Computer scientists often make unfounded bold statements in an effort to elevate their profession to the status of physics and maybe share some of the glory of that field. This is understandable but it becomes ludicrous when they equate their ability to model complex problems and arrive at a solution to that of physical reality in arriving at the same solution. It is troublesome that the educational establishment not only tolerates this type of behavior but also promotes it and rewards it.
In a paper that has won an award by the Αssociation for Computing Μachinery and titled “The Complexity of Computing a Nash Equilibrium“, Constantinos Daskalakis, Paul W. Goldberg and Christos H. Papadimitriou, provide evidence that there are games in which convergence to a mixed Nash equilibrium takes prohibitively long. The paper details are left out for the following reasons: (1) I am not familiar with this subject in depth and I cannot judge the work details and (b) the details are very technical and only those who have worked in this area can comment on the specifics.
What interests me from from the philosophy of science point of view is this statement made in the paper
“But there is a third important desideratum on equilibrium concepts, of a computational nature: An equilibrium concept should be efficiently computable if it is to be taken seriously as a prediction of what a group of agents will do. Because, if computing a particular kind of equilibrium is an intractable problem, of the kind that take lifetimes of the universe to solve on the world’s fastest computers, it is ludicrous to expect that it can be arrived at in real life.”
The above statement is ludicrous in at least two counts. Let us consider both: The first involves the first part of the statement:
“But there is a third important desideratum on equilibrium concepts, of a computational nature: An equilibrium concept should be efficiently computable if it is to be taken seriously as a prediction of what a group of agents will do.”
This first statement equates ability to model efficiently to what we should take seriously. This notion hides a constructivist view of reality, in which real is what we can model efficiently. This is ludicrous because even our planetary system, in the sense of a many-body problem in a gravity field, we cannot model efficiently because as it turns out it is a chaotic system but it is a concept that it is taken, of course, seriously.
The second statement is even more ludicrous:
“Because, if computing a particular kind of equilibrium is an intractable problem, of the kind that take lifetimes of the universe to solve on the world’s fastest computers, it is ludicrous to expect that it can be arrived at in real life”
No, it is the statement that is ludicrous. The fact that we cannot compute efficiently a particular kind of equilibrium or that it turns out to be an intractable problem does not imply necessarily that the equilibrium state cannot be reached in real life. Unless, again, those constructivists think that real life is what we can model efficiently (Logical positivists claimed once that whatever cannot be described mathematically does not exist). Reality does not use any algorithm that computes equilibrium. Reality is the algorithm itself and this algorithm may be of a different kind than the ones derived based on our axiomatic math. Reality is what is. We know of many phenomena that we cannot compute and predict but happen in real life. For example, our best theories predict a huge cosmological constant whereas observations result in a very small value (Weinberg, S. 1999. The cosmological constant problem. Reviews of Modern Physics 61: 1-23.). Actually the difference is in the order of 10100! Wouldn’t it be ludicrous for someone to claim that reality is wrong and our theories are correct?
As far as equilibrium concepts and real life, those computer scientists should remember this one: “Now you see it, now you don’t”. Many rational economic agents do not seek equilibrium nowadays and their actions in effect cause non-equilibrium because they cannot profit from mere equilibrium as margins are very small. For example, rational economic agents seek to create bubbles knowing that this is how they can make money. It is an interesting period in the history of mankind we live where meaning of terms and objectives have changed because of greed and fear. What is rational nowadays is not the same as with what was rational 30 years ago or when keynes lived or Nash developed his theory. What equilibrium means today, is not necessarily what equilibrium will mean tomorrow. Equilibrium states are as dynamic as the process that generates them and what appears to be chaos today may be the equilibrium of tomorrow. Never forget that we have transcended from pre-established harmony to chaos. This world is becoming chaos and no algorithm will be able to predict anything in the future. This is the slow death of algorithms and the beginning of a new era.