Week 1: Ruin

The course's opening claim, which you will not believe today — that a bet worth taking, taken repeatedly, ends at zero.

22 February 2027rationedhanded to you, never assessedOpen the slides

Here is a bet. Nineteen times in twenty it pays you 10%. One time in twenty it takes everything you staked.

Work out what it is worth on average. The multiplier per bet is

(19/20) × 1.10  +  (1/20) × 0
  =  0.95 × 1.10
  =  1.045

so the bet pays 4.5% per go. That is a good bet. Take it forty times and the average outcome is

1.045^40  =  5.82

nearly six times your money. Every part of that is correct and you should not take the bet.

Why not#

To finish forty bets you have to survive all forty, and the chance of that is

(19/20)^40  =  0.95^40  =  0.1285

Thirteen people in a hundred. The other eighty-seven are at zero, and zero is not a bad result you recover from next semester — it is the end of the sequence, because nothing multiplies back up out of nothing.

Both numbers are true at once, and reconciling them is the whole of this lecture. A hundred people take the bet forty times. Between them they end with 582 units of whatever they started with, which is the 5.82 average. But only 12.85 of them are holding any of it, so the survivors are sitting on

1.10^40  =  45.26

about forty-five times their stake each. Note that this is not an average over a range of good outcomes: a survivor is someone the loss never happened to, so every survivor multiplied their stake by 1.10 exactly forty times and they all hold the identical amount. There is no spread among the winners to explain.

Those thirteen will be asked how they did it. They will have answers. The answers will be sincere and they will be worthless, because the eighty-seven did the same thing.

This is called gambler’s ruin. The name is old and the arithmetic above is all of it.

What the course does with this#

The claim is not that you should never take a bet with a chance of total loss. It is narrower and harder to escape: if you take such a bet repeatedly, the expected value of the bet stops being the thing that decides your outcome. The number that decides it is how many times you are exposed, and that number is usually a choice you made without noticing you were making it.

Drag the first slider above to see the shape of that. At one bet you survive 95% of the time. At forty you are down to 12.85%. At a hundred it is 0.59%, and nothing about the bet changed — only how often you accepted it.

Watch the bottom two rows while you drag, because they move in opposite directions. At forty bets the average outcome is 5.82 times your stake. At a hundred bets it is

1.045^100  =  81.59

The bet has become fourteen times more attractive by the only measure most people use, and your chance of being there to collect has fallen from 12.85% to 0.59%. Both of those are consequences of the same act: accepting it more often. There is no version of this where the average is the thing to look at.

You will spend eleven weeks finding out that the chance of total loss is not exotic. It is in the fees, in the position size, in the borrowed money, in the door that turns out to be locked. Each of the next ten weeks names one of them and shows you the arithmetic.

You do not believe this yet#

That is expected, and it is planned for. Today the argument is abstract: it is about a bet nobody offered you, in a sequence you have not run. There is no reason it should land.

So in this session you will open a position, under the bounded-loss policy, and file an opening record you are not allowed to revise. It stays open until week 12. Over the following ten weeks each lecture will describe one specific way for it to die, and you will check whether that one is happening to you.

In week 12 this lecture runs again — the same slides, the same two numbers — and by then you will be sitting in a room where roughly nine people made money and can tell you why. That is the version of this argument that works.

Before you leave today#

Open something, and write down the four answers on the records page before you do. The fourth question is the one people leave blank, and blank is a permitted answer in week 1. It is worth noticing that it is week 1.