Understanding the 80-20 Rule

1. Simple properties of systems that lead to 80-20 type evolutions.
   Rules

   r1) Actors have normally distributed abilities I call this 'advantage'. r2) Actors are chosen randomly based on current winnings, the more you have won, the more you compete. So Pr( actor_i chosen ) = totalWorth_i / totalWorthOfAllActors I call this 'the network effect'. r3) Winner of competition wins one point from the loser
2. Simulating the above system leads to distributions seen in wealth, GDP, income statistics. The simulated distribution is similar across multiple quintiles.
   Example

   Real Data ... Shares of household income of quintiles in the United States from 1970 to 2016 1970 2016 0 4.1% 3.1% 1 10.8% 8.3% 2 17.4% 14.2% 3 24.5% 22.9% 4 43.3% 51.5% vs extracts from simulation Simulate 1000 competitors competing using rules r1,r2 and r3 for 1300000 competitions leads to a wealth distribution similar to that of 1970 US above... competitors: 1000 competitions: 1300000 0 has 6.6849% 1 has 11.4564% 2 has 16.0846% 3 has 23.2039% 4 has 42.5702% simulate a bit longer and we essentially get the 2016 wealth distribution... competitors: 1000 competitions: 1700000 0 has 4.4998% 1 has 8.8603% 2 has 13.8255% 3 has 22.2612% 4 has 50.5532% competitors: 1000 competitions: 1800000 0 has 4.0532% 1 has 8.2682% 2 has 13.215% 3 has 21.8921% 4 has 52.5715% Example: 1999 2012 Bottom quintile 5.0 3.9 Second quintile 10.9 9.7 Middle quintile 16.4 16.3 Fourth quintile 22.6 23.1 Top quintile 45.1 46.9 vs extracts from simulation (not runs in src folder)... competitors: 1000 competitions: 1400000 0 has 6.243% 1 has 10.1721% 2 has 14.8497% 3 has 22.8296% 4 has 45.9056% competitors: 1000 competitions: 1500000 0 has 5.6742% 1 has 9.5138% 2 has 14.2514% 3 has 22.5647% 4 has 47.9959% Example: Distribution of world GDP in 1989 Quintile of population Income Poorest 20% 1.40% Fourth 20% 1.85% Third 20% 2.30% Second 20% 11.75% Richest 20% 82.70% vs extracts from simulation (not runs in src folder) ... competitors: 1000 competitions: 3400000 0 has 0.6114% 1 has 1.6852% 2 has 3.9405% 3 has 10.928% 4 has 82.8349%
3. An alternate simulation, where r2 is replaced by r2' or r2'' leads to less similar distributions (see for example a run using rules r1, r2' and r3).
   r2') Actors are chosen randomly (equally likely) or r2'') One actor chosen randomly, the other is chosen based on the network effect
   Example

   Distribution of world GDP in 1989 Quintile of population Income Poorest 20% 1.40% Fourth 20% 1.85% Third 20% 2.30% Second 20% 11.75% Richest 20% 82.70% vs extracts from simulation with rules r1,r2' and r3... competitors: 1000 competitions: 9900000 0 has 0.0% 1 has 0.0% 2 has 0.0278% 3 has 22.1514% 4 has 77.8208% vs extracts from simulation with rules r1,r2 and r3 competitors: 1000 competitions: 3500000 0 has 0.5075% 1 has 1.7191% 2 has 4.088% 3 has 10.6756% 4 has 83.0098%
4. Simulate more and more of the 'wealth' is accumulated by higher quintiles.
5. If you observe the members at the top quintile, they are not necessarily those with significant advantage. Some advantage as well as 'luck' in having won early seems sufficient. As long as there is sufficient network effect, an actor will be carried.
6. It makes sense: Coke wins now because it won earlier, so now Coke is in vending machines, Coke is served in many restaurants. They initially had some advantage, then the network effect carried them.
7. The network effect is more in play now than ever before: the internet, global distribution channels, media
8. Can understand, not just how something happens, but can identify simple properties such that systems contain those properties will evolve as in the target system. In this example it seems to be 1) Advantage 2) Network effect
9. The way to randomly choose from and continually update the distribution is interesting. Possibly should be part of a first year CS assignment.

