The Biography of a Dangerous Idea

[everytwoweeks]

Another type of exponential growth is the Malthusian growth model. Sometimes called the simple exponential growth model, it is essentially exponential growth based on a constant rate of compound interest. Named after the Reverend Thomas Robert Malthus. At best, it can be described as an approximate physical law as it is generally acknowledged that nothing can grow at a constant rate indefinitely (Cassell's "Laws Of Nature," Professor James Trefil, 2002). Professor of Populations Joel E. Cohen has that the simplicity of the model makes it useful for very short-term predictions and of not much use for predictions beyond 10 or 20 years ("How Many People Can The Earth Support," 1995).

The Malthusian growth model is the direct ancestor of the logistic function. Pierre Francois Verhulst first published his logistic growth function in 1838 after he had read Malthus' "An Essay on the Principle of Population."


The sigmoid function

A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. The untrammeled growth can be modelled as a rate term +rKP (a percentage of P). But then, as the population grows, some members of P (modelled as − rP2) interfere with each other in competition for some critical resource (which can be called the bottleneck, modelled by K). This competition diminishes the growth rate, until the set P ceases to grow (this is called maturity).

A logistic function is defined by the mathematical formula:



for real parameters a, m, n, and τ. These functions find applications in a range of fields, from biology to economics.

For example, in the development of an embryo, a fertilized ovum splits, and the cell count grows: 1, 2, 4, 8, 16, 32, 64, etc. This is exponential growth. But the fetus can grow only as large as the uterus can hold; thus other factors start slowing down the increase in the cell count, and the rate of growth slows (but the baby is still growing, of course). After a suitable time, the child is born and keeps growing. Ultimately, the cell count is stable; the person's height is constant; the growth has stopped, at maturity.
 

libor, your so funny !

[naom]

Zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other participant(s). It is so named because when the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Chess and Go are examples of a zero-sum game - it is impossible for both players to win. Zero-sum is a special case of a more general constant sum where the benefits and losses to all players sum to the same value. Cutting a cake is zero- or constant-sum because taking a larger piece reduces the amount of cake available for others.

Situations where participants can all gain or suffer together, such as a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, are referred to as non-zero-sum. Other non-zero-sum games are games in which the sum of gains and losses by the players are always more or less than what they began with. For example, a game of poker played in a casino is a zero-sum game unless the pleasure of gambling or the cost of operating a casino is taken into account, making it a non-zero-sum game.

It has been theorized by Robert Wright, among others, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.

An example

A game's payoff matrix is a convenient way of representation. Consider for example the two-player zero-sum game pictured



The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.

Example: the first player chooses action 2 and the second player chose action B. When the payoff is allocated the first player gains 20 points and the second player loses 20 points.

Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Player 1 could reason as follows: "with action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, player 2 would choose action C. If both players take these actions, the first player will win 20 points. But what happens if player 2 anticipates the first player's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if the first player in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?

John von Neumann had the fundamental and surprising insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each player. This minimax method can compute provably optimal strategies for all two-player zero-sum games.

For the example given above, it turns out that the first player should choose action 1 with probability 57% and action 2 with 43%, while the second player should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Player one will then win 2.85 points on average per game.

[uber]

Indeed naom, they say if they they lose skin contact they have to touch tongues.

[harris]

Who, uber?

[triad]

tag

[nono]

you mean you're interested in this thread, that's why your posting that word here, don't ya?

[pruritis]

alternatively .. s/he may be tagging his/her username by posting on a thread

[e.]

[falg]

[abu]

I became acquainted with the psychological power of zero in doing a meditation technique called holotropic breathing, or simply breathwork. In holotropic breathing, the participant engages in deep, rhythmic breathing while listening to a carefully chosen suite of very loud music. The participant often achieves a trance-like state, with vivid images. Sometimes the images are abstract. Sometimes the images are very real. They may consist of reliving certain life experiences, having conversations with long-dead ancestors, or having fantastic experiences such as being in the midst of a Civil War battle or dancing in a harem. As the music begins to slow down and become quieter and sweeter, there is often a powerful emotional release.

Somewhere during the breathwork I've experienced, I come to a point at which time and space have totally collapsed. I am at zero: there is no time, and there is no space. What is most powerful about this experience is that, at this zero point, there are no temporal or spatial constraints -- whatever the "I" is that is experiencing this can go anywhere in time and space. In fact, the "I" has disappeared into the zero. Or it's as I have been divided by zero and have become undefinable.

We might think that, in ordinary geometrical space, three dimensions is about as real as we can get. Two dimensions constrain us to thinking about a plane; one dimension is simply a line. But when we hit zero dimensions, we're at a single, dimensionless point: all constraints fall away.