In 1988 Benatti first developed an agent-based economic model with the very peculiar characteristic that it neither included any representation of matter nor of energy, and no prices. Instead it merely represented agents and the money (capital) which was randomly exchanged between the agents. In 2000 Drăgulescu and Yakovenko developed a set of similar models, unaware of Benatti’s previous work. They are now collectively called the BDY models, or capital exchange models. Variations on these models reveal some striking connections to very fundamental dynamics studied in physics, thermodynamics, and statistical chemistry. In this presentation I will explore the history of capital exchange models, explain one of Yakovenko’s BDY models, present a definition of entropy for such models, describe one variation of a BDY, and briefly mention the remarkable connection to the work of J Willard Gibbs and Gavin Crooks.
Thank you for your presentation - I especially like employment of entropy in research problems covered by social sciences (and I am sociologist not physicist). I have question regarding the equation of CFT at 11:30:
I intuitively understand it, that if the probability of transition from A to B is higher than transition from B to A, entropy increases (delta S is positive). OK, it is in line with 2nd law of termodynamics - In the isolated system entropy increases. And now the mere question:
How we could use this theorem? For what is it useful? It seems to me just as different formulation of 2nd law… so, i.e. we might predict increase of entropy in system, if we know both transition probabilities?
Thank you, Francesco, for your question.
I will give you a short answer, and a longer answer.
I, personally, am excited about this equation, even though I do not really know where it is going to take me ultimately. I am excited, not just because it is in agreement with a very important new presentation of the second law of thermodynamics (the CFT), but because my personal mathematical development of this equation is only very loosely tied to thermodynamics. It potentially has broad applicability to any circumstance in which a histogram of some sufficiently well-conserved measurement is involved. This includes energy (obviously, as found in thermodynamics), genetics (as found in biology, the topic bridged into by the paper by Jeremy England), economics and money (as shown in capital exchange models). But, with much more generality, it implies the “arrow of time” for any social or economic process for which there is a ratio of asymmetric probabilities.
- think of buying an apple, and selling it back to the seller again;
- think of divorcing, and marrying the same person again;
- think of cleaning a floor, then intentionally throwing the dirt back on the floor.
Each of those actions has a high probability of occurrence in one direction, and a low probability of occurrence in the other direction. The reasons why we would not undo these events are both varied and obvious. The fact that some kind of socially-determined entropy is rising is not obvious. But what IS obvious is that these events have very little to do with thermodynamic entropy. They have everything to do with probable behaviour of social beings.
For a more detailed answer, keep reading.
My derivation of it is simple, and starts with a paper by Victor Yakovenko.
Victor M. Yakovenko (2010) “Statistical mechanics approach to the probability distribution of money”, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
Victor Yakovenko is an accomplished physicist, and his presentation is full of arguments that call on a deep knowledge of physics. But there is a snippet of information within it which does not need that deep knowledge. It only requires an understanding of high-school level combinatorial mathematics.
On page 2 of this cited paper, in reference to his equation # 3, we find this definition of entropy.
This quantity is given by the combinatorial formula in terms of the factorials W = N! / ( N1! N2! N3! … ) ) This logarithm of the multiplicity is called the entropy S = ln W.
The formula he calls the multiplicity (W) is the well-known multinomial coefficient of combinatorial mathematics. As a high-school teacher of mathematics, I have used it often in reference to the ways you could construct a given histogram. In my study of agent-based models I had been looking for a way to calculate entropy of a histogram, and this is the key.
Yakovenko, then demonstrates the use of this calculation in his own calculation of the entropy of his capital exchange models.
Then the following series of events led to the equation in my presentation:
- I modified one of Yakovenko’s capital exchange models to be doubly-bounded;
- I developed an equation for the probability of transformation from one state to another;
- I noticed a peculiar property of these transformations that I called an asymmetric ratio of probabilities (AROP); Essentially, that is [ P(AB)/P(BA) ]
- I noticed the cited paper by Jeremy England that contained the same AROP;
- He cited a paper by Crooks that contained the same AROP.
- I returned to the equation S=ln(W) and discovered that the definition of entropy cited by Yakovenko is mathematically equivalent to the definition delta S = ln [P(A–>B)/P(B–>A) ]
Suppose we then decide to step out of thermodynamics, and simply view S=ln(W) as a new kind of measure of any histogram. It can be any kind of well-formed histogram. Then, the question is, when does entropy calculated in this way rise? It is when we attach probabilities to the changes of state from one histogram to another, and when the probabilities are different in the two different directions. These probabilities normally, can only be calculated if the contents of the two histograms are conserved during the transformations.
If you want to see my diary notes on the topic, I would be happy to email them to anyone who is interested.
Thanks Garvin. I knew the model on matching two agents and flip a coin which leads to interesting distribution of wealth. Interesting how you have worked this out with an upper bound.
Although the simple model leads to macro patterns that are similar as observed empirically, this does not mean that the mechanisms modeled are the true dynamics of the system. I wonder what the justification is of the two persons matching and on wins the 10 dollars game. I could imagine all kind of variations of the mechanisms and wonder how this impact the macro-level outcomes: Those with higher wealth are more likely to win; agents can get a loan to stay in the game and play (why is someone allowed to play without an endowment), agents may not interact randomly, but within a network of agents or more likely to interact with those who are closed, etc. etc.
Thank you, Marco, for your comments and your questions. There are many aspects to your questions that I have had to think about. Let me try to address them one at a time.
About variations: There are several variations of the so-called BDY (Benatti-Drăgulescu-Yakovenko) capital exchange models. Drăgulescu and Yakovenko (2000) described seven of the eight models that they studied. The eighth model was said to be too detailed to describe in that paper, because it involved banking. They claimed all eight models lead to a distribution of wealth that mimicked either an exponential distribution of energy, or a Maxwell-Boltzmann-like distribution of energy. The only deciding factor was whether the exchange was symmetric in time, or asymmetric in time. For example, if A paid B $C, and then they immediately had to switch roles, did B pay back the same amount, reinstating the previous state? If the answer was “yes”, then the exponential distribution appeared. It mattered not if the amount exchanged was $1 or 10$. If the answer was “no” then the Maxwell-Boltzmann distribution appeared. An example of an exchange that is asymmetric in time is a transfer of a percentage of one’s wealth. One of their described models even used a very complex stochastic implementation of a Cobb-Douglas production function to determine who employs, who gets hired, at what wage, what widgets are built, who buys the widgets, and for how much. The model that they did not describe involved banks. EiLab contains my implementation of the seven BDY models they described. CmLab is my own version of a banking model. I can confirm their observations about time symmetry. Only two types of distribution appear as the ultimate high-entropy distributions of wealth.
Justification of Application to Economic Data – Do these BDY models demonstrate the same mechanisms as is found in real economies? These types of models are sometimes called “kinetic wealth exchange models” (KWEMs). The exchange of kinetic energy in models of ideal gases is a binary event in which a conserved quantity (energy) is exchanged. That is a very thoroughly-studied phenomenon in physics, going back to the days of Maxwell, Boltzmann and Gibbs in the late 1800s. The exchange of modern money in any economy is also a binary event in which a conserved quantity (money) is exchanged. The key elements of the exchange are identical, whether we are dealing with kinetic energy or money. The question is, what do we mean by “identical”?
• All transactions are binary, involving only two agents;
• The exchanged quantity is conserved in the binary transaction (i.e. can be described by a continuity equation);
• The determination of which two agents interact is stochastic.
In the various models tested by D&Y, and those tested by me, the amount exchanged, the nature of the exchange, the constraints on the exchange (e.g. whether an agent is bankrupt), or the other conditions of the agents selected for the exchange (e.g. levels of wealth, or roles as employer or employee) – none of these have any significant effect on the ultimate distribution of wealth, other than the time-symmetry or time-asymmetry of the exchange mechanism. The distribution of wealth always asymptotically approaches one in which entropy is maximized, for that distribution.
Are Skilled Agents Differentially Enfranchised: In the KWEMs, does a wealthy person have more access to the economy? Yes, they do, except in my “Model I”. But, this does not change the ultimate equilibrium distribution. In the real world, does an educated, skilled, or intelligent person have more access to the economy? Yes, but this does not change the ultimate equilibrium distribution by much. It has a small effect on the ultimate shape of the empirically-determined equilibrium distribution, often called a “fat tail”. There is a lot of literature discussing the “fat tails” of the empirically measured wealth distributions in which there are slightly more wealthy people than predicted by KWEMs. However, there are many who argue that this effect is minor, compared to the dominant role of the effects of maximizing entropy. Those personal attributes of individuals (education, skill, wealth, determination) might decide who rises to the top, but they do not have much effect on the ultimate shape of the distribution.
Does Network Connectivity Matter: In physics, ideal gas atoms only interact with those in a small immediate neighbourhood, determined by their ever-changing intersecting trajectories, and their speeds, and by the mean path distance between collisions. I suspect that such a constraint would be describable as a “small world network”, but I am uncertain. In the BDY models, the pairing of agents for any given exchange is not limited in this way. Any agent can interact with any agent. To be honest, I don’t know what such a network would be called, so I’ll refer to it as a “universal network”. In my model called ModEco, agents can only interact with agents in a small neighbourhood. In that case, I have implemented a kind of small world network, and the ultimate wealth distribution is a Maxwell-Boltzmann-like distribution. It would be interesting to add a “small world network” contact constraint within a BDY model to see if it has any affect on the ultimate shape of the equilibrium distribution. Based on my experience so far, I expect that it would not.
Exploitation of Symmetries (of Universals): One of the philosophical foundations of science is symmetry of application. There are universal dynamics that appear in many kinds of stochastic systems - systems that are otherwise apparently not associated. A carefully constructed ABM can demonstrate (not simulate) these universals. The “stylized statistics” that can be gathered from such universal dynamics as seen in ABMs can teach us how to understand all of those other dynamic systems that share the same universal dynamic behaviours – that share the same “stylized statistics”. This is a fundamental philosophical argument.
The exploitation of such symmetries is a means to transfer knowledge from one discipline to another – with care. The causes and effects of rising entropy in ABMs is one such universal – one such symmetry – that can shed a lot of light on the murky dynamics of ecosystems, of social systems and of economic systems.
The fact that “Model I” has shown clear evidence of sharing a universal with statistical chemistry, with statistical mechanics, and with empirical economic data, implies to me that there are bridges to be built and exploited using ABMs.