Network Effects in Coordination Games

1
Network Effects in Coordination Games

• Satellite symposium
• Dynamics of Networks and Behavior
• Vincent Buskens
• Jeroen Weesie
• ICS / Utrecht University

2
Introduction

• Actors have interactions while they are organized
  in networks
• How can we analyze the co-evolution of networks
  and behavior?
• First, fixed networks
• Second, dynamic networks
• An example using coordination games

3
Introduction

• Examples of coordination problems
• Driving on left or right side of the road
• Meeting a friend in a train station with two
  meeting points
• Smoking behavior among friends
• More generally, emergence of conventions and norms

4
The Coordination Game

• Player 2
• Player 1
• b lt c lt a lt d
• RISK (a b)/(a d b c)

X Y
X a,a c,b
Y b,c d,d
5
The Equilibria

• (X, X) and (Y, Y) are both Nash equilibria
• There is also a mixed equilibrium
• (Y, Y) is the payoff-dominant equilibrium
• (X, X) is the risk-dominant equilibrium if RISK gt
  0.5 (Y, Y) is the risk-dominant equilibrium if
  RISK lt 0.5. The mixed equilibrium is risk
  dominant if RISK .5.

6
The Problem

• Payoff-dominant equilibrium is better for both
  players, however, under some conditions the other
  equilibrium may emerge, especially when this is
  the risk-dominant equilibrium
• What is the role of the structure of the network
  in this process?

7
Theory on Local Interaction

• Depending on noise and type of learning
• either the risk-dominant equilibrium will
  emerge (Ellison 1993, Young 1998 Ch.6)
• or payoff-dominant or mixed absorbing states
  remain possible (Berninghaus and Schwalbe 1996,
  Anderlini and Ianni 1996).
• Closed neighborhood better than circle
• Neighborhood size no effect (?)
• Neighborhood overlap promotes the
  payoff-dominant equilibrium

8
The Model

• Actors located on graphs (undirected ties)
• Actors play repeatedly coordination games with
  all neighbors
• At each point in time, actors play the same move
  against all their neighbors.
• Actors receive information about the proportion
  of neighbors that played X and Y

9
The Model

• Actors start with propensity 0.5 to play Y
• After each round, this propensity increases or
  decreases with 0.1 depending on the best-reply
  against the neighbors in the last round.
• In this simulation 100 replications until
  convergence for each starting propensity.

10
The Networks and Risk

• The set of non-isomorphic connected networks with
  2 to 8 actors (N 12,112)
• Selection of networks with 9 to 25 actors (N
  100,502)
• Payoffs integer values such that 0 b lt c lt
  a lt d 20
• .095 lt a / (20 a c) RISK lt .905

11
Analytic Results

• RISK has a negative effect on reaching the
  payoff-dominant equilibrium (Y,Y) the effect is
  not linear but a step-function
• If RISK 0.5, i.e., a b d c, there are no
  network effects towards the payoff-dominant
  equilibrium
• Comparing RISK and 1 RISK, all network effects
  are reversed effects that work for RISK gt 0.5
  towards (Y,Y) work in the other direction for
  RISK lt 0.5
• We restrict ourselves to RISK gt 0.5, i.e., where
  the risk- and payoff-dominant equilibrium do not
  coincide.

12
Analyses

• Predicting the expected proportion of actors in a
  given network that play Y after convergence for
  14 categories of RISK gt .5.
• Independent variables
• Network size
• Density (proportion of ties present)
• Centralization (degree variance)
• Segmentation (P3/P2, where Pi is de proportion of
  distances in the network larger than or equal to
  i)
• Proportion of actors with an odd number of
  neighbors
• Maximal degree in the network
• Proportion of times not converged to ALL X or ALL
  Y

13
Regression for RISK-values
14
Network dynamics Why

• Actors will avoid ties in which coordination
  fails and seek ties in which coordination
  succeeds
• Networks may segmentize, with different behaviors
  in segments.
• Potentially different network effects

15
Network dynamics What limits number of ties?

• Few models adequately deal with explaining number
  of ties
• Theoretically, we should argue from goal
  attainment through ties, not through ties
  directly
• We know of no satisfactory simple solution

16
Networks dynamics Assumptions

• At each time, with some probability, actors have
  the opportunity to relocate a tie one-sidedly.
• No switch costs
• Sequential changes, in random order
• Myopic decisions relocate tie if this increases
  payoff.
• Relocate tie to actor with whom coordination
  fails to one with whom coordination succeeds
• No change in ties if payoff-irrelevant
  otherwise network would never converge
• Obviously Size and density do not change
• Unknown consequences for
• Degrees and degree-variance change
• Connectedness and segmentation

17
Simulation

• Initial networks all non-isomorphic networks of
  sizelt8, including disconnected networks
• One RISK value maximal static network effects
• For each of these networks
• Initial behavior and adaptation of propensities
  as before
• Iterate until convergence
• No actors wants to change behavior
• No actors wants to change ties
• Convergence attained in all simulations
  exceptions are possible (for instance 2-cycles)

18
Questions for analysis

• How does the proportion of Y-choices depend on
  the initial network and the tie-change rate?
• How does the probability that equilibrium
  consists of two norms (both X and Y choices)
  depend on the initial network and the tie-change
  rate?
• How does the final network depend on the initial
  network and the tie-change rate?

