Abstract
Many popular internet platforms use so-called collaborative filtering systems to give personalized recommendations to their users, based on other users who provided similar ratings for some items. We propose a novel approach to such recommendation systems by viewing a recommendation as a way to extend an agent’s expressed preferences, which are typically incomplete, through some aggregate of other agents’ expressed preferences. These extension and aggregation requirements are expressed by an Acceptance and a Pareto principle, respectively. We characterize the recommendation systems satisfying these two principles and contrast them with collaborative filtering systems, which typically violate the Pareto principle.
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More precisely, the closure of the set \(\{ \theta u + \kappa : u \in U, \theta \in {\mathbb {R}}_+, \kappa \in {\mathbb {R}} \}\) is unique.
This allows us to dispense with the closure operator in the uniqueness statement above as well as the closedness assumption of Danan et al. (2015).
While in the examples above we consider possible any rating within the allowed scale, our framework can accommodate alternative specifications (for instance, one might consider intermediate ratings only).
It is without loss of generality in the sense that we can always choose a representation U such that this holds.
Indeed, if there are lotteries x and y such that \(x \succ \!\!\succ _0 y\) whereas \(y \succsim _n x\) for all \(n=1,\ldots ,N\), then the Acceptance principle implies \(x \succ \!\!\succ y\) whereas the Pareto principle implies \(y \succsim x\), a contradiction.
This is not trivial because of our restriction to expected multi-utility preferences.
Indeed, in that case, \(U_0\) can not contain constant utility functions and, hence, any virtual guide \(u = \sum _{n=1}^N \theta _n u_n + \kappa \in U_0\) must be such that \(\theta _n > 0\) for at least one \(n=1,\ldots ,N\).
In this example, the recommendation could possibly preserve agent 0’s strict preferences and/or follow unanimous strict preferences from agents \(1,\ldots ,N\), depending on which virtual guides are selected. Other examples can be constructed where that is not possible.
Indeed, u fills agent 0’s missing ratings for alternatives c, d, and e with 3.2, 4.6, and 0, respectively, whereas all virtual guides fill agent 0’s missing rating for e with at least 2.8. Note that u cannot be obtained as a linear combination \(\theta _1 u_1 + \theta _2 u_2 + \kappa \) with \(u_1 \in U_1\), \(u_2 \in U_2\), and \(\theta _1,\theta _2,\kappa \in {\mathbb {R}}\).
Collaborative filtering systems, on the other hand, satisfy the anonymity principle but violate the two robustness principles.
For instance, any recommendation rule that selects a single (signed) virtual guide whenever \(\succsim _1,\ldots ,\succsim _N\) are complete but keeps all (signed) virtual guides otherwise satisfies the Acceptance, Pareto (resp. Pareto-Indifference) and External-Robustness principles but not the Internal-Robustness principle.
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Funding
We thank Franz Dietrich, Georgios Gerasimou, Itzhak Gilboa, Philippe Jehiel, Jérôme Lang, Antonin Macé, Tigran Melkonyan, Efe Ok, Marcus Pivato, Daniel Read, Régis Renault, Zvi Safra, two anonymous referees, and seminar participants at Düsseldorf U., Warwick Business School, PSE, U. St Andrews as well as participants to the conference Times, Uncertainty and Strategy (Dec. 2018) and the CHOp Workshop, for useful feedback. This work is part of the project “Coping with Heterogeneous Opinions”, which benefits from the support of the Grant ANR-17-CE26-0003. Danan acknowledges financial support from the Grants ANR-11-LBX-0023-01 and ANR-16-IDEX-0008. Tallon acknowledges financial support from the Grant ANR-17-EURE-001.
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Appendix: Proofs
Appendix: Proofs
We derive Propositions 1 and 2 from slightly more general results. To this end we consider the following weakening of the Acceptance principle.
Axiom
(Weak Acceptance principle) For all \(x,y \in X\), if \(x \succsim _0 y\) then \(x \succsim y\).
Lemma 1
Let U be a representation of \(\succsim \) as per Definition 1.
