we use CNS instead of the po-

sitional scoring (evaluation function) alone during the

game’s progression? The positional scoring act as a

guiding function for an individual player to determine

his/her state in the game’s progress (Buro, 1999). Al-

though it might be suitable for identifying critical po-

sitions, it still lack of the probabilistic element in ab-

stracting the player’s decision to determine the next

game state. Basically, positional scoring shows how

the game progresses (i.e. which player is leading the

game) but the CN-values potentially show its prob-

able changes. Thus, the goal of critical position is

to determine the possibility of improving or deteri-

orating the positional score value of a move (evalu-

ate for probable impact of next moves), while posi-

tional scoring makes use of the evaluation features

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

of a move (evaluate for probable gain of next moves)

(Buro, 1999).

3.1 Experimental Results and

Discussion

CNS requires two types of conspiracy numbers

needed to maintain, ↑ needed and ↓ needed. We con-

sider this as a scalar measures in the computer-game’s

search progress to analyze and justify the rationale of

the hypothesis mentioned earlier. A simple Tic-Tac-

Toe game is used as the test bed of our study.

This experiment is tested with ﬁxed-depth mini-

max algorithm applying CNS as a scalar measures for

recording the conspiracy numbers of the root values

for each game moves. For every player X with search

depth i there is an opposing player O with search

depth j, where 2 ≤ i ≤ 6 and 2 ≤ j ≤ 6. Therefore,

there are 25 total game sets (both player X and player

O have ﬁve depths). Player X is assumed to be the

ﬁrst player to start the game in every case, since if

player O starts the results will be just the reverse. The

rules of the games are as follows:

• For player O, the score is negative. For player X,

the score is positive.

• Player X tries to maximize the score, while player

O tries to minimize the score.

• For each row, if there are both X and O, then the

score for the row is 0.

• If the whole row is empty, then the score is 1.

• If there is only one X, then the score is 10. If there

is only one O, then the score is -10.

• If there are two X, then the score is 100. If there

are two O, then the score is -100.

• If there are 3 X, then the score is 1000. If there

are three O, then the score is -1000.

Table 1 and Table 2 shows the recorded maximum

conspiracy numbers (MaxCN) and minimum conspir-

acy numbers (MinCN) for every game sets, respec-

tively. The odd and even numbers (highlighted in light

gray) of the game’s progress are relevant to player X

and player O, respectively. In all cases of game sets,

the game outcomes is a draw. To counter this, we con-

sider the ﬁnal score of any player as win, lose, or draw

as ﬁnal score of 100, -100, and 0, respectively. Table

3 shows the ﬁnal scores of every game of player X

versus player O.

Observing Table 1 and Table 2, the MaxCN of

player X decreases steadily (in most cases) while the

MaxCN for player O decreases abruptly. To sim-

plify the results interpretation, the following rules are

adopted: The MaxCN of current player p is consid-

ered abruptly decreased if |MaxCN

i+1

−MaxCN

| >

|MaxCN

i+1

|, where i equals to the game progress

Table 1: Maximum Conspiracy numbers of player X against

player O with depth variations.

Maximum Conspiracy Numbers

Depth Game Progress

(p1,p2) 1 2 3 4 5 6 7 8 9

2,2 8 7 6 5 4 3 2 1 1

2,3 8

22 6 8 4 4 2 1 1

2,4 8

62 6 16 4 4 2 1 1

2,5 8

128 6 17 4 4 2 1 1

2,6 8

288 6 17 4 4 2 1 1

3,2 56 7 26 5 10 3 2 1 1

3,3 56

22 26 8 10 5 2 1 1

3,4 56

62 26 16 10 5 2 1 1

3,5 56

128 26 17 10 5 2 1 1

3,6 56

288 26 17 10 5 2 1 1

4,2 136 7 89 5 17 3 2 1 1

4,3 136

22 89 8 17 3 2 1 1

4,4 136

62 89 16 17 3 2 1 1

4,5 136

128 89 17 17 3 2 1 1

4,6 136

288 89 17 17 3 2 1 1

5,2 360 7 206 5 17 3 2 1 1

5,3 360

22 206 8 17 3 2 1 1

5,4 360

62 206 16 17 3 2 1 1

5,5 360

128 206 17 17 3 2 1 1

5,6 360

288 206 17 17 3 2 1 1

6,2 648 7 295 5 17 3 2 1 1

6,3 648

22 295 8 17 3 2 1 1

6,4 648

62 295 16 17 3 2 1 1

6,5 648

128 295 17 17 3 2 1 1

6,6 648

288 295 17 17 3 2 1 1

p1 = player X, p2 = player O

(e.g. moves). The value of MaxCN implies insta-

bility and changes in the root value is more likely. In

other words, possibility of losing or winning is high

since the likeliness of the root value to change is high.

