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## WHAT IS IT?

Skiers and snowboarders are fond of saying “There are no friends on a powder day”. This model simulates the use of new snow on a ski slope, and the amount of snow skied by skiers of varying abilities. With this model we approximate the distribution of “wealth” (how much snow each skier can ride over) under varying conditions. The distribution of “poor” red skiers, “middle class” green skiers, and “rich” blue skiers will demonstrate whether the Pareto principle holds true on the slopes, and the breakdown of overall wealth by ability level will show whether advanced skiers truly have an advantage over beginners.

This model is adapted from Wilensky's Wealth Distribution model, which is in turn modeled on Epstein & Axtell's "Sugarscape" model. Key differences incorporated here include:

• Each patch’s snow is only reduced, not totally consumed, when visited by one or more agents.
• Agents do not have a ‘metabolism’, they simply collect wealth. Instead, they get sore incrementally over time.
• Agents are split randomly among 3 ability levels to represent beginner, intermediate, and advanced skiers. Their ability level is proportional to the amount of snow they destroy and wealth they gain at each tick.
• Agents only move across, diagonally downward, or straight down the hill.
• Initial wealth is set at 0.1 for all skiers, reflecting that no one has yet enjoyed the hill at the start of that day.

## HOW IT WORKS

We begin with an even breakdown of beginner, intermediate, and advanced skiers, who are spread evenly over the slope. They ski down, covering ground and pushing snow proportional to their respective ability level (labeled next to each skier). At each tick, every skier observes the patches ahead in five directions; right, down-and-right, straight down, down-and-left, and left. Skiers look out as far as their randomly assigned vision. They then move in the direction of the best snow they have seen, moving 3 patches if they are experts, 2 for intermediate skiers, and 1 for beginners.

Skiers also have a randomly assigned level of energy and soreness, since even the best skiers may have partied at the ski lodge the night before. As time goes on skiers gain soreness, which is added equally among the ability levels. When a skier can no longer find enough snow to justify their soreness, or they are sorer than their initial energy level could accommodate, they ‘go home’ and are replaced by a new skier with randomly assigned attributes.

To answer the question of relative wealth gained on a “pow day”, simply click setup and then go. You will see that NUM-SNOW-FALL is set to zero, meaning there is no new snow falling any more (as is the case when the fresh snow fell overnight and the sun comes out in the morning, commonly referred to as a “bluebird day”). This default setup also assumes that there is a 3:1 advantage in enjoyment for an expert skier over a beginner. With these settings, the model confirms that the Pareto principle holds. As with the Wealth model on which this is based, the Lorenz curve illustrates that after approximately 500 ticks all the wealth is isolated among just a couple of skiers, with the vast majority having no more snow to ski.

Another perspective added in this model is the percent of wealth held by each group of skier abilities. If it is true that you, an expert, should abandon your beginner or intermediate friends on a day with lots of snow, then wealth held by the expert ability group should be drastically higher than that of the beginner or intermediate groups. We also track the relative numbers of beginner, intermediate, and expert skiers to ensure that their relative populations (which are random and should even out over time) aren’t accounting for the differences in the relative wealth of each ability group.

## HOW TO USE IT

Use the NUM-SKIERS slider to adjust the number of skiers on the hill. The MAX-VISION slider adjusts the maximum amount of distance each skier can be assigned to see (the actual distance each can see is randomly assigned between 1 and MAX-VISION).
The ABILITY-ADVANTAGE slider adjusts how much more snow a skier with higher ability (ranging from 1 to 3) can add to their wealth and remove from the hill. The wealth added at each patch is equal to:

snow-here * ability ^ ability-advantage

Since the ability is in the exponent, a value of 1 gives us a ratio of 3:2:1 across ability levels. This makes sense since skiers also move at a speed ratio of 3:2:1 (from expert to beginner). However, you could argue that with increased ability comes increased snow removed at each patch, so values of 2 and 3 can be selected to illustrate a more dramatic difference.

ENERGY-MIN and ENERGY-MAX are the range of “endurances” for each skier, again randomly assigned. Taken together, the above values compose the attributes of our agents.

PERCENT-BEST-LAND determines what percent of patches receive the best snow. The default is ‘total coverage’ at 100%, as is often the case after heavy snow on an open ski run. A lower setting might simulate a run later in the day, or perhaps partial tree coverage.

NUM-SNOW-FALL is the amount of snow added per patch per SNOW-FALL-INTERVAL. Using these sliders together you can simulate heavier/lighter and faster/slower snowfalls. The default NUM-SNOW-FALL is zero, to simulate ‘bluebird’ conditions. With NUM-SNOW-FALL at zero, SNOW-FALL-INTERVAL has no effect.

