Hipsters and the Cool: A Game Theoretic Analysis of Social Identity, Trends and Fads
HHipsters and the Cool: A Game Theoretic Analysis ofSocial Identity, Trends and Fads
Russell Golman ∗ , Aditi Jain , and Sonica Saraf Department of Social and Decision Sciences, Carnegie Mellon University Department of Mathematics, Carnegie Mellon University
October 30, 2019
Abstract
Cultural trends and popularity cycles can be observed all around us, yet our theories of socialinfluence and identity expression do not explain what perpetuates these complex, often unpre-dictable social dynamics. We propose a theory of social identity expression based on the opposing,but not mutually exclusive, motives to conform and to be unique among ones neighbors in a so-cial network. We then model the social dynamics that arise from these motives. We find thatthe dynamics typically enter random walks or stochastic limit cycles rather than converging to astatic equilibrium. We also prove that without social network structure or, alternatively, withoutthe uniqueness motive, reasonable adaptive dynamics would necessarily converge to equilibrium.Thus, we show that nuanced psychological assumptions (recognizing preferences for uniquenessalong with conformity) and realistic social network structure are both necessary for explaining howcomplex, unpredictable cultural trends emerge.
Keywords : Conformity | Games on Social Networks | Popularity Cycles | Social Dynamics | Uniqueness ∗ Corresponding Author; E-mail: [email protected] a r X i v : . [ ec on . T H ] O c t ntroduction Popular cultural practices come into and out of fashion. Researchers have observed boom-and-bustcycles of popularity in music, clothing styles, given names, automobile designs, home furnishings,and even management practices (Shuker, 2016; Richardson and Kroeber, 1940; Reynolds, 1968;Sproles, 1981; Berger, 2008; Berger and Le Mens, 2009; Lieberson, 2000; Lieberson and Lynn,2003; Robinson, 1961; Abrahamson, 1991; Zuckerman, 2012). Popularity cycles appear to bedriven by social influence, e.g., by people adopting the music that their friends listen to or that theyperceive as popular (Salganik et al., 2006; Salganik and Watts, 2008). At the individual level, peo-ple are constantly looking for new ways to express their preferred social identities (Hetherington,1998; Rentfrow and Gosling, 2006; Berger, 2008; Chan et al., 2012). The resultant social dynam-ics do not typically converge to equilibrium. What are the social forces that lead to such perpetualchange and novelty?Social pressure to conform is a powerful force when behavioral patterns across a society shiftin unison. Psychologists since Asch have recognized the remarkable strength of the conformitymotive, stemming from a fundamental goal to fit in as part of a social group (Asch, 1955, 1956;Cialdini and Trost, 1998). People tend to feel uncomfortable about considering, holding, andexpressing beliefs that conflict with the prevailing views around them as well as about behavingoddly, in ways that might expose oneself as an outsider to the group (Akerlof and Kranton, 2000;Golman et al., 2016). Given the conformity motive alone, we might expect to observe convergenceto an equilibrium in which society becomes monolithic, yet instead we actually observe persistentdiversity.Opposing the motive to conform is a similarly universal human need for uniqueness (Snyderand Fromkin, 1980; Lynn and Snyder, 2002). While the desire to differentiate oneself clearlyworks against the desire to blend in (Imhoff and Erb, 2009), Chan, Berger and van Boven (2012)demonstrate that people simultaneously pursue assimilation and differentiation goals, aiming tobe identifiable, but not identical (see also Leibenstein, 1950; Robinson, 1961). Preferences foridiosyncratic behavioral patterns can preserve diversity (Smaldino and Epstein, 2015). Still, the2uestion remains why behavioral patterns often do not remain in a stable equilibrium with everyonefinding an optimal balance between distinctiveness and conformity. Why instead do behavioralpatterns go through perpetual change, with particular behaviors cycling into and out of fashion ascultural trends play out?Here, we show that along with conformity and uniqueness motives, a realistic network of socialinteraction is a critical, necessary ingredient for complex social dynamics to emerge. Specifically,we show that reasonable adaptive dynamics that would necessarily converge to a static equilibriumgiven random interactions in a well-mixed pool of people instead typically enter random walks orstochastic limit cycles, and thus never converge, when interactions are restricted to individuals’local