{"title": "A Game-Theoretic Approach to Recommendation Systems with Strategic Content Providers", "book": "Advances in Neural Information Processing Systems", "page_first": 1110, "page_last": 1120, "abstract": "We introduce a game-theoretic approach to the study of recommendation systems with strategic content providers. Such systems should be fair and stable. Showing that traditional approaches fail to satisfy these requirements, we propose the Shapley mediator. We show that the Shapley mediator satisfies the fairness and stability requirements, runs in linear time, and is the only economically efficient mechanism satisfying these properties.", "full_text": "A Game-Theoretic Approach to Recommendation\n\nSystems with Strategic Content Providers\n\nOmer Ben-Porat and Moshe Tennenholtz\nTechnion - Israel Institute of Technology\n\nHaifa 32000 Israel\n\n{omerbp@campus,moshe@ie}.technion.ac.il\n\nAbstract\n\nWe introduce a game-theoretic approach to the study of recommendation systems\nwith strategic content providers. Such systems should be fair and stable. Showing\nthat traditional approaches fail to satisfy these requirements, we propose the Shap-\nley mediator. We show that the Shapley mediator ful\ufb01lls the fairness and stability\nrequirements, runs in linear time, and is the only economically ef\ufb01cient mechanism\nsatisfying these properties.\n\n1\n\nIntroduction\n\nRecommendation systems (RSs hereinafter) have rapidly developed over the past decade. By\npredicting a user preference for an item, RSs have been successfully applied in a variety of applications.\nMoreover, the amazing RSs offered by giant e-tailers and e-marketing platforms, such as Amazon\nand Google, lie at the heart of online commerce and marketing on the web. However, current\nsigni\ufb01cant challenges faced by personal assistants (e.g. Cortana, Google Now and Alexa) and mobile\napplications go way beyond the practice of predicting the satisfaction levels of a user from a set of\noffered items. Such systems have to generate recommendations that satisfy the needs of both the end\nusers and other parties or stakeholders [8, 39]. Consider the following cases:\n\u2022 When Alice drives her car, her personal assistant runs the default navigation application. When\nshe makes a stop at a junction, the personal assistant may show Alice advertisements provided by\nneighborhood stores, or an update on the stock market status as provided by \ufb01nancial brokers. Each of\nthese pieces of information \u2014 the plain navigation content, the local advertisements and the \ufb01nancial\ninformation \u2014 are served by different content providers. These content providers are all competing\nover Alice\u2019s attention at a given point. The personal assistant is aware of Alice\u2019s satisfaction with\neach content, and needs to select the right content to show at a particular time.\n\u2022 Bob is reading news of the day on his mobile application. The application, aware of Bob\u2019s interests,\nis presenting news deemed most relevant to him. The news is augmented by advertisements, provided\nby competing content providers, as well as articles by independent reporters. The mobile application,\nbalancing Bob\u2019s taste and the interests of the content providers, determines the mix of content shown\nto Bob.\nIn these contexts, the RS integrates information from various providers, often sponsored content,\nwhich is probably relevant to the user. The content providers are strategic \u2014 namely, make decisions\nbased on the way the RS operates, aiming at maximizing their exposure. For instance, to draw Bob\u2019s\nattention, a content provider strategically selects the topic of her news item, aiming at maximizing the\nexposure to her item. On the one hand, fair content provider treatment is critical for smooth ef\ufb01cient\nuse of the system and also to maintained content provider engagement over time. On the other hand,\nthe strategic behavior of the content providers may lead to instability of the system: a content provider\n\n32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montr\u00e9al, Canada.\n\n\fmight choose to adjust the content she offers in order to increase the expected number of displays to\nthe users, assuming the others stick to their offered contents.\nIn this paper, we study ways of overcoming this dilemma using canonical concepts in game theory to\nimpose two requirements on the RS: fairness and stability. Fairness is formalized as the requirement\nof satisfying fairness-related properties, and stability is de\ufb01ned as the existence of a pure Nash\nequilibrium. Analyzing RSs that satisfy these two requirements is the main goal of this paper.