Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.

中文翻译：

---

部分反馈的卡片猜测

考虑以下实验：一个带有米的副本n不同类型的牌是随机洗牌的，猜者在抽牌时尝试按顺序猜牌。每次进行猜测时，都会给出一些“反馈”。例如，可以告诉猜者他们刚刚猜到的牌的真实身份（完整的反馈模型），也可以什么都不告诉他们（无反馈模型）。在本文中，我们探索了一个部分反馈模型，在猜测一张牌时，猜测者只会被告知他们的猜测是否正确。我们在这种情况下表明，均匀地在n， 最多$m+O(m^{3/4}\log m)$可以按预期正确猜出卡片。这解决了 Diaconis 和 Graham 从 1981 年开始提出的问题，即使是$m=2$案件已开。