doi:10.1093/comjnl/bxm060 Evaluation of Economy in a Zero-sum

A zero-sum perfect information game is one where every player knows all the moves. Chess is a good example where the object is to checkmate or capture the enemy king. One important feature of checkmates, especially in chess problem composition, is economy. This paper proposes a computational function to evaluate the economy of checkmate configurations on the chessboard. Several experiments were performed comparing chess compositions and regular games to validate the function. The results suggest that the proposed evaluation function is able to correctly discern economical differences in checkmate positions to a high degree of statistical significance and correlates positively with the perception of human chess players. This evaluation function can therefore be useful in increasing the versatility of chess database search engines, as a component in aesthetic models of chess and aiding judges in chess composition tournaments.


INTRODUCTION
Zero-sum games of perfect information are those where every player knows all the moves.Examples include chess, checkers, go and noughts and crosses.Such games are particularly amenable to computation because in theory, a perfect game (i.e.where a player can either force a win or at least never lose) is possible by analyzing the game tree.In noughts and crosses for example, it is quite easy for a computer to play so that it never loses or at worst draws since the game tree is rather small.In checkers, chess and go however, it gets much larger.Even so, there is much that computers have been able to achieve there.
Checkers is getting closer to being 'solved' and one of the best players in the world is widely regarded as the program Chinook [1][2][3][4].Chess playing programs on the other hand have rivaled human grandmasters since the 1990s.The most famous event being Gary Kasparov's loss to IBM's Deep Blue in 1996 [5][6][7].Even in go, where the game tree is much larger than chess, strong programs such as Handtalk have been written for some time [8 -10].The strength of these programs usually lies in the effectiveness of their heuristic evaluation functions and search capabilities.
This research was intended to see if a primarily aesthetic but nevertheless important concept like economy could also be evaluated in a game like this.Economy essentially refers to the efficient use of resources in the game to achieve a particular objective (see Section 2.1).It is important because it usually reflects sound play and is appreciated aesthetically by both players and problem composers, particularly the latter [11].Garry Kasparov, arguably the world's best chess player, is even reported to have said, "I want to win, I want to beat everyone, but I want to do it in style!" [12].
For this research, chess (i.e.Western or International chess 1 ) was deemed the most suitable since there is plenty of literature on the game and a clear description of what economy means in that domain.The discrete properties of the chessboard and chessmen also provide a convenient and reliable metric to base the evaluation function on.

REVIEW
Developing an evaluation function for games usually involves selecting features, assigning weights to them and combining certain features to best represent a particular heuristic [13].Functions can also be tuned for better results through 1 Western or International chess is the version played internationally and regulated by the World Chess Federation or Fe ´de ´ration Internationale des E ´checs (FIDE).It is different from other variants (e.g.xiangqi, shogi, fairy chess) usually in terms of the pieces used, board type or rules of play.supervised learning (using training data) and unsupervised learning (fitting a model to observations), depending on what we are trying to evaluate [14].This usually works well for heuristics intended to drive actual game play and is based on specific results such as winning or losing.It can also be based on a reasonably accurate assessment of particular game states.A variety of different approaches (e.g.evolutionary algorithms) not necessarily related to properties of the game being studied, might be employed in the process [15 -18].
In the case of economy however, an evaluation function does not really have a winning or losing reference by which to tune itself and it is not intended to drive game play.Economy-if noted at all-is either deemed good or bad with a general idea (relating to specific features) about what makes it better or worse.It is somewhat subjective, but closely tied to the rules and properties of the game.A comparison could be made to the aesthetic features of images and how computers might go about obtaining and evaluating them [19 -21].Ideally data should be collected from a random sample of human subjects under controlled conditions, but this is not always possible given resource constraints.Alternatively, data could be obtained from a large repository at an online community where a diverse group of people have rated such objects (e.g.images).The data can then be analyzed and aesthetic features extracted.
Subsequently, classification (setting high and low thresholds) or linear regression can be used to weight them.The former is usually more reliable given the subjective nature of aesthetics.Absolute scores are therefore less meaningful even though it would be desirable if a machine could rate the objects on a scale of say 1 -7, like humans do.Economy, though similar in the sense of being an aesthetic feature in chess, is not as subjective as those in images.Aesthetic principles in chess have been confirmed experimentally (based on the opinion of master players) and in chess literature [11, 22 -28].Economy is unmistakably one of them.Subsequently within economy itself, there are many game-related properties, which directly influence its value.These will be explained in the following sections.
Weighting these properties based on subjective human judgement (as with extracted features in images) is therefore unnecessary, not to mention unfeasible since no database of economically rated checkmate positions could be obtained.For the same reason, training an economic evaluation function in the traditional sense is not possible either and this poses an interesting challenge in determining its efficacy.

