2009/ Segregation models

Extending the concept of free energy to account for individual dynamics

Residential model: the agents prefer to live in mixed blocks but still prefer very crowded blocks to almost empty ones. When the dynamics is based on collective interest, the system ends up in an optimal state where all the blocks have a density of 1/2 (central graph). When the dynamics is based on individual interest, the systems "unexpectedly" ends up in non-optimal segregated states in which some blocks are rather crowded and the other empty (right graph).

The intricate relations between the individual and collective levels are at the heart of many natural and social sciences. Different disciplines wonder how atoms combine to form solids, neurons give rise to consciousness, or individuals shape societies. This apparent similarity conceals an essential difference across disciplines. In order to define the “normal” or “equilibrium” aggregated state, physics looks at the collective level, selecting the configurations that minimize the global (collective) free energy. In contrast, economic agents behave in a selfish way, and equilibrium is attained when no agent can increase its own (individual) satisfaction, or utility. Finding the equilibrium states in that case can sometime be tricky since the satisfaction of an individual agent may depend in a complicated way of the choices of the other agents.

While it is always tempting to apply some physical approach to economic systems, one has to be very careful to handle and this difference of point of view, individualistic or collective. In a recent article (PNAS 2009), we illustrate this difference on an exactly solvable model based on Schelling’s segregation model. Our model interpolates continuously between cooperative (based on the maximization of the collective utility) and individual (based on individual satisfaction) dynamics, thanks to a “cooperativity” parameter. When this cooperativity parameter decreases, we observe a transition between an optimal state (in which all agents are satisfied) to a state in which the agents, because of a lack of coordination, do not maximize their utility anymore.

Transition with respect to the cooperativity parameter (α).

The key ingredient of our model is what we call the Link function, which is a state function (in the physics sense, ie depending on the global collective state of the systems) which keeps tracks of the individual utilities. In that sense, it can be seen as an effective hamiltonian accounting for individual dynamics.

Collective Utility function and "individual-based" Link function


For more details, refer to the following articles:

PNAS 2009, our first paper focusing on the transition between individual and collective dynamics.
ACS 2011, a paper intended for an audience of physicists focusing solely on the Link function (ie on how one can characterize an individual dynamics thanks to a global state function).
J PUBLIC ECONOMICS 2011, a paper intended for an audience of economists, focusing on the very general analytical resolution of Schelling’s segregation model provided by our framework.


  • Pablo Jensen, Remi Lemoy (IXXI, LET, ENS)
  • Eric Bertin (IXXI, ENS Lyon)
  • Florence Goffette-Nagot (GATE)


Introducing coordination in Schelling’s Segregation model

For a given utility function (agents' main preference goes toward mixed neighborhoods but they still prefer completely similar to completely dissimilar neighborhoods), segregation occurs when the agents move only regarding their own interest. But even with a little cooperation, the segregative patterns disappear.

Schelling’s segregation model simulates the moves of two types of agents in a city, with respect to the agents’ preference regarding the composition of their neighborhood. Not surprisingly, when the agents prefer to be completely surrounded by like-neighbors rather than completely surrounded by unlike-neighbors, the system ends up in segregative states. What’s intuitively unexpected is the fact that this results holds even when the agent’s main preference goes towards a mixed neighborhood. Several studies have shown the strong resilience of segregative patterns with respect to the input parameters (size of the city, number of agents, agents’ preference, noise, etc ). This robustness can be understood as the stability of segregative patterns due to fact that no agent has a personal interest to be the first to break the pattern. We investigate the impact of the introduction of different forms of cooperation between the agents, a test of robustness which has not been undergone so far in the literature. Our simulations show that even a small amount of coordination is sufficient to break segregation.

For more details, refer to the following:

Working Paper 2009, chapter 3!


  • Pablo Jensen (IXXI, LET, ENS)
  • Florence Goffette-Nagot (GATE)