SUBMIT A PRSUBMIT AN ISSUElast edit: Dec 30, 2024

Guide to Dynamic TAO

The Emissions section is preliminary

The Emissions section in this dynamic TAO guide is still not settled yet. Proceed with caution.

Follow this guide to learn the essential concepts, features and mechanisms of dynamic TAO.

looking for quick intuitions?

Skip to Intuitions of dynamic TAO.

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 $\alpha$ token reserve.
• Each subnet has its own $\alpha$ token. For example, $\alpha$ is the token for subnet $\alpha$, and $\beta$ is the token for subnet $\beta$, and so on. Note that the term $\alpha$ is also used to refer to any $\alpha$ token, not just the token for the subnet $\alpha$.
• All token exchanges between TAO and $\alpha$ occur through the subnet pool. The subnet pool's reserve ratio determines the exchange rate.

TAO reserve and alpha reserve

The TAO side of a subnet pool's reserves are denoted by $\tau_{in}$, or TAO reserve, and are expressed in units of TAO. Similarly, the $\alpha$ side of the pool reserves are denoted by $\alpha_{in}$, or alpha reserves. The terms $\alpha_{in}$ and alpha reserves are general terms and refer to the pool reserves of all subnets, including that of subnet $\alpha$.

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Rate

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

$$R = \frac{\tau_{in}}{\alpha_{in}}$$

Example

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

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

Hence,

$$\text{1 } \varepsilon = 0.0625 \text{ TAO}$$

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 works as follows. 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 $\alpha$, for the TAO that was just added to the TAO reserve side. This amount of $\alpha$ 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 $\alpha$. This is referred as Stake ($\alpha$).

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 $\alpha$ 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,

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

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 $k$.

Staking and unstaking do not change the constant product

Staking and unstaking operations do not change the constant product $k$, but result in slippage due to the fact that these are exchange (i.e., swap) operations. On the contrary, emissions into a subnet pool do change the constant product $k$ but do not result in slippage, due to the fact that these are not swap operations. See Slippage. Also see Emissions.

Slippage

When we stake we exchange TAO for the alpha token in the pool. The following occurs while we stake:

1. Prior to the staking operation, we know the expected price of the alpha token. This expected price is the reserve ratio, as described in Rate, prior to the staking operation.
2. However, the very action of our staking changes the token reserve ratio, because we are adding TAO to the TAO reserves.
3. Mathematically the pool is required to maintain the constant product $k$. As a result, the pool algorithm automatically adjusts the alpha token reserves to keep the constant product $k$ unchanged. This results in a change in the reserve ratio.
4. This means that the actual price at which our staking operation is executed is different from our original expected price of the alpha token. This difference between the original expected alpha token price and the actual actual token price at which staking operation is performed is called slippage.

Example

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 $\alpha_{in}$ reserves should decrease. These units are then taken out of the $\alpha_{in}$ reserves and sent to the validator’s hotkey in the subnet. See an example below.

The below example shows how staking 5 TAO works.

Let us assume the following initial state of TAO and $\alpha$ reserves in a subnet:

• TAO reserves: 10 TAO tokens
• $\alpha$ reserves: 100 $\alpha$ tokens

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

Without slippage

• Expected price of $\alpha$ token:

$$= \frac{\text{TAO reserves}}{\text{α reserves}} = \frac{10}{100} = \text{0.1 TAO per α}$$

• Hence, the expected number of α tokens:

$$\text{} = \frac{\text{TAO staked}}{\text{Expected α price}} = \frac{5}{0.1} = \text{50 α tokens}$$

With slippage

• Initial relative price of $\alpha$ is = reserve ratio of subnet pool $\alpha$ = (TAO token reserves of pool $\alpha$)/($\alpha$ 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 $\alpha$ reserves, in order to maintain k at 1000.
• Calculating:

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

• Hence, the $\alpha$ that is taken out of the $α_{in}$ reserve and sent to the validator’s hotkey is:

$$\text{Stake in α tokens} = 100 − 66.67 = 33.33 \text{ α}$$

Hence, when you stake 5 TAO, your stake is $33.33 \text{ α}$, taking into account the slippage. As a result:

• Slippage in terms of $\alpha$ tokens:

$$\begin{split} \text{Slippage} & = \text{Expected α tokens} - \text{Actual α tokens obtained}\\ & = 50-33.33 = 16.6667 α \end{split}$$

