To demonstrate the validity and application of the previously presented method, not only for living ecosystems, or like the example shown above with innovation ecosystems, this section shows its application to the "ecosystem" formed by the pieces of a chessboard. It is presumed that the application to the case of chess (Hypothesis 4), a limited and controlled network of actors (pieces), will allow an exhaustive analysis of a complete life cycle, from the creation of the network or ecosystem (in this case, coincident with the initial moment of positioning of all the pieces on the board), until the final moment of non-survival of the game ecosystem, where one of the players loses or both tie and the chess game ends.
In chess there are two groups of actors who compete in a finite game for their survival, where the "ecosystem of the game" will be affected by the victory of one or the other player (the game of the ecosystem is an infinite game for its survival), so its monitoring and analysis through the measures of equilibrium and survival will allow us to show revealing elements of the behavior of the actors, groups of actors (pieces of each player) and the ecosystem itself (the game), as shown next.
4.2. Calculation of equilibrium and survival measures
To carry out a test that allows the initial calculations to be made and its possible behaviors, difficulties and results to be analyzed, it has been decided to use a short game with a short life cycle. To do this, a well-known move in chess, called "scholar's mate" will be used. Thanks to this move, the total moves of this first pilot test are reduced to seven moves of white and black pieces (Fig. 2), which in PGN notation translates into the following: 1. e4 c5 2. Bc4 a5 3. Qf3 h6 4. Qxf7# 1 − 0
Below is the summary of total calculations obtained for the pilot game of "scholar's mate".
Table 4. Results of the pilot test of the game with scholar's mate (source: own elaboration).
All values in Table 4 have been calculated using the equilibrium measurements described in point 3.2. In the header of the table in green, the labels of the columns of the seven moves made in the game are shown, from M1 to M7 (which is when the game ends). In the lower part of the table, also in green, the rows where the balance values for each of the movements are stored are shown, corresponding to the two groups of pieces, white (White player equilibrium) and black (Black player equilibrium), as well as the equilibrium values of the starting ecosystem (Ecosystem equilibrium).
To define the different survival thresholds of the game ecosystem, it has been considered that at the beginning of the game (Starting Point) all the pieces are in equilibrium, as well as the two groups of pieces (white and black) and the ecosystem beginning. Based on the above, the following survival thresholds are obtained:
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Survival threshold of a piece > = 0, since in the initial starting position shown in Table 4 it is the minimum value identified (in the pieces that correspond to the rooks, such as White_Rook_a, the initial equilibrium value is 0).
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Group survival threshold (White player equilibrium / Black player equilibrium) > = 0.042
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Starting ecosystem survival threshold > = 0.020
As an example of calculating the equilibrium value of a piece, in movement M1, the value 0.20 corresponding to the piece “White_Queen” is calculated as follows:
White_Queen (M1) = (defense_in – attack_in) / Relationships = (1–0) / 5 = 0.20
Where:
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defense_in = number of white pieces defending the White_Queen piece
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attack_in = number of black pieces attacking the White_Queen piece
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Relationships = number of total relationships of the piece White_Queen:
Regarding the calculation of the equilibrium value of the groups of white and black pieces, for the example of the move M1 we would have:
White player equilibrium (M1) = [Sum of values of all white pieces] / [Number of white pieces * (Number of white pieces − 1)] = 9.73 / (16*15) = 0.041
Finally, to calculate the equilibrium value of the ecosystem, for the example of movement M1 we would have:
Ecosystem equilibrium (M1) = [Sum of values of all actors] / [Number of actors * (Number of actors − 1)] = 19.73 / (32*31) = 0.020
In the following point, the results of this pilot game of scholar's mate are analyzed in detail.
