# Evaluating the Acer Teamstory Cup Playoff Race

- November 3, 2013
- misc
- python starcraft
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These are the standings of the group stage of the Acer team story cup as of November 3^{rd}. Each team plays against all others once for a total of 9 games. The best 4 teams advance to the playoffs.

1 | WesternWolves | 6–2 | 35–23 | +12 |

2 | TeamAcer | 5–1 | 28–15 | +13 |

3 | QuanticGaming | 5–1 | 26–22 | +4 |

4 | Axiom | 3–2 | 19–15 | +4 |

5 | TeamLiquid | 3–3 | 25–25 | |

6 | mousesports | 3–3 | 21–23 | -2 |

7 | Millenium | 2–3 | 19–23 | -4 |

8 | compLexityGaming | 2–4 | 20–25 | -5 |

9 | InvictusGaming | 1–5 | 18–28 | -10 |

10 | Empire-Ence | 1–7 | 24–36 | -12 |

So with 14 outstanding matchups the question is: which teams are already qualified? Which teams are eliminated?

## Procedure

A simple method to look at these questions is the magic number. The number is very easy to calculate, but also very crude. A team might clinch the playoffs or be eliminated long before the magic number confirms it. Also, since each team has only 9 clan wars, there tend to be a lot of ties and the map differential matters. The magic number does not usually consider tie breaking.

This is why I’m using *enumeration* to determine exact rank ranges for each team. For each clan war there are 10 potential outcomes (5:0, … 5:4, 4:5, …, 0:5). Checking all those outcomes directly isn’t feasible with 14 games left, but the number of cases to consider is much smaller if the analysis is split in two phases:

- All 2
^{14}(16384) combinations of wins and losses are considered while ignoring the number of maps won by the losing team for now. For every combination the points for each team are counted, and the teams are ranked by points. No tie breaking is applied; instead the potential range of ranks is noted. - Next, for each of the 16384 outcomes separately, we determine the exact best or worst case rank of our teams adding map wins to the equation.

### Step 1: Ranges based of Wins/Losses

The procedure is as follows: Check all 2^{14} combinations of winners for the 14 matchup. For each combination calculate the ranking based on points only. If the considered team is tied with other teams, then note the best and worst rank of this group. For instance, if the team has the second most points tied with four other teams, then the true rank is between 2 and 5.

#### Result for Western Wolves

These are the resulting ranges for the leading *Western Wolves*. There are 4 combinations where they are one of six teams with the same number of wins, and 2 combinations where the team has the second most points tied with 5 other teams. On the bright side there are 1862 combinations where the team comes up at uncontested first place.

group | count | % |
---|---|---|

1 – 1 | 1862 | 11.36% |

1 – 2 | 3200 | 19.53% |

1 – 3 | 1976 | 12.06% |

1 – 4 | 908 | 5.54% |

1 – 5 | 178 | 1.09% |

1 – 6 | 4 | 0.02% |

2 – 2 | 1278 | 7.80% |

2 – 3 | 2500 | 15.26% |

2 – 4 | 1824 | 11.13% |

2 – 5 | 220 | 1.34% |

2 – 6 | 2 | 0.01% |

3 – 3 | 1376 | 8.40% |

3 – 4 | 976 | 5.96% |

3 – 5 | 16 | 0.10% |

4 – 4 | 64 | 0.39% |

### Step 2: Look at individual combinations

From the previous table we know that there are instance where *Western Wolves* has the same record as the sixth ranked team. We don’t know though if the map differential tie breaker will ever put the team there. To find out, we have a closer look at those 6 combinations where the team might be sixth.

Here is one of the two combinations where the first six teams (*Millenium*, *Team Acer*, *Western Wolves*, *Team Liquid*, *mousesports*, *Quantic Gaming*) all have 6 wins and 3 losses:

- Millenium wins against Team Acer
- Millenium wins against compLexity Gaming
- Quantic Gaming wins against Axiom
- mousesports wins against Western Wolves
- Millenium wins against Invictus Gaming
- Team Liquid wins against Axiom
- compLexity Gaming wins against Team Acer
- mousesports wins against Axiom
- Team Acer wins against Invictus Gaming
- Team Liquid wins against Quantic Gaming
- mousesports wins against Quantic Gaming
- Millenium wins against Axiom
- Invictus Gaming wins against compLexity Gaming
- Team Liquid wins against Empire-Ence

For this combination check all potential worst case scenarios for the *Western Wolves*. Even though the winner for each matchup is given, there are still 5 different ways to win, i.e., 5:0, 5:1, 5:2, 5:4, or 5:4. Fortunately, there are many cases where the worst case is obvious, or where the result has no effect on the tie breaking procedure:

**case a:***Western Wolves*loses all remaining matches 0:5**case b:**all teams that are tied in points with*Western Wolves*win 5:0 against teams that are not tied in points.**case c:**all teams that are tied in points with*Western Wolves*lose 4:5 against teams that are not tied in points.**case d:**if two teams play against each other that are tied with*Western Wolves*, check all 5 possible numbers of maps won by the losing team.**case e:**if two teams play against each other that are both not tied with*Western Wolves*, the result doesn’t matter.

