Later in their response to Hogbin and Nijdam,[44] they did agree that it was natural to suppose that the host chooses a door to open completely at random, when he does have a choice, and hence that the conditional probability of winning by switching (i.e., conditional given the situation the player is in when he has to make his choice) has the same value, 2/3, as the unconditional probability of winning by switching (i.e., averaged over all possible situations). Thus, the posterior odds become equal to the Bayes factor 1 : 2 : 0. The odds that your choice contains a pea are 1/3, agreed? For example, assume the contestant knows that Monty does not pick the second door randomly among all legal alternatives but instead, when given an opportunity to pick between two losing doors, Monty will open the one on the right. such as zero-sum games, and modern topics, such as sponsored search After choosing a box at random and withdrawing one coin at random that happens to be a gold coin, the question is what is the probability that the other coin is gold. treatment, interspersed with illuminating examples, makes it a So the player's choice after the host opens a door is no different than if the host offered the player the option to switch from the original chosen door to the set of both remaining doors. The version of the Monty Hall problem published in Parade in 1990 did not specifically state that the host would always open another door, or always offer a choice to switch, or even never open the door revealing the car. 1 The player picks one of the three cards, then, looking at the remaining two cards the 'host' discards a goat card. agents interacting and myriad opportunities for conflict and world-view. of presenting both the breadth and coherence of its underlying The typical behavior of the majority, i.e., not switching, may be explained by phenomena known in the psychological literature as: Experimental evidence confirms that these are plausible explanations that do not depend on probability intuition. The Lookahead auction is approximately optimal, Chapter 15. theory as a field. Although game theory can be and has been used to analyze parlour games, its applications are much broader. auctions, are covered. The formulation is loosely based on quantum game theory. The Three Prisoners problem, published in Martin Gardner's Mathematical Games column in Scientific American in 1959 [7][58] is equivalent to the Monty Hall problem. The problem continues to attract the attention of cognitive psychologists. But, knowing that the host can open one of the two unchosen doors to show a goat does not mean that opening a specific door would not affect the probability that the car is behind the initially chosen door. [3] In this case, there are 999,999 doors with goats behind them and one door with a prize. As previous, but now host has option not to open a door at all. MAA Member Price: $67.50. "Anything else is a different question. From this point of view, one has to remember that the player has two opportunities to make choices: first of all, which door to choose initially; and secondly, whether or not to switch. Moreover, the host is certainly going to open a (different) door, so opening a door (which door unspecified) does not change this. "Game Theory, Alive" is a wonderful book and is to be highly recommended, either for teaching or self-study...this reviewer would not be surprised if "Game Theory, Alive" becomes the standard text for an introductory course on Game Theory. Similarly, strategy A "pick door 1 then switch to door 2 (if offered), but do not switch to door 3 (if offered)" is dominated by strategy B "pick door 3 then always switch". Sperner’s Lemma in higher dimensions*, 6.3.3. [33] There, the possibility exists that the show master plays deceitfully by opening other doors only if a door with the car was initially chosen. Vos Savant commented that, though some confusion was caused by some readers' not realizing they were supposed to assume that the host must always reveal a goat, almost all her numerous correspondents had correctly understood the problem assumptions, and were still initially convinced that vos Savant's answer ("switch") was wrong. Product Code: MBK/101.E In an invited comment[40] and in subsequent letters to the editor,[41][42][43][44] Morgan et al were supported by some writers, criticized by others; in each case a response by Morgan et al is published alongside the letter or comment in The American Statistician. Intuitively, the player should ask how likely it is that, given a million doors, he or she managed to pick the right one initially. The analysis also shows that the overall success rate of 2/3, achieved by always switching, cannot be improved, and underlines what already may well have been intuitively obvious: the choice facing the player is that between the door initially chosen, and the other door left closed by the host, the specific numbers on these doors are irrelevant.

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