The concept of economy
Economy implies using no more resources than necessary to achieve to a particular objective.In chess, the objective is simply to checkmate (i.e.capture) the enemy king and the resources for this are the chessmen.Economy in chess however can have a few meanings [25].The most common is economy of material, which means not using more pieces than necessary to achieve checkmate.The other meaning is economy of space, which is using the chessboard to its fullest as opposed to cramming all the pieces into one corner.This can also be described as an aesthetic preference for sparsity on the chessboard.Finally, is economy of motivation which is keeping all lines (i.e.variations) in the solution (e.g. of a chess problem) relevant to the theme. 2It should be noted that this last interpretation of economy is rather limited to compositions since multiple variations and usage of themes are primarily chess problem conventions.Hence for this research, it is not included.
In most references, economy usually assumes its default meaning (i.e.material), but some go a step further to consider not only the number of pieces used, but the amount of material these pieces add up to include their relationship with one another [26 -28].For example, using a queen to do a pawn's job (e.g.blocking the advance of an enemy pawn) would economically be bad because not only is the queen considered more powerful in most positions, but also that it should have had more responsibilities for precisely that reason [29].In some situations, this imbalance might be necessary for checkmate but that does not improve its economy.On the other hand, several pieces working together in a combination can be seen as economical even though their cumulative material value might be higher than the queen's [28].
The final checkmate position encapsulates the concept of economy in chess most effectively since it is the culmination of any (e.g.direct-mate 3 ) chess composition and the objective of every over-the-board4 (OTB) game [30].It is in this position therefore that economy is most evident and best evaluated.Additionally, economy of material was chosen because it is the most widely used definition of economy in chess literature [31].

