1. 0 Network Effects in Coordination Games Satellite symposium “Dynamics of Networks and Behavior” Vincent Buskens Jeroen Weesie ICS / Utrecht University

2. 1 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. 2 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. 3 The Coordination Game Player 2 Player 1 b < c < a < d RISK = (a – b)/(a + d – b – c) XY Xa,ac,b Yb,cd,d

5. 4 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 > 0.5; (Y, Y) is the risk-dominant equilibrium if RISK < 0.5. The mixed equilibrium is risk dominant if RISK =.5.

6. 5 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. 6 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. 7 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. 8 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. 9 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 < c < a < d = 20.095 < a / (20 + a – c) = RISK <.905

11. 10 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 > 0.5 towards (Y,Y) work in the other direction for RISK < 0.5 We restrict ourselves to RISK > 0.5, i.e., where the risk- and payoff-dominant equilibrium do not coincide.

12. 11 Analyses Predicting the expected proportion of actors in a given network that play Y after convergence for 14 categories of RISK >.5. Independent variables –Network size –Density (proportion of ties present) –Centralization (degree variance) –Segmentation (P 3 /P 2, where P i 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. 12 Regression for RISK-values

14. 13 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. 14 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. 15 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. 16 Simulation Initial networks : all non-isomorphic networks of size<=8, 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. 17 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. 18 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---------+--------------------------------------- change rate | - - - ----------------+---------------------------------------- r2 | 0.0195 0.0229 0.0322

20. 19 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---------+--------------------------------------- Change rate | + - -

21. 20 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. 21 Associations of Initial and Final Network Properties higher tie-change rate correlation NoChange ---------------------------> 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 ---------------------------> Connected 1 0.34 0.30 0.20 0.15 0.12 %final nets 89 82 72 59 42 19

23. 22 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. 23 “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. 24 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 _cons | 0.290 0.153 0.306 ----------------+----------------------------------- r2 | 0.303 0.643 0.047

26. 25 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 ----------------+------------------------------ r2 | 0.385 0.131