Last time we looked at the break points of where your win percentage should be to play in 8-4s vs. 4-3-2-2s. There is a third draft option, however, and that is Swiss.In Swiss drafts, you play 3 matches and win a pack for each match. Figuring out the expected value of a Swiss draft is pretty easy. There are 3 independent trials and the probability of winning each trial is $p$. There are four possible outcomes for the Swiss drafts. They and the associated probabilities are:

• Win 0 matches: $-p^3+3p^2-3p+1$. $1-p$ times $1-p$ times $1-p$ gives you $(1-p)^3$. Expanding the cube out gives you $-p^3+3p^2-3p+1$.
• Win 1 match: $p^3-2p^2+p$. $p$ times $1-p$ times $1-p$ gives you $p(1-p)^2$. Expanding the square out and multiplying by $p$ gives you $p^3-2p^2+p$.
• Win 2 matches: $-p^3+p^2$. $p$ times $p$ times $1-p$ gives you $p^2(1-p)$. Multiplying the two factors gives you $-p^3+p^2$.
• Win 3 matches: $p^3$. $p$ times $p$ times $p$ is $p^3$.

Now we determine the expected value of those outcomes:

• Win 0 matches: 0 value.
• Win 1 match: $3p^3-6p^2+3p$. Since there are three different ways you can win 1 match in a Swiss draft (win the first round or win the second round or win the third round), we multiply by 3 to determine the expected value of this event.
• Win 2 matches: $-6p^3+6p^2$. Since there are three different ways you can win 2 matches in a Swiss draft (lose the first round or lose the second round or lose the third round), we multiply by 3 to determine the expected value of this event.
• Win 3 matches: $3p^3$.

$$3p^3-6p^2+3p-6p^3+6p^2+3p^3$$

$$3p$$

The expected value of $n$ independent events each with the same probability $p$ is $np$. For this case, it's $3p$. Now we need to compare the expected value of a Swiss draft to the expected values of 4-3-2-2 and 8-4 drafts. From the last article, we determined that the point where 4-3-2-2 and 8-4 drafts had the same return for a player is if their probability of winning a match is:

$$p=\frac{\sqrt{33}-3}{6}$$

Or approximately .457. Below that probability, playing in 4-3-2-2s is better. Above that, playing in 8-4s is better. So below that line, we'll compare the expected value of a Swiss event to a 4-3-2-2 event:

$$p^3+p^2+2p=3p$$

$$p^3+p^2-p=0$$

$$p(p^2+p-1)=0$$

Discarding the $p=0$ case and using the quadratic formula, we get:

$$p=\frac{-1\pm\sqrt{1^2-4(1)(-1)}}{2(1)}$$

$$p=\frac{-1\pm\sqrt{1+4}}{2}$$

$$p=\frac{-1\pm\sqrt{5}}{2}$$

We can throw out the minus case as it causes $p$ to be less than 0, so we end up with:

$$p=\frac{\sqrt{5}-1}{2}$$

This approximates out to .618! This means that across the entire range where you would play 4-3-2-2s, only one of 4-3-2-2 or Swiss is right from an EV perspective. Which is it? Let's try a sample case...

For $p=.1$:

$$3p\,?\,p^3+p^2+2p$$

$$3(.1)\,?\,.1^3+.1^2+2(.1)$$

$$.3\,?\,.001+.01+2(.1)$$

$$.3\,?\,.001+.01+.2$$

$$.3\,>\,.211$$

Very interesting! For the entire range where you would play 4-3-2-2 events, you should play Swiss events instead if your probability of winning is the same in both! Now what about where you should play 8-4 events, above the .457 probability?

$$4p^3+4p^2=3p$$

$$4p^3+4p^2-3p=0$$

$$p(4p^2+4p-3)=0$$

Discarding the $p=0$ case and using the quadratic formula, we get:

$$p=\frac{-4\pm\sqrt{4^2-4(4)(-3)}}{2(4)}$$

$$p=\frac{-4\pm\sqrt{16+48}}{8}$$

$$p=\frac{-4\pm\sqrt{64}}{8}$$

$$p=\frac{-4\pm8}{8}$$

Discarding the minus case as usual, we get:

$$p=\frac{8-4}{8}$$

$$p=.5$$

We have two segments to test, between .457 and .5 and between .5 and 1. The first segment:

For $p=.46$:

$$4p^3+4p^2\,?\,3p$$

$$4(.46)^3+4(.46)^2\,?\,3(.46)$$

$$4(.097336)+4(.2116)\,?\,1.38$$

$$.389344+.8464\,?\,1.38$$

$$1.235744\,<\,1.38$$

So up to a probability of .5, you should be playing Swiss. Above that?

For $p=.6$:

$$4p^3+4p^2\,?\,3p$$

$$4(.6)^3+4(.6)^2\,?\,3(.6)$$

$$4(.216)+4(.36)\,?\,1.8$$

$$.864+1.44\,?\,1.8$$

$$2.304\,>\,1.8$$

Above a probability of .5, you should play 8-4s.

So what we've learned so far is that, given equal probability of winning a match across all three draft formats, if your probability of winning a match is less than 50%, you should play in the Swiss queues, and above 50%, you should play in the 8-4 queues, and you should never play 4-3-2-2s. But the reality is that skill level across the queues are different. 8-4 drafts typically attract better players than 4-3-2-2s and Swiss queues. Next, we'll attempt to take a look at how much better each queue is than the other.