\nOur \ufb01rst result is that traditional RSs fail to satisfy both of the above requirements. Traditional RSs\nare complete, in the sense that they always show some content to the user, but it turns out that this\ncompleteness property cannot be satis\ufb01ed simultaneously with the fairness and equilibrium existence\nrequirements. This impossibility result is striking and calls for a search of a fair and stable RS.\nTo do so, we model the setting as a cooperative game, binding content provider payoffs with user\nsatisfaction. We resort to a core solution concept in cooperative game theory, the Shapley value [35],\nwhich is a celebrated mechanism for value distribution in game-theoretic contexts (see, e.g., [28]).\nIn our work, it is proposed as a tool for recommendations, namely for setting display probabilities.\nSince the Shapley value is employed in countless settings for fair allocation, it is not surprising that\nit satis\ufb01es our fairness properties. In addition, we prove that the related RS, termed the Shapley\nmediator, does satisfy the stability requirement. In particular, we show that the Shapley mediator\npossesses a potential function [27], and therefore any better-response learning dynamics converge to\nan equilibrium (see, e.g., [13, 17]). Note that this far exceeds our minimal stability requirement from\nthe RS.\nImplementation in commercial products would require the mediator to be computationally tractable.\nThe mediator interacts with users; hence a fast response is of great importance. In another major\nresult, we show that the Shapley mediator has a computationally ef\ufb01cient implementation. The\nlatter is in contrast to the intractability of the Shapley value in classical game-theoretic contexts [14].\nAnother essential property of the Shapley mediator is economic ef\ufb01ciency [36]. Unlike cooperative\ngames, where the Shapley value can be characterized as the only solution concept to satisfy properties\nequivalent to fairness and economic ef\ufb01ciency, in our setting the Shapley mediator is not characterized\nsolely by fairness and economic ef\ufb01ciency. Namely, one can \ufb01nd other simple mediators that satisfy\nthese two properties. However, we show that the Shapley mediator is the unique mediator to satisfy\nthe fairness, economic ef\ufb01ciency and stability requirements. Importantly, our study stems from\na rigorous de\ufb01nition of the minimal requirements from an RS, and so characterizes a unique RS.\nInterested in understanding the rami\ufb01cation on user utility, we introduce a rigorous analysis of user\nutility in (strategic) recommendation systems, and show that the Shapley mediator is not inferior to\ntraditional approaches.\nDue to space constraints, the proofs are deferred to the full version [5].\n\n1.1 Related work\n\nThis work contributes to three interacting topics: fairness in general machine learning, multi-\nstakeholder RSs and game theory.\nThe topic of fairness is receiving increasing attention in machine learning [6, 12, 30, 32] and data\nmining [23]. A major line of research is discrimination aware classi\ufb01cation [16, 19, 21, 38], where\nclassi\ufb01cation algorithms must maintain high predictive accuracy without discriminating on the basis\nof a variable representing membership in a protected class, e.g. ethnicity. In the context of RSs,\nthe work of Kamishima et al. [20, 22] addresses a different aspect of fairness (or lack thereof): bias\ntowards popular items. The authors propose a collaborative \ufb01ltering model which takes into account\nviewpoints given by users, thereby tackling the tendency for popular items to be recommended more\nfrequently, a problem posed in [29]. A related problem is over-specialization, i.e., the tendency to\nrecommend items similar to those already purchased or liked in the past, which is addressed in [1].