-
(a)
\(\succsim \) satisfies the Weak Acceptance and Pareto principles if and only if, for all \(u \in U\), there exist \(u_0 \in U_0,u_1 \in U_1, \ldots ,u_N \in U_N,\theta _0,\theta _1,\ldots ,\theta _N \in {\mathbb {R}}_+ , \kappa _0,\kappa \in {\mathbb {R}}\) such that:
$$\begin{aligned} u = \theta _0 u_0 + \kappa _0 = \sum \nolimits _{n=1}^N \theta _n u_n + \kappa . \end{aligned}$$(3) -
(b)
\(\succsim \) satisfies the Weak Acceptance and Pareto-Indifference principles if and only if, for all \(u \in U\), there exist \(u_0 \in U_0, u_1,v_1 \in U_1, \ldots ,u_N,v_N \in U_N,\theta _0,\theta _1,\omega _1,\ldots ,\theta _N,\omega _N \in {\mathbb {R}}_+ , \kappa _0,\kappa \in {\mathbb {R}}\) such that:
$$\begin{aligned} u = \theta _0 u_0 + \kappa _0 = \sum \nolimits _{n=1}^N (\theta _n u_n - \omega _n v_n) + \kappa . \end{aligned}$$(4)
Proof
-
(a)
It is obvious that if (3) holds then \(\succsim \) satisfies the Weak Acceptance and Pareto principles. Conversely, assume (3) does not hold. Then at least one \(u \in U\) must lie outside at least one of the two sets
$$\begin{aligned} S&= \{ \theta _0 u_0 + \kappa _0 : u_0 \in U_0, \theta _0 \in {\mathbb {R}}_+ , \kappa _0 \in {\mathbb {R}} \} \ \text {and} \\ T&= \left\{ \sum \nolimits _{n=1}^N \theta _n u_n + \kappa : u_1 \in U_1, \ldots , u_N \in U_N, \theta _1,\ldots ,\theta _N \in {\mathbb {R}}_+, \kappa \in {\mathbb {R}} \right\} . \end{aligned}$$Note that both S and T are closed, convex cones in \({\mathbb {R}}^A\).
Case 1. If \(u \notin S\) then, by the separating hyperplane theorem, there exists \(h \in {\mathbb {R}}^A\), \(h \ne 0\), such that \(\sum _{a \in A} h(a)u(a) > 0 \ge \sum _{a \in A} h(a)s(a)\) for all \(s \in S\). Since \(s+\beta \in S\) for all \(s \in S\) and \(\beta \in {\mathbb {R}}\), this can only be the case if \(\sum _{a \in A} h(a) = 0\), for otherwise we could make \(\sum _{a \in A} h(a)(s(a)+\beta ) = \sum _{a \in A} h(a)s(a) + \beta \sum _{a \in A} h(a)\) arbitrarily large by taking \(\beta \) sufficiently close to \(+\infty \) or \(-\infty \). Hence, letting \(\xi = \sum _{a \in A}|h(a)|\) and defining \(x,y \in {\mathbb {R}}^A\) by \(x(a) = \frac{\max \{h(a),0\}}{\xi }\) and \(y(a) = \frac{\max \{-h(a),0\}}{\xi }\), we have \(\xi >0\), \(x,y \in X\), and \(h = \xi (x-y)\). Since \(\sum _{a \in A} h(a)u(a) > 0\), it follows that \({{\,\mathrm{E}\,}}u(x) > {{\,\mathrm{E}\,}}u(y)\), so it cannot be the case that \(y \succsim x\) by Definition 1. But since \(U_0 \subset S\), it also follows that \(0 \ge \sum _{a \in A} h(a)u_0(a)\) and, hence, \({{\,\mathrm{E}\,}}u_0(y) \ge {{\,\mathrm{E}\,}}u_0(x)\) for all \(u_0 \in U_0\), so that \(y \succsim _0 x\) by Definition 1. This contradicts the Weak Acceptance principle.
Case 2. If \(u \notin T\) then, by the separating hyperplane theorem, there exists \(h \in {\mathbb {R}}^A\), \(h \ne 0\), such that \(\sum _{a \in A} h(a)u(a) > 0 \ge \sum _{a \in A} h(a)t(a)\) for all \(t \in T\). By the same argument as in Case 1, we must have \(\sum _{a \in A}h(a)=0\) and, hence, we can write \(h=\xi (x-y)\) with \(\xi >0\) and \(x,y \in X\). Since \(\sum _{a \in A} h(a)u(a) > 0\), it follows that \({{\,\mathrm{E}\,}}u(x) > {{\,\mathrm{E}\,}}u(y)\), so it cannot be the case that \(y \succsim x\) by Definition 1. But for all \(n=1,\ldots ,N\), since \(U_n \subset T\), it also follows that \(0 \ge \sum _{a \in A} h(a)u_n(a)\) and, hence, \({{\,\mathrm{E}\,}}u_n(y) \ge {{\,\mathrm{E}\,}}u_n(x)\) for all \(u_n \in U_n\), so that \(y \succsim _n x\) by Definition 1. This contradicts the Pareto principle.
