Game Theory in Social Media with Quantal Response Equilibrium

DEGREE PROJECT, IN COMPUTER SCIENCE , FIRST LEVEL STOCKHOLM, SWEDEN 2015

Game Theory in Social Media with Quantal Response Equilibrium

ERIC BLOMQUIST AND RASMUS ELMGREN

KTH ROYAL INSTITUTE OF TECHNOLOGY

CSC SCHOOL

THE ROYAL INSTITUTE OF TECHNOLOGY

Game Theory in Social Media with Quantal Response Equilibrium

by

Eric Blomquist, ericbl@kth.se Rasmus Elmgren, relmgren@kth.se

A thesis submitted in partial fulfillment for the Bachelor’s degree of Computer Science

DD143X, Degree Project in Computer Science, First Cycle

in the

Computer Science and Communication School Examiner

Orjan Ekeberg ¨ Supervised by Danupon Nanongkai

May 2015

“In theory there is no difference between theory and practice. In practice there is.”

Walter J. Savitch

“When you come to a fork in the road, take it.”

Yogi Berra

THE ROYAL INSTITUTE OF TECHNOLOGY

Abstract

Computer Science and Communication School

Bachelor in Computer Science

by Eric Blomquist, ericbl@kth.se Rasmus Elmgren, relmgren@kth.se

This paper examines the possibility to construct a Game Theory model to describe Social Media with a Quantal Response Equilibrium. It is based on a literature study.

The paper is influenced by ”A Game-theoretic Model of Attention in Social Networks”

written by Goel and Ronaghi but creates a more realistic model by replacing their Nash

Equilibrium with a Quantal Response Equilibrium. Such model is constructed in the

Result section and elaborated further in the Discussion. This paper also discusses the

difficulties of Game Theory in Social Media and the flaws of the model created in the

Result. The model helps provide an understanding of success in Social Media. It is

possible to do continued research with more emphasis on the value of different players

or how the order of content affects the level of attention.

Sammanfattning

Denna rapport unders¨ oker m¨ ojligheten att konstruera en Spelteoretisk modell f¨ or att beskriva Sociala Medier med ett Quantal Response Equilibrium. Arbetet ¨ ar baserat p˚ a en litteraturstudie. Arbetet ¨ ar influerat av rapporten ”A Game-theoretic Model of Attention in Social Networks”, skriven av Goel och Ronaghi, men skapar en mer realistisk modell genom att ers¨ atta deras Nash Equilibrium med ett Quantal Response Equilbrium.

En s˚ adan modell skapas i Resultat-delen och behandlas ytterligare i Diskussionen. Denna rapport diskuterar ¨ aven sv˚ arigheterna f¨ or Spelteori i Sociala Medier och bristerna i den skapade modellen. Modellens syfte ¨ ar att bidra till en ¨ okad f¨ orst˚ aelse f¨ or succ´ e i Sociala Medier. Vidare unders¨ okningar kan g¨ oras med fokus p˚ a v¨ ardet av olika spelare eller hur ordningen av inneh˚ all p˚ averkar uppm¨ arksamhetsniv˚ an.

iii

Acknowledgements

Thank you Danupon Nanongkai, for your help, time and supervision.

Contents

1 Introduction 1

1.1 Problem statement . . . . 2

1.2 Purpose . . . . 2

2 Background 3 2.1 Social Network . . . . 3

2.2 What is Game Theory? . . . . 3

2.3 Nash Equilibrium . . . . 4

2.4 Goel’s and Ronaghi’s Model . . . . 4

2.5 Quantal Response Equilibrium (QRE) . . . . 5

2.6 State of the Art . . . . 6

3 Method 7 4 Result 8 4.1 Assumptions . . . . 8

4.2 Model . . . . 9

5 Discussion 11 5.1 Conclusion . . . 12

5.2 Continued Research . . . 12

v

Introduction

In modern society humans interact a lot via Social Media online. Success is measured in attention by receiving likes, comments or by getting your content shared. The urge to succeed affects how you decide what to post, when to post it and one could say that it has become one of the main incentives of our generation. To grant a better understanding of the correlations leading to success a mathematical model can be of help. One way of constructing such model is by using Game Theory.

