Consider the following competitive game:
There is a pie, whose size is a random uniform variable (from 0 to 1 kg, say). No player knows the size of the pie.
Each player writes down their request - how much pie they would like, anywhere from 0 to 1 kg. Let player 1 request a1, player 2 request a2, and so on.
The size of the pie is revealed, and the players are served their requests, starting from the player with the smallest request, continuing in ascending order through to the player with the largest request (or whenever the pie runs out). In the case of exactly equal requests where there is not enough pie to serve both players, each gets an equal portion of what remains.
What is the optimal strategy?
Optimal strategy is defined as the strategy which maximises, over multiple rounds, the expected pie won. A strategy can take into account the past actions of opposing players. No communication between players is allowed.
Even considering the case of only 2 players, the situation is complex, because it is best to bid 1 (if their bid is lower than (1-1/sqrt(2))), or slightly undercut their bid (if their bid is anything higher than (1-1/sqrt(2))). This will probably lead to a somewhat chaotic set of bids.