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 Developed in 1970’s by Leslie Scott Name from kujenga, Swahilli verb “to build” Israel name Mapolet meaning “collapse” Complex Systems Assignment 1:

Jenga - The Game Game involves stacking wooden blocks Tower collapse game over Complex Systems Assignment 1:

Jenga - A Complex System? Why would Jenga be Complex? Displays properties of Complex Systems Tower collapse similar to previous work on Avalanche Theory Complex Systems Assignment 1:

Jenga - A Complex System? Emergence History Self-Adaptation Not completely predictable Multi-Scale Metastable States Heterogeneity Complex Systems Assignment 1:

Ultimate Jenga Strategy Motivation? Ultimate Jenga Strategy Complex Systems Assignment 1:

Motivation Power Law 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) Complex Systems Assignment 1:

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 Frette et al. (1996) Periods of quietness broken by bursts of activity Complex Systems Assignment 1:

Sandpile model Avalanche size: 2 Minor perturbation can lead to local instability or global collapse – ‘avalanche’ Avalanche size: 2 Complex Systems Assignment 1:

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 Complex Systems Assignment 1:

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 Complex Systems Assignment 1:

What We Did Played a LOT of games of Jenga ~400 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 Complex Systems Assignment 1:

Strategies JENGA JENGA JENGA JENGA JENGA JENGA JENGA JENGA JENGA JENGA Middles Out ZigZag JENGA JENGA JENGA JENGA Side 1 Side 2 JENGA Middle Then Sides JENGA Side 1 JENGA JENGA 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 JENGA JENGA JENGA Side 1 Side 2 Complex Systems Assignment 1:

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!!!!!!!! Complex Systems Assignment 1:

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 Complex Systems Assignment 1:

Results – Stability Regions 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:

Complex Systems Assignment 1:

Results – Different Strategies Complex Systems Assignment 1:

Maximum Distance of falling Block Results Maximum Distance of falling Block Not Enough Data to definitively rule out one distribution, Gaussian and Cauchy-Lorentz look to fit data quite well Complex Systems Assignment 1:

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? 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:

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 Complex Systems Assignment 1:

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!!!! Complex Systems Assignment 1:

Complex Systems Assignment 1:

Complex Systems Assignment 1:

Complex Systems Assignment 1:

Playing Jenga is a random walk process!!!! 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. Complex Systems Assignment 1:

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. Complex Systems Assignment 1:

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 Complex Systems Assignment 1:

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.com/jenga/ Turcotte, Self-organized Criticality, Rep. Prog. Phys. 62 (1999) Complex Systems Assignment 1: