Understanding Fashion Cycles as a Social Choice
Anish Das Sarma, Sreenivas Gollapudi, Rina Panigrahy, Li Zhang
UUnderstanding Fashion Cycles as a Social Choice
Anish Das Sarma , Sreenivas Gollapudi , Rina Panigrahy , Li Zhang Yahoo! Research, Microsoft Research [email protected] , { sreenig,lzha,rina } @microsoft.com Abstract
We present a formal model for studying fashion trends, in terms of three parameters of fashionableitems: (1) their innate utility; (2) individual boredom associated with repeated usage of an item; and(3) social influences associated with the preferences from other people. While there are several worksthat emphasize the effect of social influence in understanding fashion trends, in this paper we show howboredom plays a strong role in both individual and social choices. We show how boredom can be usedto explain the cyclic choices in several scenarios such as an individual who has to pick a restaurant tovisit every day, or a society that has to repeatedly ‘vote’ on a single fashion style from a collection. Weformally show that a society that votes for a single fashion style can be viewed as a single individualcycling through different choices.In our model, the utility of an item gets discounted by the amount of boredom that has accumulatedover the past; this boredom increases with every use of the item and decays exponentially when not used.We address the problem of optimally choosing items for usage, so as to maximize over-all satisfaction,i.e., composite utility, over a period of time. First we show that the simple greedy heuristic of alwayschoosing the item with the maximum current composite utility can be arbitrarily worse than the optimal.Second, we prove that even with just a single individual, determining the optimal strategy for choosingitems is NP-hard. Third, we show that a simple modification to the greedy algorithm that simply doublesthe boredom of each item is a provably close approximation to the optimal strategy. Finally, we presentan experimental study over real-world data collected from query logs to compare our algorithms. a r X i v : . [ c s . G T ] S e p Introduction
When an individual or a society is repeatedly presented with multiple substitutable choices, such as differentcolors of cars or different themes of musicals, we often observe a recurring shift of preferences over time,or commonly known as fashion trends . While some trends are relatively easy to explain (e.g., sweatersales increasing in the winter), some other trends may result from a variety of factors. In this paper, wefirst describe a utility model which we think may explain such trends. Then we study the computationalissues under the model and provide simple mechanisms by which consumers may make close to optimal decisions on which products to consume and when, in order to maximize their overall utility . We thenconduct experiments to show how various parameters in our model can be estimated and to validate ouralgorithm.Understanding fashion trends are of significant academic interests as well as commercial importancein various fields, including brand advertising and market economics. Therefore, there’s a large body ofwork in multiple disciplines – sociology (e.g. [4, 3]), economics (e.g. [7]), and marketing (e.g. [11, 12],on theories for evolution of fashion. Despite much study, there is a lack of a well accepted theory. Thisis probably not surprising as what makes us like or dislike an alternative and how that changes over timeinvolves economical, psychological, and social factors. Next we describe three such factors that influencefashion.First, and perhaps the most basic, cause of a product becoming trendy is its utility , intuitively capturingthe value it adds to an individual. We call this the innate utility of a product. Second, psychologically, aperson’s utility of consuming a product may be discounted by constant consumption of the same item — asone gets tired of existing products, he desires new and different ones. Third, while at an individual level, wehave certain inclinations based on our tastes, these are influenced by social phenomena, such as what we seearound us, friends’ and celebrities’ preferences.In this paper, we present a formal model that unifies the aforementioned three broad categories of factorsusing innate utility , individual boredom , and social influence (as depicted in Figure 1 and explained below).We attempt to construct a mathematical model for these factors and use the model to explain the formationof fashion trends. We use the term item to denote any product, good, concept, or object whose fashion trendwe are interested in.1. Innate utility:
The utility of an item captures the innate value the item provides to an individual. Weassume it is fixed, independent of other influences.2.
Individual boredom:
If we use any item for too long, we get bored of it, and our appreciation for itgoes down. This is modeled as a negative component added to the utility. This factor grows if onerepeatedly uses the same item and fades away when one stops consuming the item.3.
Social influence:
Our valuation of an item can change significantly by the valuation of our friends orinfluencing people. For example, when we see that many people around us like something we maystart liking it; or we may consciously want to differ from some other people around us. We modelsuch influences as a weighted linear combination from other people. To model boredom on any item at any given time t , we associate with each past usage of the item, say attime t (cid:48) , a factor in the form of (1 − r ) t − t (cid:48) for some r < . Then the total boredom on the item takes the sumof the this factor from all the past usage of the item. This definition captures the intuition that the boredomgrows if an item is repeatedly used. As we show in our experiments, such exponential decay model matcheswell people’s interests in songs and movies. The utility maximization under this model, albeit NP-hard,naturally displays cyclic patterns. We also provide a simple strategy to achieve near to optimal utility whenthe decay factor r is small.The effect of influence can be formalized using a linear model. For example, to model social influence,consider one item and a society consisting of m people. Let G denote the influence graph on these people Additional influence may come from the association of a product to things/concepts we like or dislike. For instance, someonemay be very fond of green technologies or dislike things that are scary. We may simply model such concepts as individuals.
