Evolution in Structured Populations

Abstract

<p>How does social and economic interaction of agents within large populations depend on their perception of the matching-structure?<br /> When do evolutionary dynamics with limited information processing lead to stable outcomes prescribed by rational concepts?<br /> In the chapter "Anticipated Stability in Social and Economic Networks", I model agents to meet with non-uniform probabilities. As friends or colleagues are more likely to interact frequently, this deviation seems to be plausible. A comfortable approach to model such conditional interaction is the one of network formation. I transfer Jackson & Wolinsky (1996) and its dynamic interpretation Jackson & Watts (2002) to a non-cooperative model of network formation. Unsurprisingly, a pairwise stable network results from a Nash equilibrium. My focus rather is on closed cycles. A closed cycle is a subset of networks all of whose members are active periodically. Such a set could also be interpreted as a random graph. Jackson & Watts (2002) show that the process of network formation eventually stops in a pairwise stable network or is stuck in a closed cycle. The point of my paper is that this result crucially depends on the assumption of myopic optimization. I propose that agents may hold beliefs that are consistent with actual behavior for networks that have distance less than κ to the current network and optimize given these beliefs. The parameter κ captures the computational capabilites of the agents. If κ is large enough, small cycles can be excluded if agents anticipate such cycles. I define anticipated stability as the result of optimization given consistent beliefs around the current network. If κ is very large, agents are required to hold sophisticated beliefs for any network, as for example in Dutta, Ghosal and Ray (2005). For large populations this requirement seems inplausible to me since the number of dimensions of the strategy space grows fast with the population size. My concept can flexibly be adapted to small and large populations by fixing a large or small κ. It may seem promising to apply this concept to infinite game trees in which a distance function on the set of nodes is plausible.<br /> In the chapter "Evolution and Correlated Equilibrium" I define a game in strategic form in which players receive signals and choose strategies. According to Aumann (1987), rational play induces a correlated equilibrium distribution on the set of outcomes. Players are rational if they compute conditional probabilities of signals received by other players and optimize given this information and equilibrium strategies of their opponents. I analyze a setting in which players do not know the signal generating process, are not able to apply Bayes’ rule and do not hold beliefs over the set of strategies chosen by their opponents. I approach the concept of correlated equilibrium by an evolutionary methodology and show that even if agents display extreme bounded rationality some correlated equilibria remain to be plausible (are stable with respect to imitation dynamics). The general formulation of the signal generation encompasses Lenzo & Sarver 2005 and Mailath, Samuelson & Shaked (1997). I apply the concept of strict equilibrium sets by Balkenborg (1994) and characterize hereby asymptotically stable sets of correlated equilibrium strategies with respect to convex monotonic dynamics Hofbauer & Weuibull (1996). Balkenborg & Schlag (2007) and Cressman (2003) show similar characterizations with respect to distinct dynamics. With this framework at hand I turn to characterize robust signals. In which situations agents would not influence the signal generating process if they could? I show that if one requires asymptotically stable behavior given signals, only signals inducing strict Nash equilibria yield no incentives to influence the process of signals. If one only imposes the weaker requirement of Aumann’s rational play, I show for the example of the Chicken Game that only those signals are robust to manipulation if equilibrium play yields payoffs within the convex hull of the Nash-payoffs. It remains to be studied whether this implication transfers to other games.<br /> The third chapter "Persistent Ideologies in an Evolutionary Setting" was inspired by the discussion on religious topics partially initiated by Richard Dawkins. From my point of view, religion attaches a set of unverifiable consequences to the set of material consequences of interaction. I show that a religion that views these consequences qualitatively different from the material consequences may face no disatvantage even if agents adopt this religion more frequently if the recommend behavior yields relatively high material payoffs. I hereby critizise the approach of selecting certain preferences by evolutionary methods, as my approach allows for more general interpretations as ideologies or preferences. I generalize Sandholm (2001) in which agents are biased to one of two actions in symmetric games. In my model agents hold biases for outcomes of general asymmetric games in normal form. I assume that agents choose optimal actions given their bias and given their belief of the action choice of their opponents. Biases are heterogeneous within the population and unobserved to other players. I show that if one is willing to adopt the ‘indirect evolutionary approach’ of faster growth of preferences that induce relative successful behavior, situations in which no agents hold preferences that are equivalent to material payoffs are stable for (almost all) games in strategic form if a general model is considered. This is contrary to Ok & Vega-Redondo’s (2001) result in a different setup.</p>