SUBMIT A PRSUBMIT AN ISSUElast edit: Nov 08, 2024

Guide to Dynamic TAO

Dynamic TAO is a sophisticated mechanism for both a TAO holder and a validator to manage their stake, influence, and governance across multiple subnets in the Bittensor network. More important, the dynamic TAO:

• Eliminates the role of the centralized root network.
• Enables a TAO holder to participate in a subnet of their choice.
• Creates a new measure of a subnet's performance via the subnet-specific dynamic TAO token (dTAO). A dTAO token is exchangeable with TAO.
• Incorporates the consensus power of a validator into their dTAO token stake holdings.
• Uses a subnet's dTAO to determine emissions to the subnet.

The dynamic TAO elegantly solves the problem of managing influence across different subnet economies while maintaining mathematical consistency and economic incentives.

Subnet pool

The heart of the dynamic TAO mechanism is a subnet pool, with the following properties:

• Associated with each subnet is a subnet pool.
• A subnet pool consists of two token reserves, a TAO token reserve and a subnet-specific dTAO token reserve.
• Each subnet has its own dTAO token. For example, dTAO α is the token for subnet α, dTAO β is the token for subnet β, and so on.
• All token exchanges between TAO and dTAO occur through the subnet pool. The subnet pool's reserve ratio determines the exchange rate.

TAO reserve (τ_in) and alpha reserve (α_in)

The TAO side of a subnet pool's reserves are denoted by τ_in, or TAO reserve, and are expressed in units of TAO. Similarly, the dTAO side of the pool reserves are denoted by α_in, or alpha reserves. The terms α_in and alpha reserves are general terms and refer to the pool reserves of all subnets, including that of subnet α.

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Rate (τ_in/α_in)

We use the term rate to refer to relative price of the dTAO token. This is calculated as a ratio of the subnet's pool reserves, i.e., TAO reserve(τ_in) / alpha reserves (α_in). This is commonly referred as the exchange rate between the subnet dTAO token and the TAO token.

Example

For example, if for subnet $ε$, its subnet pool contains TAO reserves of 1000 TAO units and its alpha reserves of 16000 dTAO $ε$ units, then the relative price of the dTAO $ε$ token is:

$$R = \frac{\tau_{in}}{\alpha_{in}} = \frac{1000}{16000} = 0.0625$$

Hence,

$$\text{1 dTAO ε} = 0.0625 \text{ TAO}$$

This exchange rate can change every block when staking or unstaking or emissions occur on this subnet.

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Staking

With dynamic TAO, staking would work like this. See the below diagram.

1. Under the dynamic TAO, as a TAO holder you would still stake to a validator’s hotkey, but now you can also select a subnet (netuid) of your choice. When you run btcli stake add command, the TAO you want to stake first goes into the subnet pool.
2. The TAO to be staked is added to the TAO reserves side of the subnet pool.
3. The subnet pool algorithm uses the exchange rate and calculates the equivalent units of dTAO α, for the TAO that was just added to the TAO reserve side. This amount of dTAO α is taken out of the alpha reserve of the pool and is sent to the validator’s hotkey.
4. The validator’s hotkey holds the dTAO α. This is referred as Stake (α).

Stake is always expressed in alpha units

In dynamic TAO, except for the stake held in subnet zero, the stake held by a hotkey in a subnet is always expressed in the subnet-specific dTAO α units and not TAO units.

Constant product k

The subnet pool algorithm is set up to always maintain a constant product $k$ of the two reserves. That is,

$$k = \text{(TAO reserves)}\times\text{(dTAO α reserves)}$$

Anytime either of the reserves increases, for example as a result of a random external action such as some stake TAO entering the pool, the subnet pool algorithm automatically recalculates, using the new reserves, how much the other reserve should decrease in order to maintain the same constant product $k$.