Factors influencing fashion trends that is directed where each edge is labeled with a weight that indicates the strength of this influence. Ahigh value on an edge, such as outgoing edges from celebrities indicates a strong outgoing influence; on theother hand a negative value indicates a desire to distance oneself or be different from the source node. Let A denote the corresponding influence matrix. let u i ( t ) denote the utility of the item to the i th person; let u ( t ) denote the vector of utilities. If we assume that for each time step the influence from all friends of a personadd linearly then we may write u ( t + 1) = A u ( t ) , which is similar to [11]. Note that for stability of thisiterative powers, we should assume that its top eigenvector has magnitude .. We will show that under thisinfluence model, we may treat the society as an individual making choices under the effect of boredom. Discussion of our results.
We argue that fashion trends can be viewed as not just the effect of the influenceof a privileged few but more as a democratic process that churns the social boredom and channels the innateinstinct for change. Boredom is the innate psychological force that dulls the effect of a constant stimulusover a period of time and make us look for newer stimuli. It is well known that the mind tends to growoblivious to almost all types of sensations (visual, olfactory, touch, sound) to which it is exposed for a longtime. Thus the ‘coolness’ of a fashionable item drops over time and things that we haven’t seen or used in along time begin to appear more ‘cool’.We show how several scenarios involving individual and social choices are essentially driven by thesame underlying principles. The individual choice may be as simple as choosing a restaurant to visit ona particular day. Alternatively, it may be a social choice where the market forces of a society ‘chooses’different fashions such as styles for clothing, or cars. Or, a news channel is picking the front page newsarticle to maximize readership and has to choose from different types of news articles, e.g., politics, natural-disaster, celebrity gossip. Each news item may be popular or fashionable for a period of time and thenboredom sinks in and the media may switch focus to a different event probably of an entirely different type.Boredom is thus the single most and simplest explanation for oscillations in individual and social choices.This is not at all surprising; indeed boredom is perhaps a strong influence when we make choices such asfood, clothes, fashions, governments. Social influence no doubt plays a large part in individual choices. Butwhen we look at the social system as a whole the influences across individuals are forces within the systemand in the net effect it simply gives a larger voice to the more influential individuals. We also note thatinfluence by itself is not sufficient to create fashion cycles. In fact, if all influences are positive then withoutany boredom the system u ( t + 1) = A u ( t ) converges to a fixed value resulting in a fixed fashion choice.Finally, we recognize that the factors we consider are by no means comprehensive; several other ‘exter-nal’ factors may change the values of nodes. For example a shortage of oil may increase the utility of greentechnologies, the strength of the edges in the graph may change, the structure of the graph may change withnew node and edge formations. Our decay model for boredom and linear model for influence may be toosimplistic. Nonetheless, we believe the influence graph and boredom capture several important aspects ofthe underlying psychological processes that people use to value items. Outline of the paper.
All the main theoretical results achieved by this paper are presented in Section 2, with2roofs appearing in Section 3. Section 4 presents detailed experimental results for validating our model andalgorithms; our experiments use real-world data from Google Trends [1] on the popularity of songs andmovies in the last 3 years. Related work is presented next and we conclude in Section 5.
Related work.
There are several theories of fashion evolution in various communities, e.g., sociologistshave modeled fashion trends as a collection of several social forces such as differentiation, influence, andassociation. While there have been several explanations of cycles in fashion trends [3, 4, 5, 7, 12], most pastwork does not offer a formal study. We compare our work with one notable exception [11] next. The focusof our paper is on understanding the impact of various factors—boredom, association rules, and utility—onthe fashion choices made by individuals and a society. We explain the existence of cycles based on ourformal model of fashion, and provide algorithms for making optimal choices.Reference [11] proposed a formal model of fashion based on association rules. Intuitively, an individ-ual’s utility for an item is impacted by how similar it is to items he likes, and how dissimilar it is to itemshe dislikes. Further, he is influenced by the society through other individuals’ preferences for various items.Consider a single item, whose consumption vector is given by c ( t ) at time t . Considering the recurrence c ( t +1) = W c ( t ) , where W is the weight influence matrix, [11] observed that if the matrix W has a complextop eigenvalue (corresponding to negative influences), then the item’s consumption pattern may be periodic,producing cycles in preferences. Our model of utility is similar to the consumption model in [11]. However,we consider an additional parameter of boredom that is essential to explain fashion cycles in a society withnon-negative influences as such a matrix W always has a real top eigenvalue.Some other recent work (e.g., [6]) study behavioral influences in social networks, such as in termsof information propagation. For instance, [8] studies how two competing products spread in society, [9]provides techniques for tracking and representing “memes”, which may be used to analyze news cycles,and [10] studies how recommendations propagate