\nZheng [39] surveys multi-stakeholder RSs, and highlights practical applications. Examples include\nRSs for sharing economies (e.g. AirBnB, Uber, etc.), online dating [31], and recruiting [37]. Burke\n[8] discusses fairness in multi-stakeholder RSs, and presents a taxonomy of classes of fairness-aware\nRSs. The author distinguishes between user fairness, content provider fairness and pairwise fairness,\nand reviews applications for these fairness types. A practical problem concerning fairness in multi-\nstakeholder RSs is discussed in [26]. In their work, an online platform is used by users who play two\nroles: customers seeking recommendations and content providers aiming for exposure. They report,\n\n2\n\n\fTable 1: Consider an arbitrary game, a \ufb01xed strategy pro\ufb01le X and an arbitrary user ui. TOP\nselects uniformly among the players that satisfy ui the most. The Bradley-Terry-Luce mediator\n[7, 25], or simply BTL, selects player j w.p. proportional to her satisfaction level over the sum of\nsatisfaction levels. NONE displays no item, and RAND selects uniformly among players with a\npositive satisfaction level. Both TOP and BTL satisfy F, but do not satisfy S. NONE and RAND\nsatisfy S, but do not satisfy F. The bottom line refers to the Shapley mediator, SM, which is de\ufb01ned\nand analyzed in Section 3. In contrast to the other mediators, SM satis\ufb01es both F and S.\n\nPROBABILITY COMPUTATION\n\nP (M(X, ui) = j)\nj\u2208arg max\n1\nj(cid:48) )\nj(cid:48) \u03c3i(X\n|arg maxj(cid:48) \u03c3i(Xj(cid:48) )|\n(cid:80)N\n\u03c3i(Xj(cid:48) )\n0\n(cid:80)N\n\n\u03c3i(Xj )>0\n\n\u03c3i(Xj )\n\nj(cid:48)=1\n\n1\nj(cid:48)=1\n\nMEDIATOR\n\nTOP\n\nBTL\nNONE\nRAND\n\n1\n\n\u03c3i(X\n\nj(cid:48) )>0\n\nSM\n(SECTION 3)\n\nEQUATION (2)\n\nFAIRNESS (F)\n\u221a\n\u221a\n\u00d7\n\u00d7\n\u221a\n\nSTABILITY (S)\n\n\u00d7 (THEOREM 1)\n\u00d7 (THEOREM 1)\n\u221a\n\u221a\n\n\u221a\n\n(THEOREM 2)\n\nbased on empirical evidence, that collaborative \ufb01ltering techniques tend to create rich-gets-richer\nscenarios, and propose a method for re-ranking scores, in order to improve exposure distribution\nacross the content providers.\nNote that all the work above considers traditional machine learning tasks that enforce upon the solution\nsome form of fairness, as de\ufb01ned speci\ufb01cally for each task. They suggest additional considerations,\nbut do not consider that the parties (i.e., users, content providers) will change their behavior as a\nresult of the new mechanism, nor examine the game theoretic aspects imposed by the selection of the\nRS in a formal manner. To the best of our knowledge, our work is the \ufb01rst to suggest a fully grounded\napproach to content provider fairness in RSs.\nFinally, strategic aspects of classical machine learning tasks were also introduced recently [3, 4].\nThe idea that a recommendation algorithm affects content-provider policy, and as a result must be\naccompanied by a game-theoretic study is key to recent works in search/information retrieval [2, 33];\nso far, however, such work has not dealt with the issue of fairness.\n\n2 Problem formulation\n\nFrom here on, our ideas will be exempli\ufb01ed in the following motivational example: a mobile applica-\ntion (or simply app) is providing users with valuable content. A set of players (advertisers) publish\ntheir items (advertisements) on the app. When a user enters the app, a mediator (RS/advertising\nengine) decides whether to display an item to that user or not, and which player\u2019s item to display.\nThe reader should notice that while we use that motivation for the purpose of exposition, our formal\nmodel and results are applicable to a whole range of RSs with strategic content providers.\nFormally, the recommendation game is de\ufb01ned as follows:\n\u2022 A set of users U = {u1, . . . , un}, a set of players [N ] = {1, . . . N}, and a mediator M.\n\u2022 The set of items (e.g. possible ad formats/messages to select from) available to player j is denoted\nby Lj, which we assume to be \ufb01nite. A strategy of player j is an item from Lj.\n\n\u2022 Each user ui has a satisfaction function \u03c3i : L \u2192 [0, 1], where L =(cid:83)N\n\nj=1 Lj is the set of all\n\navailable items. In general, \u03c3i(l) measures the satisfaction level of ui w.r.t. l.\n\u2022 When triggered by the app, M decides which item to display, if any. Formally, given the strategy\npro\ufb01le X = (X1, . . . , XN ) and a user ui, M(X, ui) is a distribution over [N ] \u222a {\u2205}, where \u2205\nsymbolizes maintaining the plain content of the app. That is, displaying no item at all. We refer to\n\n3\n\n\fP (M(X, ui) = j) as the probability that player j\u2019s item will be displayed to ui under the strategy\npro\ufb01le X.\n\u2022 Each player gets one monetary unit when her item is displayed to a user. Therefore, the expected\n\u2022 The social welfare of the players under the strategy pro\ufb01le X is the expected number of displays,\n\npayoff of player j under the strategy pro\ufb01le X is \u03c0j(X) =(cid:80)n\nV (X) =(cid:80)N\n\nP (M(X, ui) = j).