In competitive interactions between two individuals there is always a winner and a loser. If an individual A consistently defeats player B, then it is said that A dominates B. Such a relationship can be captured in a network in which the nodes represent players and arcs show which player is dominated. Animal behaviourists have frequently employed linear hierarchical ranking techniques to determine the dominant individuals within a community.
There are many procedures, of varying complexity, for ranking the members of a social group into a dominance hierarchy (see de Vries 1998 for a review of these techniques). In general dominance hierarchy, techniques can be divided into 2 categories. The first class of methods attempts to determine the dominance ranking by maximising or minimising some numerical criteria. The second class aims to provide a measure of overall individual success from which the rank can be directly derived. One relatively simple ranking method belonging to the second class is David’s score (David 1987, 1988).
Individual ranks calculated with David’s Score are not
disproportionately weighted by minor deviations from the main dominance
direction within dyads, because win/loss asymmetries are taken into
account by the use of dyadic dominance proportions in the calculations.
The proportion of wins/losses by individual 
in his interactions with another individual 
(
) is the number of times that 
(αij ) divided by the total number of interactions between 
and 
(
), i.e. 
. The proportion of losses by 
in interactions with
, is
. If 
then 
and 
(David 1988; de Vries 1998). The David’s Score
for each member, 
of a group is calculated with the formula:
where 
represents the sum of i’s 
values, 
represents the summed 
values weighted by the appropriate 
values, (see the worked example in the following
section) of those individuals with which i interacted, 
represents the sum of i’s 
values and 
represents the summed 
values (weighted by the appropriate 
values) of those individuals with which i interacted (David 1988: p. 108; de Vries
1998).
Table 5.1 shows a worked example with calculated w, w2, l, and
l2 values. Specifically for individual A, 
represents the sum of A’s 
values (i.e. 
), and 
represents the summed 
values (weighted by the appropriate 
values) of those individuals with which A
interacted (i.e. 
. A’s 
and 
values are calculated in a similar manner
(David 1988; de Vries 1998).
Table 5.1. Example of ranking between players according to David’s score
|
Player Losses |
|||||||||
|
A |
B |
C |
D |
E |
|
|
|
||
|
Player Victories |
A |
— |
4.5(0.75) |
4(0.8) |
7(1) |
2(0.5) |
3.05 |
2.90 |
3.20 |
|
B |
1.5(0.25) |
— |
4(0.8) |
0(0) |
5(1) |
2.05 |
2.38 |
2.45 |
|
|
C |
1(0.2) |
1(0.2) |
— |
0(0) |
4(1) |
1.40 |
1.52 |
-0.20 |
|
|
D |
0(0) |
0(0.0) |
0(0) |
— |
0(0) |
0.00 |
0.00 |
-1.95 |
|
|
E |
2(0.5) |
0(0.0) |
0(0) |
0(0) |
— |
0.50 |
1.52 |
-3.5 |
|
|
|
0.95 |
0.95 |
1.60 |
1.00 |
2.50 |
||||
|
|
1.80 |
1.03 |
1.52 |
0.95 |
3.02 |
||||
Bobby Fischer is probably the most famous chess player of all time and, in many peoples' view, the strongest. In the 1970s, Fischer achieved remarkable wins against top ranked grandmasters. His celebrated 1972 World Championship Match with Boris Spassky in Reykjavik made headline news all around the world. One of Fischer's favoured opening lines is the Ruy Lopez. The Ruy Lopez has been a potent weapon for Bobby throughout his career. Strategic play across the board suited Fischers talents. In his prime (and later in his career) Fischer was so proficient (dominant) in the main lines of the Ruy Lopez that many of his opponents chose irregular setups when attempting to defend their positions.
The Ruy Lopez was named after the Spanish clergyman, Ruy Lopez, of Safra, Estramadura. In the mid-sixteenth century, he published the first systematic analysis of the opening. The speed of development, flexibility and attacking nature, has seen the Ruy Lopez remain popular since its conception to the modern chess era. Figure 5.7 shows the board setup of the Ruy Lopez.
All the games played by Fischer were converted into a network. Each player was represented as node or vertex within the system, with each game between 2 players shown as an arc. The arcs are drawn from loser to the winner, and given a weight of 1 for each victory of a given player. In the event of a draw/tie/stalemate, an arc was drawn in both directions and given a weight of ½. In many ways, the data presented here is limited, as the dataset does not contain all top level games played within the Ruy Lopez opening system (some notable players are missing from the database), nor does it capture all the Ruy Lopez games played between players. Also, some of the games are incomplete or the result of the game is unknown. While these constraints will limit the accuracy of the result, the dataset is complete enough to detect trends and regularities, and will not greatly influence the general findings of this chapter.
In this experiment, I examine the relationships between Fischer and his nearest neighbours. The network contains a total of 65 players. Figure 5.8 shows the network relationships between Fischer and his immediate opponents, the size of the arc represents the level of dominance of that player. Application of David’s score to this network reveals that Bobby Fischer is the most successful player within his local neighbourhood. Table 5.2 lists the top 10 players in the neighbourhood.
In this section, I have been able to show that, from the network formed by the competitive interactions of individuals, key or dominant individuals can be identified. The proposed approach can also be used to determine a ‘pecking order’ between individuals within a group, in which individuals are ranked from most to least dominant. While the context of illustration is the world of chess, the framework can be applied to any network in which dominance between elements can be established. One should note, however, that hidden patterns, such as incorrect weightings and missing links or nodes, can distort the final result.