JP7680972B2 - Fairness judgment device, fairness judgment method, and fairness judgment program - Google Patents

Description

Article 30, Paragraph 2 of the Patent Act applies. (1) Published in the Proceedings of the 2022 Symposium on Cryptography and Information Security, which began distribution online on January 12, 2022. (2) Presented online at the 2022 Symposium on Cryptography and Information Security on January 19, 2022.

The present invention relates to an apparatus, method, and program for determining the fairness of matching.

The optimization problem of matching in a bipartite graph has been studied for a long time as a mathematical model for the problem of allocating workers to jobs. Matching that optimizes overall satisfaction is reduced to a minimum flow problem in a weighted minimal complete bipartite graph. The stable marriage problem in Non-Patent Document 1 is a problem that adopts the concept of stability in this matching, and an algorithm has been proposed to find a stable matching with a computational complexity of O(n ² ).

Various extension problems have been considered for the stable marriage problem, and one area of extension problems is the discussion of fairness. The stable matching found by the algorithm in Non-Patent Document 1 is the optimal matching for either the man or the woman, and the optimization of the overall cost was not taken into consideration.
Therefore, optimization problems that extend the stable marriage problem, such as minimum-cost stable matching, minimum-regret stable matching (Non-Patent Document 2), and gender-equality stable matching (Non-Patent Document 3), have been studied. These problems aim for fairness in the sense of global optimization.

However, such a matching result that satisfies fairness through global optimization may, for example, cause a relative disadvantage to participants with high abilities or high evaluations. Non-Patent Document 4 defines meritocratic fairness in problems such as male-female matching, which is not necessarily stable, and also proposes an index that can quantitatively evaluate the degree of meritocratic fairness.

D. Gale and L. S. Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, Vol. 69, pp. 9-15, 1962. R. W. Irving, P. Leather, and D. Gusfield. An efficient algorithm for the "optimal" stable marriage. Journal of the ACM, Vol. 34, No. 3, pp. 532-543, 1987. A. Kato. Complexity of the sex-equal stable marriage problem. Japan Journal of Industrial and Applied Mathematics, Vol. 10, pp. 1-19, 1993. Toru Nakamura, Shuya Nitta, Takamasa Isohara. Individualistic fairness in matching and its infringement detection. Computer Security Symposium 2021 (CSS2021), Vol. 1D1-1, 2021.

Non-patent document 4 proposed a definition of meritocratic fairness, but did not mention an efficient method for determining whether a match satisfies meritocratic fairness. As a result, when searching for a fair match, the processing load required to determine fairness each time a matching candidate is obtained was an issue.

The present invention aims to provide a fairness determination device, a fairness determination method, and a fairness determination program that can efficiently determine meritocratic fairness in matching.

The fairness determination device according to the present invention includes a preference ranking acquisition unit that acquires preference rankings of pairs in each element of two mutually prime sets from the preference ranking of the other set in each element of each of the two sets and information on pairs matched between the two sets, a popularity calculation unit that combines the preference rankings for each element of the two sets to calculate the popularity of each element, a popularity ranking calculation unit that sorts each element of each of the two sets in ascending order of the popularity and assigns the same popularity ranking to elements with the same popularity ranking, and a popularity ranking calculation unit that calculates the popularity ranking of the elements of each of the two sets in ascending order of the popularity and assigns the same popularity ranking to elements with the same popularity ranking to elements with the same popularity ranking. The system includes a judgment table generating unit that generates a judgment table that stores the minimum and maximum values of the preference rankings, a table-specific judgment unit that judges that a condition is met if there is no i in the judgment table where the maximum value of the preference rankings up to the i-th popularity ranking is greater than the minimum value + l of the preference ranking for the i+k+1th popularity ranking, and that the condition is not met if there is such a value, and a fairness judgment unit that judges that meritocratic fairness is met if it is judged that the condition is met for both of the two sets, and that it is not meritocratic fairness if it is judged that the condition is not met for either of them.

The popularity ranking calculation unit may sort each element in ascending order of the popularity and in ascending order of the preference ranking of the pair.

The judgment table generation unit may omit storing the maximum value of the preference ranking at the maximum popularity ranking for each of the two sets.

The judgment table generation unit may omit storing the minimum value of the preference ranking for the minimum popularity ranking for each of the two sets.

