  We study a general scenario of simultaneous contests that allocate prizes
based on equal sharing: each contest awards its prize to all players who
satisfy some contest-specific criterion, and the value of this prize to a
winner decreases as the number of winners increases. The players produce
outputs for a set of activities, and the winning criteria of the contests are
based on these outputs. We consider two variations of the model: (i) players
have costs for producing outputs; (ii) players do not have costs but have
generalized budget constraints. We observe that these games are exact potential
games and hence always have a pure-strategy Nash equilibrium. The price of
anarchy is $2$ for the budget model, but can be unbounded for the cost model.
Our main results are for the computational complexity of these games. We prove
that for general versions of the model exactly or approximately computing a
best response is NP-hard. For natural restricted versions where best response
is easy to compute, we show that finding a pure-strategy Nash equilibrium is
PLS-complete, and finding a mixed-strategy Nash equilibrium is
(PPAD$\cap$PLS)-complete. On the other hand, an approximate pure-strategy Nash
equilibrium can be found in pseudo-polynomial time. These games are a strict
but natural subclass of explicit congestion games, but they still have the same
equilibrium hardness results.
