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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

Jenga – A Brief History

Complex Systems Assignment 1:

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

Jenga - The Game

Complex Systems Assignment 1:

- Game involves stacking wooden blocks
- Tower collapse game over

Jenga - A Complex System?

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

Motivation?

Complex Systems Assignment 1:

Ultimate Jenga Strategy

Motivation

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)

Self Organizing Criticality

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

Sandpile model

Complex Systems Assignment 1:

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

Avalanche size:

2

Sandpile model

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

Cautious way forward…

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

What We Did

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

Strategies

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

Many Strategies So We Could …

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!!!!!!!!

What We Expected

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

Results – Stability Regions

Complex Systems Assignment 1:

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

Complex Systems Assignment 1:

Complex Systems Assignment 1:

Results – Different Strategies

Complex Systems Assignment 1:

Results

Complex Systems Assignment 1:

Not 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:

Results – Memory effects?

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!!

Modified sandpile model

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.

Fitness & The Magic Number

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:

Complex Systems Assignment 1:

Complex Systems Assignment 1:

Accordance with the data

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.

Conclusions

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

Bibliography

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)