The fairness determination method according to the present invention includes a preference ranking acquisition step for acquiring preference rankings of pairs in each element of two mutually prime sets from the preference ranking of the other set in each element of the two sets and information on pairs matched between the two sets; a popularity calculation step for combining the preference rankings for each element of the two sets to calculate the popularity of each element; a popularity ranking calculation step for sorting each element of the two sets in ascending order of the popularity and assigning the same popularity ranking to elements with the same popularity ranking; and a preference ranking calculation step for calculating the preference rankings for each popularity ranking of the two sets. The computer executes a judgment table generation step of generating a judgment table storing minimum and maximum values of the preference rankings, a table-specific judgment step of judging that a condition is satisfied if there is no i in the judgment table where the maximum value of the preference rankings up to the i-th popularity ranking is greater than the minimum value + l of the preference rankings for the i+k+1th popularity ranking, and that the condition is not satisfied if there is such a case, and a fairness judgment step of judging that meritocratic fairness is present if it is judged that the condition is satisfied for both of the two sets, and that meritocratic fairness is not present if it is judged that the condition is not satisfied for either of the sets.

The fairness judgment program according to the present invention is for causing a computer to function as the fairness judgment device.

The present invention makes it possible to efficiently determine meritocratic fairness in matching.

FIG. 2 is a diagram illustrating a functional configuration of a fairness determination device according to an embodiment. FIG. 13 is a diagram showing an algorithm by which a preference ranking acquisition unit in the embodiment obtains preference rankings of pairs. FIG. 13 is a diagram showing an algorithm used by a popularity calculation unit in the embodiment to calculate the popularity of each element. FIG. 13 is a diagram showing an algorithm used by a popularity ranking calculation unit in the embodiment to calculate the popularity ranking of each element. 11 is a diagram showing an algorithm by which a judgment table generating unit generates a judgment table in the embodiment. FIG. FIG. 13 is a diagram illustrating an algorithm used by a table-by-table determination unit in an embodiment to determine meritocratic fairness for each determination table. FIG. 13 is a diagram showing an algorithm by which the fairness determination unit 16 in the embodiment determines relaxed (k, l)-meritocratic fairness.

An example of an embodiment of the present invention will now be described.
The fairness determination method of this embodiment uses a matching obtained in a bipartite graph as an input and quickly determines whether the matching satisfies meritocratic fairness. Here, male-female matching is used as an example, but this embodiment can also be applied to similar use cases such as job matching.

A matching instance I consists of two relatively prime sets M and W of n elements, where M is the set of men and W is the set of women, and each element has a strict total order, called the preference order, with respect to the elements of the other set.
A matching X is a pair of M and W. That is, X ⊆ M × W. For simplicity, we will consider only the case where n one-to-one pairs are made.

In a matching X, a man m and a woman w are a pair, i.e., (m, w) ∈ X, then X(m) = w, X(w) = m. Also, the rank of w in the preference order of m is denoted as _Pm (w), which is called the preference rank of w in m. Similarly, the preference rank of m in w is denoted as _Pw (m).

Individualistic fairness in this embodiment refers to the idea that those with higher abilities or evaluations are more likely to get what they want. In matching between men and women, individuals who tend to be designated as having a higher (lower) preference rank by the opposite sex in the order of preferences are considered to be highly evaluated individuals.

Here, a person is quantitatively evaluated based on the sum of preference rankings from all opposite-sex individuals, and compared with others of the same sex. For any person a in a given match, if the preference ranking of the other person paired with same-sex individual b, who is rated lower than person a, is always lower (higher) or equal to the preference ranking of the other person paired with person a, then the match is considered to be individualistically fair.

For m∈M and w∈W, let G(m) = ^Σni _{=1Pwi (} m) and G(w) = ^Σni _{=1Pmi (} w) be the popularity of m and w, respectively.
Let R(m) be the rank of m when the set of men M is sorted in ascending order of popularity G(m), and call this the popular rank of m. Similarly, let R(w) be the popularity rank of w when the set of women W is sorted in ascending order of popularity G(w).
It should be noted that for a certain member m or w, the smaller the popularity G and popularity ranking R, the more popular the member is with the opposite sex.
Here, for a certain set H and a certain natural number p∈N, if a subset of H whose popularity ranking is smaller than p (having a high rating) is defined as H _<p , then individualistic fairness can be defined as follows.