-
(b)
It is obvious that if (4) holds then \(\succsim \) satisfies the Weak Acceptance and Pareto-Indifference principles. Conversely, assume (4) does not hold. Then at least one \(u \in U\) must lie outside at least one of the two sets S and:
$$\begin{aligned} T'&= \left\{ \sum \nolimits _{n=1}^N (\theta _n u_n - \omega _n v_n) + \kappa : u_1,v_1 \in U_1, \ldots , u_N,v_N \in U_N, \theta _1,\omega _1,\ldots ,\theta _N,\omega _N \in {\mathbb {R}}_+, \kappa \in {\mathbb {R}} \right\} . \end{aligned}$$We then proceed as in the proof of Part (a), except that in Case 2, since \(T'\) is a linear subspace of \({\mathbb {R}}^A\), the separating hyperplane theorem now implies that \(\sum _{a \in A} h(a)u(a) > 0 = \sum _{a \in A} h(a)t(a)\) for all \(t \in T'\). Hence for all \(n=1,\ldots ,N\), we now have \({{\,\mathrm{E}\,}}u_n(x) = {{\,\mathrm{E}\,}}u_n(y)\) for all \(u_n \in U_n\), so that \(x \sim _n y\) by Definition 1, which contradicts the Pareto-Indifference principle.
\(\square \)
Proof of Proposition 1
It is obvious that if \(\succsim \) can be represented as per Definition 1 by some set \(U \subseteq V\) of virtual guides, then it satisfies the Acceptance and Pareto principles. Conversely, assume \(\succsim \) satisfies these two principles and fix some representation U of \(\succsim \) as per Definition 1. Then for all \(u \in U\), (3) holds by Lemma 1(a). It is sufficient to show that (3) actually holds with \(\theta _0 > 0\) for some \(u_0 \in U_0\), since then the closed convex hull of the set \(\bigl \{ \frac{u - \kappa _0}{\theta _0} : u \in U \bigr \}\) also represents \(\succsim \) as per Definition 1 and is a subset of V. So suppose (3) does not hold with \(\theta _0 > 0\) for any \(u_0 \in U_0\). Note that this implies that u is a constant function whereas \(U_0\) contains no constant function. Hence, by the separating hyperplane theorem, there exists \(h \in {\mathbb {R}}^A\), \(h \ne 0\), such that \(\sum _{a \in A} h(a)u_0(a) > \kappa \sum _{a \in A} h(a)\) for all \(u_0 \in U_0\) and \(\kappa \in {\mathbb {R}}\). By the same argument as in Case 1 of the proof of Lemma 1(a), we must have \(\sum _{a \in A}h(a)=0\) and, hence, we can write \(h=\xi (x-y)\) with \(\xi >0\) and \(x,y \in X\). For all \(u_0 \in U_0\), since \(\sum _{a \in A} h(a)u_0(a) > 0\), it follows that \({{\,\mathrm{E}\,}}u_0(x) > {{\,\mathrm{E}\,}}u_0(y)\) and, hence, \(x \succ \!\!\succ _0 y\). The Acceptance principle then implies that \(x \succ \!\!\succ y\), contradicting the fact that U contains a constant function. \(\square \)
Proof of Proposition 2
Similar to the proof of Proposition 1. \(\square \)
Proof of Proposition 3
Letting \(\kappa = \kappa ^+ - \kappa ^-\) with \(\kappa ^+,\kappa ^- \in {\mathbb {R}}_+\) in (1), V is non-empty if and only if the following finite linear system has a non-negative solution:
By Farkas’ Lemma, this system has a non-negative solution if and only if there exist no \(h \in {\mathbb {R}}^K\) and \(\zeta \in {\mathbb {R}}\) such that:
or, equivalently, if there exists no \(h \in {\mathbb {R}}^K\) such that:
By the same argument as in Case 1 of the proof of Lemma 1(a), this is equivalent to the non-existence of lotteries \(x,y \in X\) such that \(x \succ \!\!\succ _0 y\) and \(y \succsim _n x\) for all \(n=1,\ldots ,N\). The fact that V is then the convex hull of finitely many functions simply follows from the finiteness and linearity of the above system as well as the fact that V is a subset of the compact set \(U_0\). \(\square \)
Proof of Proposition 4
Letting \(\psi _{n,i}=\psi _{n,i}^+-\psi _{n,i}^-\) and \(\kappa = \kappa ^+ - \kappa ^-\) where \(\psi _{n,i}^+,\psi _{n,i}^-,\kappa ^+,\kappa ^- \in {\mathbb {R}}_+\) in (2), W is non-empty if and only if the following finite linear system has a non-negative solution:
By Farkas’ Lemma, this system has a non-negative solution if and only if there exist no \(h \in {\mathbb {R}}^K\) and \(\zeta \in {\mathbb {R}}\) such that:
or, equivalently, if there exists no \(h \in {\mathbb {R}}^K\) such that:
By the same argument as in Case 1 of the proof of Lemma 1(a), this is equivalent to the non-existence of lotteries \(x,y \in X\) such that \(x \succ \!\!\succ _0 y\) and \(x \sim _n y\) for all \(n=1,\ldots ,N\). The fact that W is then the convex hull of finitely many functions simply follows from the finiteness and linearity of the above system as well as the fact that W is a subset of the compact set \(U_0\). \(\square \)
Proof of Proposition 5
It is obvious that the recommendation rule on \(D_c\) consisting in selecting all virtual guides for all profiles satisfies the Acceptance, Pareto, Internal-Robustness, Anonymity, and External-Robustness principles. Conversely, let f be a recommendation rule on \(D_c\) satisfying the Acceptance, Pareto, and Internal-Robustness principles. Consider a profile \(\bigl (\succsim _0,\succsim _1,\ldots ,\succsim _N\bigr ) \in D_c\) and let \(\succsim \ = f \bigl (\succsim _0,\succsim _1,\ldots ,\succsim _N\bigr )\). Fixing arbitrary representations \(U_0,U_1,\ldots ,U_N\) of \(\succsim _0,\succsim _1,\ldots ,\succsim _N\), respectively, let V denote the corresponding set of virtual guides. Then by Proposition 1, \(\succsim \) can be represented as per Definition 1 by some set \(U \subseteq V\).