\n\ni=1\n\nj=1 \u03c0j(X).\n\nFor ease of notation, we shall sometimes refer to \u03c3i(X) as the maximum satisfaction level of user ui\nfrom the items in X, i.e., \u03c3i(X) = maxj \u03c3i(Xj).\nWe demonstrate our setting with the following example.\nExample 1. Consider a game with two players and three users. Let L1 = {l1, l2},L2 = {l3} such\nthat the satisfaction levels of the users with respect to the items are\n\n(cid:34) u1\n\n0.1\n0.8\n0.9\n\nl1\nl2\nl3\n\n(cid:35)\n\n.\n\nu2\n0.9\n0.7\n0.8\n\nu3\n0.2\n0.9\n0.1\n\nConsider a mediator displaying each user with the most satisfying item to her taste, denoted by TOP.\nFor example, P(TOP ((l1, l3), u1) = 2) = 1, since \u03c31(l3) = 0.9 > \u03c31(l1) = 0.1. The pro\ufb01le (l1, l3)\nwill probably be materialized in realistic scenarios, since the payoff of player 1 under the strategy\npro\ufb01le (l2, l3) is \u03c01(l2, l3) = 1, while \u03c01(l1, l3) = 2. Notice that from the users\u2019 perspective,1 this\ni=1 \u03c3i(l2, l3) = 2.6;\n\ni=1 \u03c3i(l1, l3) = 0.9 + 0.9 + 0.2 = 2, while(cid:80)3\n\npro\ufb01le is not optimal, since(cid:80)3\n\nhence, the users suffer from strategic behavior of the players.\n\nAfter de\ufb01ning general recommendation games, we now present a few properties that one may desire\nfrom a mediator. First and foremost, a mediator has to be fair. The following is a minimal set of\nfairness properties:\nNull Player. If \u03c3i(Xj) = 0, then it holds that P (M(X, ui) = j) = 0. Informally, an item will not\nbe displayed to ui if it has zero satisfaction level w.r.t. him.\nSymmetry. If ui has the same satisfaction level from two items, they will be displayed with the same\nprobability. Put differently, if \u03c3i(Xj) = \u03c3i(Xm), then P (M(X, ui) = j) = P (M(X, ui) = m).\nUser-Independence. Given the selected items, the display probabilities depend only on the user: if\nuser ui(cid:48) is removed from/added to U, P (M(X, ui) = j) will not change, i.e.,\nP (M(X, ui) = j) = P (M(X, ui) = j | ui(cid:48) \u2208 U) .\nLeader Monotonicity. M displays the most satisfying items (w.r.t.\na speci\ufb01c user) with\nhigher probability than it displays other items. Formally, if j \u2208 arg maxj(cid:48)\u2208[N ] \u03c3i(Xj(cid:48)) and\nm /\u2208 arg maxj(cid:48)\u2208[N ] \u03c3i(Xj(cid:48)), then P (M(X, ui) = j) > P (M(X, ui) = m).\nFor brevity, we denote the above set of fairness properties by F. In addition, an essential property\nin a system with self-motivated participants is that it will be stable. Instability in such systems is\na result of a player aiming to improve her payoff given the items selected by others. A minimal\nrequirement in this regard is stability against unilateral deviations as captured by the celebrated\npure Nash equilibrium concept, herein denoted PNE. A strategy pro\ufb01le X = (X1, . . . , XN ) is\ncalled a pure Nash equilibrium if for every player j \u2208 [N ] and any strategy X(cid:48)\nj \u2208 Lj it holds that\n\u03c0j(Xj, X\u2212j) \u2265 \u03c0j(X(cid:48)\nj, X\u2212j), where X\u2212j denotes the vector X of all strategies, but with the j-th\ncomponent deleted. We use the notion of PNE to formalize the stability requirement:\n\nStability. Under any set of players, available items, users and user satisfaction functions, the game\ninduced by M possesses a PNE.\nFor brevity, we denote this property by S. The goal of this paper is to devise a computationally\ntractable mediator that satis\ufb01es both F and S.2\n\n1For a formal de\ufb01nition of the user utility, see Subsection 6.2.\n2One may require the convergence of any better-response dynamics, thereby allowing the players to learn the\n\nenvironment. In Section 3 we show that our solution satis\ufb01es this stronger notion of stability as well.\n\n4\n\n\f2.1\n\nImpossibility of classical approaches\n\nWe highlight a few benchmark mediators in Table 1, including TOP, which was introduced informally\nin Example 1. Another interesting mediator is BTL, which follows the lines of the Bradley-Terry-\nLuce model [7, 25]. BTL is addressed here as a representative of a wide family of weight-based\nmediators: mediators that distribute display probability according to weights, determined by a\nmonotonically increasing function of the user satisfaction (e.g., softmax). Common to TOP, BTL\nand any other weight-based mediator, is that an item is displayed to a user with probability 1.3 We\nmodel this property as follows.