FAST: A Fairness Assured Service Recommendation Strategy Considering Service Capacity Constraint

Abstract

An excessive number of customers often leads to a degradation in service quality. However, the capacity constraints of services are ignored by recommender systems, which may lead to unsatisfactory recommendation. This problem can be solved by limiting the number of users who receive the recommendation for a service, but this may be viewed as unfair. In this paper, we propose a novel metric Top-N Fairness to measure the individual fairness of multi-round recommendations of services with capacity constraints. By considering the fact that users are often only affected by top-ranked items in a recommendation, Top-N Fairness only considers a sub-list consisting of top N services. Based on the metric, we design FAST, a Fairness Assured service recommendation STrategy. FAST adjusts the original recommendation list to provide users with recommendation results that guarantee the long-term fairness of multi-round recommendations. We prove the convergence property of the variance of Top-N Fairness of FAST theoretically. FAST is tested on the Yelp dataset and synthetic datasets. The experimental results show that FAST achieves better recommendation fairness while still maintaining high recommendation quality.

A Properties of F-FAST

THEOREM 1. The sum of Top-N Fairness of all the users in each round is equal to zero, $\sum _{u_i\in U}F_i^T = 0$.

PROOF. According to Eq. (4), the sum of Top-N Fairness can be formulated as:

$$\begin{aligned} \sum _{u_i\in U}F_i^T = \sum _{u_i\in U}\sum _{s_j\in l(N)_i}\frac{p_{i,j}^T-p^{T}_j}{p^{T}_j} = \sum _{s_j\in S}\sum _{u_i\in U_j}\frac{p_{i,j}^t-p^{T}_j}{p^{T}_j} \end{aligned}$$

(7)

Since all users will receive recommendations in every round, $\delta _i^t$ will all be equal to 1, and according to Eq. (1) and Eq. (3), $p^{T}_j$ and $p_{i,j}^t$ can be re-expressed as:

$$\begin{aligned} p^{T}_j = \frac{\sum _{u_i\in U_j}\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{\sum _{u_i\in U_j}T}, p_{i,j}^T = \frac{\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{T} \end{aligned}$$

(8)

Then the sum of Top-N Fairness will be:

$$\begin{aligned} \begin{aligned} \sum _{u_i\in U}F_i^T =&\sum _{s_j\in S}\sum _{u_i\in U_j}\frac{\frac{\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{T}}{\frac{\sum _{u_i\in U_j}\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{\sum _{u_i\in U_j}T}} - \sum _{s_j\in S}\sum _{u_i\in U_j}1 \end{aligned} \end{aligned}$$

(9)

After reducing Eq. (9), we can get:

$$\begin{aligned} \sum _{u_i\in U}F_i^T = \sum _{s_j\in S}\left( \frac{\sum _{u_i\in U_j}T}{T} - \sum _{u_i\in U_j}1\right) = 0 \end{aligned}$$

(10)

THEOREM 2. Variance among Top-N fairness of all users $D(F_i^T)$ converges to 0 with the recommended round T.

PROOF. According to THEOREM 1, we can get the mean of Top-N Fairness of all users equals to zero, and the variance can be formulated as:

$$\begin{aligned} D(F_i^T) = \frac{\sum _{u_i\in U}\left( F_i^T \right) ^2}{n} \end{aligned}$$

(11)

According to Eq. (4), we know that:

$$\begin{aligned} \sum _{u_i\in U}\left( F_i^T \right) ^2 = \sum _{u_i\in U}\left( \sum _{s_j\in l(N)_i}\frac{p_{i,j}^T-p^{T}_j}{p^{T}_j} \right) ^2 \end{aligned}$$

(12)

Since every user receives a recommendation in each round, $p^{T}_j$ of each service should be a constant. We discuss this issue in the following two cases.

For services without capacity conflicts: $c_j\geqslant len(U_j)$. Each service $s_j$ can be assigned to every user in its $U_j$ in each round. So $p^{T}_j$ and $p_{i,j}^T$ will always be 1. So the addends in summation formula of Top-N Fairness are always equal to 0 and can be ignored in this discussion.

For services with capacity conflicts: $c_j < len(U_j)$. Each service $s_j$ will always be assigned to $c_j$ users, so $p^{T}_j$ will be a constant less than 1, and we call it $Const_j$:

$$\begin{aligned} p^{T}_j ={c_j}/{len(U_i)} = Const_j < 1 \end{aligned}$$

(13)

Then, Eq. (12) will be:

$$\begin{aligned} \sum _{u_i\in U}\left( F_i^T \right) ^2 = \sum _{u_i\in U}\left[ \sum _{s_j\in l(N)_i}\left( \frac{p_{i,j}^T}{Const_j} -1\right) \right] ^2 \end{aligned}$$

(14)

The only variable in Eq. (14) is $p_{i,j}^t$, and according to Eq. (8), we can get:

$$\begin{aligned} \begin{aligned} p_{i,j}^{T+1}&= \frac{\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{T+1} + \frac{In\_tn(s_j,l_i^{T+1},N)}{T+1} \end{aligned} \end{aligned}$$

(15)

According to THEOREM 1, we can divide users into two groups, users with low Top-N Fairness($F_i^T < 0$) and users with high Top-N Fairness($F_i^T \geqslant 0$).

For users with low Top-N Fairness, addends with $p_{i,j}^T < Const_j$ occupy the main influence factor in the summation formula of Top-N Fairness in this situation. As designed in our strategy, users with low Top-N Fairness will always be allotted first, that:

$$\begin{aligned} \begin{aligned} 1>Const_j&> P_{i,j}^{T+1} = \frac{\sum ^T_{t=0}In\_tn(s_j,l_i^t,N) + 1}{T+1} > \frac{\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{T} = P_{i,j}^T \end{aligned} \end{aligned}$$

(16)

According to Eq. (14), we know that

$$\begin{aligned} \left| F_i^{T+1}\right|< \left| F_i^T\right| , (F_i^{T+1})^2 < (F_i^T)^2 \end{aligned}$$

(17)

For users with high Top-N Fairness, addends with $P_{i,j}^T \geqslant Const_j$ occupy the main influence factor in the summation formula of Top-N Fairness in this situation. Also, according to our recommendation strategy, these users will always be assigned last and will most likely not be assigned under the condition that service capacity is limited, so that:

$$\begin{aligned} \begin{aligned} Const_j \leqslant p_{i,j}^{T+1}&= \frac{\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{T+1} < \frac{\sum ^T_{t=0}In\_tn(s_j,l_i^t,N)}{T} = P_{i,j}^T \end{aligned} \end{aligned}$$

(18)

According to Eq. (14), we can also get:

$$\begin{aligned} \left| F_i^{T+1}\right|< \left| F_i^T\right| , (F_i^{T+1})^2 < (F_i^T)^2 \end{aligned}$$

(19)

In both cases, $(F_i^T)^2$ becomes smaller as the round of recommendation increases. When a user’s $F_i^T$ is not equal to the average $F_i^T$ of users, F-FAST will continue to work until $F_i^T$ of all users is equal, and we can get that $D(F_i^T)$ will converge to 0, thus THEOREM 2 is true.