There are times when there is a disagreement over what time a particular event will happen and people want to turn the different in opinions into money. It is common to place “over-under” bets — if the event happens before time T, Andrew gets the money, otherwise Bob gets it. Usually the further the event is from T, the more money exchange hands.

I don’t like this style of betting because it’s simply not expressive enough. Instead, I prefer to bet by specifying my probability distribution of the timing of the event, and then using these distributions to determine payouts. My friend and I made this bet once and it was a fun activity, despite the math involved behind-the-scenes, not that frustrating and requiring very little mathematics to actually place the bet.

Essentially, each party draws a probability distribution of the timing of the event — a histogram with the time on the horizontal axis and the probability density function on the vertical axis. The latter can simply be intuited as “the relative probability that the event will happen around the time specified on the horizontal axis”. So if the histogram is twice as tall around 8pm than around 7pm, the event is twice as likely to happen around 8pm than around 7pm.

That’s all each person really needs to do. No need to worry about the area of the histogram summing up to 1 since the vertical axis can be scaled up appropriately. The two people should also agree on how money they are willing to bet — say k dollars each.

When the event actually occurs at time T, the two people compare the value of the probability density function (the height of the bar) at time T on their graphs (the height will be scaled appropriately so that the area adds up to 1–so of course you can’t cheat by making your graph taller) and pay up based on the difference in these values.

Executing such bets is a little difficult since it involves calculating areas under the graph which may be very irregular. Those who are mathematically masochistic can constrain themselves to piecewise linear functions, or, in the extremely, easily integrable functions; otherwise the graph can be scaned and a simple graphics editing application (like Photoshop) can be used to determine the area under the graph (using the flood fill and histogram tools).

This entry was posted on Sunday, June 13th, 2010 at 20:19 and is filed under clever, paramathematics, technology, things everyone should do. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.