• The percentage of slippage:

$$\begin{split} \text{Percentage slippage} & = \frac{\text{Slippage α tokens}}{\text{Expected α tokens}}\times 100\% \\ & = \frac{16.6667}{50}\times 100\% \\ & = 33.333\% \end{split}$$

Effect of slippage

• TAO reserves: 15 tokens
• $\alpha$ reserves: 66.67 $\alpha$ tokens
• Constant product 𝑘 = 15 × 66.67 = 1000 (unchanged)
• New relative price of $\alpha$ after the staking event = reserve ratio of subnet pool $\alpha$ = (TAO token reserves of pool $\alpha$)/($\alpha$ token reserves) = 15/66.67 = 0.225 TAO.
• Price impact due to slippage: As a result of the slippage, staking 5 TAO into subnet $\alpha$ changed the relative price of $\alpha$ token: it increased from 0.1 TAO to 0.225 TAO. All else being equal, slippage leads to an increase in $\alpha$ token's relative price when we stake.

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Stake or alpha outstanding

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

Hotkey's stake share

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 $\alpha$ token you want to unstake, this $\alpha$ is first taken out of the validator’s hotkey and added to the $\alpha$ reserves of the subnet pool.
2. The subnet pool algorithm then applies the latest exchange rate and calculates the equivalent TAO units for the $\alpha$ token units that were just added to the $\alpha$ 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

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

• TAO reserves: 15 tokens
• $\alpha$ reserves: 66.67 $\alpha$ tokens
• Constant product 𝑘 = 15 × 66.67 = 1000

Without slippage

Don't know what slippage is?

See Slippage if you are not familiar with it.

• Expected price of $\alpha$ token before unstaking:

$$= \frac{\text{TAO reserves}}{\text{α reserves}} = \frac{15}{66.667} = \text{0.225 TAO per α}$$

• Hence, the expected number of TAO tokens:

$$\begin{split} \text{} & = \text{α unstaked}\times{\text{Expected α price}}\\ & = 20\times0.225 \\ &= \text{4.5 TAO tokens} \end{split}$$

With slippage

• Relative price $\alpha$ is = reserve ratio of subnet pool $\alpha$ = (TAO token reserves of pool $\alpha$)/($\alpha$ token reserves) = 15/66.67 = 0.225 TAO (same as the relative price after the above staking operation)
• Unstake 20 $\alpha$: This adds 20 $\alpha$ to the $\alpha$ side of the pool. New $\alpha$ reserves = 66.67 + 20 = 86.67 $\alpha$ tokens.
• Using the new $\alpha$ 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 is taken out of the TAO reserves of the pool and sent to the TAO holder’s coldkey is: 15 − 11.54 = 3.46 TAO.

$$\text{Unstaked TAO tokens} = 15 − 11.54 = 3.46 \text{ TAO}$$

Hence, when you unstake 20 $\text{α}$, your unstaked TAO is $3.46 \text{ TAO}$, taking into account the slippage. As a result:

• Slippage in terms of TAO tokens:

$$\begin{split} \text{Slippage} & = \text{Expected TAO tokens} - \text{Actual TAO tokens obtained}\\ & = 4.5-3.46 = 1.04\text{ TAO} \end{split}$$

• The percentage of slippage:

$$\begin{split} \text{Percentage slippage} & = \frac{\text{Slippage TAO tokens}}{\text{Expected TAO tokens}}\times 100\% \\ & = \frac{1.04}{4.5}\times 100\% \\ & = 23.1\% \end{split}$$

Effect of slippage

• TAO reserves: 11.54 TAO tokens
• $\alpha$ reserves: 86.67 $\alpha$ tokens
• Constant product 𝑘 = 11.54 × 66.67 = 1000 (unchanged)
• New relative price of $\alpha$ after the unstaking event = reserve ratio of subnet pool $\alpha$ = (TAO token reserves of pool $\alpha$)/($\alpha$ token reserves) = 11.54/86.67 = 0.133 TAO per $\alpha$.
• Price impact due to slippage: As a result of slippage, unstaking 20 $\alpha$ changed the $\alpha$ price: it decreased from 0.225 TAO to 0.113 TAO. All else being equal, unstaking from a subnet leads to a decrease in its $\alpha$ token's relative price.

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Local weight or TAO Equiv

A hotkey's stake, i.e., stake(α), represents subnet-specific stake. As a result, a hotkey's stake share can only represent the validator's staking power within the subnet. Moreover, an $\alpha$ token of a subnet is not fungible with an $\alpha$ 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 ($\alpha$) = 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 TAO reserve

In any subnet, if you add up the local weights of all alpha holders, you will always get exactly the TAO reserve ($\tau_{in}$) of that subnet.