4.3. Analysis of pilot test results
Every ecosystem has a moment of equilibrium. In the case of the chess ecosystem, the equilibrium position is found at the beginning of the game, when all the pieces on the board are in place for the game to begin. Looking at the equilibrium values in Table 4 (pieces, set of player pieces and game ecosystem), we can confirm that the initial value of each of them is greater than zero in most cases (only the rooks start the game with value equal to 0). In any case, as can also be seen in Table 4, each piece has its own equilibrium point. This occurs in a similar way in all ecosystems, where each type of actor and group will have its own equilibrium point, which will vary depending on its interests, objectives and the idiosyncrasy of the ecosystem to which it belongs.
Before going into a more detailed analysis of the values obtained during the development of the game of the pilot example scholar's mate, there is an interesting fact that is worth highlighting. Although in the relationships described in Table 3, the pieces that carry out a defense against another piece are assigned a value of 0 points (in principle, they do not obtain a gain or loss when carrying out this action), in reality, this type of relationship has a negative effect on these parts. Let's analyze it in more detail.
In the initial game situation, the defense that, for example, the piece “White Horse_b” carries out against the piece “White_Pawn_d”, causes its initial equilibrium value to be 0.5 points, instead of 1 point, which is the value that could be expected that this piece should have, since it is defended by the piece “White_Rook_a” (from this positive relation it obtains 1 point) and in turn “White Horse_b” makes the defense on the piece “White_Pawn_d” (from this relation gets 0 points). The reason for obtaining this equilibrium value of 0.5 points is because the number of relationships of “White_Horse_b” is 2 at the initial starting point, and the equilibrium calculation, as shown above, is based on the average of the number of relationships of each actor, which finally means that the resulting equilibrium value is 0.5 at the initial moment.
Therefore, even with a sum of value 0 for each defense action carried out, each piece sees its equilibrium value reduced in relation to its final calculation of relationships. This fact, although it may seem strange, makes sense since, although the piece that carries out a defense does not obtain a direct damage for the fact of the defense, it does have to make a certain "effort" to be able to carry out this action. In addition, if this piece is defending a piece that is being threatened by pieces of the other color, the piece carrying out the defense will not be able to move freely around the board to carry out other actions, since surely by stopping defend that piece, this will be eaten by others; which again supposes a detriment for the piece that carries out the defense, since it prevents it from moving freely. In short, it seems that nothing is free.
This fact does not only occur in chess, but can also be observed in other ecosystems. In the business ecosystem, in the example of a public entity that helps startups with non-refundable financing, although the entity does not obtain a direct loss from this activity, it does lose equilibrium by having to dedicate its resources to those startups. In other words, the public entity is losing budget that it will eventually have to recover, in one way or another, in order to stay in operation (this money that it is allocating, as well as the personnel resources involved in the different actions, is reducing its own resources, necessary for their day to day: for their survival).
In the case of nature, something similar happens, for example, when a mother deer tries to protect her young from predators, or she has to get food and feed the young. Although the mother deer performs a good for her offspring and does not receive direct harm for it, she does lose energy in the process, in the actions necessary to be able to carry out the care and survival of the offspring. Everything has a price, as noted above, that is, everything takes its toll on the actor who performs the action and not always a reward or benefit is obtained from this action, even if it is presumably a noble and positive action.
Delving a little deeper into the analysis of the results, several colors are identified in Table 4. The “red color” defines an equilibrium value less than or equal to a predefined survival threshold, which in this case is 0 points. Different facts that can lead to this situation are identified below:
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Pieces that start from an initial equilibrium position below the predefined survival threshold. As noted above, at the beginning of the game (Starting point), the rooks are at an equilibrium threshold less than or equal to 0. Looking at these pieces on the board, it can be seen that the rooks do not receive protection from no other piece, while rooks do act in defense of others, such as pawns or knights. This effectively demonstrates the reason for a equilibrium value below the threshold, since the rooks are in an unfavorable position (they are offering “their services” to other pieces, but they are not getting anything in return). In this example, in addition, it can also be seen that an equilibrium value below the threshold for a piece negatively affects the group of pieces of the same color; if, for example, a piece of the opposite color attacked one of the rooks, there would be no other piece to defend it and, therefore, the loss of that piece could not be compensated, which would result in a decrease in the equilibrium of the group of pieces of that color.