Definition 1: For any m ∈ M and w ∈ W,
(1) For any m′∈M _{< R(m)} ,
P _m' (X(m'))≦P _m (X(m))
and,
(2) For any w′∈W _{< R(w)} ,
P _w' (X(w'))≦P _w (X(w))
A matching X is strictly meritocratically fair if

However, a strictly meritocratically fair matching that satisfies the conditions of Definition 1 does not necessarily exist for any preference ordering. In addition, there may be a matching that satisfies strict meritocratic fairness but significantly impairs global optimality.
Therefore, we present a definition of meritocratic fairness based on the following relaxed conditions.
- Allow same-sex people who are within the kth lowest popularity ranking to be matched with people of the opposite sex who have a lower preference ranking than you.
- For two people of the same sex, if the difference in preference rankings of their matched partners is within l, it is considered to be no difference and is accepted.

Definition 2: For any m ∈ M and w ∈ W,
(1) For any m′∈M _{< R(m)−k} ,
P _m' (X(m'))≦P _m (X(m))+l
and,
(2) For any w′∈W _{< R(w)−k} ,
P _w' (X(w'))≦P _w (X(w))+l
If x = 1, then the matching X is (k, l)-meritocratically fair.

Next, the functional configuration of the device for determining relaxed meritocratic fairness and the processing algorithm are shown.
In addition, strict meritocratic fairness can also be determined using this algorithm, since it is a special case of relaxed (k, l)-meritocratic fairness where k=0, l=0.

Here, the set of men is M = { _m1 , _m2 , ..., _mn }, and the set of women is W = { _w1 , _w2 , ..., _wn }. M and W are considered as arrays, and element a of these arrays is a structure having attributes a.pref_rank, a.popularity, and a.pop_rank.
a.pref_rank is the preference rank of a's pair on a, i.e., P _a (X(a)), and a.popularity and a.pop_rank are the popularity and popularity rank of a, respectively.

FIG. 1 is a diagram showing the functional configuration of a fairness determination device 1 according to the present embodiment.
The fairness determination device 1 is an information processing device (computer) such as a server device or a personal computer, and includes a control unit 10 and a storage unit 20 as well as various data input/output devices and communication devices.

The control unit 10 is a part that controls the entire fairness determination device 1, and realizes each function in this embodiment by appropriately reading and executing various programs stored in the memory unit 20. The control unit 10 may be a CPU.

The storage unit 20 is a storage area for various programs, including a fairness determination program for causing the hardware group to function as the fairness determination device 1, and various data, and may be a ROM, RAM, flash memory, or hard disk drive (HDD), etc.

The control unit 10 includes a preference ranking acquisition unit 11, a popularity calculation unit 12, a popularity ranking calculation unit 13, a judgment table generation unit 14, a table-specific judgment unit 15, and a fairness judgment unit 16.

The preference ranking acquisition unit 11 acquires the preference ranking of pairs in each element of each of the two sets M and W, which are mutually prime, from the preference ranking of the other set in each element of the two sets, and information on pairs matched between these two sets.

FIG. 2 is a diagram showing an algorithm PreferenceRank used by the preference ranking obtaining unit 11 in this embodiment to obtain the preference ranking of pairs.
The inputs are an instance of a matching, I, and a matching, X (step 1), where I is two arrays of length n (=|M|=|W|) whose elements are lists of preference orders of length n.

The preference ranking acquisition unit 11 loops n elements in number (step 2) and stores the preference ranking P M[i] (X(M[i])) of its paired element X(M[i]) in the attribute pref_rank of the i-th element _M[ i] in array M (step 3).
Similarly, the preference ranking acquisition unit 11 stores the preference ranking P _W[ i] (X(W[i])) of its paired element X(W[i]) in the attribute pref_rank of the i-th element W[i] in the array W (step 4).

The preference ranking acquisition unit 11 outputs arrays M and W in which the preference ranking of pairs for each element is stored in the attribute pref_rank (step 6).

The popularity calculation unit 12 combines the preference rankings for each element of the two sets M and W to calculate the popularity of each element.

FIG. 3 is a diagram showing the Popularity algorithm used by the popularity calculation unit 12 in this embodiment to calculate the popularity of each element.
The inputs are a matching instance I and two arrays M and W (Step 1).

The popularity calculation unit 12 performs a double loop (steps 2 and 3) for n elements, and adds the preference ranking of M[i] in each element of array W to the attribute popularity of the i-th element M[i] in array M for all elements (step 4).
Similarly, the popularity calculation unit 12 adds the preference ranking of W[i] in each element of array M to the attribute popularity of the i-th element W[i] in array W for all elements (step 5).