\n\nComplete. For any recommendation game and any strategy pro\ufb01le X,(cid:80)N\n\nP (M(X, ui) = j) =\n\n1.\nSince the goal of an RS is to provide useful content to users, satisfying Complete seems justi\ufb01ed.\nAlthough it seems unreasonable to avoid showing any content to a certain user at a certain time, it\nturns out that this avoidance is crucial in order to satisfy our requirements.\nTheorem 1. No mediator can satisfy F, S and Complete.\n\nj=1\n\nProof sketch. We construct a game with two players, three users and three strategies, and show that\nno mediator can satisfy F, S and Complete. Importantly, our technique can be used to show that any\narbitrary game does not possess a PNE or that a slight modi\ufb01cation of this game does not possess a\nPNE.\n\nConsider the following satisfaction matrix:(cid:34) u1\n\n(cid:35)\n\n,\n\nu2\ny\n0\nx\n\nu3\nx\ny\n0\n\n0\nx\ny\n\nl1\nl2\nl3\n\nwhere (x, y) \u2208 (0, 1]2. Let L1 = L2 = {l1, l2, l3} (i.e., a symmetric two-player game). By using\nthe properties of F we characterize the structure of the induced normal form game. We show that\nin this normal form game, a PNE only exists if P (M((l2, l3), u1) = 1) = 0.5 (and similarly to\nthe other users and strategy pro\ufb01les, due to User-Independence). Since this holds for every x\nand y, the mediator displays a random item for each user under any strategy pro\ufb01le. Recall that a\nrandom selection does not satisfy Leader Monotonicity; hence, no mediator can satisfy F, S and\nComplete.\n\nabilities, i.e., (cid:80)N\n(cid:80)N\n\nMoreover, Theorem 1 is not sensitive to the sum of the display probabilities being equal to\n1. One can show a similar argument for any mediator that displays items with constant prob-\nP (M(X, ui) = j) = c for some 0 < c \u2264 1. Theorem 1 suggests that\nP (M(X, ui) = j) should be bounded to the user satisfaction levels. In the next section,\n\nj=1\n\nj=1\n\nwe show a novel way of doing so.\n\n3 Our approach: the Shapley mediator\n\nIn order to provide a fair and stable mediator, we resort to cooperative game theory. Informally,\na cooperative game consists of two elements: a set of players [N ] and a characteristic function\nv : 2[N ] \u2192 R, where v determines the value given to every coalition, i.e., every subset of players.\nThe analysis of cooperative games focuses on how the collective payoff of a coalition should be\ndistributed among its members.\nOne core solution concept in cooperative game theory is the Shapley value [35].\nDe\ufb01nition 1 (Shapley value). Let (v, [N ]) be a cooperative game such that v(\u2205) = 0. According to\nthe Shapley value, the amount that player j gets is\n\n(cid:88)\n\n(cid:0)v(P R\n\n1\nN !\n\nR\u2208\u03a0([N ])\n\nj )(cid:1),\n\nj \u222a {j}) \u2212 v(P R\n\n(1)\n\n3Perhaps excluding pro\ufb01les X where \u03c3i(X) = 0. We allow M to behave arbitrarily in this case.\n\n5\n\n\fwhere \u03a0([N ]) is the set of all permutations of [N ] and P R\nj\nplayer j in the permutation R.\n\nis the set of players in [N ] which precede\n\nOne way to describe the Shapley value, is by imagining the process in which coalitions are formed:\nwhen player j joins coalition C, she demands her contribution to the collective payoff of the coalition,\nnamely v(C\u222a{j})\u2212 v(C). Equation (1) is simply summing over all such possible demands, assuming\nthat all coalitions are equally likely to occur.\nFor our purposes, we \ufb01x a strategy pro\ufb01le X, and focus on an arbitrary user ui. How should a\nmediator assign the probabilities of being displayed in a fair fashion? The induced cooperative game\ncontains the same set of players. For every C \u2286 [N ], let XC denote the strategy pro\ufb01le where all\nplayers missing from C are removed. We de\ufb01ne the characteristic function of the induced cooperative\ngame as\n\nvi(C; X) = \u03c3i(XC),\n\nwhere \u03c3i(XC) is the maximal satisfaction level a user ui may obtain from the items chosen by the\nmembers of C. Indeed, this formulation represents a collaborative behavior of the players, when they\naim to maximize the satisfaction of ui. Observe that vi(\u00b7; X) : 2[N ] \u2192 R is a valid characteristic\nfunction, hence (vi(\u00b7; X), [N ]) is a well de\ufb01ned cooperative game. Note that the selection of a\nmediator fully determines the probability of the events M(X, ui) = j, and vice versa. The mediator\nthat sets the probability of the event M(X, ui) = j according to the Shapley value of the induced\ncooperative game (vi(\u00b7; X), [N ]) is hereinafter referred to as the Shapley mediator, or SM for\nabbreviation.