# The effect of unconditional speedup on seigniorage

The round exponent adjustment strategy is currently defined in the Highway paper as:


There will likely be a nominal round exponent for the whole network, below which finalization of most blocks would fail. We end up with a repeating cycle which looks like:

1. acceleration
2. failure of more than $C_0$ out of $C_1$ rounds
3. deceleration
4. wait for the next acceleration
5. goto 1

The cycle, illustrated:

|--|--|--|--|--|--|----|----|----|----|----|----|----|
|                 |                                  |
* beginning of    |                                  |
| cycle           |                                  |
|                 * failure of more than C0 out      |
|                 | of C1  rounds and deceleration   |
|                 |                                  |
|                 |      end of cycle & acceleration *
|<-  C1*2^(m-1) ->|                                  |
|<-                      B*2^m                     ->|

where `|` represents the beginning and end of rounds. In the above example, `C_1=6` and `B=10`.

In this case, the number of successful rounds in a cycle is

$$
n_s = \frac{B\times 2^m-C_1\times 2^{m-1}}{2^m} = B - \frac{C_1}{2}
$$

and the number of failed rounds in a cycle is

$$
n_f = C_1.
$$

### Effect on seigniorage

If we mint a total era reward of $R$, then $R\times n_f/(n_s+n_f)$ tokens will be burned.

If we have a seigniorage rate of 2%, we can calculate the net seigniorage simply by multiplying with the coefficient

$$
\frac{n_s}{n_s+n_f} = \frac{2B-C_1}{2B+C_1}.
$$

For $B = 1000$ and $C_1 = 40$, the coefficient is ~0.961, which results in a net seigniorage rate of 2*0.961 ~ 1.92%. The smaller the difference $B-C_1$, the higher the drop in net seigniorage.

I'm guessing that validators would eventually implement their round exponent strategy where they continuously measure latency, determine the nominal round exponent, and not try to accelerate because they'd know it would result in burned block rewards. This is what I'd expect from a rational validator, and I'm not sure how we would prove liveness under such a behavior.

Feedback from Andreas:

Fortunately the only thing required for the liveness proof is:

- We slow down whenever we are unable to finalize a block for a while.
- But there are some other things we want to get out of our round exponent adjustment:
- The adversary shouldn't be able to make us slow down or speed up in a way that moves us too far away from the optimal speed.
- Maybe we want to make the strategy enforceable? (Not necessarily, if the economic incentives are sufficient.)

Apart from that, we are pretty free to improve the strategy. I agree that the unconditional speedup is not ideal!