Monty Halls Three Doors for Dummies

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Monty Halls Three Doors for Dummies

• Andrea Morone
• Annamaria Fiore
• (presenting author)

University of Bari
FUR XII Rome, LUISS
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Overview

• Presentation of the Game
• Previous Literature
• Our experimental design
• Results
• Discussion
• Conclusions

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? In an American once-popular TV show, Lets
Make a Deal, in the final game the host Monty
Hall asked his contestant to choose one of three
doors. Only behind one of them there was a big
prize, whereas behind the remaining two doors
there were valueless prizes (a goat, for
example). ? After the contestant chose a door
(e.g., door A), Monty Hall opened one of the
remaining two that hid a valueless prize (e.g.,
door B) and then offered the contestant the
chance to choose the other door (e.g., door C).
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1. The Game
The contestant is asked to choose one of the
three doors
B
C
A
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1. The Game
Contestant chooses door A
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1. The Game
Monty Hall opens the door B with a valueless
prize behind
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1. The Game
The contestant is asked to stick with the door A
or to switch to the door C
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WHAT IS THE BEST STRATEGY?

• Applying the Bayes rule

• if you stick with your first chosen door, you
  have only one third of chance of winning

• if you switch, you double your chance of
  winning!

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WHAT IS THE BEST STRATEGY?
A graphical explanation
John de Pillis, 2002
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WHAT IS THE BEST STRATEGY?
? The best strategy is switching ? Nevertheless,
the stylised fact is that very few contestants
took advantage of the opportunity to switch ?
The fact that many people systematically failed
to see that the switching strategy doubles the
chance of winning encouraged plentiful of
controlled experiments
 
 
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2. Experimental Evidence

• Friedman (1998)
• ?The overall switch rate over ten periods was
  only 28.7 percent (Treatment 1)
• ?At this point, three-door task appeared to be a
  robust and persistent anomaly, even if
  potentially mitigated by individual learning
  (Treatment 2 switch rate 46)

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2. Experimental Evidence

• Other experiments
• ? Page (1998) tests if "exaggerating a
  bias enables people to learn to overcome it " .
  Choice among 3 doors (11.6), 10 doors (47.1),
  100 doors (87.5)
• ? Palacios-Huerta (2003) size of monetary
  incentives, individuals initial skills, and
  communication are important determinants of
  initial choices and subsequent learning
• ? Slembeck and Tyran (2004) effect of
  competition and of group decision making is
  immediate and strong, when separately considered,
  but especially if combined together (100 of
  rational choices in 4 very late periods they had
  40 periods)

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2. Experimental Evidence

• ? Previous experiments on three-door task
  focused almost exclusively on some particular
  aspects that could be able to mitigate the
  anomaly and help people to behave rationally.
  These are treatments designed to endorse learning
  and to test other institutions recognized
  sensitive to anomalous choice behaviour

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Motivation

• If a thing is true there should be an easy way
  to explain it

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A Debiasing Test

• ? Our approach is different from all previous
  attempts to explain and correct the Monty Hall
  anomaly, inasmuch we developed a more radical
  debiasing test. Even if the game remains
  virtually the same, we completely ruled out any
  Bayesian updating, as the new framework do not
  require subjects to make any calculations.
  Consequently, having eliminated the cause, also
  the effect could disappear.

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3. Experimental Design

• 3 treatments
• CONTROL participants were proposed the same task
  as in Friedman
• INTERMEDIATE
• FOR DUMMIES
• Between design
• Task repeated for 12 periods whose only 10 paid
  for real
• 20 subjects for each treatment
• Computerized experiment (Ztree, Fischbacher,
  1999)
• The subjects could not see each other or
  communicate
• Undergraduate students in Economics not familiar
  with previous similar experiments.

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3. Experimental Design

• Learning devices
• trial periods
• subjects allowed to ask questions after trial
  periods
• stationary replication
• immediate feedback after each period
• We paid particular attention in writing
  instructions, in avoiding any possible
  misunderstanding and/or deception, and in keeping
  them as simple as possible

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3. Experimental Design

• Each treatment consisted of 2 stages
• 1st stage in common among the three
• they differentiated in the 2nd stage
• 1st stage
• to pick a card among three
• Rewards
• subjects gained 0.5 when they chose the lucky
  card and zero otherwise

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3. Experimental Design

• CONTROL Treatment 2nd stage

The computer revealed the content of an empty
card among the two unchosen cards
Subject asked if wanted to keep his/her first
choice or if he/she preferred to go for the
remaining card
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3. Experimental Design

• INTERMEDIATE Treatment 2nd stage

The computer revealed the content of an empty
card among the two unchosen cards
Subject asked if wanted to keep his/her first
choice or if he/she preferred to go for the two
unchosen cards
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3. Experimental Design

• FOR DUMMIES Treatment 2nd stage

The computer revealed no content of any unchosen
cards
Subject asked if wanted to keep his/her first
choice or if he/she preferred to go for the two
unchosen cards
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4. Results
Control
Intermediate
For Dummies
(Wilcoxon rank test CONTROL/INTERMEDIATE, p
0.4203 CONTROL/FORDUMMIES, p 0.0010
INTERMEDIATE/FORDUMMIES, p 0.0125)
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4. Results