Game Theory is a mathematical approach used to describe the interaction and decision- making of a finite number of players with a finite number of strategies. It is used to help understand behaviors and interaction in and between agents. It is most commonly applied within economics and to some extent in the fields of politics, sociology, biology and psychological behaviors.

A common notion in Game Theory is the Nash Equilibrium which within a game rep- resents that all players have chosen the best possible strategy with maximum payoff in response to other players strategies. The Nash Equilibrium always assumes that every player acts completely rational.

In 2012 Ashish Goel and Farnaz Ronaghi proposed a Game Theoretic model for the eco- nomics of producing and consuming content on Social Media using a Nash Equilibrium.

The players represents the Social Media users and the strategies include liking, comment- ing, sharing etc. In Goel and Ronaghi’s model the users selects strategies where quality contributions rewards the user with attention and utility. All players are assumed to strive for maximum utility. Goel and Ronaghi have not explicitly specified which Social Media they used in their thesis but when studied it is possible to draw the conclusion that the authors have constructed their model based on Facebook. This paper will take the same approach.

However, a Nash Equilibrium is not always representative of real world problems due to

its naivety assuming all players make completely rational decisions. Therefore Quantal

Response Equilibrium was created, granting players the option of making non-optimal

decisions. This paper will apply the Quantal Response Equilibrium to Social Media.

1.1 Problem statement

This paper aims to answer the question ’Is it possible to construct a Game Theory model in Social Media using Quantal Response Equilibrium?’.

1.2 Purpose

The purpose of this paper is to provide a more realistic model than the earlier Game Theory models in Social Media. This will help grant a better understanding of success in Social Media.

2

Background

2.1 Social Network

A social network is a platform where people can interact with other people, commu- nicate, share thoughts and form groups, etc. The social network produce a feed of content to each user, mostly user-contributed but it may also contain content from com- panies and organisations. Each user interacts with the content by commenting and/or liking/voting/sharing etc.

Users with more views, likes and comments may get a higher social status in the network.

Social status has always been highly rated and a study by Huberman, Loch and ¨ On¸ c¨ uler [1] suggests that one is willing to trade of some material gain to obtain it.

2.2 What is Game Theory?

Game theory is a study of mathematical models where agents take decisions (strategies) to obtain outcomes or payoffs [2]. The agents can act together (cooperative) or in conflict (alone). Game theory mainly appears in biology, logic and computer science. There are many different types of games. The specific agent can be a part of the decision or the game can be symmetric, meaning that it does not matter which agent made the choice, the outcome is the same. There can also be sequential or simultaneous games, depending on the order of choices.

Dixit and Skeath [3] defines the rules for a game in this quote.

“Strictly speaking, the rules of the game consist of (1) the list of players, (2) the strategies

available to each player, (3) the payoffs of each player for all possible combinations of

strategies pursued by all the players, and (4) the assumption that each player is a rational

maximizer.”

2.3 Nash Equilibrium

A Nash equilibrium appears when all players (agents) chooses the best strategy that is the best response to other players strategies. Nash’s existence theorem from 1951 states that [4]:

“Any game with a finite set of players and finite set of strategies has a Nash equilibrium of mixed strategies.”

Assumed that there is a finite set of strategies, a finite set of users and that it is possible to form a game, there is also, according to Nash’s theorem, an equilibrium.

2.4 Goel’s and Ronaghi’s Model

Goel and Ronaghi have written the paper A Game-theoretic Model of Attention in Social Networks[5]. They model the economics of producing content in online social networks such as Facebook and Twitter. A lot of assumptions had to be made to make the model work. The model describes how users perceive and exchange information. Attention and information are assumed to be the main motivation for user contribution and attention is treated as a mechanism from consuming information.

Begin by assuming every user on the network has a relationship with at least one other user. A user a produces x a units of information that her friends will see in their feed.

The user a perceives

y a = X

b∼a

q _ba x _b (2.1)

units of information from her feed. Where b ∼ a says that b is a friend of a and q _ba represents a’s interest in b’s information. So y _a is the amount of information a gets by watching her feed.

