the pairing game

Exciting news for Warmachine and Hordes players. A new edition will be announced in a matter of weeks, bringing in a more balanced and streamlined gaming system. This is awesome for gamers but horrible for my WTC analysis. While I will still have good player rankings (assuming that players remain at the same skill ranking in the new edition), it seems likely that the caster pairing advantage will be dramatically changed. Until data becomes available for the new edition, it will be tricky to make sound predictions.

For now, I’ve been investigating the team pairing process itself. While it is difficult to predict how a team will behave, it is worthwhile to consider how different team behaviours will affect round outcomes.

WTCChampionship2015

The WTC pairing process for between two teams, A and B, for a round is as follows:

  1. team B reveals their first player,
  2. team A reveals two players,
  3. team B selects one of team A’s revealed players to match with their revealed player,
  4. team B reveals their next two players,
  5. team A selects one of team B’s revealed players to match with their revealed player,
  6. team A reveals their next two players,
  7. team B selects one of team A’s revealed players to match with their revealed player,
  8. team B reveals their next two players,
  9. team A selects one of team B’s revealed players to match with their revealed player,
  10. the remaining two players are matched.

The widely held opinion is that being team A is an advantage, since this team makes the selections that define the outcomes in three of the five matchups. But how much effect does this really have?

In my initial look at this I’m going to simplify this to a player being a number, and a game being won by the player with the highest number. Each team will have players with rating 1, 2, 3, 4, and 5. When a player with rating 1 plays a player with rating 2, the latter player always wins. When a player with rating 1 plays a player with rating 1, the result is always a draw. If a team can engineer their opponents’ strongest players to play their weaker players, they will have opportunities to win the match by having the advantage in multiple games.

In this analysis:

  • there is exactly one value for each player,
  • both teams know the value of all players,
  • when two players are matched, the player with the highest score wins.

This simple model is not taking into account the fact that each player has two lists; therefore the matrix of possible outcomes is a little more complex. This is also not attempting to model the rock-paper-scissors of individual list pairings. Nonetheless, the results for this simple system are interesting.

Each of these picking strategies follows slightly different rules:

  • Pick First means select the first player presented. This is a naive approach which
    does not react in any way to the behaviour of the other team.
  • Pick Random means select any of the players at random. This is also a naive approach,
    and for randomly sorted players, should behave in the same way as picking first.
  • Pick Max means select the player which wins by the most, unless a player cannot
    win, in which case have the player defeated by the largest margin.
  • Pick Just Max means select the player which wins by the least, unless a player cannot
    win, in which case have the player defeated by the largest margin.

Which do you think is the most effective strategy? If you were to play this game, what strategy would you play? One of these, or some other? Do determine the best, I implemented each of these strategies, and the pairing game itself in R and have published this code as the package throwdown on GitHub.

Using these algorithms I was able to simulate 100,000 pairing games between teams with player value 1 to 5. A team won when it had more games where the player value was greater. A game was a draw (could have gone either way) when two players of the same value are matched. A match is a draw when both teams have the same number of wins.

Since the player values were randomly sorted, but Pick First and Pick Random should perform in the same manner. It is commonly stated that team B has an advantage in the pairing game, since they are able to dictate the last two matchups. When teams pick randomly, this advantage is lost, since the teams are as likely to pick the unfavourable matchups as the best matchups. The following tables show the proportion of 100,000 games drawn, and the proportion of non-drawn games won by team B. Pick First and Pick Random strategies have a likelihood of 50% of winning against itself or the other algorithm as Team B or A. This suggests that extra benefits given to team A may be unnecessary. But what about following a more reactive strategy?

Team B Team A Team B Wins Draws
pickerFirst pickerFirst 0.51 0.37
pickerRandom pickerFirst 0.50 0.38
pickerFirst pickerRandom 0.49 0.38
pickerRandom pickerRandom 0.50 0.38

Things look different when one of the teams plays the Pick Max strategy. Assuming that draws are 50% in favour of each player, Team B is only 40% chance of winning. This is because the Pick Max strategy is unnecessarily aggressive when revealing first player. When Team A plays the Pick Max strategy, they only attempts to beat Team B when it is possible to. When one Team is playing the Max strategy, and one Random/First strategies, it is preferred to be Team A.

Team B Team A Team B Wins Draws
pickerMax pickerFirst 0.35 0.43
pickerMax pickerRandom 0.37 0.42
pickerFirst pickerMax 0.35 0.43
pickerRandom pickerMax 0.38 0.41

The outcomes go entirely crazy when you look at the Pick Just Max strategy. When team B uses the Pick Just Max strategy against a team playing the Pick First strategy, the Pick Just Max team wins 70% of non-drawn matches. When team A uses the Just Max
strategy against a team picking First, they win 95% of non-drawn matches. Again, being team A is preferred. When team B uses the Pick Just Max strategy against a team playing the Pick Random strategy, the Pick Just Max team only wins 30% of non-drawn matches. When team A uses the Just Max strategy against a team picking Randomly, second they only win 55% of non-drawn matches. In this instance, the team using the Just Max strategy prefers to be team A, but their choices leave them open to defeat against a Random selection.

Team B Team A Team B Wins Draws
pickerJustMax pickerFirst 0.72 0.53
pickerJustMax pickerRandom 0.30 0.53
pickerFirst pickerJustMax 0.06 0.40
pickerRandom pickerJustMax 0.45 0.49

Just Max is performing very well against Pick First, but poorly against Pick Random. At the reveal step, Just Max is revealing the player which just beats the other team’s choice first, and then either the player which beats that choice more thoroughly, or if no others win, which loses by the most. This means that Pick First is always selecting the best option for Pick Just Max, whereas Pick Random is sometimes winning, sometimes losing. For the Pick Just Max strategy, Pick First is losing optimally, hence the far worse performance than Pick Random. The default opening for Pick Just Max is to reveal the strongest player. This is because the strongest player has the most possible wins. Perhaps this is a weakness, and a lower, or the lowest ranked player should open. That would provide less occasions where the other team can play the weakest player into the strongest player, and offer fewer winning combinations.

Playing combinations of Max and Just Max is perhaps the least interesting. Team B picking the Max rated player available to them against Team A picking the Max rated player always results in a win for Team B. This is because team B picks 3 games, selecting the Max victory, and loses 2 games.

When Team B picks Just Max and A picks Max, every game is a draw, since Team B selects a draw between the 5 rated players, then each team wins games in turn as a player is left exposed while the other team has a higher rated player in hand.

When Team B picks Max and A picks Just Max, team A wins every game, with no draws. At each reveal, B is too greedy, allowing A to win 3 games each match.

When both teams play the Just Max strategy, all games result in a draw, as both teams are cautious with how they reveal their players.


Team B Team A Team B Wins Draws
pickerMax pickerMax 1.00 0.00
pickerJustMax pickerMax 1.00
pickerMax pickerJustMax 0.00 0.00
pickerJustMax pickerJustMax 1.00

These simulations show that the behaviour of these simple strategies can be unexpected, even for this very simple representation of the pairing game. So far I have not found evidence that team A is advantaged. While I will be investigating this system further, these early results suggest that the organizers should not give too much benefit to team B, as while it may appear that A has an advantage, it may not be possible for them to capitalize on it.

These simulations also show that it is not trivial to create a strategy that is better than picking matchups randomly, particularly as the strategy of the opposing team may not be known, or evolve during the tournament, or even pairing process.

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One thought on “the pairing game

  1. Pingback: Rate My Captain WTC 2015 | analytical gaming

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