The group stage of the world cup is a four-team six-game round-robin from which the top two teams advance to the elimination round. Each team plays three games, one against each other team in the group, and the best two records advance - ideally. In practice, determining the 'best two records' is much more complicated. Example complications include - teams may tie, so how many ties are as good as one win? Is a 4-1 win better than a 2-0 win? Ignoring these subtleties, the system still makes for some interesting group theory.
As any college football fan knows, wins in sports are not transitive. If team a beats team b and team b beats team c, this does not mean team a will beat team c. From a group theory perspective this allows for cycles. Using > for 'defeats', we can write
With three elements (teams) this scenario is called a three cycle. Group H (Switzerland, Chile, Spain, Honduras) is set up to have a three cycle this world cup. The Swiss defeated Spain, Chile defeated the Swiss, and Spain is favored to defeat Chile. Everyone has defeated, or is favored to defeat, Honduras. This scenario is bad for the World Cup system - a series of complicated and unsatisfying tie breaks are necessary to determine which two teams are the best. Generally large cycles are bad for round robin tournaments for exactly this reason.
A natural follow up question is then, would a four cycle be worse or better than a three cycle in the World cup system? Lets stick with Group H. Say Honduras>Swiss>Spain>Chile>Honduras. We have now accounted for two games for each team (one win and one loss). Each team must then play one more game, here Swiss vs Chile and Spain vs Honduras. If we continue to ignore ties (absent other information ties are the most likely possibility) then the results of these last two games determine who advances. Winners move on. Thus ignoring ties, a three cycle may 'break' the qualifying system, but a four cycle may not†.
† Two cycles are impossible, since teams play each other only once.
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