This means:

• The total voting power in a subnet is fixed by its TAO reserve ($\tau_{in}$).
• For a given $\tau_{in}$, if a validator's hotkey in a subnet increases their local weight, it must come at the expense of others.

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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 $\alpha$ 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 the subnet-specific $\alpha$ 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.

Why global stake matters

Also see Why global stake matters.

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 $\alpha$
  • Validator's stake share = 30% × 50,000 = 15,000 $\alpha$
  • 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 $\alpha$
  • Validator's stake share = 40% × 80,000 = 32,000 $\alpha$
  • 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 $\alpha$
  • Validator's stake share = 20% × 30,000 = 6,000 $\alpha$
  • 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

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 description. This can change every block.

Compare with TAO Equiv

Whereas the TAO Equiv indicates a validator's local stake weight, this Exchange Value shows TAO you will receive if you unstake now.

Swap

This is the actual τ you will receive, after subtracting in the slippage charge, if you unstake from this hotkey now on this subnet. The slippage is calculated as:

$$\text{Slippage} = 1 - \frac{\text{Swap(α → τ)}}{\text{Exchange Value}(α \times τ/α)}$$

This can change every block.

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The Emissions section is preliminary

The below Emissions section is still not settled yet. Proceed with caution.

Emissions

Liquidity provision

Emissions are the liquidity injections into the subnet pools. The Bittensor blockchain is the liquidity provider in the dynamic TAO. Without the liquidity provider injecting reserves into a subnet pool, the pool may run out of the reserves of one or both the tokens, thereby halting the entire subnet pool operation. This mechanism of liquidity injection is normal.

While staking and unstaking operations are similar to trading on a pool, the emissions are liquidity providing operations.

Emissions change the constant product

Emissions into a subnet pool do change the constant product $k$ for that pool. On the contrary, the staking and unstaking operations do not change the pool's constant product $k$. See Staking and Unstaking.

Emissions algorithm

The below table shows how emissions are injected into the subnet pools.

Alpha price is always relative price

Note that the terms relative price, alpha token's price, alpha price are the same as Rate.

Every block, do this	If (sum) $\geqslant$ 1	Else (If (sum) $\lt$ 1)
Evaluate sum of all alpha prices	Alpha prices are high across the Bittensor network	Alpha prices are not high
Emission into tao_in reserve	Do nothing	Add a fraction of a TAO into TAO reserve. $\text{Fraction = } \frac{\text{this subnet's TAO reserve}}{\text{sum of all TAO reserves across all subnets}}$
Emission into alpha_in reserve	Add one alpha token into alpha reserve	Do nothing
Effect of emissions into pool	Increases this subnet pool’s alpha reserve, thereby decreasing this alpha token’s price	Increases this subnet pool's TAO reserve, thereby increasing this alpha token's price
Emission into alpha_out	Add one alpha token into the subnet alpha outstanding	Add one alpha token into the subnet alpha outstanding

Proportional TAO emissions vs. Flat alpha emissions

From the above table, we can see that while TAO emissions into the pool are proportional to each subnet's TAO reserve, the alpha emissions into the pool are flat 1 alpha per block. Furthermore, emissions into alpha outstanding are flat 1 alpha per block regardless of sum of prices.

See the below diagram showing how emissions flow:

Example

Consider the following subnets:

Case 1: Sum of alpha prices greater than 1

1. Gaming Subnet:

   • TAO reserve (τ_in): 10,000 TAO
   • Alpha reserve (α_in): 8,000 Alpha
   • Alpha outstanding (α_out): 50,000 Alpha
   • Hence alpha price = $R = \frac{\tau_{in}}{\alpha_{in}} = \frac{10000}{8000} = 1.25$

2. AI Subnet:

   • TAO reserve (τ_in): 15,000 TAO
   • Alpha reserve (α_in): 15,000 Alpha
   • Alpha outstanding (α_out): 80,000 Alpha
   • Hence alpha price = $R = \frac{\tau_{in}}{\alpha_{in}} = \frac{15000}{15000} = 1.0$

3. Storage Subnet:

   • TAO reserve (τ_in): 5,000 TAO
   • Alpha reserve (α_in): 10,000 Alpha
   • Alpha outstanding (α_out): 30,000 Alpha
   • Hence alpha price = $R = \frac{\tau_{in}}{\alpha_{in}} = \frac{5000}{10000} = 0.5$