The popularity calculation unit 12 outputs arrays M and W in which the popularity of each element is stored in the attribute popularity (step 8).

The popularity ranking calculation unit 13 assigns a popularity ranking to each of the two sets by sorting the elements in ascending order of popularity. At this time, the popularity ranking calculation unit 13 assigns the same popularity ranking to elements with the same popularity value.
Furthermore, the popularity ranking calculation unit 13 may sort the elements in ascending order of popularity and in ascending order of preference ranking of the pairs to make the subsequent processing more efficient.

FIG. 4 is a diagram showing an algorithm PopularRank used by the popularity ranking calculation unit 13 in this embodiment to determine the popularity ranking of each element.
The input is an array in which the preference order and popularity are assigned as attributes of each element, and M and W are input in that order.

First, the popularity ranking calculation unit 13 obtains an array A' by sorting the input array A in ascending order of popularity and preference ranking using the function Sort(A) (step 2).
Note that Sort(A) is a function that sorts the input array A in ascending order by the value of the attribute popularity, and in the case of a tie, outputs an array A' that has been further sorted in ascending order by the attribute pref_rank.

Next, the popularity ranking calculation unit 13 initializes the variables pre_score and next_rank to 0 (steps 3 and 4), and then performs a loop process n times for the number of elements (step 5).

In the loop, if the popularity of the i-th element of array A'(score=A'[i].popularity) is different (greater) than pre_score, the popularity ranking calculation unit 13 updates next_rank to i (step 8) and updates pre_score to score (step 9).Then, the popularity ranking calculation unit 13 stores next_rank in the popularity ranking (A'[i].pop_rank) (step 11).
That is, if i increases but the popularity remains the same (FALSE in step 7), next_rank is not updated and the same popularity ranking is assigned. On the other hand, if the popularity increases (TRUE in step 7), the increased popularity ranking (=i) is assigned.

The popularity ranking calculation unit 13 outputs an array A' in which the popularity ranking of each element is stored in the attribute pop_rank (step 13).

The judgment table generation unit 14 generates a judgment table that stores the minimum and maximum values of the preference ranking for each popularity ranking for each of the two sets.

FIG. 5 is a diagram showing an algorithm MakeTable for generating a judgment table by the judgment table generating unit 14 in this embodiment.
The input is an array in which the attribute of each element is assigned a preference order, popularity, and popularity order, and the arrays M' and W' sorted by the popularity order calculation unit 13 are input in that order.

First, the judgment table generating unit 14 performs loop processing for n elements (step 2) to initialize the minimum value (Table[i].min) and maximum value (Table[i].max) stored in the judgment table to 0 (steps 3 and 4).
Next, the variables pre_pop_rank, pre_pref_rank, and pre_index are initialized to 0, 0, and 1, respectively (steps 6 to 8).

Next, in the loop process of n number of elements (step 9), if the popularity ranking (A[i].pop_rank) of the i-th element of the input array A has changed (become larger) than pre_pop_rank (step 10), the judgment table generating unit 14 stores A[i].pref_rank in Table[i].min (minimum value of preference ranking in the i-th popularity ranking) in the judgment table, and stores pre_pref_rank in Table[pre_index].max (maximum value of preference ranking in the previous popularity ranking) (steps 11 and 12).
Furthermore, since the judgment table generating unit 14 has updated the i-th element (min) of the judgment table, it updates pre_index to i (step 13).

On the other hand, if the popularity ranking (A[i].pop_rank) of the i-th element of array A is unchanged from pre_pop_rank (step 10), the judgment table generation unit 14 suspends the storage of the value in the judgment table and updates pre_pop_rank to A[i].pop_rank and pre_pref_rank to A[i].pref_rank (steps 15 and 16).

The judgment table generation unit 14 outputs a judgment table Table whose index is the popularity ranking and whose attribute is the range of preference rankings for each popularity ranking (minimum preference ranking min and maximum preference ranking max) (step 18).

Note that the maximum value A[n].max of the preference ranking in the maximum popularity ranking and the minimum value A[1].min of the preference ranking in the minimum popularity ranking are not referenced in subsequent processing, so the judgment table generation unit 14 may omit storing these values.