Every user a receives an increasing cost c _a (x _a ) for producing information and also re- ceives an increased utility for consuming information f a (y a ). A user also gets a higher utility from receiving attention. The amount of attention user a receives from her friend b is t _a,b (ˆ x).

Therefore, the total amount of utility user a derives is:

u a (ˆ x) = f a (y a ) − c a (x a ) + t a (ˆ x) (2.2)

4

where

t a (ˆ x) = X

b∼a

t a,b (ˆ x) (2.3)

and ˆ x represents the strategy vector.

In other words, u a (x) is the total amount of utility. It is decided by the amount of information consumed, minus the effort of producing information, plus the attention received from relationships on previous productions.

2.5 Quantal Response Equilibrium (QRE)

Players sometimes take decisions that does not give them the best outcome. Humans are not perfect.

“In general, each player does not really know the other players’ value systems; this is part of the reason why in reality many games have incomplete and asymmetric information.

In such games, trying to find out the values of others and trying to conceal or convey one’s own become important components of strategy.” [3]

The Nash Equilibrium always assumes that the players takes the best strategy with respect to the other players. The Quantal Response Equilibrium is an extended version of Nash where there is a rationality variable which adds margin of error. If the rationality variable goes towards infinity, the equilibrium approaches Nashs. On the other hand, if the rationality variable approaches zero the players will pick their strategy with an irrational mind not knowing what strategy is the best. Each user has its own error.

Only player i observes the error  _ij .

McKelvey and Palfrey [6] defines a QRE as following. The model for a game in QRE consists of I = {1, ..., n} the set of players. Each i in I has a strategy set A _i , assumed to be finite containing J i elements. In Nash Equilibrium each player has a payoff function u _i : A → R. A is the set of probability distributions over A _i and an element s _i ∈ S _i is a mixed strategy. Let each i and each j ∈ {1, ..., J i } and for any s ∈ S be written as v _ij (s), the expected utility to i when using the pure strategy a _ij when the other players use s −i . In QRE for each pure strategy a ij player i receives a disturbance to their payoff,

 ij . So i’s payoff from strategy a ij using strategy profile s is ˆ

v ij = v ij (s) +  ij (2.4)

Player i’s disturbances  i = { i1 , ...,  iJ

} is distributed with the density function f _i ( i ).

Each player choose a strategy a _ij so ˆ v _ij ≥ ˆ v _ik for all other k ∈ {1, ..., J _i } where k 6= j.

v and f makes a distribution over the choices by each player. For any v define B _ij(v) as the set of realizations of  _i so the strategy a _ij has the highest disturbed, expected payoff. Then

P _ij (v) = Z

B

(v)

f ()d (2.5)

is the probability that player i chooses strategy j given v. Any fixed point s∗ such that s∗ = P (v(s∗)) is called a Quantal Response Equilibrium for the game (I, A, u).

2.6 State of the Art

Batzilis et al. [7] have modeled the game Rock, Paper, Scissors using Game Theory.

Using statistics from the game Roshambull (which basically is Rock, Paper, Scissors) on Facebook they could calculate equilibria. They described the model as a 3x3 zero-sum game, where zero-sum means that adding up wins and losses amounts to zero. Batzilis et al proposed that a Nash Equilibrium exists when players for each throw of every match expects their opponent to mix ¹ ₃ , ¹ ₃ , ¹ ₃ over rock, paper, scissors.

When players responded to information about previous wins and losses for opponents and themselves statistics showed that a Quantal Response model of the game is a better fit for the majority of the players. This due to the fact that players no longer made the rational strategy choice based on ¹ ₃ chance but instead tried to counter the opponents strategy based on statistics.

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Method

To be able to answer the question “Is it possible to construct a Game Theory model in Social Media using Quantal Response Equilibrium?” we had to try to create such equilibrium. To do this we started with a literature study on papers about related game theories such as Quantal Response Equilibrium, Nash Equilibrium and Game Theory in general. Having no previous knowledge in the field of Game Theory we had to do some extensive research. We probably spent more than 80 percent of the research time reading old papers to attain a basic understanding to build our thesis on.

While doing research we found the paper by Goel and Ronaghis [5] and thought it would be a good starting point to our thesis. They created a model with the Nash Equilibrium and discussed the importance of content order and symmetry in producers.