METHODOLOGY
As explained in Section 2, since chess positions are very seldom rated economically, it is difficult to collect data from (preferably master) players under controlled conditions.Economy would need to be defined in a clear and consistent way so that each player would understand how to rate the positions, but this defeats the purpose of such an experiment in the first place.A much larger database of 'rated' checkmates from an online community could have been used (if one was available) but this suffers from the same problem and that there would hardly be any exclusivity to master players, which influences the quality of the data.Therefore, the best recourse was chess literature [11, 22 -28].In its pages, recurrent and consistent properties can be found by which to judge economy in chess.The only lacking thing was a method by which to calculate it.
On the basis of the available information, certain economic features were determined.The first is the number of pieces used to achieve checkmate.If more pieces are involved, the less economical a position is considered to be.By 'involved', it means participating directly in the checkmate.Removing a piece that is involved would therefore invalidate the mate.The second feature is the value of the chessmen.Chessmen include the king, queen, rook, bishop, knight and pawn.In the course of a real game, a piece's value may fluctuate depending on how effective it is in a particular position [32].
The Shannon chess values of 9 (queen), 5 (rook), 3 (bishop/ knight) and 1 (pawn) are widely used as a standard measure of calculating material on the chessboard.The king is invaluable (losing it means losing the game) but for practical programming purposes it is often valued at 200.These values were suggested by Shannon in his seminal paper on a 'computing routine' to play chess back in 1950 [33].The methods he proposed form the basis of chess computer programs even today.In many circumstances, a large number of chessmen implies more material worth but this is not necessarily so (e.g. three pawns are worth less than one rook).
The third feature stems from the conventions employed in Bohemian problems.Chess problems can typically be divided into three 'schools' namely: Bohemian, Logical and Strategic.Unlike the others, the Bohemian school of problems places a strong emphasis on economy [26,27].The Logical and Strategic schools do not neglect economy but offer more flexibility in favor of themes that might be constricted otherwise [34].Composers of the Bohemian school are particularly interested in what are known as 'model' mates.The characteristics of a model mate typically include the following rules.
(1) No square in the mated king's field (i.e.squares surrounding the king and the one it is on) is guarded more than once or is blocked as well as guarded.
(2) All of the mating side's pieces (with the possible exception of king and pawns) participate in the mate.(3) If an enemy piece is in the king's field but is pinned, it is exempted from rule #1.
Rule #1 on its own is also known as being 'pure' while rule #2 emphasizes the economical aspect.Nevertheless, model mates are those, which are considered both pure and economical [27].In the author's opinion, there is nothing contradictory in any of these rules with regard to economy in general.The same applies to 'ideal mates' (not characteristic to any particular school) where all the pieces, including the enemy, are used in the checkmate [31].
Therefore, the third feature derived is 'piece power' or mobility.This refers to the number of squares a piece controls on the board.More powerful pieces tend to control more squares and the power of a piece at any particular time in the course of a game is often based on the number of squares it controls [35].From an economical standpoint, the third feature would mean using as much of a piece's power as possible [22].If a piece (e.g.queen) is under utilized, the position is considered less economical than if more of its power (i.e.squares it controlled) were used.

The evaluation function
On the basis of the economic features obtained from chess literature, an evaluation function was developed to generate a numerical score for any checkmate position.The higher the score, the more economical a position is considered to be and vice versa.In adapting the economic features from the previous section, more detailed parameters were introduced.The evaluation function is presented here first with explanations to follow.White is assumed to be the winning side for explanatory purposes.
where e is the economic value of a checkmate position, a n the control field of a particular active (useful) piece, f n the maximum control field of that active piece, o the number of overlapping control field squares, f k the standard king's field, s n the maximum control field of a particular passive (superfluous) piece and p the number of friendly pieces on the board (including king).
The control field of a particular piece (a n ) refers to the number of squares in the enemy king's field an 'active piece' control, i.e. one that is essential to the checkmate and cannot be removed without invalidating it.The maximum control field of that particular piece (f n ) is derived from how many squares in the king's field that piece could possibly control.These values were determined by testing each piece on the board and placing it in proximity to the enemy king's field.It was found to be as follows; king (3), queen (6), rook (4), bishop (3), knight (2) and pawn (2). Figure 1 shows two examples of how this was determined for queen and knight.
Nowhere else on the board can each individual piece control more squares in the king's field.Even though a king on the edge of the board has a field of six squares and one in the corner only four, the field for a king in the center of the board (nine squares) was chosen because it is considered more beautiful and better in terms of rule #1 of the model mate [36].The main parameter of the evaluation function therefore is the summation of active piece control fields against their respective maximum fields.The closer a piece gets to its maximum field capability, the less of its power is wasted.
Secondary parameters include overlapping control field squares and the presence of superfluous (passive) pieces on the board.The overlapping control field squares (o) are those in the king's field, which are guarded by more than a single white piece and not occupied by an enemy piece.This is because the enemy piece forms a 'blockade' on that squares against its own king and need not doubly be guarded by the winning side.A passive piece, as opposed to an active one, is a piece that can be removed from the checkmate position without invalidating the mate, regardless of what purpose it might have served in the moves that lead up to the checkmate.A brief discussion about this can be found in Section 5.4.The maximum control field of a passive piece (s n ) then, is the number of squares it could have controlled.
The standard king's domain of nine squares (f k ) was chosen as a consistent denominator for liabilities (overlapping squares and passive pieces) because it was found that a smaller denominator (e.g. six or four, the other possible king fields) tended to exaggerate these liabilities beyond their secondary nature.After the secondary parameters have been subtracted from the primary one, the result is divided by the total number of (friendly) pieces on the board.Even though the king and pawns are permissible exceptions in model mates, for consistency the evaluation function does not exclude them.
The main reason for the permissibility afforded to kings and pawns in the Bohemian school of composition (at the expense of economic purity) is that composers felt rather restricted otherwise.It is difficult, perhaps impossible, to compose novel problems featuring interesting themes if every piece must strive for economy.The evaluation function therefore deals with economy in its purest sense with no regard for other problem conventions and this makes it just as applicable to regular games.
Although the evaluation function typically returns a positive value between 0 and 1 for most checkmate positions, there are situations in which it returns a negative score.One option was to normalize all negative scores to zero, implying that an economic score of 0 would simply mean 'bad economy' but this meant being unable to compare really bad positions with each other so negative values were retained and presented as they are.