The table-specific judgment unit 15 judges that the condition of meritocratic fairness is satisfied if there is no i in the generated judgment table where the maximum value of the preference rankings up to the i-th popularity ranking is greater than the minimum value of the preference rankings for the i+k+1th popularity ranking + l, and that the condition is not satisfied if there is such a value.

FIG. 6 is a diagram showing an algorithm Judge that the table-by-table judgement unit 15 uses in this embodiment to judge meritocratic fairness for each judgement table.
The inputs are the decision table Table and constants k and l relating to the relaxed fairness condition.

First, the table-specific determination unit 15 initializes the variable prefer_max to 0 (step 2).

Next, the table-specific judgment unit 15 compares prefer_max with Table[i].max (step 4) in a loop process (step 3) that changes the index i from 1 to n-k-1. If Table[i].max is greater, the table-specific judgment unit 15 updates prefer_max (step 5). As a result, the maximum value of the preference ranking in the popularity ranking up to the i-th rank is stored in prefer_max.

In addition, in the loop process (step 3), the table-specific judgment unit 15 compares prefer_max with the minimum value of the preference ranking in the i+k+1th popularity ranking (Table[i+k+1]). If Table[i+k+1] is not 0 and the difference with prefer_max is greater than l, it is judged that the condition of meritocratic fairness is not met (steps 7-9). On the other hand, if Table[i+k+1] is 0 for all i or the difference with prefer_max is equal to or less than l, it is judged that the condition of meritocratic fairness is met (step 11).

The fairness determination unit 16 determines that the judgment tables generated from both sets M and W are meritocratic fair if the table-specific determination unit 15 determines that the conditions are met, and determines that the judgment tables are not meritocratic fair if the table-specific determination unit 15 determines that the conditions are not met.

FIG. 7 is a diagram showing the FairJudge algorithm used by the fairness determination unit 16 in this embodiment to determine relaxed (k, l)-meritocratic fairness.
The inputs are an instance of a matching I, a matching X, and constants k and l for the relaxed conditions.

First, the fairness determination unit 16 uses the preference ranking acquisition unit 11 (algorithm PreferenceRank) to store the preference ranking in the arrays M and W (step 2).
Next, the fairness determination unit 16 uses the popularity calculation unit 12 (algorithm Popularity) to obtain arrays M' and W' that further store the popularity (step 3).
Furthermore, the fairness determination unit 16 uses the popularity ranking calculation unit 13 (algorithm PopularRank) to obtain arrays M'' and W'' in which the popularity rankings are stored (steps 4 and 5).

Next, the fairness determination unit 16 uses the determination table generation unit 14 (algorithm MakeTable) to obtain the determination tables Table_M and Table_W from the arrays M'' and W'', respectively (steps 6 and 7).
Next, the fairness determination unit 16 uses the table-specific determination unit 15 (algorithm Judge) to determine whether or not the fairness conditions are satisfied for each of the judgment tables Table_M and Table_W (steps 8 and 9).
Then, if all of the judgment tables satisfy the fairness conditions (step 10), the fairness judgment unit 16 judges that the system as a whole satisfies meritocratic fairness (step 11); if at least any of the judgment tables do not satisfy the conditions, the fairness judgment unit 16 judges that the system as a whole does not satisfy meritocratic fairness (step 13).

According to this embodiment, the fairness determination device 1 generates a determination table that stores the minimum and maximum values of the preference ranking for each popularity ranking for each of the two sets, and determines that the condition is satisfied if there is no i in which the maximum value of the preference rankings up to the i-th popularity ranking is greater than the minimum value+l of the preference ranking for the i+k+1th popularity ranking in this determination table, and determines that the condition is not satisfied if there is such a value. In this way, the fairness determination device 1 determines that meritocratic fairness is present when it is determined that the condition is satisfied for both of the two sets, and determines that meritocratic fairness is not present when it is determined that the condition is not satisfied for either of the sets.
As a result, the fairness judgment device 1 can judge meritocratic fairness by performing only one loop process in the algorithm Judge using the judgment table. Therefore, in a judgment method that simply follows the definition of fairness, since each element of a set is compared with other elements, a calculation amount of O(n ² ) accompanied by a double loop is expected, whereas in this embodiment, the calculation amount is suppressed to O(n). As a result, the processing load required for searching for a match that satisfies meritocratic fairness is significantly reduced.

