Does a loser prefer the winner to subsequently lose or win? Most major league sports have some form of tournament bracket where there are several rounds of elimination before a winner is determined.  In most of them, the winning team is undefeated over the course of the whole tournament.

This question considers the losers and their fans.  Specifically, do fans of a losing team prefer that the team which beat them go on to be the overall winners (the "We lost to the best" scenario) or that the winning team lose in the next round (the "Revenge" scenario)?

Have there been any studies of this?  Is there a difference between the attitude of the losing team's players and the losing team's fans?
First, it is worth addressing what kind of tournament this is. If this is something like the Super Bowl, where the teams are generally widely-known year after year, then the answer will surely vary depending on the context. 

Most long-time sports fans have a subjective preference for the other teams, ranging from 'second-favorite' to 'most hated', and these preferences are built up over time and are determined by the individual as well as the team. After having lost the Super Bowl XLV, a Seahawks fan may hold resentment towards the Steelers for years to come, and thus may root against them on all accounts. 

On a more individual basis, fans may view another team favorably for a variety of personal reasons (e.g. a parent or close friend rooting for that team), and thus choose to root for them even if it came down to them beating their team. One writer even analyzed this idea in a FiveThirtyEight blog post.

However, if we were to remove these factors entirely -- say, we have a tournament where there is no prior opinion towards the teams that had advanced -- then the answer may shed light on both individual differences and group dynamics.

Studies have shown that the more a fan identifies with their team, the more loyalty they show to their team in the face of a loss:


  When presented with any sort of negative information regarding the group,
  highly identified group members react differently than those with lower levels of
  identification (Cohen &amp; Garcia, 2005; Ellemers, Spears, &amp; Doosje, 2002). Highly
  identified members typically reaffirm their group membership, while those with
  lower levels of identification tend to distance themselves (Cohen &amp; Garcia, 2005).
  Such behaviors are apparent in sport fans. Wann and Branscombe (1990) found
  that highly identified fans are less likely to distance themselves from the team
  when the team lost than those who were less identified. Further, highly identified
  fans exhibit biased attribution processing favoring their team (Wann &amp; Dolan,
  1994). That is, facing a win, highly identified fans ascribe the victory to internal
  factors such as the skill of the team, or the coaching, and sometimes even fan support.
  However, upon a loss, rather than conceding another team’s superiority, they
  blame the loss on more external factors such as fate or poor refereeing. Thus,
  highly identified sports fans seem to undergo some sort of biased attributional
  processes when dealing with a loss.


And another study showed that the stronger the allegiance to the team, the more likely the fan was to perceive the win as a personal victory, thereby influencing their own self-efficacy. By contrast, the likelihood increased that a loss would be perceived as a personal offense.


  Ss watched a live basketball game; then, in the context of a 2nd, unrelated experiment, Ss estimated their own performance at several tasks. Results indicated that fans' mood and self-esteem were affected by game outcome. More important, fans' estimates of both the team's and their own future performance were significantly better in the win than in the loss condition.


Whereas on the other hand, fans who identify less with their team tend to distance themselves from the team in the face of a loss.

So it could be the case that personal allegiance to a team determines how a fan responds to a loss. If the fan identifies strongly with the losing team, they may be less likely to view the winning team (opponent) positively, and thus they may demonstrate a greater tendency to root against that team in the next round. On the other hand, less team allegiance may implicate a desire to identify with the winning team, in which case the "we lost to the best" mindset may occur.