We observed that there were a lot of assumptions and while some are necessary in order for the model to be accurate we thought the Nash Equilibrium was a bit naive. With the backing of another state of the art paper, section 2.6, we concluded that replacing the Nash Equilibrium with a Quantal Response Equilibrium would present a more realistic model. We maintained most of the assumptions from Goel and Ronaghi’s (listed in the result), limited the choices of strategy to be a finite vector and replaced the Nash Equilibrium with a Quantal Response Equilibrium.

We then started constructing the model by analysing each individual term in the paper

by Goel and Ronaghi and by studying the original Quantal Response Equilibrium defini-

tion by McKelvey and Palfrey [6]. Using the knowledge we gathered in the background,

especially in section 2.5, we could transfer values from Goel and Ronaghi’s model and

fit them into a Quantal Response Equilibrium. Our greatest obstacle when constructing

the model was understanding all the different theorems and how they were affecting one

another.

Result

4.1 Assumptions

To be able to construct a model some assumptions have to be made.

Fixed network structure

The number of players is assumed to be fixed, neither increasing or decreasing, to be able to list all the players in the game and their respective strategies.

Ignoring order of content

The order of the content is also a factor not taken into account. It is hard to measure how interest and consumption of information declines over time. Information further down a players feed is likely to get less interest.

Utility-maximizing agents

All players are assumed to strive for maximum utility.

Production of information incurs increasing cost

Producing more information incurs an increased cost. Due to the player’s ambition to produce quality content, according to the previous assumption, an increased production of content results in a higher production cost. The cost c _a (x _a ) is therefor a differentiable, increasing and strictly convex function for all agents c a (0) = 0.

Consumption utility is a concave differentiable function

Consuming information results in increased utility. f _a (y _a ) is assumed to be a differen- tiable, concave and increasing function. Also f a (0) = 0.

The attention t _a,b (ˆ x) is increasing

We assume the payoff for each player is the value of attention. Attention achieved from a friend is increasing given the better strategy and t _a,b (0, ˆ x −a ) = 0, where ˆ x −a is the strategy profile without a’s contribution.

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4.2 Model

This model originates from Goel and Ronaghi’s model, 2.4, and some of the basic con- cepts will be the same.

Applying the information from the quote in section 2.2 and the assumptions in section 4.1 to the social network we have a simultaneous game:

1. Friends and active users on the social network, 2. Comments, likes, posts, shares, views,

3. The payoff for each player is the value of attention, 4. All players are utility maximizers.

The choice of strategy affects how much the user wants to contribute to the network.

The strategy does not involve what the user is consuming.

Every player a pays a cost c _a (x) for producing information based on her choice of strategy, x a . The cost can not be negative as the utility is calculated by subtracting it. The strategy can be a mix from the vector {comment, like, post, share, view}. The choice of strategy is not guaranteed to be optimal but the one the player a perceives to give the best outcome. Due to the fact that the player is not able to make a perfect choice of strategy, an error margin  has to be added to the utility.  is specific for each player a and her choice x a , denoted as  ax

. Friends of a will see this post and give attention to a.

Consumption of information gives increased utility to a user a as f a (y a ) where y a is seen as the total information from the feed, modeled in the same way Goel and Ronaghi have in 2.1. b ∼ a represents that b is a friend of a and q ba is a’s interest in b’s information.

y _a = X

b∼a

q _ba x _b

Here we propose ˆ v to be similar to the model of Goel and Ronaghi. A quantal response equilibrium exists if every player in the game aims to maximize their utility ˆ v _ax

defined by us as:

v _ax

(ˆ x) = f _a (y _a ) − c _a (x _a ) + t _a (ˆ x) +  _ax

(4.1) with subject to player a’s choice of strategy, x _a > 0. The attention function t _a is defined by Goel and Ronaghi in 2.3 and ˆ x is the strategy vector.

The total utility is equal to the combination of the rationality variable and the con- sumption, cost and attention functions. This is the main result of the paper.

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Discussion

Theoretically defining an equilibrium, like the one we have specified in the result 4.1, is not hard. Proving that the equilibrium actually applies to the game on the other hand can be very difficult. In this particular case, where the game is something as complex as Social Media with irrational players, it is nearly impossible.