in a network through social influence.The focus of our paper is on formalizing a practical theory for fashion trends with boredom, combinedwith utility and a simple social behavior. Therefore, for a large part of the paper we consider only a singleindividual and study fashion trends based on boredom, and utility. Further, in our extension to multipleindividuals, we assume a linear weighting of influences from friends’ preferences for particular items. We consider a user living in discrete time periods , , . . . and consuming one item among n substitutableitems at each time; for example, a person needs to decide which restaurant to go to every night or whichpolitical party to vote for every four years. We assume that each item i brings a base utility v i to the user.Now if we assume that the utilities are fixed then the user would always choose the same item with themaximum v i . This would be inconsistent with the observed common behavior of cycling among multipleitems, which we refer to as fashion cycles. In order to explain fashion cycles, it is necessary to model theutility dependence of the consumptions across different time periods.We propose a simple model in which the utility of an item at any time t is the base utility discounted bya boredom factor proportional to the “memory” the person has developed by using this item in the past. Themore the user has used the item, the more memory and boredom is developed for the item, and consequentlythe less utility the item has to the user.We naturally assume that the memory drops geometrically over time, and the total memory of a personis bounded. This leads to the following definition of memory. Let < r < be a memory decay rate, i.e.,the rate at which a person “forgets” about things. Let x i ( t ) ∈ { , } indicate if the user uses the item i attime t . Then the memory of i at time t is M i ( t ) = r (cid:80) t − τ =0 x i ( τ )(1 − r ) t − τ . We add the factor r so that M i ( t ) ≤ . The boredom b i ( t ) = α i M i ( t ) is proportional to the memory and depends on the item. Theutility of item i is defined as u i ( t ) = v i − b i ( t ) = v i − α i M i ( t ) . Henceforth, we will refer to v as the baseutility and α as the boredom coefficient. 3 .2 Utility optimization with boredom With the above model, one natural question is to compute the choices of the items to maximize the user’soverall utility. If we allow the user to choose at continuous time, the maximization problem becomes rela-tively easy as the best way to consume an item is to do it cyclically at regular time intervals. However, suchregular placement may not be realizable or is hard to find. As we will show below, it is NP-hard to computethe best consumption sequence.We also consider the natural greedy strategy and show that the under the greedy strategy, the utility ofeach item is always bounded in a narrow band and so each item is consumed approximately cyclically. Thegreedy strategy, however, may have produce a sequence giving poor overall utility. We provide a simpleheuristics, called double-greedy strategy, and show that it emulates the cyclic pattern of the optimal solutionon the real line and yields utility close to the optimal when r is small. In the greedy strategy, at each time t , the user consumes the item with the maximum utility u i ( t ) . Thisstrategy is intuitive and probably consistent with how we make our daily decisions. We show that the utilitygap between any two items is small all the time. We provide an example to show it has poor performance interms of utility maximization. Denote by α = max i α i . Theorem 2.1.
There exists a time T such that for any t ≥ T , u ≤ max i u i ( t ) ≤ u + O ( rα log n ) where u is the unique solution to the following system:For all items with f i > , the quantity v i − f i α i = µ ; if f i = 0 then v i < µ ; and (cid:80) f i = 1 . While the greedy algorithm has the nice property of keeping the utility gap between any items small, itmay produce a sequence with poor overall utility.
Observation 2.1.
The Greedy strategy of always picking the highest utility item each day is not optimal.
To see the non-optimality of greedy, simply consider two items for beverage, say “water” and “soda”.Assume water has low base utility say that never changes and zero boredom coefficient. Soda on the otherhand has high utility say but also a high boredom coefficent say . So if one drank soda every day itsutility would drop to below that of water. Observe that the greedy strategy will choose soda till its utilitydrops to that of water and then it is chosen whenever its utility rises even slightly over . So the averageutility of the greedy strategy is close to . A smarter strategy is to hold off on the soda even if it is a betterchoice today so as to enjoy it even more on a later day. Thus it is possible to derive an average utility thatis much higher than . For example, we can get average utility of about by alternating between water andsoda in the above example. Note that the greedy algorithm produces poor performance in the above exampleeven for small r .This naturally raises the question: what is the optimal strategy? More importantly is there an optimalstrategy that is a simple ’rule of thumb’ that is easy to remember and employ as we make the daily choices.Unfortunately it turns out that computing the optimal strategy is NP-hard. Given a period T , target utility U ∗ , and n items, it is NP-hard to determine whether thereexists a selection of items with period T such that the total utility of the selection is at least U ∗ . On the positive side we show that there is indeed a simple “rule of thumb” that gives an almost optimalsolution when r is small. The strategy “double-greedy” waits longer for items that we get bored of too4uickly. It is a simple twist on the greedy strategy: instead of picking the item that maximizes the utility u i ( t ) = u i − b i ( t ) , it picks the one which maximizes w i ( t ) = u i − b i ( t ) . Thus it doubles the boredom ofall items and then runs the greedy strategy. We show that: Theorem 2.3.