However, the fact that the outcome is inevitable (ei-

ther win or lose) but not known, this stage simulates

the critical position which is highly recommended for

applying speculative play.

For MinCN, however, a different interpretation is

needed. The abruptly decreased MinCN utilizes the

same rule as MaxCN: The MinCN of current player

p is considered abruptly decreased if |MinCN

i+1

−

MinCN

| > |MinCN

i+1

|, where i equals to the

game’s progress (e.g. moves). This situation is

the critical position for tactic changes (↑ tactic or

↓ tactic). In the case of abrupt inclining of MaxCN,

the root value stabilizes and tactic change of the cur-

rent player from better to worst (from winning to a

draw or a draw to losing) is more likely. This particu-

lar situation is when the ↓ tactic occurs. In the case of

abrupt inclining of MinCN, the root value is limited to

the available MinCN value only. Thus, this situation

implies the likelihood of identifying the tactic change

of the current player from worst to better (from losing

to a draw or from a draw to winning). This particular

CriticalPositionIdentificationinGamesandItsApplicationtoSpeculativePlay

Table 2: Minimum Conspiracy numbers of player X against

player O with depth variations.

Minimum Conspiracy Numbers

Depth Game Progress

(p1,p2) 1 2 3 4 5 6 7 8 9

2,2 1 6 1 4 1 3 0 0 0

2,3 1

4 1 4 1 1 0 0 0

2,4 1

13 1 8 1 0 0 0 0

2,5 1

13 1 5 1 0 0 0 0

2,6 1

23 1 1 1 0 0 0 0

3,2 6 6 4 4 0 0 0 0 0

3,3 6

4 4 4 0 0 0 0 0

3,4 6

13 4 8 0 0 0 0 0

3,5 6

13 4 5 0 0 0 0 0

3,6 6

23 4 1 0 0 0 0 0

4,2 4 6 3 4 1 3 0 0 0

4,3 4

4 3 4 1 1 0 0 0

4,4 4

13 3 8 1 0 0 0 0

4,5 4

13 3 5 1 0 0 0 0

4,6 4

23 3 1 1 0 0 0 0

5,2 13 6 5 4 0 3 0 0 0

5,3 13

4 5 4 0 1 0 0 0

5,4 13

13 5 8 0 0 0 0 0

5,5 13

13 5 5 0 0 0 0 0

5,6 13

23 5 1 0 0 0 0 0

6,2 13 6 5 4 0 3 0 0 0

6,3 13

4 5 4 0 1 0 0 0

6,4 13

13 5 8 0 0 0 0 0

6,5 13

13 5 5 0 0 0 0 0

6,6 13

23 5 1 0 0 0 0 0

p1 = player X, p2 = player O

Table 3: Final scores based on depths of player X against

player O.

Final Score

Player O

2 3 4 5 6

2 0 0 0 0 0

3 0 0 0 0 0

Player X 4 -100 -100 -100 -100 -100

5 -100 -100 -100 -100 -100

6 -100 -100 -100 -100 -100

situation is when the ↑ tactic occurred.

For instance, consider player X with search depth

2 against player O with search depth 3. Generally,

player O can be regarded to outperform player X

due to lookahead superiority. Figure 2 and Figure

4(a) simulates the mentioned game’s progress situ-

ation. During the ﬁrst move, player X chooses the

middle position of the board leaving player O with

limited options. The MaxCN and MinCN of player

X is currently 8 and 1 respectively. After consid-

ering the vertical, diagonal, and horizontal spaces,

player O chooses the upper left corner of the board

Table 3: Final scores based on depths of player X against

2 3 4 5 6

0 0 0 0 0

-100 -100 -100 -100 -100

of the current player from worst to better (from losing

to a draw or from a draw to winning). This particular

For instance, consider player X with search depth

2 against player O with search depth 3. Generally,

player O can be regarded to outperform player X

due to lookahead superiority. Figure 2 and Figure

4(a) simulates the mentioned game’s progress situ-

ation. During the ﬁrst move, player X chooses the

middle position of the board leaving player O with

1 2 3 4

11 1 1

0 0

Moves

Player X with search depth 2

Max

Min

1 2 3 4

4 4

Moves

Player O with search depth 3

Max

Min

Figure 2: MaxCN and MinCN for player X with search

depth 2 against player O with search depth 3

Figure 2: MaxCN and MinCN for player X with search

depth 2 against player O with search depth 3.

and currently possess MaxCN and MinCN of 22 and

4 respectively. Consequently, player X chose the up-

per middle side of the board, which effects MaxCN

to decrease steadily, while MinCN remain the same.

