The ‘Invisible Hand’ in the Game Theoretic Approach and the Limit of Computability
The game theoretic view may historically be explained by the classical Max-Min Theorem and its duality theorem of a two-person game or linear programming. This approach was originally established by von Neumann (1937), and extended to a quasi-growth model of balanced growth with multiple sectors operating side by side.
In contrast with the linearity of productive processes, the introduction of a growth rate or interest rate makes the model nonlinear, where the objective functions are rates of growth and interest, defined as ratios of the model variables. The model’s constraints are all linear. Von Neumann formulated the balanced growth model of the multi-sectoral production model in terms of the max-min game theoretic form but proved the model equilibrium by means of the Fixed-Point Theorem.Von Neumann’s game fulfills the max-min property, so the solution does not guarantee a Nash equilibrium. To derive a Nash equilibrium, even restricting the model to a simplified form like Leontief’s input-output model, another interpretation is necessary, which assumes each producer (process of production) is a player (Schwarz 1961). This meaning of the framework is very different from the world of the max-min theorem. Equilibrium can be achieved in either model, but by different mechanisms. It is therefore obvious that the meaning of the invisible hand will also be different in each.
In Robbins’ approach, the market may sometimes be replaced by a planning agency, such as in socialist regimes. In this case, the agency is a kind of computer. As was seen in the debate on planning economies in the 1930s, many economists believed that super-computers would eventually replace the market, which turned out to be completely untrue. We now know computers can replace neither markets nor planning agencies. However, computers are now vital for the successful operation of the market.
The history of computers is quite complex. As Arthur (2009, Chap. 8: Revolutions and Re-domainings) suggested, the financial market emerged rapidly because of the evolution of computers in the last century. The so-called renovation of financial business was then feasible, because computers evolved to solve the complicated risk calculation needed for option dealings (Arthur 2009, p. 154).
In the 1960s, putting a proper price on derivatives contracts was an unsolved problem. Among brokers it was something of a black art, which meant that neither investors nor banks in practice could use these with confidence. But in 1973 the economists Fischer Black and Myron Scholes solved the mathematical problem of pricing options, and this established a standard the industry could rely on. Shortly after that, the Chicago Board of Trade created an Options Exchange, and the market for derivatives took off.
The Black-Scholes equation Black and Scholes (1973) has not been found to have long-term legitimacy in the options market, because of the collapse of longterm capital management (LTCM), to which Myron Scholes was deeply committed. This means that, for obvious reasons, computation of risk cannot necessarily guarantee the correct working of the financial market. Instead of being the market itself, computers have instead become a means to operate and expand the market. Computation, however, is merely something to which the market must adapt, and is not subject to the market.
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