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 <20220711235731.GD20899@erisian.com.au>
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From: Zac Greenwood <zachgrw@gmail.com>
Date: Wed, 13 Jul 2022 20:29:53 +0200
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To: Anthony Towns <aj@erisian.com.au>, 
 Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
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Subject: Re: [bitcoin-dev] Surprisingly, Tail Emission Is Not Inflationary
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> your proof is incorrect (or, rather, relies on a highly unrealistic
assumption)

The assumption that coin are lost ar a constant rate is not required. Tail
emission will asymptotically decrease the rate of inflation to zero, at
which point the increase in coin exactly matches the amount of coin lost.
The rate at which coin are lost is irrelevant.

This is easy to see. Consider no coin are ever lost. The rate of inflation
will slowly decline to zero as the amount of coin grows to infinity.
However, lost coin ensures that the point at which the rate of inflation
becomes zero will be reached sooner.

If a black swan event destroys 90% of all coin, the constant tail emission
will instantly begin to inflate the supply at a 10x higher percentage. The
inflation expressed as a percentage will also immediately start to decline
because each new coin will inflate the total supply with a slightly smaller
percentage than the previous new coin. The rate of inflation will continue
to decline until zero, at which point it again matches the coin-loss
induced deflation rate.

Another scenario. Suppose that the number of coin lost becomes
significantly less for instance because better wallets and a more mature
ecosystem prevent many common coin loss events. A constant issuance of new
coin would increase the total supply, but each new coin would add less to
the total supply when expressed as a percentage. The rate of inflation
would decline to zero, at which point it again has matched the rate of
deflation due to coin loss.

Even when the rate at which coin are lost will not be constant, a tail
emission will tend to an equilibrium.

It must be observed that tail emission causes the total *potential* supply
to vary greatly depending on the deflation rate. In a low-deflation
scenario, the supply will have to grow much larger before an equilibrium
can be reached than in a scenario with moderate deflation rate. Not being
able to predict the ultimate total supply of coin is however seems
undesirable. But is it really?

The rate of inflation required for keeping Bitcoin useful highly depends on
the value of the token. At US$100k, a tail emission of 1 BTC per block
ensures safety within a few blocks for even large amounts. Continuing this
example, 1 BTC per block would mean 5.25m extra coin per 100 years. At 21m
coins and 1 BTC perpetual reward per block, the rate of inflation would be
0.25% per year.

This should put things a bit into perspective.


On Tue, 12 Jul 2022 at 01:58, Anthony Towns via bitcoin-dev <
bitcoin-dev@lists.linuxfoundation.org> wrote:

> On Mon, Jul 11, 2022 at 08:56:04AM -0400, Erik Aronesty via bitcoin-dev
> wrote:
> > > Alternatively, losses could be at a predictable rate that's entirely
> > > different to the one Peter assumes.
> > No, peter only assumes that there *is* a rate.
>
> No, he assumes it's a constant rate. His integration step gives a
> different result if lambda changes with t:
> https://www.wolframalpha.com/input?i=dN%2Fdt+%3D+k+-+lambda%28t%29*N
>
> On Mon, Jul 11, 2022 at 12:59:53PM -0400, Peter Todd via bitcoin-dev wrote:
> > Give me an example of an *actual* inflation rate you expect to see,
> given a
> > disaster of a given magnitude.
>
> All I was doing was saying your proof is incorrect (or, rather, relies
> on a highly unrealistic assumption), since I hadn't seen anybody else
> point that out already.
>
> But even if the proof were correct, I don't think it provides a useful
> mechanism (since there's no reason to think miners gaining all the coins
> lost in a year will be sufficient for anything), and I don't really
> think the "security budget" framework (ie, that the percentage of total
> supply given to miners each year is what's important for security)
> you're implicitly relying on is particularly meaningful.
>
> So no, not particularly interested in diving into it any deeper.
>
> Cheers,
> aj
>
> _______________________________________________
> bitcoin-dev mailing list
> bitcoin-dev@lists.linuxfoundation.org
> https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
>

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Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