Accordingly, this is one of the worst case scenarios for *Western Wolves*. The instance and number of considered results is noted in the last two columns.

Result | Type | Cases | ||||
---|---|---|---|---|---|---|

TeamAcer | 3 | : | 5 | Millenium | d | 2^{*} |

Millenium | 5 | : | compLexityGaming | b | 1 | |

QuanticGaming | 5 | : | Axiom | b | 1 | |

WesternWolves | : | 5 | mousesports | a | 1 | |

InvictusGaming | : | 5 | Millenium | b | 1 | |

Axiom | : | 5 | TeamLiquid | b | 1 | |

compLexityGaming | 5 | : | 4 | TeamAcer | c | 1 |

mousesports | 5 | : | Axiom | b | 1 | |

TeamAcer | 5 | : | InvictusGaming | b | 1 | |

TeamLiquid | 5 | : | 4 | QuanticGaming | d | 5 |

QuanticGaming | 4 | : | 5 | mousesports | d | 5 |

Axiom | : | 5 | Millenium | b | 1 | |

InvictusGaming | 5 | : | 3 | compLexityGaming | e | 1 |

Empire-Ence | : | 5 | TeamLiquid | b | 1 |

^{*} Team Acer led 3:1 when the game was stopped, hence, 0:5, 1:5, and 2:5 aren’t possible. In total there are just 3*5*5 = 75 combinations.

The resulting standings after those results are as follows.

1 | TeamAcer | 6–3 | 40–25 | +15 |

2 | Millenium | 6–3 | 39–26 | +13 |

3 | TeamLiquid | 6–3 | 40–29 | +11 |

4 | mousesports | 6–3 | 36–27 | +9 |

5 | QuanticGaming | 6–3 | 39–32 | +7 |

6 | WesternWolves | 6–3 | 35–28 | +7 |

7 | compLexityGaming | 3–6 | 28–39 | -11 |

8 | Axiom | 3–6 | 19–35 | -16 |

9 | InvictusGaming | 2–7 | 23–41 | -18 |

10 | Empire-Ence | 1–8 | 24–41 | -17 |

*Western Wolves* gets the same differential as *Quantic Gaming* in this setting, but has less wins which puts the team last of all 6:3 teams. So there is still a scenario where *Western Wolves* misses the playoffs.

### Best Case

The procedure to determine the best case is very similar:

- the considered team wins all remaining matches 5:0
- all teams that are tied in points with our team win 5:4 against teams that are not tied in points.
- all teams that are tied in points with our team lose 0:5 against teams that are not tied in points.
- if two teams play against each other that are tied with our team, check all 5 possible numbers of maps won by the losing team.
- if two teams play against each other that are both not tied with our team, the result doesn’t matter.

## Results

Applying the procedure to all 10 teams leads to those ranges:

Team | W/L | maps | diff | pot.ranks | |
---|---|---|---|---|---|

1 | WesternWolves | 6–2 | 35–23 | 12 | 1–6 |

2 | TeamAcer | 5–1 | 28–15 | 13 | 1–8 |

3 | QuanticGaming | 5–1 | 26–22 | 4 | 1–8 |

4 | Axiom | 3–2 | 19–15 | 4 | 1–9 |

5 | TeamLiquid | 3–3 | 25–25 | 1–9 | |

6 | mousesports | 3–3 | 21–23 | -2 | 1–9 |

7 | Millenium | 2–3 | 19–23 | -4 | 1–10 |

8 | compLexityGaming | 2–4 | 20–25 | -5 | 2–10 |

9 | InvictusGaming | 1–5 | 18–28 | -10 | 4–10 |

10 | Empire-Ence | 1–7 | 24–36 | -12 | 8–10 |

Even with 14 games remaining it is possible to exactly determine the worst and best case rank for all teams. Mathematically all except the last team can still clinch the playoffs, 7 teams are even contenders for the top spot. On the other hand, no team is qualified yet. Even at 6-2, the first team might fall as far as sixth place.