In an important paper, Weinstein and Yildiz (2007) show that if players have an infinite depth of reasoning and this is commonly believed, types generically have a unique rationalizable action in games that satisfy a richness condition. We show that this result does not extend to environments where players may have a finite depth of reasoning, or think it is possible that the other player has a finite depth of reasoning, or think that the other player may think that is possible, and so on, even if this so-called grain of naivetue9 is arbitrarily small. More precisely, we show that even if there is almost common belief in the event that players have an infinite depth of reasoning, there are types with multiple rationalizable actions, and the same is true for nearby types. Our results demonstrate that both uniqueness and multiplicity are robust phenomena when we relax the assumption that it is common belief that players have an infinite depth, if only slightly.
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机译：温斯坦和伊尔迪兹（Weinstein and Yildiz，2007）在一篇重要论文中指出，如果玩家具有无限的推理深度，并且人们普遍认为，在满足丰富性条件的游戏中，类型通常具有独特的合理化作用。我们证明了这个结果并没有扩展到玩家可能具有有限的推理深度，或者认为其他玩家可能具有有限的推理深度，或者认为其他玩家可能认为这样的环境，等等。即使这个所谓的naivet ue9粒度很小。更准确地说，我们表明，即使在玩家具有无限深层次的推理能力的情况下，即使人们几乎有共同的信念，也存在具有多种合理化行动的类型，附近的类型也是如此。我们的结果表明，当我们放松以下假设时，唯一性和多样性都是健壮的现象：人们普遍认为玩家的深度是无限的，即使只有很小的深度。
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