<div><div dir=3D"auto">&gt;=C2=A0<span style=3D"color:rgb(0,0,0)">your proo=
f is incorrect (or, rather, relies=C2=A0</span><span style=3D"border-color:=
rgb(0,0,0) rgb(0,0,0) rgb(0,0,0) rgb(204,204,204);color:rgb(0,0,0)">on a hi=
ghly unrealistic assumption)</span></div><div dir=3D"auto"><br></div></div>=
<div><div dir=3D"auto">The assumption that coin are lost ar a constant rate=
 is not required. Tail emission will asymptotically decrease the rate of in=
flation to zero, at which point the increase in coin exactly matches the am=
ount of coin lost. The rate at which coin are lost is irrelevant.</div><div=
 dir=3D"auto"><br></div><div dir=3D"auto">This is easy to see. Consider no =
coin are ever lost. The rate of inflation will slowly decline to zero as th=
e amount of coin grows to infinity. However, lost coin ensures that the poi=
nt at which the rate of inflation becomes zero will be reached sooner.</div=
><div dir=3D"auto"><br></div><div dir=3D"auto">If a black swan event destro=
ys 90% of all coin, the constant tail emission will instantly begin to infl=
ate the supply at a 10x higher percentage. The inflation expressed as a per=
centage will also immediately start to decline because each new coin will i=
nflate the total supply with a slightly smaller percentage than the previou=
s new coin. The rate of inflation will continue to decline until zero, at w=
hich point it again matches the coin-loss induced deflation rate.</div><div=
 dir=3D"auto"><br></div><div dir=3D"auto">Another scenario. Suppose that th=
e number of coin lost becomes significantly less for instance because bette=
r wallets and a more mature ecosystem prevent many common coin loss events.=
 A constant issuance of new coin would increase the total supply, but each =
new coin would add less to the total supply when expressed as a percentage.=
 The rate of inflation would decline to zero, at which point it again has m=
atched the rate of deflation due to coin loss.</div><div dir=3D"auto"><br><=
/div><div dir=3D"auto">Even when the rate at which coin are lost will not b=
e constant, a tail emission will tend to an equilibrium.</div><div dir=3D"a=
uto"><br></div><div dir=3D"auto">It must be observed that tail emission cau=
ses the total *potential* supply to vary greatly depending on the deflation=
 rate. In a low-deflation scenario, the supply will have to grow much large=
r before an equilibrium can be reached than in a scenario with moderate def=
lation rate. Not being able to predict the ultimate total supply of coin is=
 however seems undesirable. But is it really?</div><div dir=3D"auto"><br></=
div><div dir=3D"auto">The rate of inflation required for keeping Bitcoin us=
eful highly depends on the value of the token. At US$100k, a tail emission =
of 1 BTC per block ensures safety within a few blocks for even large amount=
s. Continuing this example, 1 BTC per block would mean 5.25m extra coin per=
 100 years. At 21m coins and 1 BTC perpetual reward per block, the rate of =
inflation would be 0.25% per year.</div><div dir=3D"auto"><br></div><div di=
r=3D"auto">This should put things a bit into perspective.</div></div><div><=
div dir=3D"auto"><br></div><div><br><div class=3D"gmail_quote"><div dir=3D"=
ltr" class=3D"gmail_attr">On Tue, 12 Jul 2022 at 01:58, Anthony Towns via b=
itcoin-dev &lt;<a href=3D"mailto:bitcoin-dev@lists.linuxfoundation.org" tar=
get=3D"_blank">bitcoin-dev@lists.linuxfoundation.org</a>&gt; wrote:<br></di=
v><blockquote class=3D"gmail_quote" style=3D"margin:0px 0px 0px 0.8ex;borde=
r-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color=
:rgb(204,204,204)">On Mon, Jul 11, 2022 at 08:56:04AM -0400, Erik Aronesty =
via bitcoin-dev wrote:<br>
&gt; &gt; Alternatively, losses could be at a predictable rate that&#39;s e=
ntirely<br>
&gt; &gt; different to the one Peter assumes.<br>
&gt; No, peter only assumes that there *is* a rate.<br>
<br>
No, he assumes it&#39;s a constant rate. His integration step gives a<br>
different result if lambda changes with t:<br>
<a href=3D"https://www.wolframalpha.com/input?i=3DdN%2Fdt+%3D+k+-+lambda%28=
t%29*N" rel=3D"noreferrer" target=3D"_blank">https://www.wolframalpha.com/i=
nput?i=3DdN%2Fdt+%3D+k+-+lambda%28t%29*N</a><br>
<br>
On Mon, Jul 11, 2022 at 12:59:53PM -0400, Peter Todd via bitcoin-dev wrote:=
<br>
&gt; Give me an example of an *actual* inflation rate you expect to see, gi=
ven a<br>
&gt; disaster of a given magnitude.<br>
<br>
All I was doing was saying your proof is incorrect (or, rather, relies<br>
on a highly unrealistic assumption), since I hadn&#39;t seen anybody else<b=
r>
point that out already.<br>
<br>
But even if the proof were correct, I don&#39;t think it provides a useful<=
br>
mechanism (since there&#39;s no reason to think miners gaining all the coin=
s<br>
lost in a year will be sufficient for anything), and I don&#39;t really<br>
think the &quot;security budget&quot; framework (ie, that the percentage of=
 total<br>
supply given to miners each year is what&#39;s important for security)<br>
you&#39;re implicitly relying on is particularly meaningful.<br>
<br>
So no, not particularly interested in diving into it any deeper.<br>
<br>
Cheers,<br>
aj<br>
<br>
_______________________________________________<br>
bitcoin-dev mailing list<br>
<a href=3D"mailto:bitcoin-dev@lists.linuxfoundation.org" target=3D"_blank">=
bitcoin-dev@lists.linuxfoundation.org</a><br>
<a href=3D"https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev" =
rel=3D"noreferrer" target=3D"_blank">https://lists.linuxfoundation.org/mail=
man/listinfo/bitcoin-dev</a><br>
</blockquote></div></div>
</div>

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