\n\n3.1 Properties of the Shapley mediator\n\nSince the Shapley value is employed in countless settings for fair allocation, it is not surprising that it\nsatis\ufb01es our fairness properties.\nProposition 1. SM satis\ufb01es F.\n\nfunction \u03a6 :(cid:81)\n\nj Lj \u2192 R such that for any strategy pro\ufb01le X = (X1, . . . , XN ) \u2208(cid:81)\n\nWe now show that recommendation games with SM possess a PNE. This is done using the notion\nof potential games [27]. A non-cooperative game is called an exact potential game if there exists a\nj Lj, any player\nj and any strategy X(cid:48)\nj, the change in her payoff\nfunction equals the change in \u03a6, i.e.,\n\nj \u2208 Lj, whenever player j switches from Xj to X(cid:48)\nj, X\u2212j) = \u03c0j(Xj, X\u2212j) \u2212 \u03c0j(X(cid:48)\n\n\u03a6(Xj, X\u2212j) \u2212 \u03a6(X(cid:48)\n\nj, X\u2212j).\n\nThis brings us to the main result of this section:\nTheorem 2. Recommendation games with the Shapley mediator are exact potential games.\n\nThus, due to Monderer and Shapley [27], any recommendation game with the Shapley mediator\npossesses at least one PNE, and the set of pure Nash equilibria corresponds to the set of argmax\npoints of the potential function; therefore, SM satis\ufb01es S.\nCorollary 1. SM satis\ufb01es S.\n\nIn fact, Theorem 2 proves a much stronger claim than merely the existence of PNE. A better-response\ndynamics is a sequential process, where in each iteration an arbitrary player unilaterally deviates to a\nstrategy which increases her payoff.\nCorollary 2. In recommendation games with the Shapley mediator, any better-response dynamics\nconverges.\n\nThis convergence guarantee allows the players to learn which items to pick in order to maximize\ntheir payoffs. Indeed, as has been observed by work on the topic of online recommendation and\nadvertising systems (e.g. sponsored search [10]), convergence to PNE is essential for system stability,\nas otherwise inef\ufb01cient \ufb02uctuations may occur.\n\n4 Linear time implementation\n\nIn Section 3 we showed that the Shapley mediator, SM, satis\ufb01es F and S. Therefore, it ful\ufb01lls our\nrequirements stated in Section 2. However, implementation in commercial products would require\n\n6\n\n\fthe mediator to be computationally tractable. The mediator interacts with users; hence a fast response\nis of great importance. In general, since Equation (1) includes 2N summands, the computation of the\nShapley value in a cooperative game need not be tractable. Indeed, the computation often involves\nmarginal contribution nets [11, 18]. In the following theorem we derive a closed-form formula for\ncalculating the display probabilities under the Shapley mediator, which allows it to compute the\ndisplay probabilities in linear time.\nTheorem 3. Let X be a strategy pro\ufb01le, and let \u03c3m\ni (X) denote the m\u2019th entry in the result of sorting\n(\u03c3i(X1), . . . , \u03c3i(XN )) in ascending order, preserving duplicate elements. The Shapley mediator\ndisplays player j\u2019s item to a user ui with probability\n\nP (SM(X, ui) = j) =\n\ni (X) \u2212 \u03c3m\u22121\n\u03c3m\nN \u2212 m + 1\n\ni\n\n(X)\n\n,\n\n(2)\n\ni (X)(cid:88)\n\n\u03c1j\n\nm=1\n\nwhere \u03c30\n\ni (X) = 0, and \u03c1j\n\ni (X) is an index such that \u03c3i(Xj) = \u03c3\u03c1j\n\ni\n\ni (X)\n\n(X).\n\nThe Shapley mediator is implemented in Algorithm 1. As an input, it receives a strategy pro\ufb01le and\na user, or equivalently user satisfaction levels from that strategy pro\ufb01le. It outputs a player\u2019s item\nwith a probability equal to her Shapley value in the cooperative game de\ufb01ned above. Note that the\nrun-time of Algorithm 1 is linear in the number of players, i.e., O(N ). A direct result from Theorem\n3 and User-Independence (see Section 2) is that player payoffs can be calculated ef\ufb01ciently.\nCorollary 3. In recommendation games with the Shapley mediator, the payoff of player j under the\n\nstrategy pro\ufb01le X is given by \u03c0j(X) =(cid:80)n\n\n(cid:80)\u03c1j\n\ni (X)\u2212\u03c3m\u22121\n\u03c3m\nN\u2212m+1\n\ni (X)\nm=1\n\n(X)\n\n.\n\ni=1\n\ni\n\nTo facilitate understanding of the Shapley mediator and its fast computation, we reconsider Example\n1 above.