# Global Liquidity, Motivated

## A formal argument for global liquidity

**Introduction**

The Paradigm Project is largely motivated by the thesis that global liquidity is generally the most efficient market structure. In this article we explore, in detail, the argument for global liquidity.

An argument for global liquidity can be motivated from fairly basic economic principles and formally substantiated via simple mathematical reasoning. In the following sections we will do exactly that, providing intuitive models and visual aids along the way.

# Convention Notes

Medium does not support rich mathematical expressions including LaTeX or subscripts. In this article I have decided to use the convention * as a substitute for subscripts where a single * represents the subscript “1”, ** represents “2” etc.

**Case Model Specification**

Let us first define a model for our market structure and then consider a specific bid/ask case. We define our model similarly to Paul Kettler, Aleh Yablonski and Frank Proske in their paper *Market Microstructure and Price Discovery¹.*

We will explore a market, 𝑚*, for a single asset in discrete time 𝑡 ∈ 𝐓 = {0,1,…,𝑇}. We can reasonably assume that at any given time, 𝑡, ∃ 𝑛* ≤ 𝐍 where 𝑛* describes a finite number of agents participating in 𝑚* (i.e.|𝑚*|) and 𝐍 describes the total, finite, number of agents participating across markets 𝐌 (i.e. ∑(∀|𝑚|⊆𝐌)). We thus define 𝑚* ⊆ 𝐌 = {𝑚*, 𝑚**,…, 𝑀} where 𝐌 describes the superset of all markets for a particular good. At each moment 𝑡 ∈ 𝐓, the agent number 𝑖, 1≤𝑖≤𝐍 proposes a bid price 𝑏(𝑖,𝑡) and an ask price a(𝑖,𝑡) for the good in a market 𝑚. It can be assumed that a(𝑖,𝑡) ≥ b(𝑖,𝑡) and 𝐴(𝑡) ≥ 𝐵(𝑡), where 𝐴(𝑡) = min{𝑎(𝑖,𝑡): 1≤𝑖≤𝑛} and 𝐵(𝑡) = max{𝑏(𝑖,𝑡): 1≤𝑖≤𝑛}. We can now assume the market process of price discovery. A trade occurs at moment 𝑡 between the 𝑖-th and 𝑗-th agents if 𝑎(𝑖,𝑡)=𝐴(𝑡)=𝐵(𝑡)=b(𝑗,𝑡) or 𝑎(𝑗,𝑡)=𝐴(𝑡)=𝐵(𝑡)=b(𝑖,𝑡).

**Case Model**

We will now consider the scenario when at time, 𝑡, there exists, at most, two agents willing to trade. This scenario is described at any time, 𝑡, where 𝑖 proposes a bid price 𝑏(1,𝑡) = B(t) and 𝑗 proposes an ask price a(2,𝑡) = A(𝑡) such that A(𝑡) = B(t) where 𝑖 ∈ 𝐌 and 𝑗 ∈ 𝐌. This trade will only occur if both 𝑖 and 𝑗 are participating in 𝑚*. We can define a simplistic probability of this trade occurring within a specific market 𝑚* as 𝑛* / 𝐍. We can establish the bounds of 2/𝐍 to 1 on the probability of such process. If the trade does not occur, we can assume both 𝑖 and 𝑗 were not participating in the same market and now consider the moment 𝑡+1. Assuming either 𝑖 or 𝑗 have a behavioral bias towards trade, one agent strengthens their bid (or ask) thus creating an arbitrage opportunity within 𝐌. This arbitrage opportunity represents a fundamental market inefficiency as a result of the disparate microstructure of 𝐌.

**Applied**

We can visualize such bid/ask mechanics between 0x relayers, Radar Relay and IDEx. To do so, we can create a simple python program using Radar Relay’s and IDEx’s public REST APIs. We will observe the best bids/asks for the ZRX-WETH market of each relayer.

We ran this program collecting 1000 data points for each line (~1hr). Shown below is the graph that was generated.