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Pieces that, starting from an equilibrium position and due to their own action, move to an unfavorable position below the predefined threshold. In general, these are movements that entail a certain risk or errors, which represent that the piece has lost its equilibrium value and is in a situation of exposure or danger. For example, in the case of the white pieces, in the movements M1 and M3 respectively, it can be seen how “White_pawn_e” and “White_bishop_f” lose equilibrium below the predefined threshold, precisely by leaving a previous position of greater security and exposing themselves in the board.
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Pieces whose equilibrium value is reduced below the predefined threshold due to other pieces on the board. In general, this fact is produced by attacking moves of pieces of the opposite color, or by moves of pieces of the same color that leave this piece in an unfavorable equilibrium situation. For example, the movement produced by white in the move M7 causes a decrease in the equilibrium value of the pieces of the other color "Black_bishop_f", "Black_horse_g" and "Black_King", with special relevance in the case of the “Black_King” piece when an endgame or checkmate situation occurs. For its part, in move M5, the action of the white piece "White_Queen" causes several of the pieces of its color group, "White_Bishop_c", "White_King" and "White_Pawn_c", to move to an equilibrium value below of the predefined threshold by leaving them unprotected.
For its part, the "green color" shows the improvement produced in the equilibrium value of a piece, with respect to the value that piece had before that movement. Here the following situations are mainly identified:
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Pieces that see their equilibrium value increased due to their own action. In the example of the movement of the white pieces M5, the action of the piece “White_Queen” produces a positive change in its own equilibrium value. This increase occurs because, in the new position, the “White_Queen” piece protects fewer pieces of its color (it goes from a situation in which it defended 4 pieces to only 3 pieces), so that the piece requires less effort.
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Pieces whose equilibrium value has increased due to other pieces on the board. In general, this situation occurs due to the effect of other pieces of the same color, or due to the effect of the movement made by pieces of the other color. In the example of the movement of the white pieces in M1, the action of the white piece “White_Pawn_e” produces a positive change in the equilibrium value of the pieces of the same color "White_Queen", "White_King", "White_Bishop_f" and " White_Horse_g". This increased equilibrium is mainly due to the fact that those pieces no longer have to invest efforts in defending the “White_Pawn_e” piece.
As for the "yellow color", with this color a loss of equilibrium value is shown, with respect to the value that that piece had before that movement. Here the following situations are mainly identified:
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Pieces whose equilibrium value is reduced due to their own action. In the example of the movement of the white pieces in M7, the action of the white piece “White_Queen” produces a negative change in its equilibrium value. This is due to the fact that in the new position the “White_Queen” piece receives more attacks, which reduces its equilibrium value.
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Pieces whose equilibrium value is reduced due to other pieces on the board. In general, this situation occurs due to the effect of other pieces of the same color, or due to the effect of the movement made by pieces of the other color. In the example of the movement of the black pieces in M4, the action of the black piece “Black_Pawn_a” produces a negative change in the equilibrium value of the black piece “Black_Queen”. This is because in the new position of the “Black_Pawn_a” piece, the “Black_Queen” piece carries out a new defense action on that piece, which therefore reduces its equilibrium value by requiring a new effort to make that defense.
Finally, the "black color" defines a piece that has been captured, or a final situation of checkmate to the king (an event that marks the end of the game):
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Piece that has been eaten. In general, this situation is produced by the effect of pieces of the opposite color. In the example of the movement of white pieces in M7, the action of the white piece “White_Queen” causes the black piece “Black_Pawn_f” to be removed from the game board or ecosystem.
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Checkmate the king. In general, this situation is produced by the effect of pieces of the opposite color. In the example of the movement of white pieces in M7, the action of the white piece “White_Queen” causes the black piece “Black_King” to be threatened, so that it has no further moves that allow it to escape from that situation. This fact produces the end of the game.