Let U denote the average utility obtained by the double greedy algorithm and U ∗ the optimalutility. Then U ≥ U ∗ − O ( rα log n ) where α = max i α i . We note that when r → , the utility produced by double greedy is close to the optimal solution. A choice is a fashion, if it is the choice of a large fraction of the society. Thus a society only supports asmall number of fashions. Industries often target one type of fashion for each market segment. Consider asituation where the entire society consists of one fashion market segment. We will see how in this case sucha society can be compared to an individual making choices to maximize utility under the effect of boredom.Each individuals utiltities depend not only on his base utility and boredom but also on the influence fromother individuals.Consider a society of n people and m possible item choices. The society needs to choose one itemout of these at every time step. We will study the problem of the makiing the optimal choice so as tomaximize welfare. This is applicable in the following scenarios: A business is launching the next fashionstyle for its market segment, or a radio channel is broadcasting songs in a sequence to maximize the welfareto its audience. Let u ij ( t + 1) denote the utility of item i to person j at time t ; let b ij ( t ) denote theboredom value; let u i ( t ) denote the vector of utitilities to the n people for item i , v i ( t ) denote the vectorof base utilities, and b i ( t ) denote the vector of boredom values. In the absence of boredom we will say u i ( t + 1) = A u i ( t ) where A is the influence matrix. Accounting for boredom we will say, u i ( t + 1) = A u i ( t ) − [ b i ( t + 1) − b i ( t )] . Note that this is consistent with the case when there is only one individualwhere u i ( t + 1) = u i ( t ) − [ b ( t + 1) − b ( t )] . Observe that ignoring the effect of boredom we simply get therecurrence u i ( t + 1) = A u i ( t ) or u i ( t ) = A t v i . This recurrence reflects the diffusion of influence throughthe social network. Note that if the largest eigenvalue of A has magnitude more than then the process willdiverge and if all eigenvalues are < it will eventually converge to . So we will assume the maximumeigenvalue of A is has magnitude . If the gap between the magnitude of the largest and the second largesteigenvalue is at least (cid:15) then this diffusion process converges quickly in about ˜Θ(1 /(cid:15) ) steps. We will focus onthe case when rate of boredom r is much slower than the diffusion rate (this corresponds to the case whereinfluences spread fast and the boredom grows slowly). We then study the problem of making social choicesof items over time so as to maximize welfare.We will assume that A is diagonlizable and has a real top eigenvalue of and all the other eigenvaluesare smaller in magnitude. In that case it is well known that for any vector x A t x converges to to a fixedpoint and the speed of convergence depends on the gap between the largest and second largest eigenvalue.We show that under certain conditions if r/(cid:15) is small then. the choices made by the society is comparableto the choices made by an individual with appropriate base utilities and boredom coefficients. Let W i ( t ) denote the welfare of the society at time t by choosing item i ; then W i ( t ) /n Theorem 2.4.
Consider a society with influence matrix A that has largest eigenvalue and second largesteigenvalue of magnitude at most − (cid:15) . For computing the welfare over a a sequence of social choicesapproximately, such a society can be modelled as a single individual with base uitilities ˜ v i and boredomcoefficients ˜ α i , where ˜ v i = c (cid:48) v i and ˜ α i = c (cid:48) α i for some vector c . Let ˜ u i ( t ) denote the utility of item i tosuch an individual at time t .More precisely, differences in the average utility of the society for the same sequence of choices untilany time | W i ( t ) /n − ˜ u i ( t ) | ≤ r(cid:15) O ( | α i | ∞ ) for any t > T for some fixed T . The O notation hides factors thatdepends on A . For a real, symmetric matrix the constant is Technical details
The following Lemma is used in the proof of Theorem 2.1.
Lemma 3.1. (cid:80) i M i ( t ) ≤ , and (cid:80) i M i ( t ) → for large t . When t = Ω(1 /r ) , (cid:80) i M i ( t ) = 1 − O (exp( − tr )) .Proof. Observe that the memory scales down by a factor of − r each time step; exactly one item ispicked and r is added to its memory. So (cid:80) i M i ( t + 1) = (1 − r ) (cid:80) i M i ( t ) + r . This recurrence gives, (cid:80) i M i ( t ) = (1 − r ) t (cid:80) i M i (0)+ r (cid:80) tj =0 (1 − r ) j +(1 − r ) t ( (cid:80) i M i (0) − . Since M i (0) = 0 , (cid:80) i M i ( t ) ≤ .Observe also that after t = Ω(1 /r ) steps this becomes O ( exp ( − tr )) We are now ready to prove Theorem 2.1.
