Received: from sog-mx-3.v43.ch3.sourceforge.com ([172.29.43.193] helo=mx.sourceforge.net) by sfs-ml-1.v29.ch3.sourceforge.com with esmtp (Exim 4.76) (envelope-from ) id 1VeWFR-0005R8-JT for bitcoin-development@lists.sourceforge.net; Thu, 07 Nov 2013 20:32:13 +0000 Received-SPF: pass (sog-mx-3.v43.ch3.sourceforge.com: domain of petertodd.org designates 62.13.148.107 as permitted sender) client-ip=62.13.148.107; envelope-from=pete@petertodd.org; helo=outmail148107.authsmtp.com; Received: from outmail148107.authsmtp.com ([62.13.148.107]) by sog-mx-3.v43.ch3.sourceforge.com with esmtp (Exim 4.76) id 1VeWFP-0007Ro-OF for bitcoin-development@lists.sourceforge.net; Thu, 07 Nov 2013 20:32:13 +0000 Received: from mail-c235.authsmtp.com (mail-c235.authsmtp.com [62.13.128.235]) by punt12.authsmtp.com (8.14.2/8.14.2) with ESMTP id rA7KVVeS097485; Thu, 7 Nov 2013 20:31:31 GMT Received: from petertodd.org (petertodd.org [174.129.28.249]) (authenticated bits=128) by mail.authsmtp.com (8.14.2/8.14.2/) with ESMTP id rA7KVOx3007816 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES128-SHA bits=128 verify=NO); Thu, 7 Nov 2013 20:31:26 GMT Date: Thu, 7 Nov 2013 15:31:23 -0500 From: Peter Todd To: Michael Gronager Message-ID: <20131107203123.GB3805@petertodd.org> References: <527B9F9B.4060808@ceptacle.com> MIME-Version: 1.0 Content-Type: multipart/signed; micalg=pgp-sha256; protocol="application/pgp-signature"; boundary="8GpibOaaTibBMecb" Content-Disposition: inline In-Reply-To: <527B9F9B.4060808@ceptacle.com> User-Agent: Mutt/1.5.21 (2010-09-15) X-Server-Quench: 93ce76cf-47eb-11e3-b802-002590a15da7 X-AuthReport-Spam: If SPAM / abuse - report it at: http://www.authsmtp.com/abuse X-AuthRoute: OCd2Yg0TA1ZNQRgX IjsJECJaVQIpKltL GxAVKBZePFsRUQkR aAdMdgUUFloCAgsB AmUbWVReU1x7W2Q7 ag1VcwRfa1RMVxto VEFWR1pVCwQmQ20G fXhhGhpydwJOeXg+ Z0ZmXnMVVUcpIEF7 ERhJEWpTN3phaTUc TRJQdwFJcANIexZF O1F6ACIKLwdSbGoL NQ4vNDcwO3BTJTpY RgYVKF8UXXNDOTh0 WxAPVTI3AUwATiF7 JRghIV1UB0cDPw07 N1dpRlIJUVcOEARY BEhGHC5eIREPSicw RQpAUAZOWCFcRCpH SgEvJVdMBSMaGi1C TFdfQhcUAihZUS5O U34NGjEsCVUyb14t NWwVMA5wcwAA X-Authentic-SMTP: 61633532353630.1023:706 X-AuthFastPath: 0 (Was 255) X-AuthSMTP-Origin: 174.129.28.249/587 X-AuthVirus-Status: No virus detected - but ensure you scan with your own anti-virus system. 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[URIs: mail-archive.com] X-Headers-End: 1VeWFP-0007Ro-OF Cc: Bitcoin Dev Subject: Re: [Bitcoin-development] On the optimal block size and why transaction fees are 8 times too low (or transactions 8 times too big) X-BeenThere: bitcoin-development@lists.sourceforge.net X-Mailman-Version: 2.1.9 Precedence: list List-Id: List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , X-List-Received-Date: Thu, 07 Nov 2013 20:32:13 -0000 --8GpibOaaTibBMecb Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: quoted-printable > Final conclusions is that the fee currently is too small and that there > is no need to keep a maximum block size, the fork probability will > automatically provide an incentive to not let block grows into infinity. Your definition of P_fork is inaccurate for a miner with non-negligable hashing power - a miner will never fork themselves. Taking that into account we have three outcomes: 1) The block propagates without any other miner finding a block. 2) During propagation another miner finds a block. (tie) 2.1) You win the tie by finding another block. 2.2) You lose the tie because someone else finds a block. We will define t_prop as the time it takes for a block to propagate from you to 100% of the hashing power, and as a simplifying assumption we will assume that until t_prop has elapsed, 0% of the hashing power has the block, and immedately after, 100% has the block. We will also define t_int, the average interval between blocks. (600 seconds for Bitcoin) Finally, we will define Q as the probability that you will find the next block. The probabilities of the various outcomes: 1) 1 - (t_prop/t_int * (1-Q)) 2) t_prop/t_int * (1-Q) 2.1) Q 2.2) 1-Q Note that to simplify the equations we have not taking into account propagation in our calculations for outcomes 2.1 or 2.2 Thus we can define P_fork taking into account Q: P_fork(Q) =3D (t_prop/t_int * (1-Q))(1-Q) =3D t_pop/t_int * (1-Q)^2 Over the range 0 < Q < 0.5 the probability of a fork decreases approximately linearly as your hashing power increases: d/dq P_fork(Q) =3D 