neighborhoods in their social networks.A natural theoretical approach for investigating social influence on decisions is to use gametheory. The conformity motive in isolation would create a Keynesian beauty contest, in which whatis cool (like what is beautiful) is just what everybody else believes is cool (Keynes, 1936). Theuniqueness motive in isolation would create a congestion game, in which the objective is simplyto be distinct from as many other people as possible (Rosenthal, 1973). Both games are knownto be potential games, for which convergence to a pure strategy Nash equilibrium is practicallyguaranteed (Monderer and Shapley, 1996b,a). When both motives co-exist and the game is playedon a realistic social network, however, the dynamics are more complex.Cultural trends can be modeled more realistically as the dynamics of a game on a social networkbecause social influence is mediated by a social network (Jackson and Zenou, 2015). Social influ-ence on expressions of individual identity is transmitted whenever an individual observes anotherperson whom he would like to identify with, so the relevant social network is defined by directedconnections corresponding to observation. The connected components of the social network maycorrespond to distinct social groups, each with its own emergent subculture.The desire for uniqueness within one’s own social group should not be conflated with a desirefor differentiation across groups (Chan et al., 2012). In models of identity signaling, membershipin one group may be preferable to membership in another, and people want to strategically distin-3uish themselves from those in the less favorable group; e.g., an upper class tries to distinguishitself from the bourgeois while the bourgeois tries to imitate them (Berger and Heath, 2007). Thedynamic of differentiation and imitation has been hypothesized to lead to fashion cycles (Karni andSchmeidler, 1990; Pesendorfer, 1995). This dynamic does not, however, preserve diversity withingroups. Desire for uniqueness is a necessary part of the explanation. Our model features in-groupconformity and uniqueness motives; it could be augmented with a desire for differentiation acrossgroups, but for parsimony we assume that people care only about their fit within their own groups.
Model 1: Social Identity Expression in a Well-Mixed Population
We model the expression of social identity as a game played by a population of N individuals. Letus say there are d aspects (or dimensions) of identity. Each person i chooses an expression of hisidentity x i ∈ { a..b } d , i.e., represented as a tuple of d integers from some interval. For example, inthe case of choosing a color to wear, three integers between and might correspond to shadesof red, green, and blue that mix together to form any color.A person’s degree of conformity in the population depends on the distance between his ex-pressed identity and the average (population mean) expression of identity, (cid:107) x i − ¯ x (cid:107) . A person’sdegree of uniqueness in the population depends on the number of others who adopt the exact sameexpression of identity as him, denoted as n i ( X ) where X is the entire population’s profile of ex-pressed identities. Putting together conformity and uniqueness motives, we model person i ’s utilitygiven the profile of expressed identities as u i ( X ) = −(cid:107) x i − ¯ x (cid:107) − λ n i ( X ) (1)where λ is a parameter that describes the strength of the uniqueness motive relative to the confor-mity motive. This utility function describes a person whose goal is to be similar to everybody, yetthe same as nobody.Over time people may change their expressions of identity to achieve higher utility. We need4ot fully prescribe this process, but assume only that people make changes that increase their ownutility, in accordance with some better-reply dynamics (Monderer and Shapley, 1996b; Friedmanand Mezzetti, 2001). Definition 1 (Better-reply dynamics) . At any given time t , one person i may consider switchingfrom x i to x (cid:48) i ; he switches if and only if u i ( X (cid:48) ) > u i ( X ) ; and for each person i and any bestresponse x ∗ i (to X ( t ) ), the expected time until person i considers switching to x ∗ i is finite. The motivation for better-reply dynamics is that people are boundedly rational and adaptive.They can see what the people around them are doing and can search for something better (my-opically), but they do not instantaneously react to changes in other people’s behavior or anticipatethese changes before they occur. Many commonly assumed adaptive learning dynamics are partic-ular specifications of better-reply dynamics.
