Voting, Success, and Superstarspdodds/teaching.html/courses/2013-01...Voting, Success, and Superstars Winning: it’s not for everyone Superstars Musiclab References 7 of 27 Superstars

Voting, Success,and Superstars

Winning: it’s not foreveryoneSuperstars

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Voting, Success, and SuperstarsPrinciples of Complex Systems

CSYS/MATH 300, Spring, 2013 | #SpringPoCS2013

Prof. Peter Dodds@peterdodds

Department of Mathematics & Statistics | Center for Complex Systems |Vermont Advanced Computing Center | University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

http://www.uvm.eduhttp://www.uvm.edu/~pdoddshttp://www.uvm.edu/~pdodds/teaching/courses/2013-01UVM-300http://www.uvm.edu/~pdoddshttp://www.uvm.edu/~cems/mathstat/http://www.uvm.edu/~cems/complexsystems/http://www.uvm.edu/~vacc/http://www.uvm.eduhttp://www.uvm.edu/~pdoddshttp://www.uvm.edu/~pdodds

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Outline

Winning: it’s not for everyoneSuperstarsMusiclab

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Where do superstars come from?

Rosen (1981): “The Economics of Superstars” [5]

Examples:I Full-time Comedians (≈ 200)I Soloists in Classical MusicI Economic Textbooks (the usual myopic example)

I Highly skewed distributions again...

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Superstars

Rosen’s theory:I Individual quality q maps to reward R(q)I R(q) is ‘convex’ (d2R/dq2 > 0)I Two reasons:

1. Imperfect substitution:

A very good surgeon is worth many mediocre ones

2. Technology:

Media spreads & technology reduces cost ofreproduction of books, songs, etc.

I Joint consumption versus public goodI No social element—success follows ‘inherent quality’

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Superstars

2. Technology:

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Superstars

2. Technology:

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Superstars

1. Imperfect substitution:A very good surgeon is worth many mediocre ones

2. Technology:

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Superstars

2. Technology:

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Superstars

2. Technology:Media spreads & technology reduces cost ofreproduction of books, songs, etc.

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Superstars

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Superstars

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Superstars

Adler (1985): “Stardom and Talent” [1]

I Assumes extreme case of equal ‘inherent quality’I Argues desire for coordination in knowledge and

culture leads to differential successI Success can be purely a social constructionI (How can we measure ‘inherent quality’?)

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Superstars

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Superstars

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Superstars

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Voting

Evidence from the web suggestions (Huberman etal.)

1. Easy decisions (yes/no) lead to bandwagoningI e.g. jyte.com

2. More costly evaluations lead to oppositional votesI e.g. amazon.com

I Self-selection: Costly voting may lower incentives forthose who agree with the current assessment andincrease incentives for those who disagree.

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Voting

Score-based voting versus rank-based voting:I Balinski and Laraki [2]

“A theory of measuring, electing, and ranking”Proc. Natl. Acad. Sci., pp. 8720–8725 (2007)

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Voting

Laureti et al. (2004): “Aggregating partial, localevaluations to achieve global ranking” [4]

I Model: participants rank n objects based onunderlying quality q

I Assume evaluation of object i is a random variablewith mean qi

I Choose objects based on votes:

pi(t) ∝ vi(t)α or pi(t) ∝ qivi(t)α.

I If α < 1, correct quality ordering is uncoveredI If α > 1, some objects are never evaluated and

mistakes are made...I Related to Adler’s approach

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Voting

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Voting

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Voting

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Voting

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Dominance hierarchies

Chase et al. (2002): “Individual differences versussocial dynamics in the formation of animaldominance hierarchies” [3]

I The aggressive female Metriaclima zebra:

I Pecking orders for fish...

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Fish forget—changing of dominance hierarchies:

(one-sided binomial test: n ! 22, P " 0.001 and P " 0.03,respectively). In this light, 27% of the groups with identicalhierarchies is very small.

Discussion. When we rewound the tape of the fish to form newhierarchies, we usually did not get the same hierarchy twice. Thelinearity of the structures persisted and the individuals stayed thesame, but their ranks did not. Thus our results differ considerablyfrom those predicted by the prior attributes hypothesis. The factthat more identical hierarchies occurred than expected by chancealone supports the hypothesis that rank on prior attributesinfluences rank within hierarchies but not the hypothesis that

rank on prior attributes of itself creates the linear structure of thehierarchies. Although 50% of the fish changed ranks from onehierarchy to the other, almost all the hierarchies were linear instructure. Some factor other than differences in attributes seemsto have ensured high rates of linearity. In the next experiment,we tested to determine whether that factor might be socialdynamics.