Computing economy
The process of evaluating a mating position's economy requires a few steps.They are summarized as follows.
(1) Determine and remove passive pieces from the board.
(2) Sum the ratio of the remaining active piece control fields against their maximum fields.(3) Count the number of overlapping control field squares (except those occupied by enemy pieces).( 4) Sum the number of theoretical maximum control field squares for the passive pieces.( 5) Add ( 3) and ( 4) and divide by the standard king's field.( 6) Subtract ( 5) from ( 2). ( 7) Divide ( 6) by the number of friendly pieces on the board.
Step 1 is accomplished by systematic removal of pieces from the chessboard starting from the most valuable (queen) and continuing in descending order, based on their Shannon value.If duplicate pieces such as two rooks or four pawns exist, the order of removal is from the upper left of the chessboard (coordinate a8) to the lower right (h8). 5  This is similar to the sequence employed in the standard Forsythe-Edwards chessboard position notation. 6Algorithmically, steps 1 -7 can be implemented as shown in Fig. 2.

EXPERIMENTAL RESULTS
A chess computer program was written incorporating the economy evaluation function.On the basis of experience during preliminary tests, manual evaluation is possible but tedious and prone to error.The program does not possess any artificial intelligence to actually play chess but is capable of facilitating a game with complete rules between two human players.This was necessary to support the evaluation function, which depends upon things such as confirming checkmate, identifying control squares and removing passive pieces.
Four experiments were performed to test the evaluation function.The first one compared 100 randomly selected checkmate positions from generic compositions (i.e.not  5 The coordinate system used in chess refers to the vertical 'files' on the board labeled a through h (left to right) and the horizontal 'ranks' labeled 1 through 8 (bottom to top). 6Forsythe-Edwards notation provides an easy way to accurately describe a chess position and is widely used in modern chess computer programs to copy and paste positions between applications.The notation starts from the upper left of the board and moves all the way down to the lower right, using uppercase letters for white pieces, lowercase for black pieces and numbers to represent empty squares.A new rank is indicated using a slash (/).Additional information such as the side to move and castling permissions may be included at the end (e.g.1k6/8/1KPN4/8/8/8/8/8 w).