Results: Social Dynamics in a Well-Mixed Population
Theorem 1.
Suppose people derive utility from both their conformity and their uniqueness in thepopulation, as in Equation (1). Then any better-reply dynamics necessarily converges to a purestrategy Nash equilibrium.
The proof is presented in the SM Appendix. It follows from Lemma 1 in the SM Appendix,which identifies an exact potential function for this game. Two examples of Nash equilibria, amongmany that exist, are shown in Figure 1.Theorem 1 says that in a well-mixed population, in the long run we will not see popularitycycles, perpetual change, or novelty. The fact that we do, in reality, observe popularity cycles,perpetual change, and novelty suggests that we should consider a more realistic model. We nowconsider the social dynamics that result from assuming that people care only about the expressedidentity of their immediate neighbors in their social network.5igure 1: Two Nash equilibria distributions of identity expression for populations of N = 100 individuals. We set λ = 1 . for this illustration. ( A ) One-dimensional identity expression over thedomain { .. } . ( B ) Two-dimensional identity expression over the domain { .. } . By symme-try, the distributions can be shifted anywhere within these (or wider) domains, and many strategyprofiles give rise to the same population distributions. Even after accounting for these symmetries,these Nash equilibria are not unique.A B Model 2: Social Identity Expression in Social Networks
A social network is described by an adjacency matrix A where a ij = 1 if person i observes, andthus cares about, person j ’s expressed identity (and equals if not). Let η ( i ) = { j : a ij = 1 } denote the set of people that person i observes, i.e., his neighbors.Conformity among one’s neighbors depends on distance from one’s neighbors’ average iden-tity, ¯ x η ( i ) . Uniqueness among one’s neighbors depends on the number neighbors who adopt thesame expression of identity as oneself, denoted ˜ n i ( X ; η ( i )) . Thus, we now model person i ’s utilitygiven the profile of expressed identities X and his set of neighbors η ( i ) as u i ( X ) = −(cid:107) x i − ¯ x η ( i ) (cid:107) − λ ˜ n i ( X ; η ( i )) . (2) Results: Social Dynamics in Social Networks
Theorem 2.
Suppose people derive utility from both their conformity and their uniqueness amongtheir neighbors in a social network, as in Equation (2) with λ > . Then there exists a social etwork adjacency matrix ˆ A such that no pure strategy Nash equilibrium exists and, thus, better-reply dynamics never converge to an absorbing state.Proof. By construction. We provide an example of a social network with N = 3 people thatillustrates the result. (Any larger social network that contains this network as an out-componentalso suffices.) Let person observe (only) person , person observe (only) person , and person observe (only) person .Observe that the best response correspondence for each person is as follows: x ∗ ∈ { x : (cid:107) x − x (cid:107) = 1 } x ∗ ∈ { x : (cid:107) x − x (cid:107) = 1 } x ∗ ∈ { x : (cid:107) x − x (cid:107) = 1 } . Each person wants to be one unit of distance away from the person he is observing. However, it isimpossible for all three people to simultaneously choose best responses because of the mathemat-ical fact that odd-length cycle graphs are not -colorable.Theorem 2 says that with only local interactions in a social network, perpetually changingidentity expression and popularity cycles become possible. Observe that the uniqueness motive iscritical for obtaining this result. If we were to eliminate the uniqueness motive by setting λ = 0 ,then any homogeneous profile of expressed identities (with x i identical for all i ) would be a purestrategy Nash equilibrium, regardless of the social network structure. The uniqueness motive alongwith the local interactions together allow for more realistic, complex social dynamics.Still, Theorem 2 only provides an existence result constructed with a highly stylized, simplis-tic social network. It does not tell us whether complex social dynamics typically emerge fromour model when people are connected by realistic social networks. We now use computationalmodeling to explore the dynamics of our model on realistic social networks.We used a variant of the Jin-Girvan-Newman algorithm (Jin et al., 2001) to create a sample of directed social networks with a high level of clustering and community structure and limited7ut-degree ( Material and Methods ). For each of these social networks, we repeatedly computedbetter reply dynamics based on the utility function in Equation (2) to see how often the dynamicsconverged to equilibrium within , time steps ( Material