• Switch rate under FOR DUMMIES treatment higher in
  each single period
• Monotonic increase in the average switch rate
  across the three treatments, as expected (from
  41.5 in CONTROL, to 45.5 for the INTERMEDIATE,
  up to 58 in the FOR DUMMIES treatment, to a max
  of 75 in period 10), but this percentage remains
  still too low

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4. Results

• Hypothesis do subjects behave randomly?
• One-sided binomial test whether switch rate
  significantly differs from 50
• H0 switch rate 50
• H1 switch rate gt 50
• Interestingly, we reject the null hypothesis only
  for the FOR DUMMIES treatment (p .01406
  CONTROL, p .99344 INTERMEDIATE, p .91052)

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4. Results
N 20
Control
Intermediate
For Dummies
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4. Results
Dep. Var. 1 in periods in which the subjects
chose to switch and 0 otherwise Switchbonus
earnings from always switching minus earnings
from always remaining Switchwon equal to 1 if
the decision maker switched and won in the most
recent period Switchlost equal to 1 iff the
decision maker switched in the preceding period
but did not win the prize
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4. Results

• Considering the significant variables
• Switchwon behaves quite exactly as in the
  previous analysis
• Switchlost has the same magnitude, but it goes
  in the opposite direction
• We cannot confirm precisely directional learning
  theory (Selten and Buchta, 1998), because our
  results are not unambiguous
• Switchbonus shows the expected direction and is
  more effective than in the previous analysis
• Our data seem to support the reinforcement
  learning theory (Erev and Roth, 1998)
• ForDummies is significantly effective in
  increasing the probability of switching (by
  12.11)
• The negative effect for Time would suggest a
  downward trend to switch, the positive effect for
  Time2 would show that we have a non linear and
  convex trend over time, but both of them are not
  significant

 

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5. Discussion

• In the FOR DUMMIES treatment, we completely ruled
  out the Bayesian updating, but the anomaly still
  survived (switch rate 75 even in the final
  period)
• gt Bayesian updating misapplication as not a
  leading explanation for this anomaly
• It should be noted that with the new framework
  the gamblers fallacy as well is ruled out as a
  possible explanation
• What more could explain this pattern?

 

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5. Discussion

• Do people follow probability matching as decision
  rule?
• (The p-values are based on two-sided binomial
  test.
• H0 Pr(Switching) 0.33 and H0 Pr(Switching)
  0.66, respectively)

 

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5. Discussion

• Were the instructions misleading?
• Our instructions were clear, simple and neutral!
• Instructions. - This experiment is designed to
  study how people make decisions.
• The experiment is very simple, and you will have
  the possibility of earning money, which will be
  paid to you in cash at the end of the experiment.
  
• This amount will depend, on the one hand, on your
  decisions and, on the other hand, on luck.
• The game is as simple as possible. 3 cards will
  appear on your screen you will be asked to
  choose one of them simply pressing a button
  placed in the bottom right angle. During our
  experiment, each time the programme will
  establish in a complete random way which card is
  the winner one, i.e. the card behind which a
  prize is hidden.
• Whenever you choose the right card, you will win
  0.5, nothing otherwise.
• Still, in each period a second chance will be
  given to you after you have chosen your first
  card, you will have two opportunities or
  sticking with your already-chosen card or
  choosing both two remaining cards.
• At the end of each period, winning card and your
  correspondent earning will be shown to you with a
  message on your screen.
• This game will be repeated for 10 times, in
  addition to the 2 trial periods at the beginning.
  At the end, you will be paid your total payoff
  (trial periods excepted) and free to leave the
  laboratory.
• Rules are very simple. Communicating with other
  participants is forbidden (you can ask some
  questions to the experimenters only during trial
  periods), otherwise you will leave out from the
  laboratory and another player will be given your
  place in the experiment.
• Good luck!
•  

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5. Discussion

•  

• Were experimental subjects suspicious?
• We make any possible effort to keep our
  reputation as experimenters high during
  experiments
• We never deceive our experimental subjects
• We try to provide our experimental subjects with
  the wider information possible
• We always try to clarify any possible doubt
• We always give experimental subjects the
  opportunity to become friendly with the
  experimental set up

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5. Discussion

•  

• Are people temporally inconsistent?
• It could provide a good explanation
• i.e., subjects decisions at the different
  decision nodes as if related to different
  selves
• Anyway, it is quite unlikely that people do the
  same mistake repeatedly over the 10 periods at
  the end, they should realise that they are given
  the double chance of winning

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5. Discussion

•  

• Finally, are people affected by the status quo
  bias?
• People seem to attach a higher value to their
  previous chosen door
• They seem to consider their already chosen door
  as their endowment
• In order to not experience loss aversion, they
  should be reluctant to switch
• Interestingly, even in the FOR DUMMIES treatment,
  at least 15 of subjects never decided to switch

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5. Conclusion

•  

• Monty Halls three doors anomaly stronger than
  peviously thought
• Having ruled out the Bayesian updating failure
  as explanation, a way to understand this choice
  anomaly could rely on some psychological
  underpinnings

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5. Conclusion

• A leading explanation seems to be the status quo
  bias
• Our analysis also have shown some support to the
  reinforcement learning (switch chosen only when
  sufficient favourable evidence has been
  accumulated)
• These results could contribute to dismiss tha
  idea that people actually use probabilities at
  all in making some kinds of decisions
• Further development SUPER FOR DUMMIES treatment
  only 2nd stage retained

•