19

• Regression of proportion of Y-choices in
  equilibrium
• Variable Initial Final
  InitialFinal
• ------------------------------------------------
  -------
• Size 0.0029 0.0037
  0.0047
• Density 0.2818 0.4601
  0.4453
• Initial-----------------------------------------
  -------
• DegreeVar 0.0494
  0.0186
• Segmentation -0.0622
  -0.0741
• MaxDegree -0.0294
  -0.0108
• PropOddDegree 0.2036
  0.1790
• Connected -0.0295
  -0.0131
• Final-------------------------------------------
  -------
• DegreeVar 0.1243
  0.1197
• Segmentation 0.0438
  0.0630
• MaxDegree -0.0696
  -0.0711
• PropOddDegree 0.1506
  0.1103
• Connected -0.0995
  -0.0976
• Dynamics-----------------------------------------
  -------

20

• Logistic regression of Multiple norms in
  Equilibrium
• Variable Initial Final
  InitialFinal
• -------------------------------------------------
  -------
• Size -0.0568 -0.0986
  -0.0833
• Density -6.9004 -0.8058
  -1.1385
• Initial -----------------------------------------
  -------
• DegreeVar 0.0848
  0.3622
• Segmentation 0.2467
  -0.6481
• MaxDegree -0.1769
  -0.1378
• PropOddDegree 0.4122
  0.3687
• Connected -0.6910
  0.1112
• Final -------------------------------------------
  -------
• DegreeVar -2.5947
  -2.6671
• Segmentation 5.8928
  6.0690
• MaxDegree -0.9720
  -0.9663
• PropOddDegree 0.1421
  0.0863
• Connected -4.9595
  -4.9886
• Dynamics-----------------------------------------
  -------

21
Properties final networks

• Size and density are constant by construction
• Degree variance slowly increases with tie change
  rate
• Segmentation stays more or less the same for
  small tie-change rates but decreases rapidy for
  larger tie-change rates
• MaxDegree does not change for any tie-change rate
• The percentage of nodes with an odd number of
  neighbors does not really change

22

• Associations of Initial and Final
• Network Properties
• higher tie-change
  rate correlation NoChange -----------------------
  ----gt
• DegreeVar 1 0.23 0.09 0.06 0.06
  0.07
• MaxDegree 1 0.65 0.60 0.59 0.60
  0.60
• PropOddDegree 1 0.09 0.02 0.00 0.01
  0.02
• Segmentation 1 0.30 0.19 0.11 0.03
  -0.02
• Tau-b ---------------------------gt
• Connected 1 0.34 0.30 0.20 0.15
  0.12
• final nets 89 82 72 59 42
  19

23
Analyses to Be Done

• Repeated simulations separate random variation
  from lack of fit/misspecification
• Larger networks, other values of risks
• Effects of other network characteristics (e.g.,
  betweenness,..)
• Non-linearities in the effects
• Interaction effects between network
  characteristics
• Sensitivity of the analyses related to the sample
  of networks and the specification of the
  statistical model

24
Methodological conclusion

• We can derive testable hypotheses of network
  effects in interactions by
• A large systematic sample of networks
• Simulating an interaction process on this network
• Calculate relevant network characteristics
• Predict characteristics of (the equilibrium
  state of) the interaction process from initial
  network characteristics (network fixed)
• Similar approach with dynamic networks
• Selection appropriate statistical models is often
  non-trivial

25

• The distribution of degrees of the final network
• Variable DegreeVar MaxDegree
  PropOddDegr
• ---------------------------------------------
  ------
• Density 0.145 0.739
  0.067
• Size -0.008 0.007
  0.008
• Initial -------------------------------------
  ------
• DegreeVar 0.311 0.060
  0.006
• Segmentation -0.063 -0.030
  -0.002
• MaxDegree -0.069 0.170
  0.000
• PropOddDegree 0.007 0.001
  0.187
• Connected 0.049 0.026
  0.004
• Dynamics ------------------------------------
  ------
• DYN2 0.022 0.004
  -0.006
• DYN3 0.060 0.012
  -0.008
• DYN4 0.100 0.017
  -0.007
• DYN5 0.137 0.019
  -0.018
• DYN6 0.176 0.021
  -0.033

26

• Regression of Properties Final Network
  (continued)
• Variable Connected
  Segmentation
• -------------------------------------------
  ---
• Density 1.289 -0.155
  
• Size 0.018 0.009
  
• Initial -----------------------------------
  ---
• DegreeVar 0.071 0.065
  
• Segmentation -0.061 0.184
  
• MaxDegree -0.041 -0.036
  
• PropOddDegree -0.040 -0.010
  
• Connected 0.237 0.036
  
• Dynamics ----------------------------------
  ---
• DYN2 -0.076 0.002
  
• DYN3 -0.173 0.000
  
• DYN4 -0.302 -0.011
  
• DYN5 -0.470 -0.038
  
• DYN6 -0.696 -0.084
  
• _cons -0.079 0.077