Editing Validation Pool (section)

== Specification ==

=== Overview ===
The VPSC is the primary component of the [[DAO Governance Framework#DGF workflow|DGF workflow]]. An instantiation of the VPSC is called a Validation Pool. A Validation Pool takes as initial input a new post's address in the Forum, a cash fee, and a type of REP token to be minted. VPSC returns newly minted REP tokens. In addition each owner of REP tokens specific to the domain of the post may encumber their tokens and vote up or down on the post. VPSC redistributes the new REP and the encumbered REP as described below.

Let <math display="inline">R_T</math> denote the total number of reputation tokens in a DAO with <math>n</math> experts <math>e_k</math> for <math>0\leq k\leq n-1</math>. An expert is defined as anyone who owns a REP token. Let <math>R_k</math> denote the number of tokens owned by expert <math>e_k</math>. So <math display="inline">R_T=\sum_{k=1}^{n-1}R_k. </math>
=== Mint new REP ===
When VPSC is called, the fee <math>$f</math> is encumbered, and <math>mf</math> new REP tokens are minted of the type specified. The parameter <math>m</math> is called the ''minting ratio'', which determines the proportional number of REP minted to the fee, as set by the DAO.

=== Vote ===
[[File:VPSCmint.png|thumb|355x355px|The original poster (O.P.) sends their post to the forum associated with the fee. The platform mints new reputation tokens proportional to the fee. ½ of the newly minted tokens are staked for the O.P. as an upvote, ½ are staked as a downvote, then the rest of the experts are allowed to evaluate the post and participate in the betting pool.]]
VPSC then encumbers default <math>c_2=1/2</math> of the newly minted tokens in the poster's name as an upvote, and encumbers <math>1-c_2=1/2</math> of the newly minted tokens as a downvote, left unassigned to any owner until the completion of the smart contract. The <math>c_2</math> fraction of the new tokens is thus given to the worker who put their evidence of work in the post. The remaining <math>1-c_2</math> fraction of the new REP tokens is given as a reward for participating in policing.

VPSC then allows each expert <math>e_k</math> to encumber any of their REP tokens (of the proper type) with an input <math>v_k</math> of upvote (<math>v_k=+1</math>) or downvote (<math>v_k=-1</math>) or default abstention (<math>v_k=0</math>). Once the voting period is finished, VPSC counts the votes by calculating the vote total <math display="inline">V_T:=\sum_{k=1}^{n-1}v_k R_k. </math> If <math>V_T\geq 0</math> that means the total REP used to upvote <math display="inline">R_U:=\sum_{ \{ i|v_i>0\}}R_{i} </math> outweighs the total REP used to downvote <math display="inline">R_U\geq R_D:=\sum_{ \{ i|v_i< 0\}}R_{i}. </math> This means <math display="inline">V_T=R_U-R_D. </math> Therefore if <math display="inline">V_T\geq 0</math> the post is validated. If <math>V_T< 0</math> then the downvote wins and the post is invalidated. (The fact that <math display="inline">V_T=0</math> results in validation reflects the almost arbitrary decision that ties will always favor the upvotes.)
[[File:VPSCvoting.png|center|thumb|433x433px|In this example the upvotes on the evidence-of-work post win with 90 tokens against 80 staked as downvotes. The original poster’s (O.P.) newly minted 50 added to a supporter’s 40, against the 50 newly minted downvote tokens added to 30 tokens cast as downvotes by detractors. The losers’ 80 tokens are split between the O.P., who receives 50/90 of the 80 lost tokens, and his supporter, who receives 40/90. Then the post holds a record of how popular it was amongst experts to determine the post’s power as precedence.]]

=== Redistribution: binding vs. non-binding ===
The newly minted tokens on the losing side are distributed to the winning side proportionally. I.e., if upvotes win, each expert voter <math>e_j</math> with <math>v_j>0</math> receives <math display="inline">\frac{R_j}{R_U} mf(1-c_2)</math> of the new tokens. If downvotes win, each voter <math>e_j</math> with <math>v_j<0</math> receives <math display="inline">\frac{R_j}{R_U} mfc_2</math> of the new tokens. In that case, those tokens were taken from the poster whose post was invalidated. This proportional arithmetic ensures that someone who risks twice as many REP tokens on an assertion earns twice as much reward if they win. 

A VPSC can be binding or non-binding. If an instantiation of the VPSC is binding, that means the REP tokens of all those who voted against the winning side will lose the tokens they encumbered to those who won, proportionally. E.g., if the post is validated, then all the REP used to downvote <math display="inline">R_D </math> is redistributed to those who upvoted, meaning each voter <math>e_j</math> with <math>v_j>0</math> receives <math display="inline">\frac{R_j}{R_U} R_D</math> new tokens.

Further, a VPSC can be <math>p</math>-binding for any <math>0 \leq p \leq 1</math>. A <math>p</math>-binding VPSC redistributes the fraction <math>p</math> of the losers' tokens to the winners.

=== REP salary ===
Finally, the Validation Pool takes the fee encumbered from the beginning and distributes it to all DAO members in proportion to their REP token holdings. The proportionate [[Reputation#REP Salary Mechanism|REP salary]] means someone who owns twice as many REP tokens gets twice as much cash reward from the fee. Specifically, expert  <math>e_k</math> who owns <math>R_k</math> tokens gets a reward of <math>fR_k/R_T</math>.