# The Local Token - Utility and Monetary Policy

The role and economics of the native Local token

In a centralized marketplace, when fees are collected, these go to the owners of the platform. The main utility of the Local token is to decentralize the ownership of the platform, aid in the redistribution of rewards based on relative influence, and provide a business model for arbiters who can resolve disputes.

**Basic principles:**

- Transaction fees are collected for every transaction between providers and users as a sybil resistance mechanism
- Rewards are distributed as a function of the strength of the connected peers. This is represented by a graph and relative influence is measured by a modified eigenvector centrality ranking
- One can become a token holder either by being a service provider, or buy purchasing tokens.
- Being a service provider should be the preferable option.

## Self-Optimizing Rewards as a Dynamic Graph

The Dynamic Transaction Graph is a bipartite graph that captures the economic relationships between producers and buyers, with weighted edges representing the cumulative fees from their past transactions. This graph-based approach allows the network to dynamically adjust token incentives based on the actual economic activity and pricing power of participants, ensuring a fair and efficient allocation of rewards that adapts to the unique properties of each market.

**The Graph:** A bipartite graph $G = (U, V, E)$ representing producers $U$ and buyers $V$ as nodes, with weighted edges $E$ capturing transactions between them. Edge weights $w(u, v)$ track the cumulative fees contributed by producer $u$ from transactions with buyer $v$.

Buyer 1 (V1) | Buyer 2 (V2) | |
---|---|---|

Producer 1 (U1) | w(U1, V1) | w(U1, V2) |

Producer 2 (U2) | w(U2, V1) | w(U2, V2) |

**Token Incentives:** Producers earn tokens through a weighted revenue sharing mechanism that distributes block rewards proportional to their relative edge weights with each buyer. This incentivizes active participation and increased contributions.

**Augmentation Fees:** Buyers can earn augmentation fees $(w)$ to participate in transactions, creating a self-regulating market. A dynamic augmentation fee refines the optimal token allocation for each of the two participants based on the pricing power of the producer. In a market where producers have high pricing power, we imagine most of the reward for a given transaction to accrue to the producer, and the opposite to be the case for markets where producers sell goods with many substitutes.

**Implicit Staking:** Tokens are locked within the graph until participants liquidate. Tokens are minted by burning the edges. When a transaction $T = (u, v, p, f, w)$ occurs between producer $u$ and buyer $v$ with payment $p$, platform fee $f$, and augmentation fee $w$, the edge weight $w(u, v)$ is updated to $w(u, v) + f + w$.

**Early Adopter Rewards:** Early adopters are rewarded through a revenue sharing mechanism where fees f are distributed among all producers, awarding $f * (w(u', v) / \sum(u'', v) \in E w(u'', v))$ tokens to each producer u' based on their relative edge weight with buyer $v$. As the network matures, rewards become proportional to revenue contributions.

## Minting Local

Local tokens can be minted by providers based on the value of their transaction graph.

**Graph to token map:** A particular producer $u \in U$ is connected to a set of buyers through a number of weighted edges in the graph.

**Graph value metric:** We need to define a metric $G_u$, that determines what is the value that provider $u$ adds to the graph.

The simplest option is to simply add the weight of all the edges that connect with u, $\sum_v w(u,v)$. That is, the value added would be given by the total amount of transaction fees the provider has brought to the network.

This metric can be refined as well to account for connectivity. It can be considered more valuable to have transactions with a larger number of real users (who in turn transact with other providers), than just transacting with one user. Connectivity is measured by an eigenvector centrality, $x_u$ ranking. Computing eigenvector centrality ranks requires us to diagonalize a large matrix, this will increase computational complexity and we will have to set a reward epoch to not mandate that we recompute the matrix during every transaction.

For now we will not fully specify what the metric $G_u$, but assume it is some composite of quantities like $\sum_v w(u,v)$ and $x_0$.

**Graph value to token valuation:** For a given value $G_u$, we need to define a function, $M_t(G_u)$ that translates this into a token value. Given graph value $G_u$, this function explains the maximum number of Local tokens the producer is allowed to mint.

We may want the function $M_t$ to be monotonically increasing with time $t$. This will ensure the following incentives:

- The current set of producers will tend to be the majority owners of the network.
- Producers will be incentivized to not mint their available tokens, but instead keep them locked in the graph, as minting the token means their value will stay constant and not increase with time as $M_t$ increases. This
*Implicit Staking Mechanism*means that less tokens should be made available for sale, providing a positive pricing pressure to the circulating supply of tokens. - We may also require a certain level of activity from a producer. If no value is added to their graph in a certain amount of time, we may burn their edge weights or redistribute tokens to highly active participants.

#### Re-balancing graph

Suppose a producer has $M_t(G_u)$ available tokens to mint, and they mint a fraction $m < M_t(G_u)$ out of those. Once they do this, their edges in the graph get rebalanced, such that their graph value transforms as

$G_u \to \frac{M_t(G_u)-m}{M_t(G_u)}G_u.$

Depending on the definition of the metric $G_u$, if it is linear on the edge weights (such as eigenvector centrality, and total sum of weights are, individually), then edge weights also transform as

$w(u,v) \to \frac{M_t(G_u)-m}{M_t(G_u)}w(u,v).$

#### Recapturing tokens

In the same way that producers can use their graph value to mint new tokens, they can also purchase new tokens, and translate them into higher graph value. The producers would choose to do so because the tokens are subject to depreciation through inflation, while the graph value is not. Suppose a producer currently has a graph value $G_u$, with valuation $M_t(G_u)$. If they purchase an additional $m$ tokens, they can increase their graph value to,

$G_u \to \frac{M_t(G_u)+m}{M_t(G_u)}G_u.$

#### Summary

Producers can gain ownership of the network, either by providing services (which increases their graph value), or by buying tokens. It is therefore necessary to define two mapping functions: a graph value metric $G_u$ uses producer $u$'s contribution to the graph and determines what is the value they added to the network, then the token valuation function $M_t$ determines how much that graph value is currently worth in terms of number of tokens. The inflationary monetary policy given by increasing function $M_t$ means that providing service is actually favorable, since this type of ownership is not subject to depreciation through inflation. Being a protocol owner, either through token holding, or providing services, entitles one to receive rewards.