Next, player O chooses the lower middle of the board

which abruptly reduces player O’s MaxCN to 8 while

MinCN remains the same. This situation imposes that

player O is in a critical position, even if its intent is

to prevent player X from winning. Therefore, player

O is in the state of ↓ tactic and is advised to apply

different tactic. In the next move, player X chooses

the lower left corner of the board which has forced

player O into another bad position. During the con-

sequent move, player O’s MaxCN reduces steadily

while MinCN is abruptly reduced to 1, which implies

that player O has applied ↑ tactic where, instead of

losing, it tries to force player X into a draw. Thus, the

ﬁnal outcome of the game is a draw.

Another example is the case of player X with

search depth 5 against player O with search depth 4

which is given in Figure 3 and Figure 4(b). During

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

1 2 3 4

200

400

360

206

0 0 0

Moves

Player X with search depth 5

Max

Min

1 2 3 4

0 0

Moves

Player O with search depth 4

Max

Min

Figure 3: MaxCN and MinCN for player X with search

depth 5 against player O with search depth 4

Figure 3: MaxCN and MinCN for player X with search

depth 5 against player O with search depth 4.

the ﬁrst move, player X chooses the middle position

of the board leaving player O with limited options.

The MaxCN and MinCN of player X is currently 360

and 4 respectively. The high value of MaxCN is be-

cause of the deeper search depths. In the next move,

player O chooses the upper left corner of the board

and obtains a MaxCN and MinCN of 62 and 13 re-

spectively. Consequently, player X chooses the upper

middle side of the board, making MaxCN and MinCN

decreases steadily. During the rest of the moves, the

evolution of MaxCN of both player X and player O

abruptly reduces, forcing both players into “dilem-

mas” where both ↓ tactic and ↑ tactic occur. This situ-

ation can be hypothetically deﬁned as the state where

both players are able to apply different tactics. On the

other hand, observing the MinCN of both player X

and player O during ﬁfth and sixth moves implies that

both player adopted the ↓ tactic. However, player O’s

leading score plays its part and forces player X to lose

the game (assumed ﬁnal score is winning).

The challenge in the study of speculative play is

to identify critical positions at which one should con-

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Figure 4: Simulation of Tic-Tac-Toe games.

sider to apply a kind of speculative strategy such as

opponent-model search in order to change the situa-

tion: from behind to even or better. Another chal-

lenge in GM-level man-machine matches is to iden-

tify (no more promising) positions to resign. There-

fore, the experiments suggested in this study provide

the main foundation for identifying the critical posi-

tions, although further investigation is expected.

4 CNS CORRELATIONS TO PNS

Application of PNS in the previous work such as

(Ishitobi et al., 2013) raised questions whether any

correlations exists in the application of CNS as a criti-

cal position identiﬁer. Theoretically, experimental re-

sults obtained from the previous section empowered

the possibility of correlation between CNS and PNS

as indicators in their respective tree search frame-

work. This section attempts to identify that correla-

tion and give a better picture on the importance of

these two indicators. The following section will give

a short description on the basis of PNS.

4.1 The Basis of PNS

PNS, like its ancestor CNS, is a best-ﬁrst search al-

gorithm in which the tree searched so far is stored

in memory. The main difference is that PNS aims

at proving the true value of root, where the interim

minimax values are not considered (Allis et al., 1994).

CriticalPositionIdentificationinGamesandItsApplicationtoSpeculativePlay

The PNS heuristic determines the most promising leaf

by selecting a most-proving node or most-promising

node (MPN), which can contribute to either a proof

or a disproof of the root if a leaf node is solved. The

MPN can be formally deﬁned as the node which, with

the least possible effort, potentially contributes most

to the establishment of the minimax value of the root.

The MPN can be found by manipulating two criteria

of the search tree: (1) its shape (determined by the

branching factor of every internal node), and (2) the

values of the leaves. The basic and un-enhanced PNS

is an uninformed search method that does not require

any game-speciﬁc knowledge beyond its rules (Kishi-

moto et al., 2012).

The PNS produces two special values for each

node n in order to ﬁnd MPN. First, the proof num-

ber (denoted as pn(n) where pn is the proof number

of node n) which is the smallest number of leaf nodes

in the subtree starting with n that have to be proven

in order to prove that n is a win. Second, the disproof

number (denoted as dn(n) where dn is the disproof

number of node n) which is the minimum number of

leaf nodes that have to be disproved in order to prove

that n is a loss.

