  We introduce a novel system of matching and scoring players in tournaments,
called Multi-Tier Tournaments, illustrated by chess and based on the following
rules:
  1. Players are divided into skill-based tiers, based on their Elo ratings.
  2. Starting with one or more mini-tournaments of the least skilled players
(Tier 1), the winner or winners -- after playing multiple opponents -- move to
the next-higher tier.
  3. The winners progress to a final tier of the best-performing players from
lower tiers as well as players with the highest Elo ratings.
  4. Performance in each tier is given by a player's Tournament Score (TS),
which depends on his/her wins, losses, and draws (not on his/her Elo rating).
  Whereas a player's Elo rating determines in which mini-tournament he/she
starts play, TS and its associated tie-breaking rules determine whether a
player moves up to higher tiers and, in the final mini-tournament, wins the
tournament. This combination of players' past Elo ratings and current TS's
provides a fair and accurate measure of a player's standing among the players
in the tournament. We apply a variation of Multi-Tier Tournaments to the top 20
active chess players in the world (as of February 2024). Using a dataset of
1209 head-to-head games, we illustrate the viability of giving lower-rated
players the opportunity to progress and challenge higher-rated players. We also
briefly discuss the application of Multi-Tier Tournaments to baseball, soccer,
and other sports that emphasize physical rather than mental skills.
