Jon Von Neumann
December 28, 1903-Febuary 8, 1957
Sean Depaula
Development of economic Thought
April 20th
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John Von Neumann, born Janos Neumann in December 28th, 1903 in Budapest,
Hungary, but he was better known as Jancsi as a child. Born into a life of prominence; the
son of a wealthy banker John Von Neumann, benefited from private education from some
of the most prestigious schools money could buy, Though his education had little to do
with his rise to prominence. He was viewed a child prodigy at an early age and could
divide eight digit numbers in his head. John Von Neumann put his genius to work,
attending both the University of Budapest and the University of Berlin at the same time.
He was also instrumental in the emergence of Game Theory, digital computing, and
cellular automata. His influence to the study of economics has received acclaim from
many of the most prominent economists, though he was not an economist himself but
rather a mathematician, His work changed economics and economic thinking forever.
Von Neumann was a brilliant mathematician, producing major mathematical
articles in his early twenties. He set out to demonstrate how math could become a widely
applicable tool, even if fields that were not easy to formalize. This belief in the relevant
role of mathematics in social sciences leads Von Neumann to the field of economics.
Von Neumann makes two major contributions to economic science while trying
to conceptualize society using mathematics. The first was the General Economic
Equilibrium Model or the Expanding Economic Model. The second, Is the Theory Of
Games and Economic Behavior.
His first written works on theoretical economics appeared in the early thirties, he
presented such work during a mathematics seminar at Princeton in 1932. The work was
originally titled “On Certain Equations of Economics and A Generalization of the
Brower’s Fixed-Point Theorem. This work would be published as part of Karl Menger’s
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Vienna Colloquium in 1937, but received greater recognition in October of 1945, when it
was published independently as “A model of General Economic Equilibrium.”
The idea of economic equilibrium was originally introduced by Leon Walras; he
stated that there existed a set of prices that would clear all markets, meaning that there
would be no surplus supply or demand. He believed this was an example of capitalism at
work. The was some controversy to this because he did not have a proof to back up his
claims. Von Neumann came up with a mathematical solution using Fixed Point Theorem
to prove that their was a set of prices that would clear all markets.
The second and most influential contribution to economics was the invention of
Game theory and its development into a recognized discipline. The mathematical theory
of games was invented by John von Neumann and Oskar Morgenstern and put into print
in 1944 in the book The Theory of Games and Economic Behavior. This book
rejuvenated cardinal utility theory and introduced game theory to economics and the
other social sciences. Von Neumann and Morgenstern devised a system of axioms to
formulate a measurable utility. The mathematical framework of Game Theory initially
made the theory applicable only under special and limited conditions. This situation has
gradually expanded as the framework of game theory has been deepened, generalized and
refinements made over the last seven decades; since at least the late 1970s it has been
possible to say with confidence that game theory is the most important and useful tool in
analysis whenever confronted with situations in which one agent's best action depends on
expectations about what one or more other agents will do, and what counts as their best
actions similarly depend on expectations about what the first agent will do.
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Classical economists, such as Adam Smith and David Ricardo, overlooked the
importance of game theory because they were mainly interested in the question of how
agents in very large markets—whole nations—could interact so as to bring about
maximum monetary wealth for themselves. This does not measure satisfaction or benefit
to the individual but put value solely in monetary gain of nations. Smith's basic insight,
that efficiency is best maximized by agents freely seeking mutually advantageous
bargains, was mathematically verified in the twentieth century. However, the
demonstration of this fact applies only in conditions of ‘perfect competition,’ that is,
when firms face no costs of entry or exit into markets, when there are no economies of
scale, and when no agents' actions have unintended side-effects on other agents' wellbeing. Economists recognized that this set of assumptions is purely an idealization for
purposes of analysis, not a possible state of affairs anyone could try, nor should try an
attain. Until the invention of game theory, economists had to hope that the more closely a
market approximates perfect competition, the more efficient it will be. However, this
can’t be mathematically or logically justified it is a strict generalization and the
assumption can be shown to be false. Game theory gives a logical and concrete solution
to how to maximize efficiency given any set of preferences. The idea that there was more
than one answer to any solution given the differences in preferences was a radical idea
but it would lead economics into modern thinking.
Originally game theory was just applicable to zero sum games, or games with a
winner and loser such as parlor games. The invention of game theory had wider
application than just determining the winner and loser of a poker game or chess it had
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several profound effects on economics and many other sciences, we will look at some of
these later on, but first we must understand what game theory is and how it works.
