Optimal strategies for the progressive Monty Hall problem

2009 ◽  
Vol 93 (528) ◽  
pp. 410-419
Author(s):  
Stephen K. Lucas ◽  
Jason Rosenhouse
Keyword(s):  
Optimal Strategies ◽  
The Other ◽  
Game Show ◽  
Monty Hall ◽  
Monty Hall Problem

In the classical Monty Hall problem you are a contestant on a game show confronted with three identical doors. One of them conceals a car while the other two conceal goats. You choose a door, but do not open it. The host, Monty Hall, now opens one of the other two doors, careful always to choose one he knows to conceal a goat. You are then given the options either of sticking with your original door, or switching to the other unopened door. What should you do to maximise your chances of winning the car?

scholarly journals One game show, two boys, two aces, three prisoners - what’s an AI to do?

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Eric Neufeld ◽  
Sonje Finnestad
Keyword(s):  
Probability Theory ◽  
Word Problem ◽  
Random Variables ◽  
Game Show ◽  
Causal Diagram ◽  
Monty Hall ◽  
Monty Hall Problem ◽  
Technical Terms

We review a quartet of widely discussed probability puzzles – Monty Hall, the three prisoners, the two boys, and the two aces. Pearl explains why the Monty Hall problem is counterintuitive using a causal diagram. Glenn Shafer uses the puzzle of the two aces to justify reintroducing to probability theory protocols that specify how the information we condition on is obtained. Pearl, in one treatment of the three prisoners, adds to his representation random variables that distinguish actual events and observations. The puzzle of the two boys took a perplexing twist in 2010. We show the puzzles have similar features, and each can be made to give different answers to simple queries corresponding to different presentations of the word problem. We offer a unified treatment that explains this phenomenon in strictly technical terms, as opposed to cognitive or epistemic.  

scholarly journals Sleeping Beauty on Monty Hall

Philosophies ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 15
Author(s):  
Michel Janssen ◽  
Sergio Pernice
Keyword(s):  
Simple Model ◽  
Philosophy Of Science ◽  
Sleeping Beauty ◽  
Simple Solution ◽  
Degrees Of Belief ◽  
Sleeping Beauty Problem ◽  
Game Show ◽  
Bayes’S Theorem ◽  
Monty Hall ◽  
Monty Hall Problem

Inspired by the Monty Hall Problem and a popular simple solution to it, we present a number of game-show puzzles that are analogous to the notorious Sleeping Beauty Problem (and variations on it), but much easier to solve. We replace the awakenings of Sleeping Beauty by contestants on a game show, like Monty Hall’s, and increase the number of awakenings/contestants in the same way that the number of doors in the Monty Hall Problem is increased to make it easier to see what the solution to the problem is. We show that these game-show proxies for the Sleeping Beauty Problem and variations on it can be solved through simple applications of Bayes’s theorem. This means that we will phrase our analysis in terms of credences or degrees of belief. We will also rephrase our analysis, however, in terms of relative frequencies. Overall, our paper is intended to showcase, in a simple yet non-trivial example, the efficacy of a tried-and-true strategy for addressing problems in philosophy of science, i.e., develop a simple model for the problem and vary its parameters. Given that the Sleeping Beauty Problem, much more so than the Monty Hall Problem, challenges the intuitions about probabilities of many when they first encounter it, the application of this strategy to this conundrum, we believe, is pedagogically useful.

The Monty Hall problem

2019 ◽  
pp. 121-124
Author(s):  
Steve Selvin
Keyword(s):  
Game Show ◽  
Monty Hall ◽  
Monty Hall Problem

A famous problem that arose from a television game show that produced issues that are widely debated, called the “Monty Hall problem.”

Connecting Research to Teaching: Good Things Always Come in Threes: Three Cards, Three Prisoners, Three Doors

2006 ◽  
Vol 99 (6) ◽  
pp. 401-405
Author(s):  
Laurie H. Rubel
Keyword(s):  
Game Show ◽  
Monty Hall ◽  
Monty Hall Problem

Readers are likely to be familiar with the infamous Monty Hall problem, played on the Let's Make a Deal game show and later addressed in the “Ask Marilyn” column in a 1990 issue of Parade.

scholarly journals The game of Double-Silver on intervals

2001 ◽  
Vol 26 (8) ◽  
pp. 485-496 ◽  
Author(s):  
Gerald A. Heuer
Keyword(s):  
Measure Zero ◽  
Optimal Strategies ◽  
The Other ◽  
Parameter Plane ◽  
Left Endpoint ◽  
Relative Placement ◽  
Special Case

Silverman's game on intervals was analyzed in a special case by Evans, and later more extensively by Heuer and Leopold-Wildburger, who found that optimal strategies exist (and gave them) quite generally when the intervals have no endpoints in common. They exist in about half the parameter plane when the intervals have a left endpoint or a right endpoint, but not both, in common, and (as Evans had earlier found) exist only on a set of measure zero in this plane if the intervals are identical. The game of Double-Silver, where each player has its own threshold and penalty, is examined. There are several combinations of conditions on relative placement of the intervals, the thresholds and penalties under which optimal strategies exist and are found. The indications are that in the other cases no optimal strategies exist.

scholarly journals Responses to Modified Monty Hall Dilemmas in Capuchin Monkeys, Rhesus Macaques, and Humans

2018 ◽  
Vol 31 ◽  
Author(s):  
Julia Watzek ◽  
Will Whitham ◽  
David A. Washburn ◽  
Sarah F. Brosnan
Keyword(s):  
Decision Making ◽  
Rhesus Macaques ◽  
The Other ◽  
Comparative Approach ◽  
Capuchin Monkeys ◽  
Monty Hall Dilemma ◽  
Decision Making Processes ◽  
Monty Hall ◽  
Text Version ◽  
Switch Strategy

The Monty Hall Dilemma (MHD) is a simple probability puzzle famous for its counterintuitive solution. Participants initially choose among three doors, one of which conceals a prize. A different door is opened and shown not to contain the prize. Participants are then asked whether they would like to stay with their original choice or switch to the other remaining door. Although switching doubles the chances of winning, people overwhelmingly choose to stay with their original choice. To assess how experience and the chance of winning affect decisions in the MHD, we used a comparative approach to test 264 college students, 24 capuchin monkeys, and 7 rhesus macaques on a nonverbal, computerized version of the game. Participants repeatedly experienced the outcome of their choices and we varied the chance of winning by changing the number of doors (three or eight). All species quickly and consistently switched doors, especially in the eight-door condition. After the computer task, we presented humans with the classic text version of the MHD to test whether they would generalize the successful switch strategy from the computer task. Instead, participants showed their characteristic tendency to stick with their pick, regardless of the number of doors. This disconnect between strategies in the classic version and a repeated nonverbal task with the same underlying probabilities may arise because they evoke different decision-making processes, such as explicit reasoning versus implicit learning.

Sign in / Sign up

Export Citation Format

Share Document