It might seem possible that ‘‘noise,’’ random fluctuations inindividuals’ attributes or behaviors, could account for the ob-served differences between the first and second hierarchies.However, a careful consideration of the ways in which fluctua-tions might occur shows that this explanation is unlikely. Forexample, what if the differences were assumed to have occurredbecause some of the fish changed their ranks on attributes fromthe first to the second hierarchies? To account for our results,this assumption would require a mixture of stability and insta-bility in attribute ranks at just the right times and in just the rightproportion of groups. The rankings would have had to have beenstable for all the fish in all the groups for the day or two it tookthem to form their first hierarchies (or we would not have seenstable dominance relationships by our criterion). Then, in three-quarters of the groups (but not in the remaining one-quarter)various numbers of fish would have had to have swapped rankson attributes in the 2-week period of separation so as to haveproduced different second hierarchies. And finally, the rankingson attributes for all the fish in all the groups would have had tohave become stable once more for the day or two it took themto form their second hierarchies.

Alternatively, instead of attribute rank determining domi-nance rank as in the prior attribute model, dominance in pairsof fish might be considered to have been probabilistic, such thatat one meeting one might dominate, but at a second meetingthere was some chance that the other might dominate. Theproblem with this model is that earlier mathematical analysisdemonstrates that in situations in which one of each pair in agroup has even a small chance of dominating the other, theprobability of getting linear hierarchies is quite low (34). Andeven in a more restrictive model in which only pairs of fish thatare close in rank in the first hierarchies have modest probabilitiesof reversing their relationships, such as the level (0.25) weobserved in this experiment, the probability of getting as manylinear hierarchies as we observed is still very low (details areavailable from the authors).

We know of only one other study (47) in which researchersassembled groups to form initial hierarchies, separated theindividuals for a period, and then reassembled them to form asecond hierarchy (but see Guhl, ref. 48, for results in whichgroups had pairwise encounters between assembly and reassem-bly). Unfortunately, their techniques of analysis make it impos-sible to compare results, because they examined correlationsbetween the frequency of aggressive acts directed by individualsin pairs toward one another in the two hierarchies rather thancomparing the ranks of individuals. With these techniques it ispossible to get a positive correlation and thus a ‘‘replication’’ ofan original hierarchy in situations in which several animalsactually change ranks from the first to the second hierarchies.

Table 1. Percentage of groups with different numbers of fishchanging ranks between first and second hierarchies (n ! 22)

No. of fish changing ranks Percentage of groups

0 27.32 36.43 18.24 18.2

Fig. 1. Transition patterns between ranks of fish in the first and secondhierarchies. Frequencies of experimental groups showing each pattern areindicated in parentheses. Open-headed arrows indicate transitions of rank.Solid-headed arrows show dominance relationships in intransitive triads; allthe fish in an intransitive triad share the same rank.

5746 ! www.pnas.org"cgi"doi"10.1073"pnas.082104199 Chase et al.

0 27.32 36.43 18.24 18.2

I 22 observations: about 3/4 of the time, hierarchychanged

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Fish forget—changing of dominance hierarchies:

0 27.32 36.43 18.24 18.2

I 22 observations: about 3/4 of the time, hierarchychanged

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Experiment TwoWe took fish from their stock tanks, weighed them, and made upgroups of four and five fish using weights at isolation; in a set offour the largest was no more than 7% heavier than the smallest,and in a set of five, no more than 9% heavier. After 2 weeks ofisolation, as in Experiment One, we formed hierarchies byround-robin competition and group assembly.

In round-robin competition we randomly selected two pairs offish from a set for the first round of encounters. Each pair wastransferred to a 21-liter observation tank separated by a partitionand given 2 h to acclimate to the tank. We then removed thepartition and observed them through a one-way mirror. Again,all instances of nips, chases, and mouth fighting were recordeduntil one fish reached dominance over the other (a total of 15consecutive, aggressive acts against the other without retalia-tion). Of these 15 we scored only nips for the first seven acts andany combination of nips and chases for the remaining eight. Asbefore, we considered mouth fighting as a mutually aggressiveact and began recounting consecutive acts by either fish aftersuch an incidence. When one fish reached the dominancecriterion, we separated them and returned them to their isolationtanks.

We continued the rounds of encounters until all fish in a sethad met one another. Each fish in sets of four had 2 days betweenencounters, and in sets of five each had 1 day except for the oddfish out, which had 2 days. Where possible, we matched winnersto winners and losers to losers.

In group assembly we simultaneously transferred all fish in aset from their isolation tanks to a 76-liter aquarium; observationsbegan 24 h later. We observed the fish and determined stabledominance relationships and hierarchies using the same proce-dures as described for experiment one.

We used six rather than 15 consecutive aggressive acts in groupassembly to determine stable dominance relationships, becauseupon first meeting, as in round-robin competition, fish oftenexchange aggressive acts before one clearly establishes domi-nance and initiates all aggressive activity; such contests requirea fairly large number of consecutive acts to ensure a stablerelationship. In contrast, after some time together, relationships

are often in place, fish do not trade acts back and forth, thusfewer acts suffice to determine which fish in a pair is dominant.

Results. As indicated earlier in the discussion of our experimentaldesign, both hypotheses predicted that the hierarchies formedthrough group assembly should be linear, but they disagreedabout the extent of linear hierarchies formed via round-robincompetition. Although the prior attributes hypothesis antici-pated linear structures, the social dynamics hypothesis fore-casted nonlinear ones, because round-robin competition did notallow interaction in a group context.

Fig. 2 shows the various hierarchy structures and frequenciesof sets of fish forming them in round-robin competition andgroup assembly. Most of the hierarchies formed under groupassembly were linear, and the few others tended to showrelatively simple structural deviations from linearity. In contrast,many hierarchies formed with round-robin competition were notlinear, including several with quite complicated structures.

In Table 2, the probabilities of linear and nonlinear hierarchiesin sets of four and five fish expected by chance alone (if each fishhad an independent 0.5 probability of dominating each other)are compared with the proportions observed in round-robincompetition and group assembly. Although round-robin com-petition only produced significantly higher proportions of linearhierarchies than expected by chance in sets of five fish, groupassembly did so in sets of both sizes (one-sided binomial tests:round robin competition, n ! 16, P ! 0.10 in sets of four fish,and n ! 12, P " 0.005 in sets of five; group assembly, n ! 25,P " 0.001 in sets of four and n ! 11, P " 0.001 in sets of five).

Fig. 2. The structure of hierarchies forming in group assembly and round-robin competition for sets of four and five fish. An animal dominates all those listedbelow it except as indicated by heavy arrows; three fish in an intransitive triad sharing the same rank are placed on the same level in a hierarchy. Frequenciesof experimental groups showing each structure are indicated in parentheses.

Table 2. Percentage of linear structures expected in randomhierarchies and observed in round-robin competition and groupassembly in sets of four and five fish

Size of set

Method of forming hierarchy

Random, % Round robin, % Group assembly, %

4 37.5 56.2 (n ! 16) 92.0 (n ! 25)5 11.7 50.0 (n ! 12) 90.9 (n ! 11)

Chase et al. PNAS ! April 16, 2002 ! vol. 99 ! no. 8 ! 5747

SOCI

AL

SCIE

NCE

S

I Group versus isolated interactions produce differenthierarchies

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Music Lab Experiment

48 songs30,000 participants

multiple ‘worlds’Inter-world variability

I How probable is the world?I Can we estimate variability?I Superstars dominate but are unpredictable. Why?

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Salganik et al. (2006) “An experimental study of inequalityand unpredictability in an artificial cultural market” [6]

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Experiment 1 Experiments 2–4

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112243648

1

12

24

36

48

Experiment 1

Rank market share in indep. world

Ran

k m

arke

t sha

re in

influ

ence

wor

lds

112243648

1

12

24

36

48

Experiment 2

Rank market share in indep. worldR

ank

mar

ket s

hare

in in

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orld

s

I Variability in final rank.

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0 0.01 0.02 0.03 0.04 0.050

0.05

0.1

0.15

0.2Experiment 1

Market share in independent world

Mar

ket s

hare

in in

fluen

ce w

orld

s

0 0.01 0.02 0.03 0.04 0.050

0.05

0.1

0.15

0.2Experiment 2

Market share in independent world

Mar

ket s

hare

in in

fluen

ce w

orld

s

I Variability in final number of downloads.