As described in the Staking section, a staking event results in the staked TAO being added to the τ_in reserves of the subnet pool. The subnet pool algorithm calculates the number of units by which the α_in reserves should decrease. These units are then taken out of the α_in reserves and sent to the validator’s hotkey in the subnet. See an example below.

Example

slippage not considered in this example

In order to convey the key idea of staking in a simple way, slippage is not considered in this example.

The below example shows how staking 5 TAO works.

Let us assume the following initial state of TAO and dTAO α reserves in a subnet:

• TAO reserves: 10 TAO tokens
• dTAO α reserves: 100 dTAO α tokens

Hence, the constant product 𝑘 = 10 × 100 =1000.

• Initial relative price of dTAO α is = reserve ratio of subnet pool α = (TAO token reserves of pool α)/(dTAOα token reserves) = 10/100 = 0.1 TAO.
• Stake 5 TAO: This adds 5 TAO into the TAO reserves of the pool. Hence, the new TAO reserve = 10 + 5 = 15 TAO tokens.
• Using the new TAO reserve of 15, the subnet pool algorithm calculates what should be the new dTAO α reserves, in order to maintain k at 1000.
• Calculating:

$$\begin{split} & 15 \times\text{new dTAO α reserves} = 1000\\ & \implies \text{new dTAO α reserves} = 1000/15 = 66.667 \text{ dTAO α}. \end{split}$$

• Hence, the dTAO α that should be taken out of the $α_{in}$ reserve and sent to the validator’s hotkey is:

$$100 − 66.67 = 33.33 \text{ dTAO α}$$

New state of the reserves in the subnet pool after 5 TAO is staked

• TAO reserves: 15 tokens
• dTAO α reserves: 66.67 dTAO α tokens
• Constant product 𝑘 = 15 × 66.67 = 1000 (unchanged)
• New relative price of dTAO α after the staking event = reserve ratio of subnet pool α = (TAO token reserves of pool α)/(dTAO α token reserves) = 15/66.67 = 0.225 TAO.

Effect of staking

Note that as a result of staking 5 TAO into subnet α, the relative price of dTAO α increased from 0.1 TAO to 0.225 TAO. All else being equal, staking into a subnet leads to an increase in its dTAO token relative price. This can be understood in two ways:

• By staking your TAO into this subnet, you are expressing an increase in demand for this dTAO α, thereby increasing the price of the dTAO α.
• When your TAO stake is added to the pool, the TAO reserves in the pool have increased, with a net effect that the dTAO reserves have become scarcer relative to the new TAO reserves. The dTAO price is calculated as (TAO reserves / dTAO reserves), hence the denominator decreased, leading to increase in dTAO price.

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STAKE (α_out) or alpha out (α_out)

Total stake in the subnet is referred as $α_{out}$. This is the sum of all the Stake (α) present in all the validator hotkeys in this subnet. This is often referred as α outstanding. Compare this with α reserve, which is the amount of $α_{in}$ in the subnet pool. The α outstanding can change every block. See a conceptual diagram below:

Hotkey's stake share (α / α_out)

A hotkey's share of the total outstanding alpha tokens in the subnet is calculated as:

$$\text{A hotkey's stake share} = \frac{\text{hotkey's stake}(\alpha)}{\text{Total stake}(\alpha_{out})\text{ in the subnet}}$$

Local staking power within the subnet

The hotkey's stake share represents the local stake power of the hotkey within the subnet.

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Unstaking

Unstaking works, not surprisingly, as a reverse of the staking operation. See the below diagram.

1. When you issue an unstake command, btcli stake remove, and specify the units of α token you want to unstake, this dTAO α is first taken out of the validator’s hotkey and added to the α reserves of the subnet pool.
2. The subnet pool algorithm then applies the latest exchange rate and calculates the equivalent TAO units for the α token units that were just added to the α reserves of the pool.
3. These equivalent TAO units are then taken out of the TAO reserves of the subnet pool and are sent to the TAO holder’s coldkey.