I was walking down Deerfield Hall at the University of Toronto, Mississauga, thinking about why the 80-20 rule happens. I actually asked Statistics Faculty about this. Their explanation involved reference to the Pareto distribution etc. which did not really answer my question. So I thought, what do I actually mean by this question. Newton gave up on describing why things happen and instead satisfied himself with how they happen. Now I thought what could I mean by why. Well, my answer is the following:

Can I come up with some simple properties of systems, so that, if a system had those properties, then it would display the behavior I was interested in.

So for the 80-20 rule, I set out to come up with a simple system that gives rise to 80-20 like distributions. In fact I was interested in something more 'real'. Could I come up with a simple system which would evolve distributions of wealth like

Distribution of world GDP in 1989 Quintile of population Income Poorest 20% 1.40% Fourth 20% 1.85% Third 20% 2.30% Second 20% 11.75% Richest 20% 82.70% see source) Share of net worth (wealth) held by each income quintile, 1999 and 2012 1999 2012 Bottom quintile 5.0 3.9 Second quintile 10.9 9.7 Middle quintile 16.4 16.3 Fourth quintile 22.6 23.1 Top quintile 45.1 46.9 see source Shares of household income of quintiles in the United States from 1970 to 2016 1970 2016 4.1 3.1 10.8 8.3 17.4 14.2 24.5 22.9 43.3 51.5 see source Income distribution by income quintiles, King County 1979 2010 4 3 11 9 17 15 24 23 44 49 see source Have and Have Nots Share of total U.S. net worth owned by percentage of the population 0 0 3 9 89 see source

Other similar statistics can be found at :

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To create the system, I considered a few different versions of the 80-20 rule.

Version 1: 20% of the population holds 80% of the wealth. Version 2: 20% of athletes hold 80% of the medals. For the simulation, I considered a collection of agents competing against each other for 'money'. The agents have abilities, chosen from a normal distribution. That is, most have abilities around the mean, with a few at the tails of the distribution. Each player starts out with 1 win.

Initially I randomly chose competitors c1 and c2 and had them compete. The competitor with the most ability (with a bit of randomness thrown in for good measure) won $1. I then compared the quintiles of this to things like the wealth distribution above. Funny thing, the wealth distribution did not match.

Next I tried winner takes all of losers money. Again, the distribution of wealth did not match. Finally, I thought, why do companies like Coke, McDonalds, Apple, etc. win. Well Coke actually 'competes' more than less successful companies, partially because they already have accumulated some winnings. That is, they are present in vending machines, available in many restaurants, at most grocery stores. Similarly for athletes, those that have won are able to enter more contests.

Lets call this a 'network effect'. The more you win, the more you interact. The final simulation at does this. That is instead of choosing competitors c1 and c2 uniformly, they are chosen randomly based on their current winnings. Those that have won more are chosen to compete more often. c1 is chosen with probability c1.wins/total_wins. Similarly for participant c2. This adds a network effect, that is, if we are dealing with income, the wealthier an individual is, the more they are able to extend their wealth through more interactions with others. Similar for competitions, the more an individual has won, the more they compete (ie local, area, regional, provincial, national, international). Resutls:

1) If you let the simulation evolve until the top quintile matches publicly available wealth distribution data, then the other quintiles seem to essentially match. 2) Necessarily, wealth moves inexorably to the upper quintiles. 3) If you examine the upper quintile, it will have agents with better than average ability, but not necessarily extreme ability. That is, it takes some ability (enough so that there is a base to support them), and some initial luck (early winnings) and then the network effect takes over.

May 9, 2017: Was so excited to see the results I sent an email to the department about it!