This monitoring of relationships, carried out through equilibrium measures, therefore helps ecosystem decision-making, allowing potential problems and situations to be anticipated. As an example, in the move M5, the value of the piece “Black_Pawn_e” drops to a value of -0.33 points (well below its survival threshold), which indicates a clearly unfavorable and threatening position. In this specific example, this situation finally translates not only into the loss of the “Black_Pawn_e” piece in move M7, but also the corresponding loss of the game by the player with the black pieces, when a checkmate situation. Therefore, a first conclusion that can be drawn is that the monitoring and management of these early signals of equilibrium could help prevent future problems in a given ecosystem, as has been observed with the example of the case of chess.
The results obtained from the analysis of this first pilot are very revealing. However, to verify its confirmability, a set of 15 games by grandmasters in chess history have been subsequently analyzed, games downloaded in PGN notation from the Chess.com website. To facilitate the analysis of the games, the software application shown in Fig. 3 has been used.
In the application directory of Fig. 3 it can found preloaded grandmaster games for analysis, or there is also the possibility of loading any other game in PGN notation, thus being able to verify the operation of the equilibrium measures and survival thresholds. The games analyzed in this research is shown below, in Table 5, of grandmasters in chess history such as Magnus Carlsen, Judit Polgar, José Raul Capablanca, Bobby Fischer or Garry Kasparov, among others.
Table 5
Chess games analyzed in the research
(source: own elaboration from Chess.com)
Sample of games analyzed
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Links to PGN games
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Abhimanyu Mishra vs Baadur Jobava 2021.07.12
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https://www.chess.com/games/view/16074762
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Anatoly Karpov vs Garry Kasparov 2021.08.24
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https://www.chess.com/games/view/16103752
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Anish Giri vs Hikaru Nakamura 2021.12.06
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https://www.chess.com/games/view/16162819
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Bobby Fischer vs Boris V Spassky 1992.__.__
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https://www.chess.com/games/view/584192
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Daniil Dubov vs Magnus Carlsen 2021.05.24
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https://www.chess.com/games/view/16045828
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Garry Kasparov vs Jan-Krzysztof Duda 2021.07.11
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https://www.chess.com/games/view/16074360
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Hikaru Nakamura vs Magnus Carlsen 2021.05.02
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https://www.chess.com/games/view/16032773
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Hikaru Nakamura vs Mikhail Demidov 2021.06.01
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https://www.chess.com/games/view/16053762
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Jose Raul Capablanca vs Karff May N 1941.__.__
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https://www.chess.com/games/view/35893
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Jose Raul Capablanca vs Lluis R 1941.__.__
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https://www.chess.com/games/view/35954
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Jose Raul Capablanca vs Rodriguez Carnero A 1941.__.__
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https://www.chess.com/games/view/35955
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Maxime Vachier-Lagrave vs Garry Kasparov 2021.07.10
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https://www.chess.com/games/view/16074168
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Peter Szekely vs Mikhail Tal 1992.__.__
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https://www.chess.com/games/view/4082955
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Tanguy Ringoir vs Judit Polgar 2014.06.19
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https://www.chess.com/games/view/13571387
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Vladimir Kramnik vs Fabiano Caruana 2021.06.22
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https://www.chess.com/games/view/16064778
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From the analysis of the items shown in Table 5, it has been possible to verify the same patterns, behaviors and results obtained in the pilot example. However, it should be noted that it has also been possible to identify that sometimes one piece is eaten by another, that is, it ceases to exist in the context of the game, even when that piece that is eaten had a good level of equilibrium (i.e. had positive relationships that predicted their survival). This fact is interesting to analyze as it is an exception, but also because it can occur in other ecosystems. For example, in natural ecosystems, when a leopard threatens a wildebeest calf surrounded by its herd, it eventually hunts and eats the calf, even in opposition to the mother and the rest of the wildebeest herd (the fact that the calf was surrounded by other members of its species, that is to say that it was well related and protected, does not finally avoid the fatal outcome). In business ecosystems, we could find similar cases, for example, in the case of a "hostile takeover bid”.