Example

slippage not considered in this example

In order to convey the key idea of unstaking in a simple way, slippage is not considered in this example.

The below example shows how unstaking 20 dTAO α works. Let’s continue with the state of the TAO and dTAO α reserves after the above staking operation:

• TAO reserves: 15 tokens
• dTAO α reserves: 66.67 dTAO α tokens
• Constant product 𝑘 = 15 × 66.67 = 1000
• Relative price dTAO α is = reserve ratio of subnet pool α = (TAO token reserves of pool α)/(dTAOα token reserves) = 15/66.67 = 0.225 TAO (same as the relative price after the above staking operation)
• Unstake 20 dTAO α: This adds 20 dTAO α to the dTAO side of the pool. New dTAO α reserves = 66.67 + 20 = 86.67 dTAO α tokens.
• Using the new dTAO α reserve of 86.67, the subnet pool algorithm calculates what should be the new TAO reserve in order to maintain $k$ at 1000.
• Calculating:

$$\begin{split} & 86.67 \times\text{new TAO reserves} = 1000\\ & \implies \text{new TAO reserves} = 1000/86.67 = 11.54 \text{ TAO}. \end{split}$$

• Hence, the TAO that should be taken out of the TAO reserves of the pool and sent to the TAO holder’s coldkey is: 15 − 11.54 = 3.46 TAO.

New state of the reserves in the subnet pool after 20 dTAO α is unstaked

• TAO reserves: 11.54 TAO tokens
• dTAO α reserves: 86.67 dTAO α tokens
• Constant product 𝑘 = 11.54 × 66.67 = 1000 (unchanged)
• New relative price of dTAO α after the unstaking event = reserve ratio of subnet pool α = (TAO token reserves of pool α)/(dTAOα token reserves) = 11.54/86.67 = 0.133 TAO.

Effect of unstaking

Note that as a result of unstaking 20 dTAO α from subnet α, the dTAO α price decreased from 0.225 TAO to 0.113 TAO. All else being equal, unstaking from a subnet leads to a decrease in its dTAO token's relative price. This can be understood in two ways:

• By unstaking your TAO from this subnet, you are expressing a decrease in demand for this dTAO, thereby decreasing the price of the dTAO.
• When your stake TAO is removed from the subnet pool, the TAO reserves in the pool have decreased, with a net effect that the dTAO reserves have become more abundant relative to the new TAO reserves. The dTAO price is calculated as (TAO reserves / dTAO reserves), hence the denominator increased, leading to decrease in the dTAO price.

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Local weight or TAO Equiv (τ_in x α/α_out)

A hotkey's stake, i.e., stake(α), represents subnet-specific stake. As a result, a hotkey's stake share (α / α_out) can only represent the validator's staking power within the subnet. Moreover, a dTAO token of a subnet is not fungible with a dTAO token of another subnet. For this very reason, when a hotkey is validating in multiple subnets, the hotkey's stake share in one subnet cannot be simply added to the same hotkey's stake share in a different subnet.

This is where the TAO-equivalent value of the hotkeys stake share (α / α_out) comes into play. As we saw in the Staking section, any TAO staked into a subnet is added to the TAO reserves of the subnet pool, i.e., added to τ_in. Similarly any TAO unstaked from a subnet is removed from the TAO reserves of the subnet pool. As a consequence, τ_in reserve represents the total voting power of the subnet as a whole. Hence,

$$\text{a subnet's total voting power}\times\text{hotkey's stake share in the subnet}$$

represents the hotkey's local voting power proportional to its stake share in the subnet. It is calculated as:

$$\tau_{in}\times\text{hotkey's stake share} = \tau_{in}\times\frac{\alpha}{\alpha_{out}}$$

and is expressed in TAO units. This can change every block.

Local voting power is also referred as local weight

This local voting power or local weight represents the hotkey's influence in the subnet.