Proof. ( Theorem 2.1 ) To see that the solution to the given system is unique, note that f i = ( v i − uα ) + (where x + denotes max ( x, , and so (cid:80) i ( v i − uα ) + = 1 . This must have a unique solution as (cid:80) i ( v i − uα ) + isdecreasing function of u and strictly decreasing as long as the sum is positive. Let u denote the solution tothe above system.We now show max i u i ( t ) ≥ u for any t . This is done by contradiction. Suppose that for all i u i ( t ) < u .We have that (cid:80) i ( v i − uα i ) + < (cid:80) i v i − u i ( t ) α i . But (cid:80) i v i − u i ( t ) α i = (cid:80) i M i ( t ) ≤ . We have that (cid:80) i ( v i − uα i ) + < ,a contradiction.Let S g denote the set of all the items ever picked by the greedy algorithm. Let T be the time by whicheach item in S g has been used at least once. By Lemma 3.1, after some steps (cid:80) i M i ( t ) converges toarbitrarily close to . Lets assume for simplicity of argument that it is exactly with sufficiently large T .To show the upper-bound on max i u i ( t ) , we show that for t ≥ T and any i ∈ S g , max j u j ( t ) − u i ( t ) = O ( αr log n ) .Denote by x ( t ) the item that has the maximum utility at time t . It suffices to show that u x ( t ) ( t ) ≤ u i ( t )+ O ( αr log n ) . We recursively compute a decreasing sequence of t j as follows. Let t = t . For j > , supposewe have computed t j − . Let S j − = { x ( t ) , x ( t ) , · · · , x ( t j − ) } . Now let t j = max t (cid:48)
Since any item picked by the greedy algorithm in [ t j + 1 , t j − ] is in S j − , we have that for t (cid:48) ∈ [ t j + 1 , t j − ) , (cid:80) (cid:96) ∈ S j − M (cid:96) ( t (cid:48) + 1) = (1 − r ) (cid:80) (cid:96) ∈ S j − M (cid:96) ( t (cid:48) ) + r ≥ (cid:80) (cid:96) ∈ S j − M (cid:96) ( t (cid:48) ) . The last inequality isby (cid:80) (cid:96) M (cid:96) ( t (cid:48) ) ≤ . Therefore (cid:80) (cid:96) ∈ S j − M (cid:96) ( t j − ) ≥ (cid:80) (cid:96) ∈ S j − M (cid:96) ( t j +1) . By (2), we have A ( S j − , t j − ) ≤ A ( S j − , t j + 1) . 6 laim 2. A ( S j − , t j + 1) ≤ A ( S j , t j ) + r . Proof.
Since t j / ∈ S j − is the item picked by the greedy algorithm at t j , (cid:80) (cid:96) ∈ S j − M (cid:96) ( t j + 1) = (1 − r ) (cid:80) (cid:96) ∈ S j − M (cid:96) ( t j ) . Thus (cid:80) (cid:96) ∈ S j − M (cid:96) ( t j + 1) − (cid:80) (cid:96) ∈ S j − M (cid:96) ( t j ) = r (cid:80) (cid:96) ∈ S j − M (cid:96) ( t j ) ≤ r . Again by (2),we have A ( S j − , t j + 1) ≤ A ( S j , t j ) + r . Claim 3. A ( S j − , t j ) ≤ B ( S j − ) B ( S j ) A ( S j , t j ) . Proof.
Immediately follows from u t j ( t j ) ≥ u (cid:96) ( t j ) for (cid:96) ∈ S j − .Repeating (1), we have that A ( S j − , t j − ) ≤ B ( S j − ) B ( S j ) A ( S j , t j ) + r ≤ B ( S j − ) B ( S j ) (cid:18) B ( S j ) B ( S j +1 ) A ( S j +1 , t j +1 ) + r (cid:19) + r = B ( S j − ) B ( S j +1 ) A ( S j +1 , t j +1 ) + r · B ( S j − ) B ( S j ) + r · · · ≤ B ( S j − ) B ( S k ) A ( S k , t k ) + r · k − (cid:88) (cid:96) = j − B ( S j − ) B ( S (cid:96) ) . Hence, we have that u x ( t ) ( t ) = α A ( S , t ) ≤ α (cid:32) B ( S ) B ( S k ) A ( S k , t k ) + r · k − (cid:88) (cid:96) =1 B ( S ) B ( S (cid:96) ) (cid:33) = 1 B ( S k ) A ( S k , t k ) + r · k − (cid:88) (cid:96) =1 B ( S (cid:96) ) . Since i = x ( t k ) , for any i (cid:48) , u i (cid:48) ( t k ) ≤ u i ( t k ) . Therefore A ( S k , t k ) = (cid:80) j ∈ S k u j ( t k ) α j ≤ u i ( t k ) (cid:80) j ∈ S k α j = u i ( t k ) B ( S k ) . By that α = max i α i , we have B ( S (cid:96) ) ≥ (cid:96)/α . Hence u x ( t ) ( t ) ≤ B ( S k ) A ( S k , t k ) + r · k − (cid:88) (cid:96) =1 B ( S (cid:96) ) ≤ u i ( t k ) + αr k − (cid:88) (cid:96) =1 /(cid:96) = u i ( t k ) + O ( αr log n ) . Since item i is not used during the interval of [ t k + 1 , t ] , we have u i ( t k + 1) ≤ u i ( t ) , and hence u i ( t k ) ≤ u i ( t k + 1) + α i r ≤ u i ( t ) + α i r . Therefore we have that, max j u j ( t ) = u x (1) ( t ) ≤ u i ( t ) + O ( αr log n ) .On the other hand, we know that there exists i ∈ S g such that u i ( t ) ≤ u because otherwise it wouldbe the case that (cid:80) i ( v i − uα i ) + > (cid:80) i v i − u i ( t ) α i = (cid:80) i M i ( t ) ≈ , a contradiction. Hence max j u j ( t ) = u + O ( αr log n ) . A selection Y is periodic with period T , if for any t , y ( t + T ) = y ( t ) , where y ( t ) is the item chosenat time t . Clearly, in a periodic selection, the utility of the item chosen at time t is the same as the onechosen at time t + T . For utility maximization, it suffices to consider those items chosen in [0 , T ) . Let U ( Y ) = (cid:80) T − t =0 u y ( t ) ( t ) denote the total utility of Y in [0 , T ) . Theorem 3.2.
