α-Rank: Unified Item-Fair Ranking from A Cooperative Game Theory View

• The use of word utilitarianism, dealism, and egalitarianism is a bit problematic: -- This article employs obscure and convoluted vocabulary to redefine some properties that have already been well-defined in the fairness and economic literature, i.e., individual fairness, group fairness, and social welfare. -- The reviewer has personally checked the literature, the word "egalitarianism" appeared only once in [4].

• Citation Issue: -- some proper literature on optimal transport has not been cited, e.g., [2]. -- In the related work section, the majority of the works mentioned date back to before 2000, and this issue is particularly pronounced in the paragraph related to cooperative games and fairness principles. For instance, [3] should be included in the paragraph discussing cooperative games.

• Problem Formulation Issue: -- $\mathbf{v}$ is defined both as a vector and a function of $\mathbf{x}_u$. -- For the definition of $\mathcal{D}$, the correct formulation should be $\mathbf{1}^\top \mathbf{x}_u \leq K$? -- The original characterization of OT doesn't have the entropy term $H(x)$.

• Minor presentation issues: -- two "in such games" in the intro. -- In paragraph item fairness methods, the sentence "For different fairness aspects, Current mainstream", the "C" should be in lowercase. -- In the second paragraph of introduction, the sentence "which equalizes the utilities of all item utilities", duplicate use of the word "utility". -- It would be better to place the definitions of "CTR" and "CVR" in the caption of Table 1. -- In paragraph cooperative games, the sentence "Another approach, proposed by (Nash Jr, 1950), is rooted in the concept of bargaining. " has an extra parathesis. --The word charging can be replaced by other words, e.g., payment, bid, etc.

[2] Monge, G. (1781). Mémoire sur la théorie des déblais et des remblais. Mem. Math. Phys. Acad. Royale Sci., 666-704.

[3] De Clippel, G., & Rozen, K. (2019). Fairness through the lens of cooperative game theory: An experimental approach. Available at SSRN 3527069.

[4] Bertsimas, D., Farias, V. F., & Trichakis, N. (2011). The price of fairness. Operations research, 59(1), 17-31.

Some notations are confusing, e.g., in equation (1) the v is a vector, then what does v($x_u$) mean?

I have a major concern about the paper, which is how well it fits the ICLR community. Indeed, I expect that most of the researchers interested in ICLR are not familiar with basic notions of cooperative game theory. Thus, they would the paper difficult to read.

I also believe believe that the authors should make a better job selling the paper, making it clear from the beginning which are the possible applications of the framework and methods they provide in the paper.

• It is not clear to me what fairness properties and axioms the final result of the \alpha-rank algorithm is guaranteed to satisfy. Could the authors please comment on this?
• I am not sure I understand the relationship between the value of the fairness objective of the output of \alpha-rank and that of the optimal solution to the original integer linear program in equation 2, i.e., between W(\alpha) and the fairness of L_K(u). Could you please help clarify? Is there a theoretical result I am missing? Alternatively, could you point to specific empirical results that illustrate this relationship? What are the axiomatic properties of the ranking?
• Is the objective to provide item fairness for each individual user decision or over all user decisions for fixed set of users? Could you please help clarify? The source of my confusion is the definitions of the fairness objectives starting at the end of Page 3 and the top of Page 4.
• The main theorems are stated too informally, and I do not understand what is being claimed just from reading them. It is possible to dig deeper and understand what is being proved by reading the proof, but it was not obvious from reading the theorem statement or the surrounding discussion. E.g. what does it mean for a fairness property X to adhere to an axiom Y? Do you mean that every feasible solution satisfying fairness property X (e.g. maximizing egalitarian objective) satisfies Axiom Y? Or does it mean that a feasible solution satisfy X and Y always exists? Could you please help clarify?
• Some technical terms used in the discussion need to be clarified in order to understand the paper properly. E.g. The authors state that the axioms in Section 4.1 show "how item fairness principles will behave when there are changes in available resources". There are several things that are not clear at this point. The term "resources" are never formally defined and it is not obvious from the discussion. I assume it refers to the number of items a user can be recommended. Is this right? Could you please clarify what is the difference between resources, attention, slots, cake, and other such terms used in the paper? How does Pareto optimality show how fairness principles behave as there are changes in available resources?
• There are several major typos and grammatical errors that make the paper hard to read.

Other comments placed here for the lack oxf another dedicated space:

• P1: Is "dealism" a commonly used term in the literature? Does this refer to the Nash bargaining solution?
• P1: ... papers advocate item from distinct principles ... -> item fairness
• P1: Please consider specifying what is an exposure slot.
• P1: utilitarianism objective ... maximizing summarization ... -> utilitarian objective ... maximizing summation?
• P1: dgalitarianism -> egalitarianism
• P2: we introduce the concept of \alpha-fairness to achieve a well balanced equilibrium among item fairness notions. What is meant by equilibrium here? Could you please elaborate?
• P3: vector x\in \mathbb{R}^{n\times m}. Do you mean matrix x? Is this a typo?
• P3: Please consider discussing what is a "resource" in Section 3.
• P3: It is not obvious from the text what v(x_u) or v_i(x_u) are. I assume v_i(x_u) is the utility for item from a decision x_u for user u. Is this right?
• P4, Axiom 5: The notion of a utility set has not been introduced if I remember correctly.
• In Fig. 1, what are the dashed lines?
• Overall, it might help to clarify whether resources refer to the K recommendation slots for each user. Are these indivisible or divisible? What does "cake" refer to?
• P6: each user can only be ranked among the top K items. Please consider rephrasing.