Example

For example, for a validator's hotkey in a given subnet:

• Total outstanding alpha (α_out) = 30,000 alpha tokens.

• The hotkey's alpha stake (α) = 6,000 alpha tokens.

• TAO reserve in the subnet's pool (τ_in) = 5,000 TAO.

• Then, this validator's hotkey stake share = (6,000/30,000) = 0.2 or 20%.

• Hence, this validator hotkey's local voting power, i.e., local weight = τ_in x hotkey's stake share = 5000 x 0.2 = 1000 TAO.

Stake share → TAO-denominated voting power

Hence, while the hotkey's stake share in this subnet is 20%, its actual local weight (local voting power) in the subnet is 1,000 TAO units. This is the power of dynamic TAO mechanism. It converts a hotkey's alpha stake share into an equivalent TAO-denominated voting power in the subnet. This makes alpha stakes comparable across different subnets, even though these subnets might have very different amounts of alpha tokens outstanding.

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Local weights vs τ_in

In any subnet, if you add up the local weights of all alpha holders, you will always get exactly the TAO reserve (τ_in) of that subnet.

This means:

• The total voting power in a subnet is fixed by its TAO reserve (τ_in).
• For a given τ_in, if a validator's hotkey in a subnet increases their local weight, it must come at the expense of others. That is, you cannot create "extra" voting power, it's a zero-sum game within each subnet.

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Subnet zero

In dynamic TAO, subnet zero is a special subnet. It is designed to provide the following benefits:

• The subnet zero is the only subnet that does not have any subnet pool associated with it. TAO holders can only stake into subnet zero in TAO token denominations. Hence, TAO holders and validators who prefer not to think about either subnet-specific staking or about dTAO tokens can stake in subnet zero. Their stake will remain in TAO denomination in the validator hotkeys as there is no notion of alpha in subnet zero.
• The subnet zero is also the only subnet that does not have any validating or mining defined on it. Incentive mechanisms cannot be run in subnet zero. Nevertheless, a validator's hotkey registered in subnet zero will accept stake TAO.
• Because there is no subnet pool attached to the subnet zero, a validator hotkey's local weight in subnet zero is simply the raw TAO units that exist as stake in this hotkey.

Staking in subnet zero

As a rule, you can only stake in TAO denomination in subnet zero, and only in alpha (dTAO) denomination in all other subnets. Hence, stake in subnet zero is always expressed in TAO units.

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Global weight

Global weight of a validator hotkey is the sum of the local weights for all the subnets, including subnet zero, where this hotkey is validating.

Example

Let's say that a validator's hotkey has the following positions in four different subnets, including in subnet zero.

• Subnet zero:

  • validator's stake: 1000 TAO
  • validator's stake share: 10% (i.e., out of all TAO stake in subnet zero, this validator holds 10% of it)

• Gaming subnet:

  • validator's stake share = (α / α_out) = 30%
  • τ_in in this subnet pool = 10,000 TAO

• AI subnet:

  • validator's stake share = (α / α_out) = 40%
  • τ_in in this subnet pool = 15,000 TAO

• Storage subnet:

  • validator's stake share = (α / α_out) = 20%
  • τ_in in this subnet pool = 5,000 TAO

Hence, the local weights, or local voting power of this hotkey in each subnet are as below:

• For subnet zero: 1000 TAO (TAO number used as is, without any multiplier, see subnet zero)
• Gaming subnet: $\tau_{in}\times\text{hotkey's stake share}$ = 0.30 × 10,000 = 3,000 TAO
• Similarly, for AI subnet: 0.40 × 15,000 = 6,000 TAO
• For Storage subnet: 0.20 × 5,000 = 1,000 TAO

Hence the global weight of this hotkey is: 1000 + 3000 + 6000 + 1000 = 11,000 TAO.

