Game theory social networks cmccabe-12

Game Theory & Social Network
Models

Connor McCabe

PhD Candidate in Web Science

University of Southampton

cm7e09@ecs.soton.ac.uk Agents, Interaction &
May 2012 Complexity (AIC) Group

Overview
Talk: Game Theory & Social Network Models

• Introduction of basic concepts & models

• Examples of social network models using game theory

• Discussion of using game theory as a method for
investigating social scenarios.

Game theory: basic concepts & models

• Game theory (GT) is used to model situations in which
multiple participants (players) interact or affect each other’s
outcomes.

• The origins of GT are from the field of economics, (and
remains most active in that area) although it has been
applied elsewhere in fields including sociology, psychology
and complexity science.

Payoff Matrices
• Normal Form Extended Form

Player 1

A B
A 1,1 0,0
Player 2
B 0,0 1,1

Payoff Matrix for a co-ordination game

Exempli Gratia (e.g): Prisoner’s Dilemma
Player 1

Cooperate Defect
Cooperate R=1, R=1 S=10, T=0
Reward for Sucker’s
mutual payoff, and
Player 2 cooperation temptation
to defect
=
Defect T=0, S=10 P=5, P=5
Temptation to Punishment
defect and for mutual
sucker’s payoff defection

For PD, T(temptation) > R(reward) > P(punishment) > (S)sucker

See http://plato.stanford.edu/entries/prisoner-dilemma/ for full description

Models of network formation
There are 2 key aspects of game theoretic approach to
modelling network formation:

• (i) agents get some utility from the network, and there is an
overall societal welfare corresponding to any network that
might arise, and

• (ii) links are formed by the agents themselves, and the
resulting networks can be predicted through notions of
equilibrium or dynamic processes

Research case 1: Satifysing
• What is Satisfycing? (Satisfy + Sacrifice)
– Similar to the idea of ‘structural balance’ (see chapter 5,
Easley & Kleinberg, Networks, Crowds, and Markets, 2010)
• An example is the co-ordination game, played among many
participants with conflicting constraints.
– Won’t be able to co-ordinate with everyone most likely
(because players have different friends / strategies)
– Hence, the problem is then to identify the subset of the
network the player can gain most from co-ordinating their
actions with.
• The example that we discuss here is Davies et al. (2011)

Adam P. Davies et al. (2011) "if you can't be with the one you love, love the one you're with"
Artif. Life 17, 3 167-181.

Core mechanisms & results
N=100 actors (players)

Uij = symmetric payoff matrix, which defines for actors i and j either :

(i) a coordination game ( x =1, y =0), or

(ii) anti–coordination (x=0, y=1).

Players are assigned to play different type of games with others with equal
probability.

Uij =

• Adding up the payoffs for a single player i,

e.g. Ui(t) = sum(1 + 0 + 1 + 1 …) for games with player j = (1, 2, 3 …n)

• and the whole social network G(t):

e.g. G(t) = sum(50 + 49 + 53 + 40 ….) representing combined outcome for
every player’s games with their network contacts.

Core Mechanisms and Results
• Players flip their current strategy if doing so means they
can co-ordinate with most of their ‘friends’, and to anti-
coordinate with ‘non-friends’, in order to received a positive
payoff from these different social ties.

• Then, a dynamic social structure is modelled, by varying
weighting assigned to each connection as agents learn who
they most often co-ordinate with, (and who they don’t).

• Ties now represent continuous values between 0 and 1,
strongly weighted connections represent the interactions
(games).

1. Non-Habitual Agents

t=0 R
L

Player 1
L

Player 2
R
Player 2
L R
Player 1

L 0 5
R 5 0
True Utility
Coordination game (+5 utility for being the same)
AntiCoordination game (+5 utility for being different)


t=0 R
Utility=5
L
Utility=10

L
Utility=10

R
System Utility = 30 Utility=5



t=1 R
L
Utility=5
Utility=10
L
Utility=5

L
Utility=15

R



t=2 R
L
Utility=5
Utility=15
R
L
Utility=5
Utility=10

L
Utility=15

R



t=4 R
L
Utility=5
Utility=15
R
L
Utility=5
Utility=10

L
Utility=15

R



t = 1000… R
L
Utility=5
Utility=15
R
L
Utility=5
Utility=10

L
Utility=15

R
Utility=10

End of relaxation


• Now we add a preference matrix Pij so that agents
perceive satisfying some connections and sacrificing
others.

