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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

Complex Systems Assignment 1:

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

Complex Systems Assignment 1:

- Game involves stacking wooden blocks
- Tower collapse game over

Complex Systems Assignment 1:

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

Complex Systems Assignment 1:

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

Complex Systems Assignment 1:

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

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

Power Law

Frette et al. (1996)

Turcotte (1999)

Frette et al. (1996)

Complex Systems Assignment 1:

- 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

Complex Systems Assignment 1:

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

Avalanche size:

2

Complex Systems Assignment 1:

- 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

Complex Systems Assignment 1:

- “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

Complex Systems Assignment 1:

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

JENGA

JENGA

JENGA

JENGA

JENGA

JENGA

JENGA

JENGA

JENGA

JENGA

JENGA

Complex Systems Assignment 1:

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

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!!!!!!!!

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

Complex Systems Assignment 1:

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

Results – Different Strategies

Complex Systems Assignment 1:

Complex Systems Assignment 1:

Maximum Distance of falling BlockNot Enough Data to definitively rule out one distribution, Gaussian and Cauchy-Lorentz look to fit data quite well

Results – Step Size Blocks Removed

Complex Systems Assignment 1:

Results – Step Size Blocks Remaining

Complex Systems Assignment 1:

Results – Step Size Maximum Distance

Complex Systems Assignment 1:

Complex Systems Assignment 1:

Modeling – Another Spherical Cow?

Complex Systems Assignment 1:

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!!

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.

Complex Systems Assignment 1:

- 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.

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

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!!!!

Complex Systems Assignment 1:

- 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.

And the magic number emerged…..

Complex Systems Assignment 1:

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:

- 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

Complex Systems Assignment 1:

- 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)

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