# Example Staking Dynamics

To illustrate the dynamics of this system, consider a toy scenario with three delegators, Alice, Bob, and Charlie, and two validators, Victoria and William. Tendermint consensus requires at least four validators and no one party controlling more than of the stake, but this example uses only a few parties just to illustrate the dynamics.

For simplicity, the base reward rates and commission rates are fixed over all epochs at and , . The PEN and dPEN holdings of participant are denoted by , , etc., respectively.

Alice starts with dPEN of Victoria’s delegation pool, Bob starts with dPEN of William’s delegation pool, and Charlie starts with unbonded PEN.

• At genesis, Alice, Bob, and Charlie respectively have fractions , , and of the total stake, and fractions , , of the total voting power.

• At epoch , Alice, Bob, and Charlie’s holdings remain unchanged, but their unrealized notional values have changed.

• Victoria charges zero commission, so . Alice’s dPEN(v) is now worth PEN.
• William charges commission, so . Bob’s dPEN(w) is now worth , and William receives PEN.
• William can use the commission to cover expenses, or self-delegate. In this example, we assume that validators self-delegate their entire commission, to illustrate the staking dynamics.
• William self-delegates PEN, to get dPEN in the next epoch, epoch .
• At epoch :

• Alice’s dPEN(v) is now worth PEN.
• Bob’s dPEN(w) is now worth PEN.
• William’s self-delegation of accumulated commission has resulted in dPEN(w).
• Victoria’s delegation pool remains at size dPEN(v). William’s delegation pool has increased to dPEN(w). However, their respective adjustment factors are now and , so the voting powers of their delegation pools are respectively and .
• The slight loss of voting power for William’s delegation pool occurs because William self-delegates rewards with a one epoch delay, thus missing one epoch of compounding.
• Charlie’s unbonded PEN remains unchanged, but its value relative to Alice and Bob’s bonded stake has declined.
• William’s commission transfers stake from Bob, whose voting power has slightly declined relative to Alice’s.
• The distribution of stake between Alice, Bob, Charlie, and William is now , , , respectively. The distribution of voting power is , , , respectively.
• Charlie decides to bond his stake, split evenly between Victoria and William, to get dPEN(v) and dPEN(w).
• At epoch :

• Charlie now has dPEN(v) and dPEN(w), worth PEN.
• For the same amount of unbonded stake, Charlie gets more dPEN(w) than dPEN(v), because the exchange rate prices in the cumulative effect of commission since genesis, but Charlie isn’t charged for commission during the time he didn’t delegate to William.
• William’s commission for this epoch is now PEN, up from PEN in the previous epoch.
• The distribution of stake between Alice, Bob, Charlie, and William is now , , , respectively. Because all stake is now bonded, except William’s commission for this epoch, which is insignificant, the distribution of voting power is identical to the distribution of stake.
• At epoch :

• Alice’s dPEN(v) is now worth PEN.
• Bob’s dPEN(w) is now worth PEN.
• Charlies’s dPEN(v) is now worth PEN, and his dPEN(w) is now worth PEN.
• William’s self-delegation of accumulated commission has resulted in dPEN(w), worth PEN.
• The distribution of stake and voting power between Alice, Bob, Charlie, and William is now , , , respectively.

This scenario was generated with a model in this Google Sheet.