Quote from: smooth on December 17, 2015, 07:21:43 PM

Quote from: monsterer on December 17, 2015, 07:11:27 PM

Quote from: smooth on December 17, 2015, 06:51:40 PM

Your degree of control is superlinear in the amount of your stake.

Very interesting - can you prove it?

It's difficult to say exactly since we can't really consider all exploit strategies nor external factors.

But taking the model of "honest" staking at face value, at 49% you can only hope to maintain control for a limited number of blocks. At 50%+, you control the chain forever. That's clearly more than 4% increase.

Again assuming an "honest" staking model you can consider stake as votes and look to voting coalition models such ShapleyShubik, where staking is viewed as voting between competing chains, and that is trivially superlinear in terms of voting power relative to stake share.

I was thinking of a point like that too. Proportional is clearly not correct.

+1 on the ShapleyShubik reference.

Note that reaffirms the point that the cost in paid only once in PoS instead of unbounded, unless of course PoS devolves to PoW as you contemplate below...

Quote from: smooth on December 17, 2015, 07:21:43 PM

Of course we know that most (all?) PoS systems allow voting on multiple chains, so this may break down. Such systems tend to devolve to PoW though, since stakers are then competing with other stakers to find the combinatorially most-favorable chain state, in which case again control is superlinear in hash power. So I think it is correct.

That is the "nothing-at-stake" issue.

Or competing on propagation and P2P Sybil attack advantages? In any case, still not proportional since distribution of wins is not likely uniform w.r.t. to resources applied given that the longest (or what ever attribute) chain wins in capitulation by the lesser resources, i.e. by definition a chain is asymptotically (some where near majority) a winner-take-all paradigm and it is as you point out non-linear due to Shapley-Shubik.

Astute insight. Thanks. I probably would have thought of that too had I focused on it. I haven't expended much effort thinking in detail about how PoS works in its many variants.