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A. IQBAL THE COMPUTER JOURNAL, 2007 belonging to any one school of compositions) against the same number of randomly selected OTB game checkmate positions (refer Appendix).For the compositions, where more than one checkmate variation existed, only the main line was chosen and in cases where a real-game did not actually end in mate (i.e. one player resigned), the chess engine Fritz 9 was used to determine the forced line of play to checkmate that would otherwise have taken place [37].Only games between skilled players (ELO rating !2300) were included to minimize imbalances due to amateurish play.The ELO rating is a widely used method for calculating the relative skill of chess players where a minimum rating of 2300 usually qualifies for an FIDE Master title [38].Grandmasters typically have a rating of 2450 and above.Highly rated players are more likely than amateurs to make better use of their pieces and this sets a higher standard for real games in terms of economy.The result of the experiment is shown in Fig. 3.It has been sorted in descending order for clarity.
The compositions scored a mean economic value of 0.361 (SD 0.20) when compared with the tournament games that averaged only 0.025 (SD 0.11) using a two-sample t-test assuming unequal variances; t(150) ¼ 14.6, P , 0.001.This statistically significant difference in the means suggests that the evaluation function is able to correctly discern between the economic value of typical checkmate positions in compositions and regular games.However, this could be attributed mainly to the preponderance of passive pieces found in regular games.
A second experiment was conducted but this time with positions from both groups 'improved' (i.e.removing passive pieces from the board).These would better contrast the finer aspects of economy between the two groups, if any (Fig. 4).
Compositions in this case scored an improved mean of 0.523 (SD 0.14) and regular games a much larger improvement to 0.409 (SD 0.11) using a two-sample t-test assuming unequal variances; t(185) ¼ 6.4,P , 0.001.The difference is still statistically significant even though the regular games had an average increase of 0.38 economic points compared to just 0.16 points for the compositions.All of the regular games economically benefited from the removal process but only 63% of the compositions did.The result of this experiment concurs with the expectation that even without passive pieces, chess problems (having the benefit of human composers) still exhibit better overall economy than regular games.Figure 5 shows a few examples of evaluated positions taken from the first two experiments.
The third experiment tested to see if the minor difference in means (0.048) between the improved OTB positions (0.409) and unimproved compositions (0.361) was statistically significant.A two-sample t-test assuming unequal variances showed that it was; t(150) ¼ 22.1, P , 0.05.This suggests the feasible idea that passive pieces are a severe economic liability even to compositions and that without them, OTB positions are of comparable economic value.
However, any real-world application of the evaluation function would have to take OTB games and compositions as they are (refer Experiment 1).The fourth experiment tested the assessments of the evaluation function against users to see if a positive correlation existed.A survey was created for this purpose and given to chess players 7 who have been actively playing (at the club level, in tournaments, etc.) for at least the last 5 years.This was done to minimize the risk of faulty evaluations by novice and rusty players.Highly rated players with an official rating (e.g.FIDE, USCF) were not available to the author and more importantly less suitable for this experiment since average and unrated players are far more common as potential users of a system that might incorporate the evaluation function.
To keep it from being burdensome, the survey consisted of just 10 pairs of randomly selected positions-taken from the mixed pool of 200 evaluated positions (compositions and OTB games) in Experiment 1-that had a decreasing FIGURE 2. Pseudo-code of the economy evaluation function. 7These included both staff and students of the Tenaga Nasional University and University of Malaya chess clubs in Malaysia.discrepancy in their scores between each pair starting from a difference of 0.5 all the way down to 0.1.Larger discrepancies were scarce and difficult to obtain randomly.This means that there were two sets of paired positions with an approximate difference of 0.5, 0.4, 0.3, 0.2 and 0.1 between each pair.This information was kept from the respondents.Two sets were used to see if different positions with the same economic score would be evaluated by humans with similar consistency.
Economy was defined in the survey as 'White using its pieces as efficiently as possible in the checkmate position' and respondents were asked to select which position in each pair they considered to be more 'economical'.They were also asked to rate both positions between 0 (very poor economy) and 10 (perfect economy).Fractional values such as 6.25 were also accepted.An 11th 'control' position (presented first in the survey) was included to ensure respondents had a reasonable interpretation of the definition provided.This control position contrasted an unequivocally economical rook and king against king checkmate against an uneconomical king, two knights, two bishops and three pawns mate, several pieces of which were redundant.If this pair was not evaluated correctly for whatever reason, chances are none of the other evaluations by that respondent were reliable.
From a total of 53 respondents, five were disqualified because they failed in evaluating the control position.The results of the remaining 48 respondents showed that the majority tended to agree with the computer's assessment of which of the two pairs in each position was the more economical one.Taking into account that human players would probably  have a superficial understanding of economy (unlike in chess literature) and use perhaps just one or two of the economic features presented in Section 3, a majority vote was considered sufficient support.There was a decreasing trend in this majority as the disparity between positions narrowed.The following scatter plot diagram illustrates this (Fig. 6a).
The y-axis shows the percentage of respondents who agreed with the computer's assessment of which of the two positions was more economical.The x-axis shows the increasing economic score difference between each pair of positions.Both sets of 10 positions displayed a similar and good positive correlation (set 1, r ¼ 0.83; set 2, r ¼ 0.86) with the increasing disparity suggesting that, it is easier for humans to tell which one is more economical if the score gap is wider.However, two pairs with the same discrepancy might not necessarily receive the same level of agreement by users with regard to the more economical position in each pair.
For example, sets 1 and 2 at the 0.3 discrepancy show a difference in agreement of .20%.The author suspects this due to the nature of the positions presented.It turns out that the pair of positions at the 0.2 discrepancy in set 1 and the pair at the 0.3 discrepancy in set 2 happened to have significantly more pieces on their boards than the other set.This might have made it more difficult for users to tell which of the two (crowded) positions was more economical, hence the lower level of agreement.
The ratings given by the respondents are shown in Fig. 6b.The average rating (i.e.evaluation score) attributed to each position by the respondents is plotted here along with the corresponding computer rating.For illustrative purposes, the positions have been arranged in descending order based on the average human ratings.To normalize, the computer evaluations were multiplied by 10.Even though the average human evaluations never went below 5 for even the worst positions, there was still a reasonable and statistically significant positive correlation (r ¼ 0.81) between them and the computer's evaluations; t(18) ¼ 5.86, P , 0.05 (Fisher's t-test).
This suggests that there is a consistency between human chess player economy evaluation of checkmate positions and that of the evaluation function proposed.The reason human evaluation of the positions appears more 'forgiving' (i.e.no position scored below the average of 5) than the computer's evaluation is probably because the idea of 'very poor economy' is more difficult to conceptualize than 'good' or 'perfect' economy since poor positions can more easily be made worse than goods ones improved.

