Token Exchange Games

What is an abstract ledger?

Last year I worked on a crypto-token project that is different to most token systems and currencies: The token was issued in rounds and every day user’s wallets would be updated according to a global rule that affected the entire ecosystem. Sounds like a UBI doesn’t it?

But it got me thinking, all crypto-currency and token systems have issue mechanisms, a few have mechanisms by which tokens dissipate or get inflated, and locally there are rules by which the movement of tokens incur transaction costs. In any case, I had a Harajuku moment in which I convinced myself I needed to write a research paper that answered the question: “What is a Ledger?” And so, I wrote a paper, on ledgers as an abstract mathematical structure.

Here is the (short) synopsis of the research

(1) We exchange tokens all the time, not just monetary tokens but tokens that get us on board buses and airplanes, tokens that prove our identities and tokens that give us access to our bank accounts. Each time an agent exchanges a token with another agent a network is formed. The agents are the nodes in the network while exchanges between agents are links in the network.

Here are some different examples: A non-exchangeable token (such as a tattoo) forms a special case of an empty network. In monetary and crypto-token systems the coins in your wallet are doing a random walk on the agent space, they hop from person to person like a web-surfer moving from site-to-site. I pass you an apple from my lunch box so you become the new owner of the apple, and it moves between us as a token in a token exchange game.

(2) Both the nodes (the agents) and the links (the exchanges) in the network have weights. In the “crypto economy” and the “real economy” agents have wallets and wallets have balances. Exchanges also have “value” in the sense that a fixed number of identical tokens are exchanged.

Monetary systems rely on tokens that are identical (bar the encoding on different media), transferrable, scarce and durable. This suggests a game in which a finite supply of tokens is distributed amongst agents for exchange, the tokens never expire, and the measurable object of the game is the token distribution. In these types of games it’s not the tokens in your wallet that matter but the proportion of tokens you own relative to the tokens that exist.

(3) We know that exchanges happen at different times and that balances get updated. A token exchange game has to proceed round by round, and to identify an particular exchange there are three primary points required: the start point, the end point and the time of the exchange. Edge weights and balances are data, but start points, end points and time of the exchange are meta-data.

This gives us a network that evolves round by round and has weightings on the nodes and the edges. It’s a very basic model of a ledger. It’s also a powerful way to describe single crypto-currency and crypto-token systems that evolve according to mathematical rules. But now what?

(4) Well, we don’t just use one network for exchanges, we use multiple networks. Playing monopoly involves two obvious ledgers: The “money” ledger and the “property” ledger. Each ledger has its own token set and the items within the token set are fungible with each other by “rules” that define the relative value of each token. (Our undsertanding of this is that five $1 bills is fungible with one $5 bill and so forth.) Then, depending on how the players define their agreements, Property A is worth Property B plus Property C. Further, property is fungible with money and visa versa: If the bank owns it you pay the full price to buy it, but if you need to mortage it you get the half price special.

This gives a four parameter model for a system involving multiple ledgers. In order to identify a transaction in this space the start point, end point, time of the exchange and the token type are required. The rules about how exchanges are correlated between distinct token types, and how tokens are issued and destroyed within token types depends on the players. But they can be modelled mathematically as a token exchange game.

Here is what I think the research implies

(1) Crypto-token and crypto-currency systems are examples of token exchange games and they can be modelled accordingly within this framework.

(2) Complex economies can be understood within the ‘price is information’ paradigm as mechanisms for allocating and distributing tokens. Economic rules that act iteratively such as tax and transaction costs can be incorportated directly. Inflation, savings rates and the velocity of money are more complex, but they can be constructed as metrics using the connection between linear algebra and networks.

(3) Monetary systems evolve as a direct consequence of no-arbitrage arguments.

(4) Metcalfe’s law has a very logical explanation within this framework.

Here is is a powerful argument that supports the research

Google’s search engine is based on the PageRank algorithm that models the web as a network. This network problem is equivalent to a problem in linear algebra which, in turn, produces as output the PageRank of a webpage that gives it a value relative to other websites. Each website owns its PageRank with respect to a particular query. In a way, Google is a ledger created from a network, so if we run this in reverse it makes perfect sense to go from a ledger to a network. Over and above this analogy, models of the Lightning Network, the Circles cryptotoken, and the iterative Prisoner’s dilemma are developed within the paper.

If you’ve got the background in Linear Algebra and Graph theory, have a look for yourself. Have a look even if you don’t have the background. The paper may be found here.