Some may argue that as value transfers become as arbitrary as digital signatures on messages, arbitrage will function to create what is effectively global liquidity, but this is empirically false. We can observe a spread (arbitrage opportunity) between the best bids and best asks between the agents participating on relayers IDEx and Radar Relay. In this time period, buy-side agents would have paid less for liquidity by participating in IDEx’s market, but sell side agents would have been better suited participating in Radar Relay’s market. This phenomenon is likely the result of the fact that even 0x orders are not simply digital signatures on messages, but instead specify a fee recipient (generally the relayer), thus maintaining some level of propriety for relayers. These relayers are inherently incentivized to fragment liquidity (prisoner’s dilemma). For global liquidity to be possible, the “relay” function of relayers must be abstracted such that market makers can effectively own their own liquidity.

**Assumptions**

The construction of this model is relatively specific, but nonetheless functions well as an illustration of underlying inefficiencies within a fragmented market structure. This model makes a few prominent assumptions, primarily:

- Each bid/ask is for the same volume
- At time 𝑡 there exists only two agents willing to trade (ie fairly illiquid market)

Both of these assumptions are relatively insignificant and made only in an attempt to simplify such model. The first assumption can easily be ignored, as this assumption can be negated in the construction of such model where a base unit (in the case of an ERC20 token = 1*10^-18) can be established. The second assumption creates simplicity within our model, but can also be ignored as the alternative—a very liquid marketplace—will still observe similar phenomenon in regards to an incongruent distribution of bids/asks.

**Network Effects**

From the motivation above, we can reason that arbitrage — which itself represents fundamental market inefficiencies — can be greatly (or completely) reduced via a deep global liquidity pool. We will now explore the network effects around such global liquidity. To do this, we will first consider how network effects are defined for a financial exchange, or in our case, an order network.

In most exchange systems, transaction costs are a function of brokerage fees and the bid-ask spread. In the case of Paradigm, the brokerage fees are essentially the fees charged by matchers. Due to the nature of decentralized settlement, another fee, the smart contract transaction (settlement) fee must also be considered. Fortunately, in our model we can assume settlement costs are constant and thus disregard them. One may also argue that the matcher fees in Paradigm’s model trend toward 0 due to the low barrier to entry, but regardless, market liquidity remains the largest contributor to transaction cost.

We can define the cost of liquidity in this case to be the bid-ask spread that exists in a market for an asset. This is the fee required for an immediate trade within the market (an urgent buyer/seller taking a maker order). As the number of agents participating in the market increases, liquidity increases, consequently decreasing transaction costs, thus attracting more agents to the market. This process describes network effects within an exchange system. In context of our model, as 𝑛* increases, the price of liquidity decreases and network utility increases.

In order to quantify these direct network effects, we will refer to a network effects model proposed by Kyle Samani in his article titled “On the Network Effects of Stores of Value.” ³ Kyle proposed using a modified, sublinear version of Zipf’s Law to quantify the strength of network effects in markets for liquid, fungible goods. Zipf’s Law states that “given a large sample of words used, the frequency of any word is inversely proportional to its rank in the frequency table. So word number 𝑛 has a frequency proportional to 1/𝑛.” From this, a network effect of 𝑛log(𝑛) can be recognized.² Samani begins here, but suggests in practice a superlinear growth is still impractical stating “as each additional user increases daily liquidity, the marginal value of that extra liquidity becomes increasingly worthless to all existing users” and instead suggests a sublinear log(𝑛) model for the network effect quantification.

We can visualize these quantifications of network effects using a basic python program, observing the following graph, where ‘y=log(x)’ defines the only sublinear network effect.

In context of fungible goods, this quantification of network effects makes sense. Samani points out strong empirical evidence of this identifying the vast number of cryptocurrency exchanges, as qualification.

