Network Effects in Coordination Games

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

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 Introduction

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

The Coordination Game Player 2 Player 1 b < c < a < d RISK = (a – b)/(a + d – b –c)

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.

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?

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

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

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.

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

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 networkeffects 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.

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 (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

Regression for RISK-values

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

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

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

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)

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?

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

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 | + - -

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

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

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

“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

Network Effects in Coordination Games