Root weight

Notice that in the above calculation, we used the hotkey's subnet zero stake of 1000 TAO as it is while calculating the global weight of the hotkey. However, in dynamic TAO it is normal to multiply the subnet zero stake TAO number by a factor called root_weight that varies from 0 to 1.

Hence, for root_weight of 0.5, the subnet zero stake of the hotkey will now be root_weight x 1000 = 0.5 x 1000 = 500 TAO. Hence, under this condition, the updated global weight of this hotkey is: 500 + 3000 + 6000 + 1000 = 10,500 TAO.

Global weight vs. local weight

The two quantities defined above for a validator hotkey, i.e., the hotkey's global weight (across all subnets where it is validating) and its local weight (per subnet), are critically important.

global weight appears in every subnet

In addition to the local weight of a hotkey in a subnet, this hotkey's global weight also appears in this subnet. This is why global weight of a validator's hotkey is critical.

Note that in the above, both local weight and global weight of a hotkey are expressed in TAO units. However, to represent a validator hotkey's overall stake weight in a subnet, instead of using the two individual TAO units (one for global weight and second for local weight), a new quantity called validator's stake weight in a subnet, is defined as follows:

• The local weight of a hotkey is normalized with respect to the sum of all the local weights of all other hotkeys in this subnet, so that they all sum to 1.0. This normalized local weight represents the hotkey's relative proportion of its influence in the subnet.
• Similarly the global weight of the hotkey is normalized with respect to the sum of all the global weights of all other hotkeys for the subnets in question, to sum to 1.0. This normalized global weight represents the hotkey's relative proportion of its influence in the subnets in question.

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Validator stake weight

A validator hotkey's stake weight is defined for a subnet. It varies from 0 to 1. It is defined as the sum of the hotkey's normalized global and local weights, as follows:

For any subnet $i$, the validator hotkey's stake weight is:

$$\begin{split} & = \text{(global\_split}\times\text{normalized global weight)} + \text{(1-global\_split}\times\text{normalized local weight)}\\\\ & = \text{global\_split}\times\frac{\text{hotkey's global weight}}{\text{sum of all global weights of all hotkeys in the subnets in question}}\\ \\ & + \text{ (1-global\_split)}\times\frac{\text{hotkey's local weight}}{\text{sum of all local weights in the subnet}} \end{split}$$

Global split

A parameter called global_split, which varies between 0 and 1, controls the balance between the normalized global and local weights. In effect, the global_split parameter controls the balance between the validator hotkeys local and global influence.

Example

We will use the example from Global weight section and extend it to show the validator stake weight.

Assumptions

• Assume root_weight is 0.5.

• Subnet zero:

  • Total TAO outstanding = 10,000 TAO (this is the total TAO stake held by all the hotkeys in subnet zero)
  • Hence, sum of all global weights in subnet zero = 10,000 TAO
  • Validator's stake share = 10% × 10,000 = 1,000 TAO (also calculated in the above example)

• Gaming Subnet:

  • Assume total α_out = 50,000 α
  • Validator's stake share = 30% × 50,000 = 15,000 α
  • TAO reserve = 10,000 τ
  • TAO reserve is also the sum of all global weights in this gaming subnet = 10,000 TAO
  • Local weight = (15,000/50,000) × 10,000 = 3,000 τ (also calculated in the above example)

• AI Subnet:

  • Assume total α_out = 80,000 α
  • Validator's stake share = 40% × 80,000 = 32,000 α
  • TAO reserve = 15,000 τ
  • TAO reserve is also the sum of all global weights in this AI subnet = 15,000 TAO
  • Local weight = (32,000/80,000) × 15,000 = 6,000 τ (also calculated in the above example)

• Storage Subnet:

  • Assume total α_out = 30,000 α
  • Validator's stake share = 20% × 30,000 = 6,000 α
  • TAO reserve = 5,000 τ
  • TAO reserve is also the sum of all global weights in this storage subnet = 5,000 TAO
  • Local weight = (6,000/30,000) × 5,000 = 1,000 τ (also calculated in the above example)

Hence, sum of all global weights in the all the above subnets is = (root_weight x subnet zero's total global weight) + sum of all global weights in gaming subnet + sum of all global weights in AI subnet + sum of all global weights in storage subnet

= (0.5 x 10,000) + 10,000 + 15,000 + 5,000 = 35,000 TAO. This is the global weights in all the subnets where this validator's hotkey is validating.