2(Q-1) Q=3D0 -> d/dq P_fork(Q) =3D -2 Q=3D1/2 -> d/dq P_fork(Q) =3D -1 With our new, more accurate, P_fork(Q) function lets re-calculate the break-even fee/KB using your original approach: t_prop =3D t_0 + \alpha*S E_fee =3D f*S E(Q) =3D Q*(1 - P_fork(Q))*(E_bounty + E_fee) E(Q) =3D Q*[1 - (t_0 + k*S)/t_int * (1-Q)^2]*(E_B + f*S) d/dS E(Q) =3D Q*[ -2fSk/t_int*(1-Q)^2 - f*t_0/t_int*(1-Q)^2 + f - E_b*k/t_i= nt*(1-Q)^2 ] Again, we want to choose the fee so that the more transactions we include the more we earn, dE/dS > 0 We find the minimum fee to include a transaction at all by setting S=3D0, thus we get: d/dS E(Q, S=3D0) =3D Q*[ f - f*t_0/t_int*(1-Q)^2 - E_b*k/t_int*(1-Q)^2 ] > 0 f(1 - t_0/t_int*(1-Q)^2) > E_b*k/t_int*(1-Q)^2 f > [E_b*k/t_int(1-Q)^2] / [1 - t_0/t_int*(1-Q)^2] f > [E_b*k*(1-Q)^2] / [t_int - t_0*(1-Q)^2] With Q=3D0: f > E_b*k / (t_int - t_0) ~ E_b*k/t_int This is the same result you derived. However lets look at Q !=3D 0: df/dQ =3D 2*E_b*k * [t_int*(q-1)] / [t_int - t_0(q-1)^2]^2 With negligible latency we get: df/dQ, t_0=3D0 =3D 2*E_b*k*(q-1)/t_int So what does that mean? Well in the region 0 < q < 1/2, df/dQ is always negative. In other words, as you get more hashing power, the fee/KB you can charge and still break even decreases linearly because you will never orphan yourself. Lets trythe same assumptions as your first analysis, based on the work by Decker et al Based on the work by Decker et al, lets try to calculate break-even fee/KB for negligible, 10%, 25% and 40% hashing power: t_0 =3D 10s t_int =3D 600s k =3D 80ms/kB E_b =3D 25BTC Q=3D0 -> f =3D 0.0033 BTC/kB Q=3D0.1 -> f =3D 0.0027 BTC/kB Q=3D0.25 -> f =3D 0.0018 BTC/kB Q=3D0.40 -> f =3D 0.0012 BTC/kB Let's assume every miner is directly peered with every other miner, each of those connections is 1MB/s, and somehow there's no latency at all: k =3D 1mS/kB Q=3D0 -> f =3D 0.000042 BTC/kB Q=3D0.1 -> f =3D 0.000034 BTC/kB Q=3D0.25 -> f =3D 0.000023 BTC/kB Q=3D0.40 -> f =3D 0.000015 BTC/kB Regardless of how you play around with the parameters, being a larger miner has a significant advantage because you can charge lower fees for your transactions and therefor earn more money. But it gets even more ugly when you take into account that maybe a guy with 0.1% hashing power can't afford the high bandwidth, low-latency, internet connection that the larger pool has: k =3D 10mS/kB, t_0=3D5s, Q=3D0.01 -> 0.000411 BTC/KB k =3D 1mS/kB, t_0=3D1s, Q=3D0.15 -> 0.000030 BTC/KB So the 1% pool has an internet connection capable of 100kB/s to each peer, taking 5s to reach all the hashing power. The 15% pool can do 1MB/s to each peer, taking 1s to reach all the hashing power. This small different means that the 1% pool needs to charge 13.7x more per KB for their transactions to break even! It's a disaster for decentralization. Businesses live and die on percentage points, let alone orders of magnitude differences in cost, and I haven't even taken into account second-order effects like the perverse incentives to publish your blocks to only a minority of hashing power.(1) This problem is inherent to the fundemental design of Bitcoin: regardless of what the blocksize is, or how fast the network is, the current Bitcoin consensus protocol rewards larger mining pools with lower costs per KB to include transactions. It's a fundemental issue. An unlimited blocksize will make the problem even worse by increasing fixed costs, but keeping the blocksize at 1MB forever doesn't solve the underlying problem either as the inflation subsidy becomes less important and fees more important. 1) http://www.mail-archive.com/bitcoin-development@lists.sourceforge.net/ms= g03200.html --=20 'peter'[:-1]@petertodd.org 00000000000000054eeccf3ac454892457bf4919d78efb275efd2ddd1a920c99 --8GpibOaaTibBMecb Content-Type: application/pgp-signature; name="signature.asc" Content-Description: Digital signature -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.11 (GNU/Linux) iQEcBAEBCAAGBQJSe/ibAAoJEBmcgzuo5/CFiLAH/Rq/O7J61kSi4WZOM30DA0EI Hv4jhK3kkT9NGTl2MFgGLtz1eKZC37Y3TNnuVei0Q0muyb2K30fHvk9mAOBXOnbY bSaPoiV4lG06t0HrMUcOD2ISC7VrB/Gm59ScV3eWW7qFlndkC2NeJvJUMrYg3PRN R2XhNwTtPXKEEg0sSmoTJ3RRQ8FX3fcXrFiWrXBK2KZWJDtLcK81Ua4npuQdjaON WI3ozXFsPK+ptDTn8bSouWPYg95FYSMLTA5xoN0tmmrdvz+DzYdhrL6Ium+VYcWS guFMC+OOHp6nLO0KWT7LEUh0dgpRCk6e0l0syPrpUphjLJHpjKxJEkk9pWbX9Pg= =7pgc -----END PGP SIGNATURE----- --8GpibOaaTibBMecb--