Saturday, October 07, 2006

"Map Select" versus "Map Concede"

Format definitions:
"Map Select" - In a best of 3 match, player A chooses the first map from the current pool of maps to play, then Player B chooses a map to play second. The third map if needed will be randomly drawn between the remaining maps.

"Map Concede" - In a best of 3 match, player A choose a map to eliminate from the pool, reducing the pool size by 1. Player B then also eliminates a map from the pool. If there are more than 3 maps remaining in the pool, the players continue to drop another map each until there is 3 maps left (or 2 maps in the case of an even amount of maps). These 3 maps are then played in either random order, or in the odrer of Player A's selection, player B's selection, map remaining. In the case of an even map count upon starting there will only be 2 maps remaining. Player A chooses the first, then the remaining map. If a 3rd map is required, reinstate all maps and run another round of map concede down to 2 maps, then randomly choose between those by a coin toss.

Eliminating potential bias:
The main reason for using map concede over map select is that any bias in the maps is reduced rather than being exploited. This gives the player a more balanced playing field to play on. As the aim of any tournament is to find the winner by merit alone, any methods of eliminating bias helps a player's true skill to shine through. This is why most grand finals for sport are played at a neutral venue, to eliminate any perceived bias due to the venue. In terms of map choice, ideally we would want all games to be played on the most neutral map possible. If players could mutually agree on a map to play on, this would also be acceptable as both parties are then comfortable that the map selected is not going to adversely affect the outcome. "Map Concede" formalizes the mutual map selection method by putting each player in the position of revealing which maps they believe WOULD affect the outcome, finally leaving the last map as the one with the least amount of perceived bias.

In contrast, "Map Select" allows players to exploit any maps in the pool that favour their particular play style, race or strategy. This creates the opposite effect where you are creating a "home & away" scenario, giving each player the ability to play in their back yard. Even though both players get the advantage of choosing where to play, the potential is there that the bias gained from playing on your preferred map overwhelms any other difference in skill. This leads to a greater chance that both players win their "home" games and you are no closer to finding a winner.

Eg: If each player is 60% confident in beating a player of his own skill on the map he selects, the chance of a draw after 2 rounds is:
draw chance = Player A wins first map AND Player B wins 2nd map OR PlayerB wins first map AND PlayerA wins 2nd map
50% home win = 0.5 * 0.5 + 0.5 * 0.5 = 0.50 => 50%
60% home win = 0.6 * 0.6 + 0.4 * 0.4 = 0.52 => 52% ***
70% home win = 0.7 * 0.7 + 0.3 * 0.3 = 0.58 => 58%
80% home win = 0.8 * 0.8 + 0.2 * 0.2 = 0.68 => 68%

So a 2% greater chance that it will be a draw after 2 rounds, jumping to 8% with 70% confidence that they can win on their preferred map. My guess would be that the player's favourite map would give confidence between these 2 values, so an average of 5% more times the match will go to a 3rd map.

When combined with a player's true skill, it means that there is also less chance for the proper winner to win outright. If Player A has a 60% chance of winning each map due to skill, combining with the home and away effect gives:
win chance = PlayerA wins first map AND Player A wins 2nd map
win chance = PlayerA skill + Home map bias AND PlayerA - home map bias.
50% home win (+0% map bias) = (0.6 + 0) * (0.6 - 0) = 0.36 => 36%
60% home win (+10% map bias) = (0.6 + 0.1) * (0.6 - 0.1) = 0.35 => 35%
70% home win (+20% map bias) = (0.6 + 0.2) * (0.6 - 0.2) = 0.32 => 32%
80% home win (+30% map bias) = (0.6 + 0.3) * (0.6 - 0.3) = 0.27 => 27%

As the home ground advantage increases, it impacts on the chance of PlayerA to win the match in 2 maps. Considering that to win in 2 maps for a player in an even matchup is 25%, dropping from 36% to an average of 33.5% is substantial (~20% drop in effect of skill).

Placing all these effects into a real world scenario, lets assume that PlayerA has a 60% advantage over PlayerB, and that there are 4 maps on offer with PlayerA home advantages of +15% ('Ma'), +5%('Mb'), -5%('Mc') and -15%('Md').
Using "Map Select" we will have a map rotation of 'Ma', 'Md' and randomly 'Mb' or 'Mc':
win chance = (PlayerA wins 1st & 2nd map) OR (PlayerA draws AND PlayerA wins 3rd map)
= (PlayerA wins 1st & 2nd map) OR (((PlayerA wins 1st & loses 2nd) OR (PlayerA loses 1st& wins 2nd)) AND (PlayerA wins 3rd map))
with 'Mb' = ((0.6 + 0.15) * (0.6 - 0.15)) + ((((0.6 + 0.15) * (0.4 + 0.15)) + ((0.4 - 0.15) * (0.6 - 0.15))) * (0.6 + 0.05))
= 0.3375 + ((0.4125 + 0.1125) * 0.65)
= 0.3375 + ((0.525) * 0.65)
= 0.67875 => ~68%
with 'Mc' = ((0.6 + 0.15) * (0.6 - 0.15)) + ((((0.6 + 0.15) * (0.4 + 0.15)) + ((0.4 - 0.15) * (0.6 - 0.15))) * (0.6 - 0.05))
= 0.3375 + ((0.4125 + 0.1125) * 0.55)
= 0.62625 => ~62.5%

