An Analysis of Jenga Using Complex Systems Theory

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An Analysis of Jenga Using Complex Systems Theory
Avalanches
Wooden Blocks
Spherical Cows
By John Bartholomew, Wonmin Song, Michael
Stefszky and Sean Hodgman
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Jenga A Brief History

• Developed in 1970s by Leslie Scott
• Name from kujenga, Swahilli verb to build
• Israel name Mapolet meaning collapse

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Jenga - The Game

• Game involves stacking wooden blocks
• Tower collapse game over

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Jenga - A Complex System?

• Why would Jenga be Complex?
• Displays properties of Complex Systems
• Tower collapse similar to previous work on
  Avalanche Theory

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Jenga - A Complex System?

• Emergence
• History
• Self-Adaptation
• Not completely predictable
• Multi-Scale
• Metastable States
• Heterogeneity

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Motivation?
Ultimate Jenga Strategy
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Motivation
http//landslides.usgs.gov/images/home/LaConchia05
.jpg
http//www.ffme.fr/ski-alpinisme/nivologie/photava
l/aval10.jpg
Power Law
Frette et al. (1996)
Turcotte (1999)
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Self Organizing Criticality

• Theory Proposed by Bak et al. (1987)
• Dynamical systems naturally evolve into self
  organized critical states
• Events which would otherwise be uncoupled
  become correlated

• Periods of quietness broken by
• bursts of activity

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Sandpile model
Minor perturbation can lead to local instability
or global collapse avalanche
Avalanche size 2
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Sandpile model

• Jenga cannot be modelled using the Sandpile Model
  because
• We have removed the memory affects
• A more suitable model involves assigning a
• fitness to each level which is altered
  dependant
• on the removal of a block

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Cautious way forward

• Experimental results have been quite ambiguous
• Turcotte 1999
• Quasi-periodic behaviour for large avalanches
  Evesque and Rajchenbach 1989, Jaeger et al 1989
• Power law behaviour
  Rosendahl et al 1993, 1994, Frette et
  al 1996

• Large periodic
• Small power law
• Bretz et al 1992

• Small periodic
• Large power law
• Held et al 1990

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What We Did
From This
To This

• Played a LOT of games of Jenga 400
• Chose 5 different strategies to play
• Recorded 3 observables
• Number of bricks that fell in avalanche
• Last brick touched before avalanche
• Distance from base of tower to furthest brick
  after the tower fell

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Strategies
Middles Out
ZigZag
Side 1
Side 2
Middle Then Sides
Side 1
AND FINALLY An optimal game strategy where we
would start from the bottom and work our way up,
pulling out any bricks which were loose enough to
pull out easily
Side 1
Side 2
All Outside Bricks
Side 1
Side 2
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Many Strategies So We Could

• Compare strategies to see if any patterns were
  emerging
• Compare more ordered methods of pulling bricks
  out to the random optimal strategy
• See if strategies used had a large impact on the
  data obtained.

Whoooooaaaaaaa!!!!!!!!
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What We Expected

• We hoped to see at least some emerging signs of a
  complex system as more data was taken
• We assumed the distance of blocks from base would
  be Gaussian to begin with but maybe tend towards
  a power law
• Perhaps some patterns relating to strategies used
  and observables

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Results Stability Regions

• Analysed number of blocks before tower collapse
• Separately for each strategy and combined
• Results show stability regions for many strategies

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Results Different Strategies
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Maximum Distance of falling Block
Results
Not Enough Data to definitively rule out one
distribution, Gaussian and Cauchy-Lorentz look to
fit data quite well
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Results Step Size Blocks Removed
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Results Step Size Blocks Remaining
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Results Step Size Maximum Distance
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Results Memory effects?
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Modeling Another Spherical Cow?

• Universality of network theory
• Topology of networks explains various kinds of
  networks.
• Social networks, biological networks, WWW

Why not Jenga?
Look at Jenga layers as nodes of a network with
specified fitness values assigned to
each layer, and each layer is
connected to the layers above it. This
simplifies the picture for us to look at 18
layers, not at all 54 pieces!!
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Modified sandpile model

• - As mentioned before, the sandpile model
  eliminates least fit cells of sand Selection
  law life is tough for weak and poor!
• - The whole system self-organizes itself to
  punctuated equilibriums due to the memory effect.

- Our case is a bit different.
Sand-pile model Toy model
Attack the least fit cell Attack the fittest layer
Neighbors to the least fit cell attacked subsequently Layers above the attacked layer are attacked subsequently
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Fitness The Magic Number

• We describe stability of each layer by fitness
• Fitness 1 indicates stability, and fitness
  below a threshold value is unstable.

• Algorithm
• We tested values for - threshold
  fitness between 0.2-0.3 -
  strength of attack 0.3-0.5 with randomness added
  i.e. human hands apply attack
  with uncertainty in strength value (shaky hands).
  
• Each attack affects the layers above with
  decreasing attack power.
• Repeat the attack until a layer appears with
  fitness lower than the threshold.
• Stack a layer on the top for every 3 successions
  of attack.

Outcomes? Distributions for Maximum
height layer index number average fitness
Magic number!! - There is always some magic
number turn that you are almost guaranteed to
have a safe pass at the turn!!!!
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Accordance with the data

• No indication of power-law behavior because of
  the
• absence of memory
• Gaussian, and Poisson distributions emerge
  instead.

Playing Jenga is a random walk process!!!! Real
data analysis shows the random walk process by
exhibiting Gaussian features in fluctuation
plots.
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And the magic number emerged..
In the case of the model Whoever takes the 7th
turn is almost guaranteed a safe pass.
The Toy Model mimics the emergence of stability
regions and gives an indication about the gross
behavior of the Jenga network.

• Allows us to see the Jenga tower as a cascade
  network.

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Conclusions

• Randomness in all strategies
• Step size structure due to artificial memory
• Modified sandpile model directed network
• Model mimicking real situation Emergence of
  stability regions
• Complex structure identified but more data needed

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Bibliography

• Bak et al., Self-organized Criticality,
  Phys. Rev. A. 31, 1 (1988)
• Bak et al., Punctuated Equilibrium and
  Criticality in a simple model of evolution,
  Phys. Rev. Lett. 71, 24 (1993)
• Bak et al., Complexity, Contingency, and
  Criticality, PNAS. 92 (1995)
• Frette et al., Avalanche Dynamics in a pile of
  rice, Nature, 379 (1996)
• Jenga, Available online at http//www.hasbro.co
  m/jenga/
• Turcotte, Self-organized Criticality,
  Rep. Prog. Phys. 62 (1999)