\nExample 2. Consider the game given in Example 1. According to the Shapley mediator, the display\nprobabilities of player 1 under the strategy pro\ufb01le X = (l2, l3) are\n\nP(SM (X, u1) = 1) =\n\nP(SM (X, u2) = 1) =\n\nP(SM (X, u3) = 1) =\n\n2\n\n1 (X) \u2212 \u03c30\n\u03c31\n2 (X) \u2212 \u03c30\n\u03c31\n3 (X) \u2212 \u03c30\n\u03c31\n\n2\n\n1 (X)\n\n2 (X)\n\n3 (X)\n\n2\n\n=\n\n=\n\n+\n\n= 0.4,\n\n2\n\n0.8 \u2212 0\n0.7 \u2212 0\n2\n3 (X) \u2212 \u03c31\n\u03c32\n\n= 0.35,\n\n3 (X)\n\n1\n\n0.1 \u2212 0\n\n2\n\n+\n\n=\n\n0.9 \u2212 0.1\n\n1\n\n= 0.85.\n\nIt follows that \u03c01(l2, l3) = 8\n10, and the pro\ufb01le to be materialized is (l2, l3).\nIndeed, it can be veri\ufb01ed that this is the unique PNE of the corresponding game. Moreover, while the\nunique PNE under TOP (see Example 1 in Section 2) results in a user utility of 2, the unique PNE\nunder the Shapley mediator results in user utility of\n\n5 while \u03c01(l1, l3) = 7\n\n(\u03c3i(l2)P(SM ((l2, l3), ui) = 1)) + (\u03c3i(l3)P(SM ((l2, l3), ui) = 2)) = 2.145 > 2.\n\ni=1\n\nHence, the users bene\ufb01t from the Shapley mediator is greater than from the TOP mediator. This is in\naddition to the main property of the Shapley mediator, probabilistic selection according to the central\nmeasure of fair allocation.\n\n3(cid:88)\n\nAs analyzed in Subsection 2.1, Theorem 1 suggests that a mediator cannot satisfy both F and S if\nP (M(X, ui) = j) is constant. One way of determining\n\nEf\ufb01ciency. The probability of displaying an item to ui is the maximal satisfaction level ui may\n\nP (M(X, ui) = j) = \u03c3i(X).\n\n5 Uniqueness of the Shapley mediator\n\nit sets the probabilities such that(cid:80)N\n(cid:80)N\nobtain from the items chosen in X. Formally,(cid:80)N\n\nP (M(X, ui) = j) is de\ufb01ned as follows.\n\nj=1\n\nj=1\n\nj=1\n\n7\n\n\fAlgorithm 1: Shapley Mediator\nInput: A strategy pro\ufb01le X = (X1, . . . , XN ) and a user ui\nOutput: An element from {\u2205, X1, . . . , XN}\n1 Pick Y uniformly at random from (0, 1)\n2 if Y > maxj\u2208[N ] \u03c3i(Xj) then\n3\n4 else\n5\n\nreturn \u2205\nReturn an element uniformly at random from {Xj | j \u2208 [N ], \u03c3i(Xj) \u2265 Y }\n\nEf\ufb01ciency (for brevity, EF) binds player payoffs with the maximum satisfaction level of ui from the\nitems chosen by the players under X. It is well known [15, 35] that the Shapley value is uniquely\ncharacterized by properties equivalent to F and EF, when stated in terms of cooperative games. It is\ntherefore obvious that the Shapley mediator satis\ufb01es EF. 4 Thus, one would expect that the Shapley\nmediator will be the only mediator that satis\ufb01es F and EF. This is, however, not the case: consider a\nmediator that runs TOP w.p. \u03c3i(X) and NONE otherwise. Clearly, it satis\ufb01es F and EF. In fact,\ngiven a mediator M satisfying F and Complete, we can de\ufb01ne M(cid:48) such that\nP (M(cid:48)(X, ui) = j) = P (M(X, ui) = j) \u00b7 \u03c3i(X),\n\n(3)\nthereby obtaining a mediator satisfying F and EF. The question of uniqueness then arises: is S\nderived by satisfying F and EF? Or even more broadly, are there mediators that satisfy F, S and EF\nbesides the Shapley mediator? Had the answer been yes, this recipe for generating new mediators\nwould have allowed us to seek potentially better mediators, e.g., one satisfying F, S and EF while\nmaximizing user utility. However, as we show next, the Shapley mediator is unique in satisfying F, S\nand EF.\nTheorem 4. The only mediator satisfying F, S and EF is the Shapley mediator.\n\n6\n\nImplications of strategic behavior\n\nIn this section we examine the implications of strategic behavior of the players on their payoffs and\nuser utility. Comprehensive treatment of the integration of multiple stakeholders into recommendation\ncalculations was discussed only recently [9], and appears to be challenging. As our work is concerned\nwith strategic content providers, it is natural to consider the Price of Anarchy [24, 34], a common\ninef\ufb01ciency measure in non-cooperative games.\n\n6.1 Player payoffs\n\nscenario and the social welfare of the worst PNE. Formally, if EM \u2286(cid:81)\n\nThe Price of Anarchy, herein denoted P oA, measures the inef\ufb01ciency in terms of social welfare, as a\nresult of sel\ufb01sh behavior of the players. Speci\ufb01cally, it is the ratio between an optimal dictatorial\nj Lj is the set of PNE pro\ufb01les\nmaxX\u2208(cid:81)\nminX\u2208EM V (X) \u2265 1. We use the subscript M to stress\ninduced by a mediator M, then P oAM =\nthat the P oAM depends on the mediator, through the de\ufb01nition of social welfare function V and\nplayer payoffs. Notice that the P oA of a mediator that does not satisfy S can be unbounded, as a PNE\nmay not exist. Quantifying the P oA can be technically challenging; thus we restrict our analysis to\nP oASM, the P oA of the Shapley mediator.