CF626G Raffles

Johnny is at a carnival which has $ n $ raffles. Raffle $ i $ has a prize with value $ p_{i} $ . Each participant can put tickets in whichever raffles they choose (they may have more than one ticket in a single raffle). At the end of the carnival, one ticket is selected at random from each raffle, and the owner of the ticket wins the associated prize. A single person can win multiple prizes from different raffles. However, county rules prevent any one participant from owning more than half the tickets in a single raffle, i.e. putting more tickets in the raffle than all the other participants combined. To help combat this (and possibly win some prizes), the organizers started by placing a single ticket in each raffle, which they will never remove. Johnny bought $ t $ tickets and is wondering where to place them. Currently, there are a total of $ l_{i} $ tickets in the $ i $ -th raffle. He watches as other participants place tickets and modify their decisions and, at every moment in time, wants to know how much he can possibly earn. Find the maximum possible expected value of Johnny's winnings at each moment if he distributes his tickets optimally. Johnny may redistribute all of his tickets arbitrarily between each update, but he may not place more than $ t $ tickets total or have more tickets in a single raffle than all other participants combined.

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In the first case, Johnny only has one ticket to distribute. The prizes are worth $ 4 $ and $ 5 $ , and the raffles initially have $ 1 $ and $ 2 $ tickets, respectively. After the first update, each raffle has $ 2 $ tickets, so Johnny has expected value  of winning by placing his ticket into the second raffle. The second update adds a ticket to the second raffle, so Johnny can win  in the first raffle. After the final update, Johnny keeps his ticket in the first raffle and wins . In the second case, Johnny has more tickets than he is allowed to spend. In particular, after the first update, there are $ 7 $ , $ 6 $ , and $ 6 $ tickets in each raffle, respectively, so Johnny can only put in $ 19 $ tickets, winning each prize with probability . Also, note that after the last two updates, Johnny must remove a ticket from the last raffle in order to stay under  the tickets in the third raffle.