An Analysis of Jenga Using Complex Systems Theory

By John Bartholomew, Wonmin Song, Michael Stefszky and Sean Hodgman

Wooden Blocks

Avalanches

Spherical Cows

Complex Systems Assignment 1:

• Developed in 1970’s by Leslie Scott

• Name from kujenga, Swahilli verb “to build”

• Israel name Mapolet meaning “collapse”

Jenga – A Brief History

Complex Systems Assignment 1:

• Game involves stacking wooden blocks

• Tower collapse game over

Jenga - The Game

Complex Systems Assignment 1:

• Why would Jenga be Complex?

• Displays properties of Complex Systems

• Tower collapse similar to previous work on Avalanche Theory

Jenga - A Complex System?

Complex Systems Assignment 1:

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

Jenga - A Complex System?

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Motivation?

Ultimate Jenga

Strategy

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http://www.ffme.fr/ski-alpinisme/nivologie/photaval/aval10.jpg

http://landslides.usgs.gov/images/home/LaConchia05.jpgMotivation

Frette et al. (1996)Turcotte (1999)

Power Law

Self Organizing CriticalityFrette et al. (1996)

• 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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Complex Systems Assignment 1:

Sandpile modelMinor perturbation can lead to local instability or global collapse – ‘avalanche’

Avalanche size:

2

Complex Systems Assignment 1:

Sandpile modelJenga 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

Complex Systems Assignment 1:

Cautious way forward…

• Large: periodic

Small: power lawBretz et al 1992

• Small: periodic

Large: power lawHeld et al 1990

“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

• 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

From This To This

Complex Systems Assignment 1:

What We Did

Strategies

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Middles OutMiddles Out

Middle Then SidesMiddle Then Sides

ZigZagZigZag

Side 1 Side 2

JENGA JENG

A

JENGA

Side 1

JENGAJENGA

Side 1 Side 2

JENGA JENG

A

JENGA

All Outside BricksAll Outside Bricks

Side 1 Side 2

JENGA JENG

A

JENGA

AND FINALLY…AND FINALLY…

An optimal game strategy An optimal game strategy where we would start from where we would start from

the bottom and work our way the bottom and work our way up, pulling out any bricks up, pulling out any bricks

which were loose enough to which were loose enough to pull out easilypull out easily

Complex Systems Assignment 1:

• 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!!!!!!!!

Many Strategies So We Could …

Complex Systems Assignment 1:

• 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

What We Expected

• 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 – Stability Regions

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Results – Different Strategies

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Maximum Distance of falling Block

Not Enough Data to definitively rule out one distribution, Gaussian and Not Enough Data to definitively rule out one distribution, Gaussian and Cauchy-Lorentz look to fit data quite wellCauchy-Lorentz look to fit data quite well

Results

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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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Universality of network theory:Topology of networks explains various kinds of

networks. • Social networks, biological networks, WWW Why not Jenga?

Modeling – Another Spherical Cow?

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!!

Complex Systems Assignment 1:

- 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 leastfit cell attackedsubsequently

Layers above theattacked layer are attacked subsequently

Modified sandpile model

Complex Systems Assignment 1:

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

• AlgorithmWe 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!!!!

Fitness & The Magic Number

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Complex Systems Assignment 1:

Complex Systems Assignment 1:

Playing Jenga is a random walk process!!!!Real data analysis shows the random walk process by

exhibiting Gaussian features in fluctuation plots.

Complex Systems Assignment 1:

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

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.

Complex Systems Assignment 1:

And the magic number emerged…..

• 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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Conclusions

• 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.com/jenga/• Turcotte, Self-organized Criticality,

Rep. Prog. Phys. 62 (1999)

Complex Systems Assignment 1:

Bibliography