Suppose \(\succsim \) cannot be represented as per Definition 1 by V. By the uniqueness of expected multi-utility representation and our finiteness assumption, we must then have \(\{ \theta u + \kappa : u \in U, \theta \in {\mathbb {R}}_+, \kappa \in {\mathbb {R}} \} \ne \{ \theta v + \kappa : v \in V, \theta \in {\mathbb {R}}_+, \kappa \in {\mathbb {R}} \}\) and, hence, there must exist some \(u \in U\) that does not belong to \(\{ \theta v + \kappa : v \in V, \theta \in {\mathbb {R}}_+, \kappa \in {\mathbb {R}} \}\). By the same argument as in Case 1 of the proof of Lemma 1(a), it follows that there exist \(x,y \in X\) such that \({{\,\mathrm{E}\,}}u(x) > E u(y)\) whereas \({{\,\mathrm{E}\,}}v(y) \ge {{\,\mathrm{E}\,}}v(x)\) for all \(v \in V\). Hence, letting \(\succsim '_0\) be represented as per Definition 1 by \(\bigl \{u_0\bigr \}\), we have \(x \succ \!\!\succ '_0 y\). On the other hand, letting \(\succsim '\ = f(\bigl (\succsim '_0,\succsim _1,\ldots ,\succsim _N\bigr ) \), we have \(y \succsim ' x\) by the Internal-Robustness principle. This contradicts the Acceptance principle. \(\square \)
Proof of Proposition 6
Similar to the proof of Proposition 5. \(\square \)
Proof of Proposition 7
Assume A contains at least three alternatives a, b, c, let \(w,w' : A \rightarrow {\mathbb {R}}\) be such that \(w(a)=w'(b)=1\) and \(w(b)=w'(a)=w(c)=w'(c)=0\), and let U denote the convex hull of \(\{w,w'\}\). Note that for all \(u,v \in U\), if \(u \ne v\) then there exist \(x,y \in X\) such that \({{\,\mathrm{E}\,}}u(x) > {{\,\mathrm{E}\,}}u(y)\) whereas \({{\,\mathrm{E}\,}}v(y) > {{\,\mathrm{E}\,}}v(x)\). Now let f be a recommendation rule on \(D_c\) (resp. \(D_{wc}\)) satisfying the Acceptance and Pareto (resp. Pareto-Indifference) principles that yields a complete recommendation for all profiles. Let \(\succsim \) be represented as per Definition 1 by U and \(\succsim '\ = f\bigl (\succsim ,\succsim ,\ldots ,\succsim \bigr )\). Then by Proposition 1 (resp. 2), \(\succsim '\) can be represented as per Definition 1 by \(\{u\}\) for some \(u \in U\). Let \(\succsim ''\) be represented by some \(v \in U \setminus \{u\}\) and \(\succsim '''\ = f\bigl (\succsim ,\succsim '',\ldots ,\succsim ''\bigr )\). Again by Proposition 1 (resp. 2), \(\succsim '''\) can be represented as per Definition 1 by \(\{v\}\). Hence \(\succsim '''\) does not refine \(\succsim '\), so f does not satisfy the External-Robustness principle. \(\square \)
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Danan, E., Gajdos, T. & Tallon, JM. Tailored recommendations. Soc Choice Welf 60, 15–34 (2023). https://doi.org/10.1007/s00355-020-01295-7
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DOI: https://doi.org/10.1007/s00355-020-01295-7