Basins of Attraction

In the field of dynamic systems, an attractor is a set of states toward which a system tends to evolve.

Here is a visual example of an attractor. The vertical axis measures the frequency with which points in the complex plane are visited. The whole system here is attracted to the point at which the vertical axis is highest.

A related idea is a basin of attraction - the region of the phase space in that region will be drawn into the attractor.

In the below example, each the pink, yellow, green, and blue areas are different basins of attraction.

As objects fall into the basin of attraction, then are naturally drawn towards the attractors.

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In the physical world, an example of this is mini golf.

A mini golf hole can be in a physical basin of attraction.
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or a physical basin of unattraction.

Attractors can be not only physical, but also psychological or crypto-economic.

An extremely Colorado-native example of a psychological attractor is a campfire at dusk. When you’re in the wilderness, a campfire’s warmth is an attractor to agents who are cold, and it’s light is a visual attractor in the dark.

Physical psychological attractors are one thing, but digital psychological attractors are a thing too.

As digital psychological or crypto economic decision spaces are not as easy to visualize as physical spaces, they are tougher to conceive of as an attractor. But IMO it is a worthwhile endeavor for the budding mechanism designer or protocol strategist working in web3 to think about attractors in visual terms. A picture is worth 1000 words after all.

An elegant example of crypto-economic attractor is the Proof of Stake Protocol in Ethereum. Proof of Stake is a stable attractor that is designed to attract honest validators for the network.

Heres how it works. Stakers are attracted by the economic incentive of a promise of an APY on their ETH if they stake 32 ETH + setup a node and secure the network with that node.

This attracts people to staking and securing the network.

POS was designed to make the network attractive to stake and unattractive to attack. This is what slashing risk is in POS. The protocol is designed to be much cheaper to defend than it is to attack. If one were to be discovered to be causing a fault in the network (cheating), then they would be slashed (lose 16 ETH).

In an ecosystem with many diverse stakers who are all in the basin of attraction of the POS economic incentives, the Ethereum network is secure.

Next post: How can we leverage this new knowledge of attractors, basins of attraction, and cryptoeconomic attractors to stimulate the regen x web3 movement?


Love the connections to different basins of attractions, and from topology to game theory. There used to be a lot more cross-pollination between these two fields of math back in the day.