In regards to the liquidity network effects exhibited by the Paradigm Protocol, a key distinction from Samani’s log(𝑛) curve estimation is that the protocol aims to enable the exchange of both fungible and non-fungible tokens. We consider the various synthetic markets and positions for contract settlement structures to be non fungible because of customized differences in time frame and contract type (forward contracts), and plan to support ERC-721 tokens as well as other non fungible tokenized asset types in the future. We hypothesize that these additional marketplace types will allow the protocol to exhibit two-sided network effects with faster growth rates than log(𝑛). Most likely closer to 𝑛log(𝑛) or the “Lazy S Shape” often used to quantify network effects.⁴

The protocol also aims to create indirect network effects by leveraging its open OrderStream as a pool of global liquidity. Indirect network effects can be defined as when, “increases in usage of one product or network spawn increases in the value of a complementary product or network, which can in turn increase the value of the original.” ³ Increased usage of the Paradigm Protocol, which is represented as order volume, creates additional liquidity for matchers and improves market efficiency for traders. For developers building matchers, additional liquidity on the network decreases barriers to entry and incentivizes the creation of higher quality matchers and exchange products.

**Liquidity Abstraction**

Paradigm aims to create global liquidity, driven primarily by maker/taker dynamics. We make the distinction between networked liquidity and global liquidity. Global liquidity is more fundamental being maker/taker driven, allowing any agent to solicit orders via the network, whereas networked liquidity is driven by relayer and exchange entities. Global liquidity faces a challenge considering it aims to disrupt the traditional model of proprietary liquidity. The model of global liquidity will be more efficient over the long run and allow a more targeted, open and demographic specific trading experience. Liquidity abstraction, combined with settlement abstraction, will shift exchange dynamics significantly. With that said, the idea that all financial markets may soon operate on a single global liquidity network is unrealistic. Whales may always favor dark pools and more proprietary venues. The project Republic Protocol is working hard to support this demographic in a decentralized system.

**Conclusion**

Shared infrastructure and the prominence of hybrid decentralized settlement logic (financial primitives) are building the basis for a more efficient and inclusive global economy. Relayers built on shared infrastructure (0x, Dharma, dydx etc.) are uniquely positioned with a clear path towards networked, and ultimately, global liquidity. Due to the canonical order schema established by the underlying settlement logic, global liquidity is a realistic opportunity. Global liquidity can create liquidity pools larger than that of any single exchange, and promote a new level of efficiency and liquidity within exchange markets.

**Glossary**

**Global liquidity- **openly accessible liquidity created by a decentralized venue for order solicitation, specifically an event-stream based order book.

**Market structure**- organisational and other characteristics of a market.

**Market arbitrage-** refers to purchasing and selling the same security at the same time in different markets to take advantage of a price difference between two separate markets.

**Matchers- **are entities build on the Paradigm Protocol that are responsible for trade execution.

**Network effect**- a phenomenon whereby a product or service gains additional value as more people use it.

**Perfect competition**- is a theoretical market structure in which competition is at its greatest possible level.

**Price discovery**- is the market driven process of finding the point where supply and demand intersect.

# Citations

- P. Kettler, A. Yablonski and F. Proske, “Market Microstructure and Price Discovery,”
*Journal of Mathematical Finance*, Vol. 3 №1, 2013, pp. 1–9. doi: 10.4236/jmf.2013.31001. - Briscoe, Bob, et al. “Metcalfe’s Law Is Wrong.”
*IEEE Spectrum: Technology, Engineering, and Science News*, IEEE Spectrum, 1 July 2006, spectrum.ieee.org/computing/networks/metcalfes-law-is-wrong. - Samani, Kyle. “On The Network Effects Of Stores Of Value — Hacker Noon.”
*Hacker Noon*, Hacker Noon, 11 May 2018, hackernoon.com/on-the-network-effects-of-stores-of-value-4286f6c98cdc. - Acemoglu, Daron, and Asu Ozdaglar. “6.207/14.15: Networks.” Lectures 17 and 18: Network Effects. 2009.

**Acknowledgements**

Special thanks to Jonathan Itzler and Henry Harder for their contributions to this post and to Saar Yalov, Nir Kabessa, Stephen Fiser, Thomas Aslanian, and Alex Hart for their feedback and revisions.