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From: Daniele Pinna <daniele.pinna@gmail.com>
Date: Sun, 30 Aug 2015 22:01:00 +0200
Message-ID: <CAEgR2PE35K6kt1sZ0iDu2Y9vv+6+Omgg_n4n3gtMggYuE3YD-g@mail.gmail.com>
To: bitcoin-dev@lists.linuxfoundation.org
Content-Type: multipart/alternative; boundary=001a11c2432cc6113d051e8cc7a9
Subject: [bitcoin-dev] ERRATA CORRIGE + Short Theorem
Precedence: list

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Since my longer post seems to be caught in moderator purgatory I will
rehash its results into this much smaller message. I apologize for the
spamming.

I present a theorem whose thesis is obvious to many.

*THESIS: All hashrates* *h' > h generate a revenue per unit of hash v' >
v. *

Let us absurdly[1] assume that an optimal hashrate *h* exists where the
average revenue for each hash in service is maximized. This will result
from perpetually mining blocks of size *q,* is *v. *All larger hashrates *h'
> h* will generate an average revenue per hash *v' < v*(effectively the
conclusion of my paper) due to the higher orphan risk carried from having
to mine blocks of size *q' > q*. Leading from Peter's model and my
analysis, the origin of this balance lies in the fact that larger miners
must somehow be forced to mine larger blocks which in turn carry a larger
orphan risk.

What happens if a large miner *h'* chooses not to mine his optimal block
size *q' *in favor of a seemingly "sub-optimal" block size* q*?
Since he mines a block of identical size as the smaller miner, they will
both carry identical orphan risks[2], and win identical
amounts*R+M(q)* whenever
they successfully mine a block. Since the larger miner can statistically
expect to win *h'/h* more blocks than the smaller miner, they will each
earn an identical revenue per unit of hash *R+M(q)/h*.

This however directly contradicts the assumption that an optimal hashrate
exists beyond which the revenue per unit of hash *v' < v*if  *h' > h. *
*Q.E.D *

This theorem in turn implies the following corollary:

*COROLLARY: **The marginal profit curve is a monotonically increasing of
miner hashrate.*

This simple theorem, suggested implicitly by Gmaxwell disproves any and all
conclusions of my work. Most importantly, centralization pressures will
always be present.

Furthermore,

[1] https://en.wikipedia.org/wiki/Reductio_ad_absurdum
[2] Orphan risks will actually favor the larger hashrate miner leading to
greater revenues per unit of hash.

I thank the dev-list for its valuable time and exchange on the subject
matter. I stand by for any further comments and questions.