It is NP-hard to decide, given T , and U ∗ , and n items, whether there exist a assignment Y with period T such that U ( Y ) ≥ U ∗ .Proof. The reduction is from the Regular Assignment Problem and is detailed in the appendix.7 .3 Optimality of double-greedy algorithm
Using the exactly same argument in the proofs of Theorem 2.1, we have that
Lemma 3.3.
There exists a time T such that for any t ≥ T , µ ≤ max i w i ( t ) ≤ µ + O ( α log nr ) where µ isthe unique solution to the following system:For all items with f i > , the quantity v i − f i α i = µ ; if f i = 0 then v i < µ ; and (cid:80) f i = 1 . By using the above theorem, we can prove Theorem 2.3 as follows.
Proof. ( Theorem 2.3 ) For < f ≤ , write ∆( f ) = r · (1 − r ) /f − (1 − r ) /f .Let U ∗ be the optimal value of the following program. max U = (cid:88) i f i ( v i − α i f i ) s.t. (cid:88) i f i ≤ and f i ≥ . (3)Let OP T denote the optimal average utility. We have that
OP T ≤ U ∗ + αr . This is by observing thatfor any < f < , placing an item /f apart gives an upper bound on the utility of consuming the itemwith frequence f . The bound is v − α ∆( f ) ≤ v − α ( r − f ) by observing that ∆( f ) > r − f .The objective of (3) is maximized when there exists λ such that ∂U∂f i = λ for f i > and ∂U∂f i < λ for f i = 0 , and (cid:80) i f i = 1 . Since ∂U∂f i = v i − α i f i , λ is exactly the same as µ in the statement of Lemma 3.3.This explains the intuition of the double greedy heuristics — it tries to equalize the marginal utility gain ofeach item. Denote the optimal solution by f ∗ i . Then for f ∗ i > , v i − α i f ∗ i = µ . Hence, U ∗ = (cid:88) i f ∗ i ( v i − α i f ∗ i ) = (cid:88) i f ∗ i ( µ + α i f ∗ i ) = µ + (cid:88) i α i f ∗ i . Let k i denote the number of times item i is used in [0 , T ] by the double-greedy algorithm, and f i = k i /T .Let M i denote the average memory on i at the times when i is picked. Then we have that U = T (cid:88) t =0 u x ( t ) ( t ) /T = (cid:88) i (cid:88) x ( t )= i u i ( t ) /T = (cid:88) i (cid:88) x ( t )= i ( w i ( t ) + α i M i ( t )) /T ≥ (cid:88) i (cid:88) x ( t )= i ( µ + α i M i ( t )) /T (by Lemma 3.3, w i ( t ) ≥ µ ) ≥ µ + (cid:88) i α i f i M i . (4)Write δ = αr log n . By Lemma 3.3, w i ( t ) = µ + O ( δ ) for each i, t . We will show that Claim 1. α i f i = α i f ∗ i − O ( δ ) . Observe that for any item i which is picked k i times in [0 , T ] , min ≤ t ≤ T M i ( t ) ≤ ∆( k i /T ) ≤ k i /T = f i . Hence, max t w i ( t ) ≥ v i − α i M i ( t ) ≥ v i − α i f i . On the other hand, w i ( t ) = µ + O ( δ ) . We have v i − α i f i = µ + O ( δ ) . But µ = v i − α i f ∗ i . Therefore α i f i ≥ α i f ∗ i − O ( δ ) . Claim 2. α i M i = α i f ∗ i − O ( δ ) . Since v i − M i ≤ µ + δ , we obtain the bound by following the sameargument as in the proof of Claim 1. Now, plugging both claims into (4), we have that U ≥ µ + (cid:88) i α i ( f ∗ i − O ( δ ) α i ) ≥ µ + (cid:88) i α i f ∗ i − O ( δ ) = U ∗ − O ( δ ) = OP T − O ( δ ) . This last equality follows from U ∗ = OP T − αr . This completes the proof.8 ong v α r The Climb 12.3 9.9 0.097Lucky 2.6 1.58 0.114Snow Patrol - Chasing Cars 10.7 6.8 0.127I know you want me 7.95 6.5 0.077Viva la vida 12.4 9.1 0.16Stop and stare 10.5 9.4 0.092Disturbia 8 7.2 0.092Pocket full of sunshine 7.6 6.3 0.14Supernatural superserious 24.2 22 0.15One step at a time 9.35 8.5 0.075
Table 1: v, α, r for the set of songs.
Movie v α r
Godfather 6.15 5.15 0.123Hancock 9.6 8.8 0.128The Bucket List 13.1 11.8 0.102Quantum of Solace 29.8 29 0.111Tropic Thunder 25.6 24.8 0.082
Table 2: v, α, r for the set of movies.