SETUP is used to set the initial values selected and should be used before increasing NUM-SKIERS to avoid an error. GO ONCE moves one tick ahead in time and GO CONTINUOUSLY moves ahead at the speed selected with the slider at the top.

Use the CLASS PLOT to see the number of agents in the red (bottom third), green (middle third), and blue (upper third) of the wealth distribution. A CLASS HISTOGRAM gives a more easily interpreted relative class breakdown at each tick. The LORENZ CURVE, with the Wealth Distribution model on which this one is based, illustrates the Lorenz curve at each moment relative to the ‘ideal’ 45-degree line, and the GINI-INDEX V. TIME provides a longitudinal look at the same ratio, with 0 being ideal equality and 1 being total inequality. Additionally, the DISTRIBUTION OF ABILITIES provides an instantaneous look at the relative number of beginner, intermediate, and advanced skiers. This gives useful background for the WEALTH BY ABILITY HISTOGRAM which can help answer our question of whether advanced skiers really do get significantly more snow, and by how much under varying conditions.

## THINGS TO NOTICE

Can we answer our key question: do some skiers really get significantly more snow? If so, by how much, and under which conditions? Importantly, is this inequality in wealth isolated among the more advanced and faster skiers, or are the wealthy chosen by chance alone?

Under most conditions we will see that Pareto’s Law holds and, over time, a few skiers have all the wealth. The wealthy skiers are usually, but not always, advanced level, so there is some chance involved as well. Also important to our question is the ‘bluebird’ scenario; if no more snow is falling, the inequality is dramatic and is created very quickly. This means it’s important to get on that first chairlift!

There is also a scenario where Pareto’s Law does not apply. When enough snow is falling, the number of skiers is low enough, and the ABILITY-ADVANTAGE is set to 1, a relatively equal wealth distribution can be found. Does this mean, if the snow is good and still falling, you should not abandon even your beginner friends?!

## THINGS TO TRY

Experiment with different conditions, varying numbers of skiers, and adjust the ABILITY-ADVANTAGE to the level you think reflects the real world. How does this change the outcome?

Check the weather report at your local resort before you go and try to simulate the expected conditions here. Does this help inform your plans?

For a more in-depth analysis, take a look at the Code tab. Do you think the arbitrary wealth function in the ski-forward procedure accurately reflects the relative enjoyment of the various ability levels? You can also try setting energy and soreness levels by ability, since beginners often tire by lunch time while experts ski from the first to last chair.

## EXTENDING THE MODEL

Add a plot of the wealth of each ability level over time.

Add switches to tie energy, soreness, and vision to ability level.

Fix Y index on Gini-Index v. Time. Lower limit should be 0,0 not -0.1, 0. This is likely due to adjusting the initial wealth to 0.1.

Add Chooser for various wealth algorithms to simulate different ways of accounting for ability-advantage.

Incorporate Poulhès and Miral values and algorithms wherever possible.

## NETLOGO FEATURES

In addition to the scale-color reporter from Wilensky's Wealth Distribution model, this model includes the update-wealth-ability procedure to report the global values which break down accumulated wealth by ability.

## CREDITS AND REFERENCES

This model is based on Wilensky's Wealth Distribution NetLogo model, which in turn is based on Epstein, J. & Axtell R. (1996). Growing Artificial Societies: Social Science from the Bottom Up. Washington, DC: Brookings Institution Press.

The Pareto principal can be understood here: https://en.wikipedia.org/wiki/Pareto_principle

The phrase “No Friends on a Powder Day” is colloquial, and is examined subjectively here: https://www.backcountry.com/explore/no-friends-on-a-powder-day-strategies-for-when-its-deep .

Of particular interest is a rigorous Agent-based model of skiers in a ski area by Poulhès and Mirial (2017), which aims to “offer a decision-making tool for the operator and design engineering” of a ski resort:

Alexis Poulhès, Paul Mirial, Dynaski, an Agent-based Model to Simulate Skiers in a Ski area,
Procedia Computer Science, Volume 109, 2017, Pages 84-91, ISSN 1877-0509, https://doi.org/10.1016/j.procs.2017.05.298. (https://www.sciencedirect.com/science/article/pii/S1877050917309559)

This model was created using the NetLogo software:

• Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

## RELATED MODELS

• Wilensky, U. (1998). NetLogo Wealth Distribution model. http://ccl.northwestern.edu/netlogo/models/WealthDistribution. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

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