DISCUSSION
Even though the experiments show promising results, there are a few issues that still need to be addressed.These include minor economical differences, paradoxical positions, perfect economy, economic evaluation other than in the checkmate position and finally, economy in other games.

Minor economical differences
Minor economical differences between two positions are difficult to confirm experimentally as in Section 4. In most cases, they are probably irrelevant but it is still important for a formalization of economy as presented in this paper to account for them.This is why several checkmate positions were chosen and tested by systematically making minor modifications  that are consistent with chess literature in terms of improving the position economically or otherwise.
In each case, the evaluation function reflected the improvement (or otherwise) one would expect from the modification, as shown in Fig. 7.In Fig. 7a, the original tournament game checkmate can be seen as quite inefficient which is why it scores poorly.In Fig. 7b, the passive rooks are removed which means less pieces are on the board so the score improves.In Fig. 7c, the unnecessary pawns are removed and the economic score increases even further.
Finally in Fig. 7d, the enemy bishop on g7 is removed and the king is made to participate, increasing the score to 0.57407.The bishop was removed so the king has a more obvious role to play by guarding the g7 square.Even if the bishop was not removed and the king became unnecessary to the checkmate, the position would still score better than Fig. 7c because the king simply cannot be removed from the board and in the strictest sense of economy is better contributing something to the checkmate (e.g.doubly guarding the g8 flight square) for its presence than sitting idly on g2.

Paradoxical positions
There are situations in which economic improvement might not be as obvious as they are in Fig. 7. Figure 8a shows a constructed position where White has material on the board worth 11 points (queen þ two pawns) whereas in Fig. 8b, the pawn on b5 is replaced with a bishop on d7, totaling 13 points in material.On the surface it would appear that Fig. 8a should be more economical than Fig. 8b because the material on the board is less while the number of pieces remains unchanged.
However, the evaluation function attributes a marginal advantage to Fig. 8b on account of the ratio of power used by the bishop (2/3 squares controlled in the king's domain) compared to the b5 pawn in Fig. 8a which uses less of its power (1/2 squares) even though the bishop in Fig. 8b does introduce an extra overlapping control square on b5 that was not present in Fig. 8a.This example illustrates that there is actually more to economy than simply having less material on the board despite how paradoxical it may seem even though human players might find it difficult to make this distinction especially when the difference between two such positions is minor.

Perfect economy
The economy evaluation function was designed around the benchmark of 'perfect economy' as a theoretical maximum for any position.Even though it is less common but nevertheless possible for a position to score less than 0 economically, it is not possible to exceed 1, the perfect score.There are some other things to consider as well with regard to how positions are scored.Looking at the four main variables of the evaluation function (i.e.active pieces, passive pieces, overlapping squares and number of friendly pieces on the board) it can be seen that they are not entirely independent of one another.
For example, introducing an extra piece to the position would necessarily increase either the active pieces on the board (and affect its related active to maximum control field ratio) or increase the number of superfluous pieces (and increase its cumulative maximum control field to king's standard field ratio).It may also introduce more overlapping control squares or reduce them (e.g. by blocking another piece).Additionally, different pieces and their locations tend to have unique effects on the economic value of a particular position.Still, the highest economic value is obtained from positions that use all the power of all the pieces with no overlapping control fields.Figure 9 shows the highest economic  value discovered in a Bohemian composition during experimentation and the highest one that the author managed to construct.Figure 9b is an example of 'perfect economy' for White.The squares c8, c7 and c6 are all controlled by the White king whereas the knight checkmates the enemy king and prevents its move to a6.Both pieces are contributing all of their available power to the checkmate and there are no more friendly pieces than necessary for it to work in this position.With the exception of pinned pieces that would keep active an otherwise passive white piece, black pieces do not affect the economic score.

Economy outside checkmate
As mentioned in Section 2.1, economy needs to have an objective before it can be evaluated and the ultimate objective is of course to checkmate the opponent.Smaller objectives such as winning material or securing a good position are difficult to determine computationally because it often goes to the intention of the player and there is no guarantee the supposed objective will result in victory.This affects its tangibility and is why economy in chess literature is almost exclusively with reference to forced checkmates or clearly 'won' positions that invariably lead to checkmate.
A more reasonable concern is therefore the economy of moves just prior to the checkmate, which may be part of a forced sequence.Given a direct mate-in-3 combination for example, some pieces on the board may have served a relevant purpose on the first or second move but appear redundant on the next or final one.If the role of such pieces could be taken into account somehow, it might provide a better estimation of the economy of the checkmate sequence in addition to the final mating position.However, doing so would also be difficult because the role of any piece in a prior move would have to be based on the evaluation of a slightly different position than the final one.
In other words, the economic factors discussed in this paper would have to change since the enemy king is not yet checkmated.The new factors would probably relate more to chess themes or tactics that a piece managed to exemplify and variations of play that were spawned or prevented as a result (refer 'economy of motivation' in Section 2.1).This in turn would require the formalization of said themes and analysis of all variations possible to the extent that would probably not be obvious to most players until they studied the mating sequence carefully and understood it for themselves.
Finally, all the different economic factors involved (i.e. in positions leading up to and including the final checkmate) will need to be reconciled in a meaningful and balanced way.All of this is quite possible since themes can be formalized and multiple variations of play only a few ply deep are computationally feasible.Unfortunately, there would be little to draw on from chess literature for this purpose.Similar research that involves chess themes, for example, while scarce, have had to rely on the opinion of one or two master players to arbitrarily attribute fixed values to them (e.g.Grimshaw ¼ 45, Pickaniny ¼ 25, direct battery ¼ 15) [39][40][41].
Although this sets a useful yet slippery precedent, a scalable formalization of each theme would probably be better.The reason being that even a single theme can and does have a wide range of possible implementations (different pieces, configurations, etc.) that are by no means equally artistic or effective and this should be accounted for in a more objective and systematic way [42].

