Zero Sum Game
(noun)
The idea that if group A acquires any given resource, group B will be unable to acquire it.
Examples of Zero Sum Game in the following topics:
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Conflict
- Conflict theory relies upon the notion of a zero sum game, meaning that if group A acquires any given resource, group B will be unable to acquire it.
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Bi-partite data structures
- The cross-product method takes each entry of the row for actor A, and multiplies it times the same entry for actor B, and then sums the result.
- The sum of the cross-products is a valued count of the preponderance of positive or negative ties.
- For binary data, the result is the same as the cross-product method (if both, or either actor is zero, the minimum is zero; only if both are one is the minimum one).
- The two actor-by-event blocks of the matrix are identical to the original matrix; the two new blocks (actors by actors and events by events) are usually coded as zeros.
- The value to fill within-mode ties usually zero, so that actors are connected only by co-presence at events, and events are connected only by having actors in common.
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Optimization by Tabu search
- Tabu search does this by searching for sets of actors who, if placed into a blocks, produce the smallest sum of within-block variances in the tie profiles.
- So, the partitioning that minimizes the sum of within block variances is minimizing the overall variance in tie profiles.
- The overall correlation between the actual scores in the blocked matrix, and a "perfect" matrix composed of only ones and zeros is reasonably good (.544).
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Valued relations
- Pearson correlations range from -1.00 (meaning that the two actors have exactly the opposite ties to each other actor), through zero (meaning that knowing one actor's tie to a third party doesn't help us at all in guessing what the other actor's tie to the third party might be), to +1.00 (meaning that the two actors always have exactly the same tie to other actors - perfect structural equivalence).
- The Euclidean distance between two vectors is equal to the square root of the sum of the squared differences between them.
- This is then repeated across all the other actors (D, E, F, etc.), and summed.
- The square root of the sum is then taken.
- This distance is simply the sum of the absolute difference between the actor's ties to each alter, summed across the alters.
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Univariate descriptive statistics
- For the information sharing relation, we see that we have 90 observations which range from a minimum score of zero to a maximum of one.
- The sum of the ties is 49, and the average value of the ties is 49/90 = .544.
- Since the relation has been coded as a "dummy" variable (zero for no relation, one for a relation) the mean is also the proportion of possible ties that are present (or the density), or the probability that any given tie between two random actors is present (54.4% chance).
- The sums of squared deviations from the mean, variance, and standard deviation are computed -- but are more meaningful for valued than binary data.
- The Euclidean norm (which is the square root of the sum of squared values) is also provided.
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Transforming data values
- For example, if I wanted to recode all values 1, 2, and 3 to be zero; and any values of 4, 5, and 6 to be one, I would create to rules.
- >Sum characterizes the strength of the symmetric tie between A and B as the sum of AB and BA.
- So, relationships that are completely reciprocal end up with a value of zero; those what are completely asymmetric end up with a value equal to the stronger relation.
- If reciprocity is necessary for us to regard a relationship as being "strong" then either "sum" or "product" might be a logical approach to symmetrizing.
- Our examples use the "marginal" total (row or column) and set the sum of the entries in each row (or column) to equal 1.0.
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The "adjacency" matrix
- That is, if a tie is present, a one is entered in a cell; if there is no tie, a zero is entered.
- Binary choice data are usually represented with zeros and ones, indicating the presence or absence of each logically possible relationship between pairs of actors.
- Sometimes the value of the main diagonal is meaningless, and it is ignored (and left blank or filled with zeros or ones).
- For example, if I summed the elements of the column vectors in this example, I would be measuring how "popular" each node was (in terms of how often they were the target of a directed friendship tie).
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Doing mathematical operations on matrices
- Representing the ties among actors as matrices can help us to see patterns by performing simple manipulations like summing row vectors or partitioning the matrix into blocks.
- To multiply two matrices, begin in the upper left hand corner of the first matrix, and multiply every cell in the first row of the first matrix by the values in each cell of the first column of the second matrix, and sum the results.
- To perform a Boolean matrix multiplication, proceed in the same fashion, but enter a zero in the cell if the multiplication product is zero, and one if it is not zero.
- A one represents the presence of a path, a zero represents the lack of a path.
- We will treat "self-ties" as zeros, which, effectively, ignores them.
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Making new kinds of graphs from existing graphs
- Those who made no contribution are coded zero.
- The Exact Matches choice will produce a "1" when two actors have exactly the same score on the attribute, and zero otherwise.
- The Sum choice yields a score for each pair that is equal to the sum of their attribute scores.
- The Minimums method examines the entries of A and B for campaign 1, and selects the lowest score (zero).
- It then does this for the other campaigns (resulting in 0, 1, 0, 1) and sums.
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Factions
- This count (27 in this case) is the sum of the number of zeros within factions (where all the ties are supposed to be present in the ideal type) plus the number of ones in the non-diagonal blocks (ties between members of different factions, which are supposed to be absent in the ideal type).