Prize money for the ‘big winner’ = 100 * 2^x ( while n >= 2^x )In the above, n is the initial number of risk-takers, x is the number of betting rounds, 100 is a constant that represents the size of the initial wager, and 2 is a constant that roughly represents the available number of betting options in a given round.
The first important property above is the relationship between n and x within the constraint. In the above construction, the number of betting rounds x is proportional to the initial number of risk takers n. Therefore, as the number of risk takers rises, the number of betting rounds increases.
This means that over time societal changes that make it easier to take risks will increase the number of betting rounds available to risk takers. This, in turn, will increase the ‘prize money’ for the biggest winners. Therefore, over time I would expect an increasing number of the mega-rich to derive their wealth from being risk takers. I can’t say for sure if this is happening (or has happened, or will happen) but at the very least I’m pretty confident in my math.
This also suggests that there will be more big losers. If we assume risk taking is a function of appetite for risk (r) and potential reward (p) we could setup a general risk function as follows:
y = r * pwhere y is a binary variable that determines if someone takes a risk or not.
If we track the math all the way through, we can argue that an initial change that makes it easier to take risk (such as the invention of the internet, which makes it easier to start a business) will lead to a rise in n, the total number of risk takers. Since n is related to x (the number of betting rounds) and an increase in x leads to an increase in p (the potential reward), we conclude that as more people decide to take risk, more people who were not interested in risk at a lower prize level p will be tempted to join the game as p rises. This, in turn, feeds back into the formula by increasing n, and the whole cycle repeats again…
The other important property is how increases in the size of the initial wager or a rise in the number of betting options available can accelerate this process by increasing the size of the payout. This is yet again simple casino math – the more money wagered or the less likely a given outcome, the higher the payout. As with above, the feedback loop of societal changes influencing risk taking decisions takes a number of steps but is unambiguous – changes that increase either constant in the original formula increase the potential prize pool which influences more risk takers into the initial pool n.
