# Bridge Tokenomics

For the Bitcoin-Ethereum bridge, in the bootstrapping phase, the rate control we implement is exposed to ETH-BTC price ratio fluctuations. To protect against this, an overcollateralized wallet capacity and an optimal wallet duration is selected (note that this is not subject to any disadvantages of overcollateralization since no new capital is required, already existing stake in EigenLayer is put to use). These values are selected on the basis of in-depth study of ETH-BTC ratio.

Say BTC and ETH prices at the time of a bridging request are $100 and $10 respectively. ETH-BTC price ratio would thus be 10/100. Say overcollateralization of 140% is selected. This means that for every 14 ETH (i.e. $140) of stake that secures our bridge, 1 BTC (i.e. $100) can be accepted as wallet capacity to be deposited and bridged. Now if the price of ETH drops relative to the price of BTC, the dollar value of the collateral at stake drops too. This is dangerous if the dollar value of the collateral drops to the point that it is worth less than the BTC in custody of the wallet, in which case corrupting the bridge would be profitable. Therefore, the collateralization ratio must be selected carefully and the duration of the wallet must also be optimal.

At any given time, the drop in the dollar value of collateralization is exactly equal to the drop in ETH-BTC price ratio relative to the ratio’s value at the time of deposit. A script is run to consider each day in approximately the past 3.5 years (from 1 Jan 2020 to 20 Jun 2023) as the time of deposit. The price ratio on that day is compared to that of its subsequent days for different wallet durations in consideration. For eg. Fig T.1 (below) is the plot for a wallet duration of 7 days. Therefore the plot shows percentage changes between the price ratio on day of deposit and price ratio on each of its subsequent 7 days for all consecutive 7 day periods possible within the selected dates (a total of 7566 data points plotted as a histogram of 250 bins).

Fig T.1 Fig T.2

The data in Fig T.1 can be approximated by the Log-Laplace distribution with the following parameters:

shape (c) = 9.579714393630999

location = -33.32190861061157

scale = 33.33428331114358

Fig T.2 shows the cumulative distribution function for this data. The point marked with the red arrow shows the percentage change in price ratio corresponding to 1% probability. This means that 99% of the times the percentage change within a 7 day period would not be larger than 11.16% in magnitude on the negative side. Thus the overcollateralization requirement to cover 99% of occurrences would be 112.56% (Solving for x in equation x - 0.1116x = 100). In practice, a much higher overcollateralization than what the study suggests must be selected in order to ensure safety.

Naturally, the overcollateralization requirement increases with an increase in the wallet duration (for eg. 30 day periods would capture higher volatility than 7 day periods). An optimal wallet duration must be selected such that it is in tune with the other system parameters to ensure economic security. In the exceptionally rare cases that the dollar value of the collateral drops to a point where it is profitable to corrupt the bridge, conditional threshold signing provides an additional layer of security. Conditional threshold signing will be implemented for all signatures until the previously mentioned EigenLayer features are up and audited.

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