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January 20 - February 1, 2022
All modeling requires the faith that, as Borges expresses it, we can occasionally turn the sand of the real world into stone. Effective models require a real world that has enough structure so that some of the details can be ignored.
Systems, whether scientific models or real-world entities, that do not have sufficient underlying structure are very unstable, difficult to understand, and hard to control.
We would like to be able to develop a theory that helps us understand how states of the world (composed of lower-level entities and interaction rules) are transformed into higher-level entities. Some initial work on this topic has been done with cellular automaton models, where it has been shown that under some conditions a variety of seemingly different interaction rules imply only a few distinct types of high-level behavior (Wolfram, 2002). 3.4
In chapter 9 we explore a model of sand piles in which we randomly drop grains of sand onto a table. A pile forms as the sand falls, and eventually grains begin to run off the edges of the table in avalanches of various sizes. The distribution of avalanche sizes follows a power law that implies behavior that is quite different from that arising from a normal distribution. Agent intention can also alter the patterns that emerge in complex systems. In the case of the Sand Pile model, if we give the falling grains of sand a bit of control on where they land and some desires (like maximizing the
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recently the area of experimental economics has experienced tremendous growth and acceptance. The legitimacy of using experimental results to challenge existing theories is also gaining acceptance, and it is even starting to drive the theoretical enterprise down new avenues of exploration, such as learning models in games and behavioral economics.
While the acceptance of an experimental component to theory creation and testing is likely to continue, it is also the case that experimental methods on human subjects are inherently limited. Some of the systems of most interest to economists, like complex macroeconomic systems composed of hordes of heterogeneous agents, are not easily captured in a standard laboratory setting.
Perhaps we could argue that an axiomatic proof is still superior because it may provide some additional insight into the underlying processes or new theoretical directions in other domains. Whatever the merits of these types of justifications, they implicitly assume that no such insights or directions will be forthcoming from the enumerative approach—an assumption that does not hold in practice.
Computational models using agent-based objects are a very natural way to explore the dynamic behavior of a system. Regardless of the presence of equilibria, such behavior is often the most interesting part of the system. As Ursula Le Guin (1969, 220) said, “It is good to have an end to journey toward; but it is the journey that matters, in the end.” In situations in which equilibria are a possibility, understanding the dynamics is likely to be insightful. In situations where equilibria are nonexistent or transient paths are long, understanding the dynamics is critical.
Whether we actually need to model heterogeneity is an important research question. It may be the case that given sufficient agent heterogeneity, the aggregate behavior of the system may no longer depend on the various details of each agent, and abstracting this behavior into a single representative agent is feasible.
We know from some mathematical models that simple asymmetries in, say, information can alter our prediction of a single, well-behaved equilibrium point to one where there are multiple equilibria linked to, say, agent expectations. Computational models that use agent-based objects can easily accommodate asymmetries.
The intermediate cases are too difficult to solve analytically and must be solved computationally. In economics, we have good mathematical models of industrial behavior with monopolies, duopolies, and perfect competition. Once we begin to analyze systems of oligopolies, however, we are confronted with a lot of theoretical ambiguity.
Agent-based object models inherently provide constructive “proofs” to propositions. In particular, once we specify an agent-based object model and find that it leads to a coherent macrophenomenon, we have thereby found at least one set of microconditions that is sufficient to generate the macro-observations.
Unlike proofs by contradiction, the formulation of a constructive proof often provides new avenues from which to venture forth with new propositions.
A two-dimensional (think about trees growing on a checker board) version of the model displays a much more dramatic connection between production and growth. In such a model there is a very dramatic change in production as growth rates are altered.6 In physics terms, such a dramatic change is known as a phase transition, and it can be shown (via percolation theory) that there is a “critical value” of g that results in the system going from a largely disconnected collection of trees to one in which all the trees are connected together as one.
The adaptive solution to the model has an interesting implication: the system adapts to a precipice. Recall that the maximum productivity of this system is associated with a critical value and that such values imply that small deviations can result in a substantial decrease in yield.7 Thus, adaptation leads the system to a state that is both optimal and fragile.
The result here has a similar flavor to these ideas; adaptation leads the system to a very interesting state that is rich in performance yet rather exposed—nothing ventured, nothing gained. Indeed, the potential for adaptation to drive systems to such a precipitous state is a compelling reason for why adaptation matters in complex systems.