The Quantal Response Equilibrium give the players an option of not choosing the best possible strategy. This is implemented with a rationality variable for each individual player, affecting the players strategy when both consuming and producing content. This variable is the main reason why there is no best strategy in the QRE. Due to the fact that the variable makes a player react differently to the same strategy, there is no way of pointing out the best strategy. In QRE there are only better and worse strategies.

This is one of the main reasons why it is hard to create an accurate simulation or to apply our model in practice. Even if the subject in hand, in this case Social Media, is narrowed down to a smaller group of people it is complex to apply the model due to the irrationality of the human mind. While the model is applicable to achieve an understanding of correlations between consumption and production of information/content within the group, the rationality variable and the fundamental values of a player may result in a (from the models point of view) totally illogical choice of strategy. If one could create an isolated environment where the players only consumed each others contributions, with no information from separate sources interfering, the model could be implemented.

The strategy vector in our model is written as {comment, like, post, share, view} but there are an infinite number of ways to construct a comment or a post. It would be possible to group comments into categories but we think that may be too inaccurate as people react in different ways to different sorts of comments. We have therefore defined comments and posts to be discrete (as opposed to continuous), otherwise we can not have a Quantal Response Equilibrium. This meaning that a comment is accounted as the same amount of attention independently of it’s message.

In this report we have built our model primarily around the social network Facebook.

In the model we defined that the strategy vector is a mix of the vector {comment, like,

share, post}. In other social networks some elements of this vector is not available. We also made the interpretation that Goel and Ronaghi’s paper originated with Facebook as the focal point which made our choice even more natural.

We consider the reports and books we have used throughout the paper as reliable. The Game Theory classbooks we have used are modern textbooks and they verify each other in the mathematical perspective. Most of our result are based on the work of Goel and Ronaghi and McKelvey and Palfrey.

5.1 Conclusion

Given our result we can answer the problem statement. It is possible to construct and define a Game Theory model in Social media using Quantal Response Equilibrium but it is very hard to prove that the model actually is the best way to describe the game.

Our purpose is also fulfilled as the Quantal Response Equilibrium is a more realistic model than the Nash Equilibrium, as stated in the background, 2.5.

5.2 Continued Research

We have not discussed how the author of the activities affects the interest, q _ba , of others.

We avoided this problem as it makes the game much harder. De facto, famous peoples activities are perceived as more interesting than regular peoples. Goel and Ronaghi talked briefly about the Shapley value and that is something that could add another layer of realism to our model. The Shapley value is a concept where some players are more important and possess extra bargaining powers compared to the rest.

As previously discussed, it would be nearly impossible to compute an equilibrium, both Nash and Quantal Response. However, if our model is applied to a smaller network, say a small number of people inside Facebook group, the computations will be fewer and easier to measure.

The order of the content is also a factor we avoided. It is hard to measure how interest and consumtion of information declines over time. Information further down a persons feed is likely to get less interest. A future study might want to look into the factor.

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Bibliography

[1] Bernardo A. Huberman, Christoph H. Loch, and Ayse ¨ On¸ c¨ uler. Status As a Valued Resource, 2013. URL http://spq.sagepub.com/content/67/1/103.abstract.

[2] Roger B. Mayerson. Game Theory, Analysis in Conflict. Harvard University Press, 1991.

[3] Avinash Dixit and Susan Skeath. Game of Strategy. W. W. Norton & Company, 2 edition, 2004.

[4] Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani. Algorithmic Game Theory. Cambridge University Press, 2007.

[5] Ashish Goel and Farnaz Ronaghi. A Game-theoretic Model of Attention in Social Networks, 2012.

[6] Richard D. McKelvey and Thomas R. Palfrey. Handbook of Experimental Economic Results, volume 1. Elsevier B.V, 2008.

[7] Dimitris Batzilis, Sonia Jaffe, Steven Levitt, John A List, and Jeffrey Picel. How Facebook Can Deepen our Understanding of Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games. 2014. URL http://soniajaffe.com/

articles/RPS.pdf.

www.kth.se