Let denote the vector with all coordinates set to and α i denote the vector of boredom coefficients α ij . Observation.
For any diagonolizable matrix A with largest eigenvalue and the second largest eigenvalueis at most − (cid:15) , there is a vector c so that. (cid:48) A t x − c (cid:48) x ≤ (1 − (cid:15) ) t √ nO ( | x | ) . The O notation hides factorsthat depends on A . For a real, symmetric matrix the constant is . Proof.
We will sketch the proof for real symmetric matrices. The same idea holds for non-symmetricmatrices. If p , . . . , p n denote the eigenvectors of A and λ , . . . , λ n denote the eigenvalues then A t x = (cid:80) j λ tj p i v (cid:48) i x = p v (cid:48) x + (cid:80) j> λ tj p (cid:48) i p i x . Now,— (cid:80) j> λ tj p (cid:48) i p i x | ≤ (1 − (cid:15) ) t | x | . So | (cid:48) ( A t x − p p (cid:48) x ) | ≤| | (1 − (cid:15) ) t | x | = √ n (1 − (cid:15) ) t | x | Setting c = (cid:48) v v (cid:48) completes the proof.We will now prove theorem 2.4 Proof. ( Theorem 2.4 ) Let ∆ b ( t ) denote b ( t ) − b ( t − . Now u i ( t ) = A u i ( t −
1) + ∆ b ( t ) . This gives, u i ( t ) = A t v i + (cid:80) t − j =0 A j ∆ b ( t ) . Note that W i ( t ) = (cid:48) u i ( t ) = (cid:48) A t v + (cid:80) t − j =0 (cid:48) A j ∆ b ( t − j ) Note ∆ b ij ( t ) = α ij ((1 − r ) M ij ( t ) + rI i ( t ) − M ij ( t )) = α ij r ( x i ( t ) − M ij ( t )) . So | ∆ b ij ( t ) | ≤ r | α i | .Now | (cid:48) A t v i − c (cid:48) v i | ≤ (1 − (cid:15) ) t √ nO ( | v i | ) . For t > (1 /(cid:15) )Ω(log( nr | v i | ) , this is at most r . Also | A j ∆ b i ( t − j ) − c (cid:48) ∆ b i ( t − j ) | ≤ (1 − (cid:15) ) j √ nO ( | ∆ b i ( t − j ) | ) ≤ (1 − (cid:15) ) j √ nrO ( | α i | ) . So, | (cid:80) j A j ∆ b i ( t − j ) − (cid:80) j c (cid:48) ∆ b i ( t − j ) | ≤ ( r/(cid:15) ) √ nO ( | α i | ) . So, | (cid:80) j A j ∆ b i ( t − j ) − c (cid:48) b i ( t ) | ≤ ( r/(cid:15) ) √ nO ( | α i | ) . There-fore | W i ( t ) − ( c (cid:48) v i − c (cid:48) b i ( t )) | ≤ ( r/(cid:15) ) O ( √ n | α i | ) . Dividing by n completes the proof. In this section, we provide experimental results to study the techniques presented in the paper. Our primaryobjectives is to evaluate the quality of greedy and double-greedy algorithms for choosing items based onutility and boredom parameters estimated from the real data.
We obtain data on the popularity of songs and movies from Google Trends [1]. We collected weekly aggre-gate counts from query logs for popular songs from the last 3 years. Similar data was collected for popularmovies. While the popularity of songs and movies depends on additional factors such as awards won by analbum or a movie, our goal was to perform a controlled experiment only based on overall utility and bore-dom. Therefore, for each item we collected weekly aggregate counts starting from the highest peak in logstill there was an “artificial peak” due to an external event such as an award. Further, we compare the utilityobtained by our model with a baseline in which the user selects an item simply based on its utility withoutany discounting from boredom. We describe how we compute the values of α , v , and r in the appendix.9 ong Greedy Double-GreedyThe Climb . .
17) 11 . . Snow Patrol - Chasing Cars - (0) . . Viva la vida . .
17) 11 . . Stop and stare - (0) . . Supernatural superserious . .
67) 17 . . Table 4:
Avg. utilities (frequencies) for selected songs.
Movie
Greedy Double-GreedyHancock - (0) . . The Bucket List - (0) . . Quantum of Solace . .
55) 20 . . Tropic Thunder . .
45) 18 . . Table 5:
Avg. utilities (frequencies) for selected movies.
Table 1 shows the v , α , and r values for a set of 10 songs used in our experiments while the correspondingdata for the movie data set is shown in Table 2; here we allow different values of r , but we notice that all r -values within the domain of songs and movies are similar. We ran a set of experiments to verify the effectiveness of the greedy and double-greedy heuristics. We ranthe experiments over steps for both the data sets. The average utility obtained by the user for boththe data sets was computed and is shown in Table 3. We also show results for the baseline approach thatalways picks the same item with the highest base utility. Tables 4 and 5 illustrate the average utility obtainedby the user over the selected songs and movies respectively. The corresponding normalized frequencies areshown in parenthesis. As expected, in the baseline case where the user selects an item according to its baseutility, the movie
Quantum of Solace (with a base utility of . ) is always selected while in the caseof songs, the song supernatual superserious (with a utility of . ) is selected. Unsurprisingly,the average utility discounting boredom for this case is very low (see Table 3). Dataset
Greedy Double-Greedy BaselineSongs .