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0

0.2

0.4

0.6

Social Influence Indep.

Gin

i coe

ffici

ent G

Experiment 1

Social Influence Indep.

Experiment 2

I Inequality as measured by Gini coefficient:

G =1

(2Ns − 1)

Ns∑i=1

Ns∑j=1

|mi −mj |

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0

0.005

0.01

0.015

SocialInfluence

Independent

Unp

redi

ctab

ility

U

Experiment 1

SocialInfluence

Independent

Experiment 2

I Unpredictability

U =1

Ns(Nw

2

) Ns∑i=1

Nw∑j=1

Nw∑k=j+1

|mi,j −mi,k |

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Sensible result:I Stronger social signal leads to greater following and

greater inequality.

Peculiar result:I Stronger social signal leads to greater

unpredictability.

Very peculiar observation:I The most unequal distributions would suggest the

greatest variation in underlying ‘quality.’I But success may be due to social construction

through following.

(so let’s tell a story... [7, 8])

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greater inequality.

unpredictability.

through following.

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greater inequality.

unpredictability.

through following.

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greater inequality.

unpredictability.

through following.

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greater inequality.

unpredictability.

through following.

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greater inequality.

unpredictability.

through following.

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greater inequality.

unpredictability.

through following. (so let’s tell a story... [7, 8])

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Music Lab Experiment—Sneakiness

0 400 7520

100

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500

Dow

nloa

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Exp. 3

Song 1

Song 481200 1600 2000 2400 2800

Exp. 4

Subjects

Song 1

Song 48

Song 48Song 48

Unchanged worldInverted worlds

0 400 7520

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Dow

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Song 2

Song 47

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Song 47

Song 47Song 47

I Inversion of download countI The pretend rich get richer ...I ... but at a slower rate

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0 400 7520

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Song 48Song 48

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Song 2

Song 2Song 2

Song 47

Song 47Song 47

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0 400 7520

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Exp. 3

Song 1

Song 481200 1600 2000 2400 2800

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Song 48

Song 48Song 48

0 400 7520

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1200 1600 2000 2400 2800

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Song 2

Song 2Song 2

Song 47

Song 47Song 47

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References I

[1] M. Adler.Stardom and talent.American Economic Review, pages 208–212, 1985.pdf (�)

[2] M. Balinski and R. Laraki.A theory of measuring, electing, and ranking.Proc. Natl. Acad. Sci., 104(21):8720–8725, 2007.pdf (�)

[3] I. D. Chase, C. Tovey, D. Spangler-Martin, andM. Manfredonia.Individual differences versus social dynamics in theformation of animal dominance hierarchies.Proc. Natl. Acad. Sci., 99(8):5744–5749, 2002.pdf (�)

http://www.uvm.eduhttp://www.uvm.edu/~pdoddshttp://www.uvm.edu/~pdodds/research/papers/others/1985/adler1985a.pdfhttp://www.uvm.edu/~pdodds/research/papers/others/2007/balinski2007a.pdfhttp://www.uvm.edu/~pdodds/research/papers/others/2002/chase2002a.pdf

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References II

[4] P. Laureti, L. Moret, and Y.-C. Zhang.Aggregating partial, local evaluations to achieveglobal ranking.Physica A, 345(3–4):705–712, 2004. pdf (�)

[5] S. Rosen.The economics of superstars.Am. Econ. Rev., 71:845–858, 1981. pdf (�)

[6] M. J. Salganik, P. S. Dodds, and D. J. Watts.An experimental study of inequality andunpredictability in an artificial cultural market.Science, 311:854–856, 2006. pdf (�)

[7] C. R. Sunstein.Infotopia: How many minds produce knowledge.Oxford University Press, New York, 2006.

http://www.uvm.eduhttp://www.uvm.edu/~pdoddshttp://www.uvm.edu/~pdodds/research/papers/others/2004/laureti2004b.pdfhttp://www.uvm.edu/~pdodds/research/papers/others/1981/rosen1981a.pdfhttp://www.uvm.edu/~pdodds/research/papers/others/2006/salganik2006a.pdf

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References III

[8] N. N. Taleb.The Black Swan.Random House, New York, 2007.

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