• The preference matrix contains a value for each player
pairing; value is initially set to zero, and is adjusted each
time step

2. Habitual Agents

R
R

Player 1
R

Player 2
Player 2
L R L
Player 1

Player 2
L 0 0
L R
Player 1

R 0 0
L 5 0
Perception Transformation R 0 5
True Utility

2. Habitual Agents

R
R

Player 1
R

Player 2
Player 2
L R L
Player 1

Player 2
L -0.1 0.1
L R
Player 1

R 0.1 -0.1
L 5 0 Player 2
Perception Transformation R 0 5 L R

Player 1
True Utility L 4.9 0.1
R 0.1 4.9
Perceived Utility

2. Habitual Agents

R
R

Player 1
R

Player 2
L
• Habitual agents use perceived utility
to make strategy decisions
Player 2
L R

Player 1
L 4.9 0.1
R 0.1 4.9
Perceived Utility

2. Habitual Agents

R
R

Player 1
R

Player 2
L
• Habitual agents use perceived utility
to make strategy decisions
Player 2
L R
• But system utility is always measured

Player 1
using true utility L 4.9 0.1
R 0.1 4.9
Perceived Utility

Research case 2: El Farrol bar model
• The El Farol bar model involves N people (N=100), each
have to decide each evening, at same time, whether they
want to go out to a bar, or else stay in.

• If less than 60% of the population go to the bar, they'll all
have a better time than if they stayed at home.

• If more than 60% of the population go to the bar, they'll all
have a worse time than if they stayed at home.

• This model represents a case of inductive reasoning, since
deterministic / pure strategies are guaranteed to fail.

Core mechanisms and results
• The actors make decisions based on probability of certain
outcomes occurring.

• Assume 100 actors each can individual form predictors /
hypothesis, of the past d week’s attendance figures.

• If for example, recent attendance might be:

• … 44 78 56 15 23 67 84 34 45 76 40 56 22 35

• Predictors of attendance might be:
– same as last week’s[35]
– a rounded average of last four weeks [49]

Core mechanisms and results
• Actors decide to go or stay based on most accurate
predictor they have found so far (active predictor)

• Once decisions are made, the actor updates the accuracies
of their predictors.

• Good predictors are kept, while those found evaluated as
not presently useful are not selected. A whole ecology
containing the active predictors of actors emerges.

Results of the model
• Bar attendance in the first 100 weeks.

• Notice how there are no persistent cycles,

• Interesting, mean attendance always converges to 60

• This is because the predictors self-organize into a pattern / equilibrium.

Core Mechanisms and Results
• Permit each player to use a mixed strategy, where a choice
is made with a particular probability.

• For the El Farol Bar problem there exists a Nash
equilibrium where a mixed strategy involves
– each player deciding to go to the bar with a certain
probability that is a function of the number of players,
and
– the relative utility of going to a crowded or an
uncrowded bar compared to staying home

Final note on mixed strategies
• Following a pure strategy, will enable other players to guess your move

• Following a pure strategy, will enable other players to guess your move.

Lisa: Look, there's only one way to settle this.
Rock-paper-scissors.

Lisa's brain: Poor predictable Bart. Always takes `rock'.
Bart's brain: Good old `rock'. Nothing beats that!

Bart: Rock!
Lisa: Paper.
Bart: D'oh!

• Following a pure strategy, will enable other players to guess your move.

Lisa: Look, there's only one way to settle this.
Rock-paper-scissors.

Lisa's brain: Poor predictable Bart. Always takes `rock'.
Bart's brain: Good old `rock'. Nothing beats that!

Bart: Rock!
Lisa: Paper.
Bart: D'oh!

• Hence the need for mixed strategies involving players randomising
their moves.

• To do well in these games involves players finding the optimal
probability with which to choose each strategy.

Research case 3: Co-evolution of
cooperation
• A model of co-evolution of a social network captures the
interplay between dynamics (games) on the network, and
structural changes of the network that influence the
dynamics(games).

• Van Segbroek et. al’s (2010) model of prisoner’s dilemma
considers how players strategies change and evolve
alongside which games are being played between whom.

• In this study they varied the payoff matrices between the
different linking strategies updates
Van Segbroeck S et. al. (2010).
Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure. Games. 1(3):317-337.

Time scale Ta denotes evolution of the network structure,
and Ts denotes strategy evolution

The impact of network dynamics on the strategy dynamics
depends on the ratio: W = Ts / Ta

Where W <<1 represents fast linking dynamic,

and W>> 1 a slow linking dynamic.

For values upwards of W=0.1, fixation of

cooperation is certain

How does the network of players evolve to co-operation?

• Heterogenous actors with different link strategies:
– Slow cooperators (SC’s), and defectors (SD’s) whose adverse
interactions last longer before they switch
– Fast coperators (FC’s) and defectors (FD’s)
• Actors can change both their strategy of co-operate or defect, and their
link strategy to fast or slow.

• How does the temptation payoff (T) affect the stability of co-operation
(graph a)

• How does speed at which links are adjusted between others (Y) affect
evolution of the network? (graph b)

• Let M represent the number (types) of linking strategies.

