Define requirements of the proposer selection procedure

docs/specification/new-spec/proposer-selection.md

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+# Proposer selection procedure in Tendermint
+
+This document specifies the Proposer Selection Procedure that is used in Tendermint to choose a round proposer. 
+As Tendermint is “leader-based protocol”, the proposer selection is critical for its correct functioning.
+Let denote with `proposer_p(h,r)` a process returned by the Proposer Selection Procedure at the process p, at height h
+and round r. Then the Proposer Selection procedure should fulfill the following properties:
+
+`Agreement`: Given a validator set V, and two honest validators,
+p and q, for each height h, and each round r,
+proposer_p(h,r) = proposer_q(h,r)
+
+`Liveness`: In every consecutive sequence of rounds of size K (K is system parameter), at least a 
+single round has an honest proposer. 
+
+`Fairness`: The proposer selection is proportional to the validator voting power, i.e., a validator with more 
+voting power is selected more frequently, proportional to its power. More precisely, given a set of processes 
+with the total voting power N, during a sequence of rounds of size N, every process is proposer in a number of rounds 
+equal to its voting power. 
+
+We now look at a few particular cases to understand better how fairness should be implemented.
+If we have 4 processes with the following voting power distribution (p0,4), (p1, 2), (p2, 3), (p3, 4) at some round r,  
+we have the following sequence of proposer selections in the following rounds:
+
+`p0, p1, p2, p3, p0, p0, p1, p2, p3, p0, p0, p1, p2, p3, p0, p0, p1, p2, p3, p0, etc`
+
+Let consider now the following scenario where a total voting power of faulty processes is aggregated in a single process
+p0: (p0,3), (p1, 1), (p2, 1), (p3, 1), (p4, 1), (p5, 1), (p6, 1), (p7, 1).
+In this case the sequence of proposer selections looks like this:
+
+`p0, p1, p2, p3, p0, p4, p5, p6, p7, p0, p0, p1, p2, p3, p0, p4, p5, p6, p7, p0, etc`    
+
+In this case, we see that a number of rounds coordinated by a faulty process is proportional to its voting power.
+We consider also the case where we have voting power uniformly distributed among processes, i.e., we have 10 processes 
+each with voting power of 1. And let consider that there are 3 faulty processes with consecutive addresses, 
+for example the first 3 processes are faulty. Then the sequence looks like this:
+
+`p0, p1, p2, p3, p4, p5, p6, p7, p8, p9, p0, p1, p2, p3, p4, p5, p6, p7, p8, p9, etc`
+
+In this case, we have 3 consecutive rounds with a faulty proposer. 
+One special case we consider is the case where a single honest process p0 has most of the voting power, for example:
+(p0,100), (p1, 2), (p2, 3), (p3, 4). Then the sequence of proposer selection looks like this:
+
+p0, p0, p0, p0, p0, p0, p0, p0, p0, p0, p0, p0, p0, p1, p0, p0, p0, p0, p0, etc
+
+This basically means that almost all rounds have the same proposer. But in this case, the process p0 has anyway enough 
+voting power to decide whatever he wants, so the fact that he coordinates almost all rounds seems correct.
+