\nTheorem 5. P oASM \u2264 2N\u22121\nHence, under the Shapley mediator the social welfare of the players can decrease by at most a factor\nof 2, when compared to an optimal solution.\n\nN , and this bound is tight.\n\nj Lj\n\nV (X)\n\n6.2 User utility\n\nWe now examine the implications of using the Shapley mediator on the users. For that, we shall\nassume that the utility of a user from an item is his satisfaction level from that item. Namely, when\n\n4 See the proof of Proposition 1 in the appendix. Leader Monotonicity, as opposed to the other fairness\n\nproperties, is not one of Shapley\u2019s axioms but rather a byproduct of Shapley\u2019s characterization.\n\n8\n\n\fn(cid:88)\n\nN(cid:88)\n\nUM(X) =\n\nn(cid:88)\n\nitem l is displayed to ui, his utility is \u03c3i(l). As a result, the expected utility of the users under the\nstrategy pro\ufb01le X and a mediator M is de\ufb01ned by\n\nP (M(X, ui) = j) \u03c3i(Xj) +\n\nP (M(X, ui) = \u2205) \u03c3i(\u2205).\n\ni=1\n\nj=1\n\ni=1\n\nNote that the \ufb01rst term results from the displayed items, and the second term from the plain content of\nthe app (displaying no item at all). To quantify the inef\ufb01ciency of user utility due to sel\ufb01sh behavior\nof the players under M, we de\ufb01ne the User Price of Anarchy,\n\nU P oAM =\n\nmaxM(cid:48),X\u2208\u03a0N\n\nj=1Lj UM(cid:48)(X)\n\nminX\u2208EM UM(X)\n\n.\n\nThe U P oA serves as our benchmark for inef\ufb01ciency of user utility. The nominator is the best possible\ncase: the user utility under any mediator M(cid:48) and any strategy pro\ufb01le X. The denominator is the\nworst user utility under M, where EM is again the set of PNE pro\ufb01les induced by M. Note that the\nnominator is independent of M. We \ufb01rst treat users as having zero satisfaction when only the plain\ncontent is displayed, i.e., \u03c3i(\u2205) = 0, and consider the complementary case afterwards. The following\nis a negative result for the Shapley mediator.\nProposition 2. The User PoA of the Shapley mediator, U P oASM, is unbounded.\nProposition 2 questions the applicability of the Shapley mediator. An unavoidable consequence of\nits use is a potentially destructive effect on user utility. While content-provider fairness is essential,\nusers are the driving force of the RS. Therefore, one may advocate for other mediators that perform\nbetter with respect to user utility, albeit not necessarily satisfying S. If S is discarded and a mediator\nsatisfying Complete adopted, would this result in better user utility? Unfortunately, other mediators\nmay lead to a similar decrease in user utility due to strategic behavior of the players, so there appears\nto be no better solution in this regard.\nProposition 3. The User PoA of TOP, U P oATOP, is unbounded.\nUsing similar arguments, one can show that U P oABTL is unbounded as well.\nIn many situations, it is reasonable to assume that when no item is displayed to a user, his utility is\n1. Namely, \u03c3i(\u2205) = 1 for every user ui. Indeed, this seems aligned with the ads-in-apps model: the\nuser is interrupted when an advertisement is displayed. We refer to this scenario as the optimal plain\ncontent case. From here on, we adopt this perspective for upper-bounding the U P oA. Observe that\nuser utility is therefore maximized when no item is displayed whatsoever. Nevertheless, displaying\nno item will also result in zero payoff for the players. Here too, U P oATOP is unbounded, while\nU P oANONE = 1. The following lemma bounds the User P oA of the Shapley mediator.\nLemma 1. In the optimal plain content case, it holds that U P oASM \u2264 4.\nIn fact, numerical calculations show that U P oASM is bounded by 1.76, see the appendix for further\ndiscussion.\n\n7 Discussion\n\nOur results are readily extendable in the following important direction (which is even further elabo-\nrated in the appendix). In many online scenarios, content providers typically customize the items they\noffer to accommodate speci\ufb01c individuals. Indeed, personalization is applied in a variety of \ufb01elds in\norder to improve user satisfaction. Speci\ufb01cally, consider the case where each player may promote a\nset of items, where different items may be targeted towards different users, and the size of this set is\ndetermined exogenously (e.g., by her budget). In this case, a player selects a set of items which she\nthen provides to the mediator. 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