and Methods ). (We chose the cutoffat , time steps based on first computing the dynamics in the full, well-mixed population, forwhich Theorem 1 tells us that they must converge, and finding that across trials, the dynamicsalways converged within time steps.) If the dynamics did not converge within , timesteps, we classified them as non-convergent (for that trial).Figure 2 shows snapshots of the dynamics on the first social network in our sample between , and , time steps. Very quickly (i.e., within just a few hundred time steps) everybodyadopts identities in the range { .. } , but individuals continually change thereafter. We can seeconsiderable change in individual expressions of identity in each snapshot. The dynamics do notconverge.On average, across all social networks in our sample, the dynamics were non-convergentfor . of our trials. Figure 3 presents the results of total trials for each of the socialnetworks in our sample, showing the number of social networks having particular frequencies ofnon-convergence. For each social network in our sample, the dynamics were non-convergent forat least out of the trials. For of the social networks, the dynamics never converged.These results tell us that with local interactions on realistic social networks, the interplay ofconformity and uniqueness motives produces social dynamics for identity expression that are in-deed typically non-convergent. People continually change their expressed identities, and certainforms of expression come into and out of fashion in unpredictable cycles. Popularity cycles areinherently unpredictable in the model because people typically have multiple better replies (andeven multiple best responses) to choose from in the face of most profiles of their neighbors’ iden-tity expression. The multiplicty of paths the dynamics could take leaves room for idiosyncrasy.The pattern of widespread non-convergence across the entire sample of social networks appearsto be robust to variations in the process of search for a better response (i.e., it can be random orsequential), variations in the distribution and average level of out-degree in the social network8
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Figure 2: Identity expression in social network changing over time. Arrows point from anindividual to the people he observes. The shading of the nodes corresponds to each person’sexpression of identity, from x i = 0 if node i is white to x i = 3 if node i is black. Shown from leftto right at , , , and time steps.9igure 3: Histogram showing the number of social networks for which we observe particularfrequencies of non-convergent trials.(short of being fully connected, of course), and variations in the preference parameter λ (over therange λ > ), based on additional trials reported in the SM Appendix. The social network with ID is the one that most frequently permits convergence to equilibrium. Figure 4 shows this socialnetwork and one example of a Nash equilibrium on it.Directed connections in the social network appear to play an important role in obtaining typi-cally non-convergent dynamics. We explored the better-reply dynamics after inserting reciprocalconnections in all of our directed social networks and found that on these (now) undirected socialnetworks, the dynamics converged to equilibrium in . of our trials. (The dynamics con-verged within time steps in over of our trials, providing reassurance that findings ofnon-convergence are fairly robust to allowing the dynamics more time to converge.) Intuitively,directed connections in the social network make it possible that an individual’s changing expres-sion of his identity imposes a negative externality on people who observe him, but who he does notnotice. The ripple effects may persist or fade, and in more realistic, more complex social networks,they tend to persist indefinitely. 10 Figure 4: Social network in a Nash equilibrium. Arrows point from an individual to thepeople he observes. The shading of the nodes corresponds to each person’s expression of identity,from x i = 0 if node i is white to x i = 3 if node i is black. Discussion