Game Theory by definition is a branch of applied mathematics that studies the
strategic situations were players choose different actions as they attempt to maximize
returns. In lemans terms game theory studies a person’s optimal behavior when costs and
benefits of each option are not fixed but depend upon the preferences of the individuals.
Von Neumann and Morgenstern came up with a cardinal utility function and with the
minimax theorem in order to determine what the optimal behavior of an individual should
be. The utility function gives a concrete ranking to a person’s preferences, while the
minimax theorem states that a person will decide to take what ever action minimizes the
maximum loss for that individual. This is best understood using a player matrix
Table 1.
Move A
Move B
Player A
2,-2
-1,1
Player B
1,-1
3,-3
In this zero sum game the Equilibrium would be 2,-2 or to choose move be A
because player B would never choose Move B because it is the potentially the least
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amount of welfare that the player would receive, therefore this move will never be
played. This is an example of the minimax theorem in a zero sum game.
Von Neumann and Morgenstern also game up with some assumptions that had to
be made about the players in game theory for the game to be measured correctly. The
players are assumed to be economically rational. That is, a player can, assess outcomes;
calculate paths to outcomes; and choose actions that yield their most-preferred outcomes,
given the actions of the other players. This rationality might in some cases be internally
computed by the agent. In other cases, it might simply be embodied in behavioral
dispositions built by natural, cultural or economic selection. In particular, in calling an
action ‘chosen’ we imply no necessary deliberation, conscious or otherwise. This means
merely that the action was taken when an alternative action was available, in some sense
of ‘available’ normally established by the context of the particular analysis. Each player
in a game faces a choice among two or more possible strategies. A strategy is a
predetermined ‘programme of play’ that tells what actions to take in response to every
possible strategy other players might use. A crucial aspect of the specification of a game
involves the information that players have when they choose strategies. The simplest
games (from the perspective of logical structure) are those in which agents have perfect
information, meaning that at every point where each agent's strategy tells the other agent
what action to take, and the agent knows everything that has happened in the game up to
that point. A board-game of sequential moves in which in which both players watch all
the action (and know the rules in common), such as chess, is an instance of such a game.
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Later on in the development of game theory Von Neumann game up with
backward-induction using game trees. The point of representing games using trees can
best be grasped by visualizing the use of them in supporting backward-induction
reasoning. Just imagine the player (or analyst) beginning at the end of the tree, where
outcomes are displayed, and then working backwards from these, looking for sets of
strategies that describe paths leading to them. Since a player's utility function indicates
which outcomes are prefers to which, we also know which paths will be preferred. Of
course, not all paths will be possible because the other player has a role in selecting paths
too, and won't take actions that lead to less preferred outcomes.
Figure 4
In this example the Equilibrium or the players optimal behavior is (3, 3), using minimax
theorem and backward induction to eliminate choices the efficient outcome would be (3,
3).
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This proof by Von Neumann and Morgenstern laid the framework for a science
that has transcended not only economics but just about every other social science. In
biology game theory has been used to understand evolution, animal communication,
along with fighting behavior and territoriality using a famous version of game theory the
Hawk-Dove game. In computer science and philosophy game theory has been used to
understand logic, even political science has used game theory. Game theory is one of the
most influential inventions in the Modern economic era and is a huge contribution to
society, by a man who was not an economist but a man who briefly stepped into the
economic realm. John Von Neumann’s impact in economics and the social sciences is
immeasurable.
This is not the only contributions to society than John Von Neumann has made.
He helped invent the modern computer, which in itself has changed society. He also
impacted society as one of the men credited with inventing the atomic bomb. He was one
of the top experts in explosives, his calculations led to the discovery that bombs of large
dimension were more devastating in they were to detonate before reaching the ground.
This discovery was used on August 6th and 9th 1945, when two nuclear weapons were
detonated above the soils of Hiroshima and Nagasaki, at the altitude calculated by Von
Neumann. He helped calculate the Lenses of implosion, which is the stratification of
explosives around a mass of plutonium which initiates the chain reaction in a nuclear
explosion. Von Neumann also used his expertise in game theory to advise the President
what ‘targets” should be selected in the bombing of Japan. His recommendation was a
first strike against Kyoto, this was dismissed.
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As you can see John Von Neumann changed the way the world would interact
with his inventions and fore thought. He was a brilliant man that should be revered with
the like minds of Elbert Einstein and Isaac Newton. His work in mathematics was crucial
in converting social theory, and the way society would evolve into the modern era. His
contributions to economics, though vast, are only a minor part of his legacy. He was one
of the smartest men on the planet until his death in 1957.
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