Calculating the values of pn and dn for each node

in the tree is performed in a bottom-up manner. Usu-

ally, In a terminal node t, the game-theoretic value

is known or the corresponding position has no legal

moves. If t is a win, then pn(t) = 0 and dn(t) = 1.

If t is a loss, then pn(t) = 1 and dn(t) = 0. If t is un-

known, then pn(t) = dn(t) = 1. In this case, thetermi-

nal node t is called a temporary terminal node. For the

internal MAX node, it is sufﬁcient to have one child

that proves the value of v. The pn of a MAX node is

equal to the minimum of the pn of its children. For

dn, the only way to disprove v is to disprove v for all

its children. So, the dn for MAX node is equal to the

sum of the dn of all its children. It is the reverse for

the internal MIN node.

However, PNS is famously known for searching

the AND/OR tree framework instead of the minimax

tree framework. The AND/OR tree is a type of tree

where the nodes have only three possible values: true,

false, and unknown(Ishitobi et al., 2013). Using PNS,

the pn for an AND/OR tree represents the minimum

number of unsolved leaf nodes that need to be solved

in order to win in the root. Similarly, the dn for an

AND/OR tree represents the minimum number of un-

solved leaf nodes that need to be solved in order to

lose in the root. The PNS always considers the MPN

in which the internal nodes can be decided recursively

if the terminal node value was decided. Thus, PNS

can be used to decide the value of the root node by

deciding values of other nodes as soon as possible

(Ishitobi et al., 2013).

4.2 General Correlation of the Elements

of CNS and PNS

As described by Ishitobi et al. (Ishitobi et al., 2013),

pn is related to the difﬁculty, which relates to the

minimum number of unsolved nodes that need to be

solved. So, maximum pn shows the complexity to

solve these unsolved nodes. On the other hand, dn

is related to the minimum number of unsolved nodes

that need to be disproved. Therefore, maximum dn

shows the complexity to disprove. In other word, both

the maximum pn and maximum dn are an effective

measures of difﬁculty to solve nodes as soon as pos-

sible for the AND/OR tree framework.

The MaxCN and MinCN show a correlation to

maximum pn and maximum dn in the minimax tree

framework. In the CNS context, MaxCN and MinCN

identify the critical position in the game’s progress

for an expected outcome. MaxCN indicates the un-

likeliness of root value in achieving a value. There-

fore, high value of MaxCN implies the high likeliness

of winning or losing. MinCN indicates that the like-

liness of root value to achieve a value is limited to

the value of MinCN. So, high value of MinCN im-

plies the possibility of target change of the possible

estimated outcome. Considering the relation to max-

imum pn and maximum dn, we ﬁnd that high values

of MaxCN and MinCN are an effective measures of

difﬁculty of a particular value to be likely. Figure 5

depicts the correlation of CNS and PNS.

AND/OR

tree

framework

Minimax

tree

framework

Prove Disprove

Proof Numbers

Measures of Complexity

Maximum Maximum

Conspiracy Numbers

Measures of

Critical

Positions

Maximum

Minimum

Instability to

Lose

(Resignation)

Stability to

Win

(Surrender/

Resisting)

Difficulty to Solve

Difficulty to be

achieve likeliness

Figure 5: Correlation between CNS and PNS.

5 CONCLUDING REMARKS

In this article, we presented two main contributions:

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

• The application of conspiracy numbers as the crit-

ical position identiﬁer in the context of tic-tac-toe

game (two-players perfect information games).

• The theoretical correlation of maximum and min-

imum conspiracy numbers to maximum proof

numbers and maximum disproof numbers.

In the results presented, we successfully identi-

ﬁed the critical position through maximum and mini-

mum conspiracy numbers in most cases of game sets.

However, not all game sets support the ﬁndings antic-

ipated from the value of the maximum and minimum

conspiracy numbers. Furthermore, further evidences

are needed in order to conclude the conspiracy num-

bers as the most suitable critical position identiﬁer.

In addition, by determining this critical position, pos-

sible application of speculative play can be explored

and exploited to improve the overall game excite-

ment. Further explorations on larger and more com-

plex game searches (othello, chess, and endgames of

difﬁcult positional chess) is the outlook which we will

focuses in future studies in order to give a more accu-

rate and extensive understanding on the role of CNS

as critical position identiﬁer. Similarly, future work

on CNS as critical position identiﬁer will potentially

reinforce claims made on its correlation to PNS.

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