• When M = 2, time spent in co-operation was lower (only 7.2%), and
most actors switched to slow defecters (SD)

• Increasing M had a positive effect on increasing selection of
cooperation (59.8% of time was spent co-operating).

Discussion
• The simplicity of game theory, using strategies and payoffs,
can become very analytical when considering a large
number of players representing a social network. This is
where computer simulation can aid.

• In practice, there are some decisions to be made when
designing a game theoretic model of a network, one key
issues which we now discuss: interactions over time

Interactions over time
• Repeated games are very sensitive to the order in which
players make their choices.

• One of careful considerations is whether all the players
make their decision at the same time (synchronous) or not
(asynchronous).

• Synchronous player updates often create coupled
dynamics,
– such as how coupled oscillators sync their rotations /
frequency over time.

Interactions over time
• Dynamics may disappear entirely with asynchronous
updates. E.g. a model of local co-operation was not stable
when asynchronous updates were used.

Source: Huberman, B. A. and Glance, N. S. (1993). Evolutionary games and computer simulations.
Proceedings of the National Academy of Sciences USA, 90(16):7715–7718.

Game theory as a research method
• For social science and Web Science:
– Provides a means to describe exactly a set of actions
and outcomes for social interactions.
– Amenable to simulation modelling
– Offers a link to investigate micro-macro behaviour
– Evolutionary game theory involving repeated games on
a network are useful to model evolution of social
systems / networks

Investigating macro from the micro-level
Game theoretic models encourages
finding the simple micro rules that
help understand the evolution of
complex macro phenomena, like the
Web, and emergent systems on it
such as ‘Web 2.0’ and the
‘blogosphere.’

Further extensions for social models
• Bounded rationality for humans (limits on cognitive
processing, imperfect information)
– Recognise costs of gathering and processing
information
– More realistic, multi-valued utility function
• Most simple game theoretic models involve agents
changing strategy on a fixed network structure.

• Some, however, are complex adaptive system models, in
which both the agent strategies and network structure co-
evolve.

Summary
• In this talk we discussed 3 game theoretic models involving simulated
social networks and their results involving
1) Satisfycing / Structural balance
• Aim to satisfy relations that often paid off in the past
2) Mixed strategies in a social decision problem
• Use heuristics to make a best guess, and keep a record of
which guesses were most often correct
3) Evolution of co-operation in dynamic social network
• Responding promptly to adverse social ties promotes evolution
of co-operation
• The discussion also addressed some of the assumptions practitioners
need to deal with in applying game theory in social network modelling

References
• Arthur, W. B. (1994). Inductive Reasoning and Bounded Rationality (The El Farol
Problem). Amer. Econ. Review (Papers and Proceedings), 84(406).

• Davies, A. P. et al. (2011) if you can't be with the one you love, love the one you're with:
How individual habituation of agent interactions improves global utility. Artif. Life 17, 3
167-181.

• Easley., D. and Kleinberg., J. Networks, Crowds, and Markets (2010) Cambridge
University Press

• Simon, H. A. (1956). Rational choice and the structure of the environment. Psychological
Review, Vol. 63 No. 2, 129-138

• Huberman, B. A. and Glance, N. S. (1993). Evolutionary games and computer
simulations. Proceedings of the National Academy of Sciences USA, 90(16):7715–7718.

• Van Segbroeck S et. al. (2010). Coevolution of Cooperation, Response to Adverse Social
Ties and Network Structure. Games. 1(3):317-337.

• Zu Erbach-Schoenberg, Elisabeth, McCabe, Connor and Bullock, Seth (2011) On the
interaction of adaptive timescales on networks. ECAL 2011, Paris, 08 - 12 Aug 2011

References (continued)
• The two ‘magics’ of web science: www.w3.org/2007/Talks/0509-www-keynote-
tbl/ (Accessed on 29/04/2012)

• Prisoner’s dilemma <http://plato.stanford.edu/entries/prisoner-
dilemma/>(Accessed on 29/04/2012)

Other useful sources

• M. O. Jackson, Social and Economic Networks (2008)

• R. A. Axelrod, Complexity of Cooperation: Agent-Based Models of Competition
and Collaboration, Princeton Studies in Complexity (1997)

Question for discussion
What sort of useful role (or not) can game theory provide as a
tool to investigate the theory and practice of Web Science?

Question for discussion
What sort of useful role (or not) can game theory provide as a
tool to investigate the theory and practice of Web Science?

Perhaps it can be viewed as:

-too simplistic? toy models?

-non realistic? mostly utilises selfish, maximising behaviour

+ good way to look at link micro-level and macro-level

+ useful for analysis and prediction (sometimes) of outcomes
of social scenarios

Game theory social networks cmccabe-12

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