Resources are limited, so how you allocate resources are very important and have big impact on you.

To be able to efficiently manage resources, one should make the best decision with given information.

If there is an investment/gambling/opportunity that will give you back £100 if you put £100 in (odd is 1 to 1), and it only have 60% of chance giving you back and 40% chance you get nothing.

What percentage of your momeny should be put in the opportunity? Turns out there is a proven theory for it and works, here is my understand of it - I might make mistakes.

What can I expect?

Exptected Values tell you what output should be expected and you should always maximising your expected value. Its calculted by multiply the outcome of sucess by the percentage success

e.g If 80% of chance to win £100, then the expected value is £80

This is what you should be expect if you doing this over and over again (on average - see Law of large number), however this is NOT the output if you only doing one or small number of times.

As you can see there are two parameters to make up the exptected value - the size of the output and the probability for the output to happen, so the following will have the same expted value

100% of chance to win £100
1% of chance to win £10000

Majority of the people will choose the first option, because that’s a sure thing, however from expected value point of view its the same, hence you choice should be neutral.

Take another example

95% of chance for £100
20% of chance for £500

Here option 2 have the higher expected value, hence it should be choosen, but most of the people prefer option 1.

This is explined by Exptected Utility - what is worth for you (what utility does it have for you) which is subjective, everyone can have different expected utility for the same things/options/expected value.

For the above choice most people will think 1 is pretty much sure thing and 2 is a gamble, for them £100 is more satisfy then a gamble to win £500. Expected utility method allow you think about what’s important for you.

Better idea

One of the strategy is to match how much you should allocate base on the chance of winning. e.g

If there is 60% of winning and 40% of losing, then you should allocate 60% of your money.

This is a form of Probabilty Matching, this is way better than allocate all money and blow up in one go and allow you recover.

If the wager is £100, then the expected value from the probabilty matching is

60*0.6 + 40*0.4 = 52

Maximise in long run

Now the proven most efficent way to allocate your money/bet/resources - Kelly Criterion

According to it the percentage of your resources should be put towards it is:

Expected Winning amount / Winning amount if you win

This make intuitive sense - the portion you should put in should be the same portion as expected winning amount to the amount you will get if win.

The expected winning amount is odd * probability of win - probability of lose e.g: 1 * 0.6 - 0.4 = 0.2

The Winning amount if you win is the odd itself e.g 1

so according to the formula you should put 0.2 or 20% of you total resource into it.

Note

As in most of the math formulas apply to real world, there are things need to be bearing in mind, as you can’t just apply blindly.

One important one is that the strategy assume there is infinte number of chances like that (or at least large number of times).

The strategy will be different if its only one game, so use Bayesian/Commen sense thinking and always have margin of error - practice is different from theory!