Validator stake weight for each subnet

• Hotkey's global weight = 10,500 TAO (from the above Root weight section).
• Total global subnet weights = 35,000 TAO (from above).
• Assume global_split is 0.3.

1. Gaming Subnet:

   • Local weight of the hotkey: 3000 TAO (from the above example)
   • Sum of all local weights in this subnet = TAO reserve = 10,000 TAO
   • Hence, the validator's stake weight in this gaming subnet = 0.3 × (10,500/35,000) + 0.7 × (3,000/10,000) = 0.3 × 0.30 + 0.7 × 0.30 = 0.09 + 0.21 = 0.30 (30% influence in the gaming subnet)

2. AI Subnet:

   • Local weight of the hotkey: 6000 TAO (from the above example)
   • Sum of all local weights in this subnet = TAO reserve = 15,000 TAO
   • Hence, the validator's stake weight in this AI subnet = 0.3 × (10,500/35,000) + 0.7 × (6,000/15,000) = 0.3 × 0.30 + 0.7 × 0.40 = 0.09 + 0.28 = 0.37 (37% influence in the AI subnet)

3. Storage Subnet:

   • Local weight of the hotkey: 1000 TAO (from the above example)
   • Sum of all local weights in this subnet = TAO reserve = 5,000 TAO
   • Hence, the validator's stake weight in this storage subnet = 0.3 × (10,500/35,000) + 0.7 × (1,000/5,000) = 0.3 × 0.30 + 0.7 × 0.20 = 0.09 + 0.14 = 0.23 (23% influence in the storage subnet)

4. Subnet zero: (special case - only local weight matters)

   • Validator's stake weight = 1,000/10,000 = 0.10 (10% influence in the subnet zero)

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Exchange Value (α x τ/α)

This is the potential instantaneous TAO you will receive, without considering slippage, if you unstake at this instant from this hotkey on this subnet. See Swap(α → τ) column description. This can change every block.

Compare with TAO Equiv

Whereas the TAO Equiv(τ_in x α/α_out) indicates a validator's local stake weight, this Exchange Value shows TAO you will receive if you unstake now.

Swap (α → τ)

This is the actual τ you will receive, after factoring in the slippage charge, if you unstake from this hotkey now on this subnet. The slippage is calculated as 1 - (Swap(α → τ)/Exchange Value(α x τ/α)), and is displayed in brackets. This can change every block.

Emission (α/block)

Shows the portion of the one α per block emission into this subnet that is received by this hotkey, according to YC2 in this subnet. This can change every block.

Emission (τ)

Shows how the one τ per block emission is distributed among all the subnet pools. For each subnet, this fraction is first calculated by dividing the subnet's TAO Pool (τ_in) by the sum of all TAO Pool (τ_in) across all the subnets. This fraction is then added to the TAO Pool (τ_in) of the subnet. This can change every block.

Tempo (k/n)

The tempo status of the subnet. Represented as (k/n) where "k" is the number of blocks elapsed since the last tempo and "n" is the total number of blocks in the tempo. The number "n" is a subnet hyperparameter and does not change every block.

Local weight coeff (γ)

A multiplication factor between 0 and 1, applied to the relative proportion of a validator's stake in this subnet. Applied as (γ) × (a validator's α stake in this subnet) / (Total α stake in this subnet, i.e., Stake (α_out)). This γ-weighted relative proportion is used, in addition to other factors, in determining the overall stake weight of a validator in this subnet. This is a subnet parameter.