King maker

Background

This topic arised after an analysis on the APS rating system, and a lengthy discussion. Also, the article on Wikipedia helped a lot.

Have you ever wondered what would happen if only top bots were in the league? Have you ever preferred Premier League over APS but didn´t know exactly why? Here is the concept which explains why those thoughts are relevant.

What is it?

King-maker scenarios are situations when weaker competitors have the power to influence the position of stronger competitors. The first place (king) is made by someone else. The concept doesn´t apply to first place alone, it also applies to any position influenced by lower positions in a given ranking.

Examples

For example, if the rumble was reduced to only 3 competitors, A with strength 4, B with strength 3, and C with strength 2. Competitor A and B have a match and the score is 57% for A and 43% for B. Then they both have matches against C and the scores are 67% for A, 33% for C, and 60% for B, 40% for C.

• APS for A would be (57%+67%)/2 = 62%
• APS for B would be (43%+60%)/2 = 51,5%
• APS for C would be (33%+40%)/2 = 36,5%

Good! The APS ratings have the same order as the strengths, showing the system is accurate. But what would happen if competitor C performs worse against B on purpose, or because of some bug only B is exploiting? Then the score for B would be 100% and C would be 0%:

• APS for A would be (57%+67%)/2 = 62%
• APS for B would be (43%+100%)/2 = 71,5%
• APS for C would be (33%+0%)/2 = 16,5%

Competitor C crappy performance overweights the "final match" between competitors A and B, and decides who is the king.

A more bizarre scenario can emerge if competitor C leaves the rumble:

• APS for A would be 57% = 57%
• APS for B would be 43% = 43%

The positions are swapped back, without any of the 2 competitors involved doing anything.

What would happen if only wins/draws/losses are taken in account?

• Winning rate for A would be 2/0/0 = 100%
• Winning rate for B would be 1/0/1 = 50%
• Winning rate for C would be 0/0/2 = 0%

If competitor C performs even worse against B, it would still be 0% for C and 50% for B:

• Winning rate for A would be 2/0/0 = 100%
• Winning rate for B would be 1/0/1 = 50%
• Winning rate for C would be 0/0/2 = 0%

If competitor C leaves the rumble:

• Winning rate for A would be 1/0/0 = 100%
• Winning rate for B would be 0/0/1 = 0%

The only way for B to take the throne from A is defeating it. And the only way for C to put B in the throne is also defeating A. In a king-maker resistant system, you need to be the king to make another king.

Winning rate is by no means the only way to fight king-maker scenarios, but it is a simple and popular one. Premier League system also fights king-maker scenarios in an effective way.

Why should I bother?

King-maker scenarios usually disrupts the competitive mood in games as the outcomes are not dictated by a competitor's own performance, but by other competitor's performances. But it is good in games where diplomacy should be a factor. In the example above, B and C joining forces against A is a kind of diplomacy. If diplomacy is desirable in the rumble is another matter entirely.

In RoboRumble, after competitor C screws up the ranking in the example above, the consequences would be:

• Either competitor A becoming pissed off and leaving the rumble
• Or competitor A joining the king-maker fest starting to overvalue matches against C, and undervaluing an otherwise decisive and elaborate match between A and B.

In another scenario where a king-maker scenario wasn't happening, competitor B would be having a hard time figuring out how to defeat competitor A. After that, competitor A would be doing the same to regain the throne. And a lot more elaborate algorithms would emerge.