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Probability vs expected return vs variance


#1

Probability, chance and luck all mean basically the same thing (ignoring the supernatural definition of luck): How likely is an event to occur?

For example in a 30 card randomised r&d with 6 agendas in it, the chance of the top card being an agenda is 6 out of 30 which is 1/5.

But to talk about expected return and variance we need to actually assign a value to each outcome as well as knowing the probability of each outcome.

So if all those agendas are worth 3 points and all non agenda cards are worth 0 points, the expected return (expected value) of running r&d is 1/5 times 3 points = 0.6 points.

What is variance then? It is the square of the standard deviation, which are both mathematical concepts related to how far apart the results are spread.

Imagine two ways of rolling a number from 2-12 on dice. The first way is to roll a single die and double the answer. In this case there is an even spread of possibilities - the results 2, 4, 6, 8, 10, and 12 are all equally likely. The second way is to roll two dice and add the results. In this case results close to the middle like 6, 7 and 8 are more likely while 2s and 12s are less likely.

Both of these choices have the same average roll (expected return) of 7. If player A plays this game against player B, each player is equally likely to win regardless of the method chosen. But the first method has a higher standard deviation. In general, if we have only a few random events but the events are of high importance then we get more of a spread, whereas many events of low importance gives less of a spread (the results on the extreme ends are still possible, but less likely).

Say if player A got to start with 3 points to reflect his higher skill level and then the dice result was added in. Player B would not prefer the second method that gives less spread out results, since then player A would probably still be about 3 points ahead. Player B would prefer the first method that allows luck to have a bigger influence on the game, reducing the effect of A’s skill advantage.

Now let’s do a netrunner example. There are 30 cards in r&d with 18 agenda points in total, and the runner is about to access the top 5. Since the runner is accessing 1/6th of the deck, their expected return is 3 agenda points regardless of the composition. But what gives a higher standard deviation - six 3 pointers or nine 2 pointers?

This table shows the probability of the runner scoring a particular number of points on access.

For six 3 pointers
0 points - 29.8%
3 points - 44.7%
6 points - 21.3%
9 points - 3.9%
12 points - 0.2%
15 points - 0.0%
Standard deviation = 2.5 points

For nine 2 pointers
0 points - 19.3%
2 points - 37.8%
4 points - 33.6%
6 points - 12.4%
8 points - 1.9%
10 points - 0.0%
Standard deviation = 1.9 points

We can see that the second composition is more consistent, with the runner scoring 2 or 4 points most of the time. Whereas the first composition has the runner scoring 0 or 6 points more of the time.

Ignoring the fact that the runner needs 7 points to win (which is the strength of the seven 3 pointers agenda composition) we can say that the 3 point composition has a higher variance - while it’s the same on average, it’s more likely to result in the runner either getting very lucky or very unlucky.


#2

So, even if two decks have the same expected value of victory (say they win 60% of their games) a stronger player will prefer the deck with lower variance because it will magnify the benefit of their superior skill’s influence on the result of the game.

Thank you, even though I knew the ideas I hadn’t understood so clearly before.