Using "Map concede" we will have a map rotation of 'Mb', 'Mc' and randomly 'Mc' or 'Md':
win chance = (PlayerA wins 1st & 2nd map) OR (PlayerA draws AND PlayerA wins 3rd map)
= (PlayerA wins 1st & 2nd map) OR (((PlayerA wins 1st & loses 2nd) OR (PlayerA loses 1st& wins 2nd)) AND (PlayerA wins 3rd map))
with 'Mb' = ((0.6 + 0.05) * (0.6 - 0.05)) + ((((0.6 + 0.05) * (0.4 + 0.05)) + ((0.4 - 0.05) * (0.6 - 0.05))) * (0.6 + 0.05))
= 0.3575 + ((0.2925 + 0.1925) * 0.65)
= 0.3575 + ((0.485) * 0.65)
= 0.67275 => ~68%
with 'Mb' = ((0.6 + 0.05) * (0.6 - 0.05)) + ((((0.6 + 0.05) * (0.4 + 0.05)) + ((0.4 - 0.05) * (0.6 - 0.05))) * (0.6 - 0.05))
= 0.3575 + ((0.2925 + 0.1925) * 0.55)
= 0.62425 => ~62.5%

As a baseline, if all maps had no home advantage:
win chance = (PlayerA wins 1st & 2nd map) OR (PlayerA draws AND PlayerA wins 3rd map)
= (PlayerA wins 1st & 2nd map) OR (((PlayerA wins 1st & loses 2nd) OR (PlayerA loses 1st& wins 2nd)) AND (PlayerA wins 3rd map))
with 'Mb' = (0.6 * 0.6) + (((0.6 * 0.4) + (0.4 *0.6)) * 0.6)
= 0.36 + ((0.24 + 0.24) * 0.6)
= 0.648 => ~65%

What this shows, rather suprisingly, is that even though there is a greater chance for a map to go to a decider for "Map Select", the player's skill on the third map should shine through enough to give essentially the same odds. With a greater than 5% swing in the overall outcome, the map selection for the decider seems to be the biggest 'random' contribution to this whole scenario; one that "map concede" tries to avoid by winning in 2.

Faster tournament play:
Due to the reduction of the 3rd map being played (by ~4% in the example scenario), this also helps tournaments become less prone to running over time by having extended best-of-3 matches. Although it is ancilliary to providing an unbiased result, it is still an issue that helps tournament organisation.

Improvement of the meta-game:
With "Map Select" the player is rewarded by having a specific map that they practice hard on so that they can maximize their chances of winning at least one game. The map selected is chosen more by the player's efforts on the map than by looking at the specific map advantages for an individual matchup. This is essentially inward looking and stagnates the metagame.
With "Map Concede", it is detrimental to focus on one map as this opens the opportunity for your opponent to concede the map, either by luck or by knowing the player's preferences. This encourages players to not only spread their own practice across different maps and playing styles, but also enhances the meta-game by encouraging players to seek out opponent's strategies and weaknesses to use in the map concede process. As a player you can choose to use the "map concede" process to simply not practice the maps you know you are going to drop, however this gives an advantage to your opponent that knows your decision because they can drop maps from the remaining pool, knowing you are forced to drop your own bad ones. Against a person with good meta-game skills, you are likely to play on your worst map of the remaining maps that you did practice.

Map Pool Selection:
Using a "Map Select" method, the tournament organisers need to be very careful with the maps provided for the players in the map pool as any map with an inherent bias toward a specific play style will be exploited. Eg: If there is a map pool of 3 balanced maps and one biased map toward a certain race, the players of that race gain an advantage while others do not. Using "Map Concede", the map pool can be arbitrarily large as the players eventually shrink the map pool down to the most neutral maps on offer. This allows some leeway for experimental maps or new maps to be brought in without a major impact on the final outcome. As a tournament director you still need to consider balance, and all maps should be as balanced as possible, however there is the flexibility that some maps can be in the pool as the best balanced for specific matchups without being used for every match. A pool of 5 or 7 is common for map concede.

Positive Play:
One thing that can hinder "map concede" is that it has a negative connotation. "Concede", "drop", "eliminate" are all negative words and it seems unintuitive to a beginner (or a spectator) that players should start with a negative mentality. Even though the players are emulating the natural process of selecting an even battlefield, this is not obvious from the name. Added to this is that "Map Select" has a better feel toward players when someone does win 2-0 (even though it is less likely). If your opponent has beaten you on your own home map, it is more likely that you will concede they are the better player. During a tournament players can have heated exchanges if they think they are being treated unfairly, so having a method that causes less stress in a tournament is helpful. If the match does go to a decider though, this effect is negated.

Personally I prefer the "Map Concede" method for its ability to find a neutral playing ground and it's ability to enhance the meta-game. I'm surprised that the statistics didn't back up what feels like an obvious advantage for playing on neutral soil, however this scenario of 4 maps with best-of-3 is possibly the worst for the format. If there is only one map required (due to time restraints) from a pool, this method is still available whereas map select is not. with 5 maps on offer, "map concede" offers 3 matches on the 3 most neutral maps while "map select" continues to widen the gap.

I'm working on a tournament format evaluator at the moment, so I'll have some figures for the other map pool sizes shortly. Hopefully I can also use it to demonstrate the anticipated stablility of "map concede" when introducing new maps. I've also looked at moving it to Ruby on Rails to let anyone be able to construct different tournament formats and see the results of the different tournament biases.