94 13 .
53 5 . Movies .
12 17 .
30 4 . Table 3: Average utility over time steps. Figure 2: Change in average utility over time.In another experiment, we measured the change in the average utility with time. Figure 2 illustratesthe change in average utility as the user selects different items at each time step for movies. Naturally, theutility is highest at the very beginning as the user picks an item with the highest base utility and decreasessubsequently as she picks items with highest discounted utility at each time step.
As we mentioned, our model is by no means comprehensive. For example, boredom may come from con-suming similar items, or there may be a cost when switching from item to item. Taking into account thesefactors raises some interesting algorithmic issues. Fully incorporating these extensions is left as future work.
We thank Atish Das Sarma for useful discussions. 10 eferences [1] Google Trends. .[2] A. Bar-Noy, R. Bhatia, J. Naor, and B. Schieber. Minimizing service and operation costs of periodicscheduling.
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A Computing the model parameters
Figure 3 shows the trend observed for a specific song from our dataset,
I Know You Want Me , overa 45-week period starting August 2, 2009. The first natural observation we make is that the total numberof queries do indeed display a steady decline, which we attribute to boredom. From the data, we use themaximum count as the peak utility, v peak , and let the final count be denoted v final . We set α = v peak − v final .Let X ( t ) denote the aggregate count for the week t , we obtain the boredom parameter r using the followingequation: e − rt = 1 − v peak − X ( t ) v peak − v min We plot rt = − ln(1 − v peak − X ( t ) v peak − v min ) , and fit a linear line on the resulting curve and obtain r from the slope.Figure 4 shows the curve for I Know You Want Me , from which we obtain the r value. B NP-hardness of item selection
Restatement of Theorem 3.2:
It is NP-hard to decide, given T , and U ∗ , and n items, whether there exist aassignment Y with period T such that U ( Y ) ≥ U ∗ . Proof.
The reduction is from the following problem.11igure 3: Weekly aggregate query counts for
I Know You Want Me for a 45-week periodfrom Google trends. Figure 4: Approximate linear trend for
I KnowYou Want Me , with slope giving r . Regular assignment problem (RAP).
Given positive integers p , p , · · · , p n , determine if there exists asequence y , y , · · · where y t ∈ { , , · · · , n } such that for any i (cid:54) = 0 , two consecutive appearances of i inthe sequence are exactly p i apart.It is shown in [2] that the regular assignment problem is NP-complete. Note that for RAP, a regularassignment exists if and only if it does so on a cycle with length T = (cid:81) i p i . We will now reduce it to theoptimal fashion selection problem.Given p , · · · , p n , we create n + 1 items such that a regular assignment, if exists, maximizes the utilityof any periodic selection with period T . Hence we can reduce RAP to the optimal selection problem. Item is a special item with v = 1 and α i = 0 . For ≤ i ≤ n , we assign v i = Tp i and α i = 1 . Further let r i = 1 /T for ≤ i ≤ n . We claim that there exists U ∗ and (cid:15) ≥ /T such that for a regular assignment Y , U ( Y ) ≥ U ∗ , and U ( Y ) < U ∗ − (cid:15) otherwise.Consider the case when there is only item and when the selections are made on the real line. Given T and an item with parameters v, α, r , let Y k ( v, α, r ) be the set of all the selections which have period T andchoose the item exactly k times on the real interval [0 , T ) . Denote by U k ( v, α, r ) = max Y ∈Y k ( v,α,r ) U ( Y ) and δU k ( v, α, r ) = U k ( v, α, r ) − U k − ( v, α, r ) . The correctness of the reduction follows from the followingclaims. Claim 1. U k ( v, α, r ) = kv − kα (1 − r ) T/k − (1 − r ) T/k , and the maximum is achieved with the regular assignment.
Claim 2.
For ≤ v ≤ n , U k ( v i , , /T ) = kv i − ( k − k + + o (1 /k )) , and δU k ( v i , , /T ) = v i − (2 k − + o (1 /k )) . Claim 3.
For any non-regular integral selection Y ∈ Y k ( v i , , /T ) , U ( Y ) < U k ( v i , , /T ) − /T .Claim 1 holds because the total memory is minimized when the k assignments are regularly spaced.Claim 2 is a direct consequence of Claim 1 by Taylor expansion on those particular parameters. Claim 3follows by comparing the memory caused by adjacent items between regular and non-regular assignments.From Claim 2, we can see that δU k ( v i , , /T ) ≥ / for k ≤ T /p i and < for k ≥ T /p i + 1 for ≤ i ≤ n , and δU k ( v , , /T ) = 1 . Combining it with Claim 3, we have that the utility gap between aregular and non-regular assignment is at least /T2