Next, in relation to this detected fact, the different cases identified from the analysis of the fifteen games of grandmasters are described:
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A piece of lesser value or degree of importance eats another piece of greater value or degree of importance that had a good level of equilibrium. In most of the games analyzed, cases have been identified in which a piece of lesser value or importance (e.g. a pawn), which threatened another piece of greater value or importance (e.g. a knight), finally executed its threat, that is, it ate the piece of higher value, even when the piece of higher value was protected by other pieces, that is, its equilibrium value was high or very high (so its survival seemed to be assured). This specific fact is due to the fact that there is a pre-established classification of the importance of the pieces, which allows the pieces to be distinguished and compared according to their value or importance on the board. Thus, at a general level, pawns are the pieces with the least value, knights and bishops would have a higher degree of importance or value than pawns, rooks a slightly higher value or degree of importance than knights and bishops, queens a higher value or degree of importance than the rooks and, finally, the kings a higher value or degree of importance than the rest of the pieces on the board.
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A piece eats another piece of equal or lesser value or degree of importance that had a good level of equilibrium. Only in some of the analyzed games, it has been identified that a piece of equal or greater value than another that it threatened, finally ate that piece, even when the eaten piece enjoyed a good level of equilibrium that seemed to ensure its survival on the board. This specific fact, for the case in which the piece that it eats is of greater value than the piece that is eaten, is called a "sacrifice" (since the piece that eats a piece that was protected, foreseeably will be eaten in turn by one of the pieces that protected that piece and, therefore, it will also cease to exist in the context of the game). This type of action in chess usually occurs due to a player's mistake, or when the act itself can be of great benefit to the strategy of that group of pieces to win the game, so it is worth the sacrifice of the piece.
A solution that makes it possible to identify and manage this type of situation consists of adding the variable "relationship risk" for each existing relationship, thus modifying the equilibrium formula for actors, as shown below:
[Equilibrium of the actor] = ∑ (relationship value * relationship risk) / [Number of relationships of the actor within the ecosystem]
In its simplest version, the “relationship risk” variable could vary between the values 1 (minimum risk), 2 (medium risk) and 3 (maximum risk). Positive-type relations would have an assigned value of 1 (since in principle they do not pose a risk, although as we have seen previously, there are occasions in which a positive relation of defense or help can cause a negative effect on the part or actor that performs it, which may also put their survival at risk; in these cases, if contextual information is available that allows evaluating this potential risk of the aid action to the detriment of the piece that performs it, the "relationship risk" variable would receive then the value of 2 or 3 points, depending on the inherent risk involved in that action). On the other hand, negative-type relationships would vary between values 2 and 3, depending on the risk of the relationship (more or less important impact for the actor).
In any case, the use of the “relationship risk” variable entails the need to have greater knowledge of the analyzed ecosystem and its actors. In the case of chess, in order to identify the first case detected from the analysis of the fifteen games of grandmasters, that is, when "a piece of lesser value or degree of importance eats another of greater value or degree of importance", it would be necessary to previously know the comparison table of the importance or value of all the pieces (the associated risk would be assigned here based on the importance of each piece and its comparison with the rest); while in the second case, when "a piece eats another piece of equal or lesser value or degree of importance", it would be necessary to know the global state of the game, that is, of all the pieces/actors as a whole, to assess the appropriateness or otherwise of a sacrificial action.
As can be anticipated, based on the foregoing, on many occasions, depending on the ecosystem being analyzed, the information necessary to assess the risk associated with a certain relationship could be difficult to obtain, difficult to specify or even not be accessible. Therefore, depending on the case study and the level of information available on the ecosystem and actors, it will be convenient to use one version or another of the actor equilibrium formula.