Economy in other games
The work presented in this paper may be exclusive to chess but it can be extended at least to other reasonably complex zero-sum perfect information games.The important thing is that the game needs to have a concept of economy that is beneficial in some way.In Western chess, economy is both practical and beautiful.Hence, other variants of chess (by certain estimates over 1000 of them) could also apply a similar function since they differ only in certain aspects [43].
If the rules are similar and the variant differs in, say, only the types of pieces used (e.g.fairy chess), then minor modifications to the maximum control fields (see Section 3.1) would probably suffice.If there was a difference in board type (e.g.hexagonal chess), then an adjustment to the maximum control fields and standard king's field should work.A difference in rules however (e.g.checkers chess) would be more difficult to accommodate.This brings the discussion to other games such as go that are not variants of chess.
Go is promising not only because it is more challenging to program than chess and has a much larger game tree but also more importantly because there is a composition or problem domain like in chess, where the notion of economy (whatever that may be for go) can be refined.Since there are only black and white stones in go as opposed to six unique piece types in (Western) chess, the concept of beauty-and there definitely is one-is actually quite different [44].Following that, 'economy' would also be different since the objective in go is to control more territory on the board and capture enemy stones by surrounding them, unlike the objective in chess.Hence, the evaluation function presented in this article cannot really be adapted to go and other significantly varying games.It is not difficult to imagine a similar economy evaluation function with entirely unique parameters being developed for go once there is sufficient literature on the subject to base it on.

CONCLUSIONS
In this article, an economic evaluation function for chess was proposed based on economic features explained in chess literature.Properties of the chessboard and chess pieces were used as a metric for the function.Four experiments conducted suggest that the function is able to correctly discern between good and poor economy in checkmate positions for both compositions and in real-games.It also had a good positive correlation with human perception of economy.Further testing with minor modifications on individual positions showed that the function mirrors these economic changes consistently as well.However, due to the many different aspects of economy taken into account, this is sometimes less obvious in certain positions.
The evaluation function can easily be implemented in any chess program to automatically identify checkmate positions that are economical.Such positions usually result from sound play and are useful to human players not only for instruction but also aesthetic appreciation.Chess databases that contain millions of professional games are a good resource for this purpose.Even though most of these games end with one player resigning rather than actually being checkmated, game engines are often able to trace the line of forced play that would otherwise have ensued and still return an economic search query.
The evaluation function could also serve as a component in larger aesthetic models of chess.New research suggests that computers are able to recognize aesthetics in the game of chess by evaluating numerous aesthetic principles that include heuristic violations, themes and economy [42,45].Subsequently, this could contribute to automatic problem composition by computers, which currently employ only certain chess problem conventions and do not address economy or aesthetics specifically [39,46,47].There has also been research into using beauty heuristics as a more effective approach to computers playing chess.Although the results are promising in terms of solving chess problems, it remains speculative whether it could actually drive an actual game [48].The concept and evaluation of economy in that research however had only been incorporated rudimentarily and it is possible a better implementation such as the one presented in this paper would help.
Finally, the proposed evaluation function could be of use to chess problem tournament judges [49].There have been attempts to replace judges altogether using mechanized evaluation methods but composers have been reluctant to accept these methods mainly because they are perceived as not holistic of what constitutes a chess problem.Despite the often-arbitrary nature by which judges evaluate compositions (which composers also complain about), a human judge is still preferred [36,41,50].Therefore, the proposed function is certainly not intended to replace judges but it could be of assistance to them in terms of evaluating just the one aspect of economy, which is seldom neglected in compositions.