Hence, sum of all alpha prices $= 1.25 + 1.0 + 0.5 = 2.75 \gt 1.0$

As a result, the emissions to the subnets are as follows:

1. Emissions to the Gaming Subnet:

   • Emission into TAO reserve (τ_in): 0 TAO
   • Emission into alpha reserve (α_in): 1.0 Alpha
   • Emission into alpha outstanding (α_out): 1.0 Alpha

2. Emissions to the AI Subnet:

   • Emission into TAO reserve (τ_in): 0 TAO
   • Emission into alpha reserve (α_in): 1.0 Alpha
   • Emission into alpha outstanding (α_out): 1.0 Alpha

3. Emissions to the Storage Subnet:

   • Emission into TAO reserve (τ_in): 0 TAO
   • Emission into alpha reserve (α_in): 1.0 Alpha
   • Emission into alpha outstanding (α_out): 1.0 Alpha

Case 2: Sum of alpha prices less than 1

Let's change the state of the subnets to the below:

1. Gaming Subnet:

   • TAO reserve (τ_in): 10,000 TAO
   • Alpha reserve (α_in): 40,000 Alpha
   • Alpha outstanding (α_out): 50,000 Alpha
   • Hence alpha price = $R = \frac{\tau_{in}}{\alpha_{in}} = \frac{10000}{40000} = 0.25$

2. AI Subnet:

   • TAO reserve (τ_in): 15,000 TAO
   • Alpha reserve (α_in): 50,000 Alpha
   • Alpha outstanding (α_out): 80,000 Alpha
   • Hence alpha price = $R = \frac{\tau_{in}}{\alpha_{in}} = \frac{15000}{50000} = 0.3$

3. Storage Subnet:

   • TAO reserve (τ_in): 5,000 TAO
   • Alpha reserve (α_in): 25,000 Alpha
   • Alpha outstanding (α_out): 30,000 Alpha
   • Hence alpha price = $R = \frac{\tau_{in}}{\alpha_{in}} = \frac{5000}{25000} = 0.2$

Hence, sum of all alpha prices $= 0.25 + 0.3 + 0.2 = 0.75 \lt 1.0$

As a result, the emissions to the subnets are as follows:

1. Total TAO reserves across all 3 subnets: 10,000 + 15,000 + 5,000 = 30,000 TAO

2. Emissions to the Gaming Subnet:

   • Emission into TAO reserve (τ_in): $\; \frac{10,000}{30,000} = 0.3333 \text{ TAO}$
   • Emission into alpha reserve (α_in): 0 Alpha
   • Emission into alpha outstanding (α_out): 1.0 Alpha
   • New $k$ = (10,000 + 0.3333) × 40,000 = 400,013,320

3. Emissions to the AI Subnet:

   • Emission into TAO reserve (τ_in): $\; \frac{15,000}{30,000} = 0.5 \text{ TAO}$
   • Emission into alpha reserve (α_in): 0 Alpha
   • Emission into alpha outstanding (α_out): 1.0 Alpha
   • New $k$ = (15,000 + 0.5) × 50,000 = 750,025,000

4. Emissions to the Storage Subnet:

   • Emission into TAO reserve (τ_in): $\; \frac{5,000}{30,000} = 0.1667 \text{ TAO}$
   • Emission into alpha reserve (α_in): 0 Alpha
   • Emission into alpha outstanding (α_out): 1.0 Alpha
   • New $k$ = (5,000 + 0.1667) × 25,000 = 125,004,175

• Note that all the TAO emissions in the block sum to 1.0 (0.3333 + 0.5 + 0.1667).
• Also note that the emissions change the constant product $k$ for each subnet.

Emission (alpha)

Represents the portion of the one $\alpha$ per block emission into a subnet that is received by this hotkey in this subnet, according to YC2 in the subnet. This can change every block.

Subnet zero does not receive any emissions

Subnet zero does not receive any emissions, either of TAO or $\alpha$. This is consistent with the fact that subnet zero does not have a pool associated with it.

Emission (TAO)

Shows how a fraction of 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.

Subnet zero does not receive any emissions

Subnet zero does not receive any emissions, either of TAO or $\alpha$. This is consistent with the fact that subnet zero does not have a pool associated with it.

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Tempo

The tempo status of the subnet. Represented as $\frac{k}{n}$ 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.

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Intuitions of dynamic TAO

This section presents a summary of intuitions about dynamic TAO. It is intended to help in conceptually grasping the dynamic TAO mechanism.