Our findings help us understand the role of social networks and local interaction in the dynamicsof cultural trends. Popularity cycles, perpetual change, and novel expressions of social identityshould be expected when people observe their neighbors in realistic, directed social networks andcare about being unique as well as fitting in. Such complex social dynamics of identity expressionare incompatible with simplistic assumptions disregarding social network structure or reducingsocial influence to mere conformity pressure absent a desire to individuate oneself.Recognition of conformity and uniqueness as opposing, but not mutually exclusive, motivesis also part of optimal distinctiveness theory (Brewer, 1991; Leonardelli et al., 2010). However,optimal distinctiveness theory posits that people form collective identities by choosing to asso-ciate themselves with social groups, whereas our concept of social identity operates at the levelof the individual. In our view, collective identities emerge at the level of the group based on theirmembers’ individual identities. From the alternative, similarly valid perspective, we could propose11hat individual identities emerge from a psychological process of finding consonance between thecollective identities of the many groups that an individual affiliates with at any point in time. Con-necting these perspectives requires deeper understanding of how people choose to associate withor withdraw from social groups, how social network structure endogenously evolves. While thisintegration remains beyond our present grasp, we find it useful to have complementary theoriesaimed at different levels of social identity.We use game theory and computational modeling here to describe social dynamics with math-ematical precision. Social phenomena do not always reflect individual preferences (Schelling,1969, 1971). Mathematical modeling helps us understand the relationship between individual mo-tives and aggregate social dynamics when interactions generate nontrivial feedbacks. Our workhere is part of a tradition of formal modeling of social identity and fashion (Bikhchandani et al.,1992; Miller et al., 1993; Strang and Macy, 2001; Bettencourt, 2002; Tassier, 2004; Acerbi et al.,2012; Smaldino et al., 2012; Smaldino and Epstein, 2015; Smaldino et al., 2015). This approachyields us deep theoretical insight, and we hope it inspires more research leading to further insightsinto social dynamics and identity expression.
Materials and Methods
The Social Networks
We borrow Jin, Girvan, and Newman’s Model II algorithm for growing undirected social networks(Jin et al., 2001) and modify it to generate directed social networks with N = 100 people, each ofwhom can observe up to a maximum of neighbors. The network is initialized with all peopleand no connections. The following three steps are then repeated times:1. Choose pairs of individuals uniformly at random. For each pair i and j , if i observes lessthan people and does not already observe j , then i begins to observe j ; else, if j observesless than people and does not already observe i , then j begins to observe i .12. Randomly select triads i , j , and k such that i observes k and k observes j or that i and j bothobserve k . If i observes less than people and does not already observe j , then i begins toobserve j . (Real social networks exhibit both patterns of directed closure (Brzozowski andRomero, 2011).)3. Randomly select and break . of connections (rounded up).All social networks and the Python source code used to create them will be made available inthe SM Appendix. The Game
Our computational model adopts the following specification of parameter values for the game: d = 1 ; { a..b } = { .. } ; λ = 1 . . The Better-Reply Dynamics
Our computational model adopts a specification of the better-reply dynamics in which at each timestep, one individual searches for (and upon discovery, adopts) a better reply to the current popu-lation profile. Initial strategies are randomly (uniformly) distributed. We check for convergenceafter every time steps by sequentially checking whether any individual can find a better reply.In the other time steps, the individual searching for a better reply is randomly selected. The Pythonsource code and complete output data will be made available in the SM Appendix.
Acknowledgments
Funding:
This research did not receive any specific grant from funding agencies in the public,commercial, or not-for-profit sectors.
Competing Interests:
The authors declare that they have no competing financial interests.13 eferences
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Formal Definitions
We can express person i ’s neighbors’ average identity as ¯ x η ( i ) = 1 | η ( i ) | (cid:88) j ∈ η ( i ) x j . We can express the number of i ’s neighbors who adopt the same expression of identity as person i as ˜ n i ( X ; η ( i )) = (cid:88) j ∈ η ( i ) δ ( x i , x j ) , where δ is the Kronecker delta function. In a well-mixed population, we set η ( i ) = { j : j (cid:54) = i } torecover n i ( X ) for all i . Supplementary Results and Proofs
Lemma 1.
In a well-mixed population with utility functions given in Equation (1), the game hasan exact potential function: Φ( X ) = − N (cid:88) i =1 N − N (cid:107) x i − ¯ x (cid:107) + 12 λ n i ( X ) . Proof.