2017 Putnam A5

Each of the integers from $1$ to $n$ is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, $A$, $B$, and $C$, take turns in the order $A,B,C,A,\dots$ choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered $1$.

Show that for each of the three players, there are arbitrarily large values of $n$ for which that player has the highest probability among the three players of winning the game.

从 $1$ 到 $n$ 的每个整数都写在一张单独的卡片上，然后将卡片组合成一副牌并洗牌。三名玩家 $A$、$B$ 和 $C$ 按 $A、B、C、A、\dots$ 的顺序轮流从牌堆中随机选择一张牌。 (牌组中的每张牌被选择的可能性相同。)选择一张牌后，该牌和所有编号较大的牌都会从牌组中移除，剩余的牌在下一轮之前重新洗牌。游戏继续进行，直到三名玩家中的一位通过抽出编号为 $1$ 的卡片赢得游戏。

证明对于三名玩家中的每一位，都有任意大的 $n$ 值，使得该玩家在三名玩家中赢得比赛的概率最高。