The idea that imperfection is a productive way to navigate multiple equilibria has been shown in many contexts, such as in simulated annealing. This research indicates that allowing mistakes (especially if they are not too costly or occur early in the search process) helps systems escape less productive outcomes and converge on more productive ones. Less perfection is often more in these types of systems.
Consider an atoll of size 20, where each site is occupied by an agent that has two possible actions {0, 1}. We assume that each agent’s behavior is controlled by the identical rule, and that this rule uses the most recent action of the agent in question and its two nearest neighbors to determine the next action.
This simple rule results in some interesting systemwide behavior. As can be seen in table 8.2, coherent macrostructures in the form of downward facing triangles composed of 0s emerge throughout the diagram. The scale of these triangles goes well beyond the scale of the behavioral rules.
the edge of chaos had the capacity for emergent computation. The intuition behind this claim has tremendous appeal: systems that are too simple are static and those that are too active are chaotic, and thus it is only on the edge between these two behaviors where a system can undertake productive activity.
This raises the possibility of emergent cognitive depth: adaptive agents may learn to think just deeply enough to promote coordination in the system.
Interesging hypothesis
in self-organized critical systems, the agents throughout the system tend to be poised in critical states where small disturbances can trigger large relaxation events that may encompass any number of agents.
With adaptive agents, the system configures itself in a way that mitigates the overall risk by preventing criticality from emerging. In essence, the adaptive actions of the individual agents lead the system away from the critical regime and more toward what an omniscient designer attempting to balance risk and stability would create.
In the original work on genetic algorithms, this performance measure came from evaluating the solution on an exogenous objective function. In many social systems, however, performance is often endogenous as it depends on the actions of the other agents in the system. When we have endogenous performance, we are in a coevolutionary world, as changes in the behavior of one agent alter the environment and performance of all of the other agents.
Bednar and Page (2006) have constructed a games-theoretic model of culture based on this idea.
The general result of this research is that context matters: how agents play a particular game depends on the collection of the other games in the ensemble. The ultimate implication of this result is that in worlds in which agents have limited cognitive capacity and face multiple games, we should predict very different behavior than that suggested by the standard game-theoretic models.
To put this another way, game theorists have ignored the distinction between partial and general equilibrium analysis that is considered so fundamental in the practice of economics. Agent-based models allow for a natural extension to multiple contexts that takes into account the full cognitive demands placed on agents. 10.5 EVOLVING COMMUNICATION Communication between
Try to understand this!
There is a constant dance among defectors, cooperators, and mimics, and this interplay becomes more elaborate as the processing ability of the automata and the number of communication tokens increase. Under such conditions, agents are able to develop more elaborate handshakes that, while still vulnerable to mimicry, tend to support alternative handshake pathways that can take over once a particular path has been compromised.
Within genetic algorithms, we find that details of parameter values or design choices often do not seem to matter in terms of the behavior of the algorithm. Moreover, other adaptive algorithms, like replicator dynamics and neural networks, seem to yield similar results on similar problems.
Arthur, W. Brian. 1994. “Inductive Reasoning and Bounded Rationality.” American Economic Review Papers and Proceedings 84:406–11.
on the beach problem
Bednar, Jenna, and Scott E. Page. 2006. “Can Game(s) Theory Explain Culture? The Emergence of Cultural Behavior within Multiple Games.” Rationality and Society 18:345–73.
Glaeser, Edward L., Bruce Sacerdote, and Jose A. Scheinkman. 1996. “Crime and Social Interactions.” Quarterly Journal of Economics 111:507–48.
criminality modeling
Kollman, Ken, John H. Miller, and Scott E. Page. 1992. “Adaptive Parties in Spatial Elections.” American Political Science Review 86:929–37.
complex adaptive system analysis - case democracy
Miller, John H. 1988. “The Evolution of Automata in the Repeated Prisoner’s Dilemma.” In Two Essays on the Economics of Imperfect Information, 49–97. Ph.D. dissertation, University of Michigan.
1996. “The Coevolution of Automata in the Repeated Prisoner’s Dilemma.” Journal of Economic Behavior and Organization 29:87–112.
Miller, John H., Carter Butts, and David Rode. 2002. “Communication and Cooperation.” Journal of Economic Behavior and Organization 47:179–95.
Communication allowing co-operation to emerge. Then mimicry destroying it
Miller, John H., and Scott Moser. 2004. “Communication and Coordination.” Complexity 9:31–40.
Communication in staghunt
Moore, Christopher, and Martin Nilsson. 1999. “The Computational Complexity of Sandpiles.” Santa Fe Institute Working Paper.
On self-organization