TAO and alpha

• The entire TAO token emitted from the coinbase in the lifetime of Bittensor network exists:
  • In coldkeys: In a TAO holder’s own coldkeys, for example, as subnet zero emissions into a subnet owner’s coldkey, and in exchange-owned coldkeys.
  • In subnet zero hotkeys: In the validator hotkeys of subnet zero as staked TAO and dividends.
  • In TAO reserves of subnet pools.
  • TAO exists nowhere else.
  • Furthermore, for the Bittensor network operations, only staked TAO matters, i.e., only the staked TAO in the validator hotkeys and in TAO reserves act as fuel in the Bittensor network operations. This means that for any discussions on consensus power, only the sum total of the entire staked TAO is considered—the TAO that is in the coldkeys or in exchanges does not contribute directly to Bittensor network operations.
• Similarly, the entire $\alpha$ token emitted from the coinbase in the lifetime of Bittensor network exists as follows:
  • An $\alpha$ token does not exist in a coldkey. It only exists in the hotkeys of subnet validators, subnet miners and subnet owners.
  • In the subnet pool reserves.
  • A $\alpha$ token of one subnet is not fungible with a $\alpha$ token of another subnet.

Staking and unstaking

• Staking and unstaking operations do not change the constant product $k$. On the contrary, emissions into a subnet pool do change the constant product $k$. Also see Emissions.
• Staking and unstaking operations are always local to a subnet and its subnet pool.
• Stake is always held in $\alpha$ token denominations. Furthermore, this $\alpha$ stake exists only in the subnet validator hotkeys.

Constant product vs. relative price

The constant product $k$ is a critical concept to understand how the relative price of an alpha token works. See the following:

• A subnet pool algorithm operates purely on the basis of maintaining the constant product $k$. The pool algorithm does not have any built-in mechanism to either target or maintain specific prices for the alpha tokens of a subnet.
• The relative price of an alpha token is only indirectly determined by the ratio of the token reserves. Furthermore, controlling these relative prices is not the goal of the pool algorithm.
• Hence, the relative price of a token is a result, not a target, of the token exchange activity.

Pool reserves

• No one directly owns the subnet pool reserves. These reserves exist to provide liquidity to the subnet pools.
• However, as we saw in Local weights vs TAO reserve , a validator who holds X% of the $\alpha$ stake in a subnet is said to own the same X% of the TAO reserve pool of that subnet.
• In dynamic TAO the sum of α_in (also called alpha reserve) and α_out (all the alpha stake in the subnet) is treated as the the sum total of all alpha. This is not not strictly true. This sum total only represents the alpha associated with staking and unstaking. There is the another source of alpha, which is the emissions alpha. This emissions alpha is awarded to validators, subnet owners and miners.
• More important, these emissions alpha bypasses the subnet pool and gets into the subnet directly from the coinbase. Furthermore, the only way for this emissions alpha to come out of the subnet is via unstaking, even though they did not participate in staking.

Subnet zero

• Even though subnet zero does not run any incentive mechanism, a hotkey can be registered in subnet zero and can accept stake TAO. This stake TAO can be either from a TAO holder or from another hotkey.

Why global stake matters

When a validator’s stake is global it protects the Bittensor network much better. It does so by making it hard for a rogue validator (or for a cabal of rogue validators) to acquire 51% of the consensus voting power. Here is a simple example showing how it works:

Let's say we have 52 subnets and hypothetically a total 52,000 staked TAO tokens distributed amongst all the validators across these 52 subnets.

• When the stake is global, every validator’s stake is 100% global, hence every validator’s stake will appear in every subnet. A rogue validator would have to hold at least 51% of the total staked TAO, i.e., at least 26,500 TAO (51% of 52,000 TAO), to take control of the consensus power. This consensus power would appear in every subnet, giving the rogue validator a control over all the subnets.

• Let’s now make stake 100% local. This means that only the stake the validator has in a subnet is applicable for that subnet. This validator's stake in other subnets is not taken into account in this subnet. For simplicity, assume that all the validators’ stake is evenly spread among these 52 subnets. Then each subnet will have 1000 TAO tokens (52,000/52) as a combined stake of its validators.

• But notice this: In this case, when the stake is 100% local, a validator in a subnet only needs 500 TAO tokens + 1 to take over that subnet. Hence shifting stake from 100% global to 100% local has resulted in severely degrading the security of subnets.

As we saw in Validator stake weight, the dynamic TAO scheme makes use of both global and local characteristics of the stake to strengthen the overall consensus security of the Bittensor network.