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<div dir=3D"ltr">Since my longer post seems to be caught in moderator purga=
tory I will rehash its results into this much smaller message. I apologize =
for the spamming.<br><br>I present a theorem whose thesis is obvious to man=
y.<br><br><div style=3D"font-size:12.8000001907349px"><b>THESIS: All hashra=
tes</b>=C2=A0<span style=3D"font-size:12.8000001907349px"><b><i>h&#39; &gt;=
 h=C2=A0</i>generate a revenue per unit of hash<i>=C2=A0v&#39; &gt; v.=C2=
=A0</i></b></span></div><div style=3D"font-size:12.8000001907349px"><br></d=
iv><div style=3D"font-size:12.8000001907349px">Let us absurdly[1] assume th=
at an optimal hashrate=C2=A0<i><b>h</b></i><b><i>=C2=A0</i></b>exists where=
 the average revenue for each hash in service is maximized. This will resul=
t from perpetually mining blocks of size=C2=A0<i><b>q,</b></i>=C2=A0is=C2=
=A0<i><b>v.=C2=A0</b></i>All larger hashrates=C2=A0<i><b>h&#39; &gt; h</b><=
/i>=C2=A0will=C2=A0<span style=3D"font-size:12.8000001907349px">generate</s=
pan><span style=3D"font-size:12.8000001907349px">=C2=A0an average revenue p=
er hash=C2=A0</span><i style=3D"font-size:12.8000001907349px;font-weight:bo=
ld">v&#39; &lt; v</i><span style=3D"font-size:12.8000001907349px">(effectiv=
ely the conclusion of my paper) due to the higher orphan risk carried from =
having to mine blocks of size=C2=A0</span><i style=3D"font-size:12.80000019=
07349px"><b>q&#39; &gt; q</b></i><span style=3D"font-size:12.8000001907349p=
x">. Leading from Peter&#39;s model and my analysis, the origin of this bal=
ance lies in the fact that larger miners must somehow be forced to mine lar=
ger blocks which in turn carry a larger orphan risk.=C2=A0</span></div><div=
 style=3D"font-size:12.8000001907349px"><br></div><div style=3D"font-size:1=
2.8000001907349px">What happens if a large miner=C2=A0<i><b>h&#39;</b></i>=
=C2=A0chooses not to mine his optimal block size=C2=A0<i style=3D"font-weig=
ht:bold">q&#39;=C2=A0</i>in favor of a seemingly &quot;sub-optimal&quot; bl=
ock size<i style=3D"font-weight:bold">=C2=A0q</i>?</div><div class=3D"gmail=
_extra" style=3D"font-size:12.8000001907349px">Since he mines a block of id=
entical size as the smaller miner, they will both carry identical orphan ri=
sks[2], and win identical amounts<i><b>R+M(q)</b></i>=C2=A0whenever they su=
ccessfully mine a block. Since the larger miner can statistically expect to=
 win=C2=A0<i><b>h&#39;/h</b></i>=C2=A0more blocks than the smaller miner, t=
hey will each earn an identical revenue per unit of hash=C2=A0<i><b>R+M(q)/=
h</b></i>.=C2=A0</div><div class=3D"gmail_extra" style=3D"font-size:12.8000=
001907349px"><br></div><div class=3D"gmail_extra" style=3D"font-size:12.800=
0001907349px">This however directly contradicts the assumption that an opti=
mal hashrate exists beyond which the revenue per unit of hash=C2=A0<i style=
=3D"font-size:12.8000001907349px;font-weight:bold">v&#39; &lt; v</i><span s=
tyle=3D"font-size:12.8000001907349px">if</span><i style=3D"font-size:12.800=
0001907349px;font-weight:bold">=C2=A0</i><span style=3D"font-size:12.800000=
1907349px">=C2=A0</span><i style=3D"font-size:12.8000001907349px"><b>h&#39;=
 &gt; h.=C2=A0</b></i></div><div class=3D"gmail_extra" style=3D"font-size:1=
2.8000001907349px"><i style=3D"font-size:12.8000001907349px"><b>Q.E.D=C2=A0=
</b></i></div><div class=3D"gmail_extra" style=3D"font-size:12.800000190734=
9px"><i style=3D"font-size:12.8000001907349px"><b><br></b></i></div><div cl=
ass=3D"gmail_extra" style=3D"font-size:12.8000001907349px"><span style=3D"f=
ont-size:12.8000001907349px">This theorem in turn implies the following cor=
ollary:</span></div><div class=3D"gmail_extra" style=3D"font-size:12.800000=
1907349px"><i style=3D"font-size:12.8000001907349px"><b><br></b></i></div><=
div class=3D"gmail_extra" style=3D"font-size:12.8000001907349px"><span styl=
e=3D"font-size:12.8000001907349px"><b>COROLLARY:=C2=A0</b></span><span styl=
e=3D"font-size:12.8000001907349px"><b>The marginal profit curve is a monoto=
nically increasing of miner hashrate.</b></span></div><div class=3D"gmail_e=
xtra" style=3D"font-size:12.8000001907349px"><span style=3D"font-size:12.80=
00001907349px"><b><br></b></span></div><div class=3D"gmail_extra" style=3D"=
font-size:12.8000001907349px"><span style=3D"font-size:12.8000001907349px">=
This simple theorem, suggested implicitly by Gmaxwell disproves any and all=
 conclusions of my work. Most importantly, centralization pressures will al=
ways be present.=C2=A0</span></div><div class=3D"gmail_extra" style=3D"font=
-size:12.8000001907349px"><span style=3D"font-size:12.8000001907349px"><br>=
</span></div><div class=3D"gmail_extra" style=3D"font-size:12.8000001907349=
px"><span style=3D"font-size:12.8000001907349px">Furthermore,=C2=A0</span><=
/div><div class=3D"gmail_extra" style=3D"font-size:12.8000001907349px"><spa=
n style=3D"font-size:12.8000001907349px"><b><br></b></span></div><div class=
=3D"gmail_extra" style=3D"font-size:12.8000001907349px">[1]=C2=A0<a href=3D=
"https://en.wikipedia.org/wiki/Reductio_ad_absurdum" target=3D"_blank">http=
s://en.wikipedia.org/wiki/Reductio_ad_absurdum</a></div><div class=3D"gmail=
_extra" style=3D"font-size:12.8000001907349px">[2] Orphan risks will actual=
ly favor the larger hashrate miner leading to greater revenues per unit of =
hash.</div><div class=3D"gmail_extra" style=3D"font-size:12.8000001907349px=
"><br></div><div class=3D"gmail_extra" style=3D"font-size:12.8000001907349p=
x">I thank the dev-list for its valuable time and exchange on the subject m=
atter. I stand by for any further comments and questions.</div><div><div cl=
ass=3D"gmail_signature"><div dir=3D"ltr"><br></div></div></div>
</div>

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