In addition, the fairness determination device 1 sorts each element of the set in ascending order of popularity and ascending order of pair preference ranking, thereby making the algorithms MakeTable and Judge more efficient and further reducing the processing load.

Furthermore, the fairness judgment device 1 can improve efficiency by omitting storage of data that is not used in the algorithm Judge from the judgment table. Specifically, for each of the two sets, storage of the maximum value of the preference ranking in the maximum popularity ranking and/or the minimum value of the preference ranking in the minimum popularity ranking may be omitted.

As a result, for example, it will be possible to efficiently provide a meritocratic and fair matching service over the Internet, which will contribute to Goal 9 of the United Nations-led Sustainable Development Goals (SDGs) which states, "Build resilient infrastructure, promote sustainable industrialization and foster innovation."

Although the embodiments of the present invention have been described above, the present invention is not limited to the above-described embodiments. Furthermore, the effects described in the above-described embodiments are merely a list of the most favorable effects resulting from the present invention, and the effects of the present invention are not limited to those described in the embodiments.

The fairness determination method by the fairness determination device 1 is realized by software. When realized by software, the programs constituting this software are installed in an information processing device (computer). These programs may be recorded on removable media such as CD-ROM and distributed to users, or may be distributed by being downloaded to the user's computer via a network. Furthermore, these programs may be provided to the user's computer as a web service via a network without being downloaded.

REFERENCE SIGNS LIST 1 Fairness determination device 10 Control unit 11 Preference ranking acquisition unit 12 Popularity calculation unit 13 Popularity ranking calculation unit 14 Determination table generation unit 15 Table-specific determination unit 16 Fairness determination unit 20 Storage unit

Claims (6)

a preference ranking acquisition unit that acquires preference rankings of pairs in each element of each of two disjoint sets from a preference ranking of the other set in each element of each of the two sets and information on pairs matched between the two sets;
a popularity calculation unit that combines the preference rankings for each element of the two sets and calculates the popularity of each element;
a popularity ranking calculation unit that sorts each element of each of the two sets in ascending order of the popularity and assigns the same popularity ranking to elements having the same popularity ranking;
a judgment table generating unit that generates a judgment table storing the minimum and maximum values of the preference ranking for each of the popularity rankings for each of the two sets;
a table-based determination unit that determines that a condition is satisfied when there is no i in the determination table in which the maximum value of the preference ranking up to the i-th popularity ranking is greater than a minimum value+l of the preference ranking of the (i+k+1)th popularity ranking, and that the condition is not satisfied when there is such a i;
a fairness determination unit that determines that the system is meritocratic fair if it is determined that the condition is satisfied for both of the two sets, and that the system is not meritocratic fair if it is determined that the condition is not satisfied for either of the sets. 2 . The fairness determination device according to claim 1 , wherein the popularity ranking calculation unit, when sorting the elements in ascending order of the popularity, sorts the elements in ascending order of the preference ranking of the pair when the popularity rankings have the same value . The fairness determination device according to claim 1 or 2, wherein the determination table generation unit omits storing the maximum value of the preference ranking at the maximum popularity ranking for each of the two sets. The fairness determination device according to any one of claims 1 to 3, wherein the determination table generation unit omits storage of the minimum value of the preference ranking in the minimum popularity ranking for each of the two sets. a preference ranking acquisition step of acquiring preference rankings of pairs in each element of each of two disjoint sets from the preference ranking of the other set in each element of each of the two sets and information of pairs matched between the two sets;
a popularity calculation step of calculating the popularity of each element by combining the preference rankings of each element in each of the two sets;
a popularity ranking calculation step of sorting each element of each of the two sets in ascending order of the popularity and assigning the same popularity ranking to elements having the same popularity value;
a judgment table generating step of generating a judgment table in which the minimum and maximum values of the preference rankings are stored for each of the popularity rankings for each of the two sets;
a table-by-table judging step for judging that a condition is satisfied when there is no i in the judgment table in which the maximum value of the preference ranking up to the i-th popularity ranking is greater than the minimum value+l of the preference ranking of the (i+k+1)th popularity ranking, and that the condition is not satisfied when there is such a i;
a fairness determination step of determining that the system is meritocratic fair if it is determined that the condition is satisfied for both of the two sets, and determining that the system is not meritocratic fair if it is determined that the condition is not satisfied for either of the two sets. A fairness judgment program for causing a computer to function as a fairness judgment device according to any one of claims 1 to 4.

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