TLDR: In token-based voting systems that are used in Token Curated Registries, a user can split his tokens over the alternatives and get guaranteed rewards without doing any useful work. But a rational user will not choose to do so, because she has a strategy that is more profitable (and just as easy), namely to delegate her votes to an expert.

A token-based coordination game is played between a (possible large) group of agents that cannot communicate or observe each other's actions. Each player has a number of tokens, and, in the game’s simplest form, can distribute these tokens over two choices. The choice that ends up with the most tokens "wins", and each of the players that placed their tokens on the winning option gets a reward that is proportional the amount of his tokens.

Such games can lead groups of self-interested players to make useful decisions: it lies at the basis of Vitalik Buterin’s “Schelling Coin” (a proposal to “crowdsource the truth”) and it is how voting in Token Curated Registries is rewarded.

There is a potential problem with this game, sometimes called the "vote splitting problem", formulated here by Mike Goldin with his usual conciseness:

Because there is no penalty for voting incorrectly, why should otherwise disinterested token holders not split their tokens in every vote and collect a steady revenue stream for doing no work? This amounts to a grief against diligent token voters, whose voting weight is diluted and so go home with less reward for their work.

The problem is posed as a rhetorical question, but I'd argue it has a clear answer: token holders will not split their tokens if they can get a better deal, at a similar cost, by delegating their tokens.

Stakhanov mining the Truth.

A well-behaved voter, let us call him Stakhanov, will optimize his payout by studying the situation to find out where others will most likely place their tokens. This is the “work” that needs to be done, it has a cost c and as a result of that work, Stakhanov assigns a probability, P_c > 0.5, that a certain choice (say choice A) will be the winning choice. Stakhanov has T tokens, each winning token gets a reward, r, and so Stakhanov expects, on average, to get the following reward:

(1) P_c ·T · r − c

Our vote splitter, who is called Oblomov, will put half his tokens on choice A, and half on choice B. So half of the tokens will always be eligible for the reward, and he is guaranteed to be paid as follows:

(2) 0.5 · T · r − c'

A cost parameter c' is added here, because also our vote splitter will have to do some work. For example, in the TCR case, Oblomov will have to pay transaction fees, and there are opportunity costs and monetary risks while the tokens are locked in for voting. In addition, if he indeed intends to turn this into a “steady revenue stream”, Oblomov will likely want to automatize this process, i.e. to delegate the work to some third party (a bot or a person).

Oblomov delegating work.

Note that a comparison between (1) and (2) gives a precise point in which a rational player should switch from vote splitting to doing the work instead: when the difference in cost c − c' is lower than the expected improvement in the total reward from doing that work (P_c − 0.5) · T. This means that the more tokens you have, the more it pays off to do the work.

It is likely that as TCRs become popular, token holders will delegate their vote to token pools where expert active voters (or software programs) vote in their place in exchange for a fee.

Now let's see what happens if Oblomov decides to delegate his votes. Oblomov knows from past performance of this delegate that he predicts the right outcome with a probability P_d > 0.5, and she will take a percentage f of the reward as a fee. So if Oblomov invests c'' in choosing a delegator, she can expect to get the following payout:

(3) P_d · T · (r−f·r) − c''

We can assume that the cost c'' of choosing a delegate and subsequently delegating your vote is similar to the cost c' of setting up a "steady revenue stream" from vote splitting — both will require transferring tokens to some system that will place the votes for him. So that means that it becomes more attractive to delegate rather than split when the following is true:

(4) 0.5 · T · r < P_d · T · (r−f·r)

This condition can be simplified to:

(5) (P_d−0.5) / P_d > f

Meaning that if the delegate does better than chance, and if the delegate charges a reasonable fee that is less than "the percentage that the delegate does better than chance” (the left hand side of the equation), Oblomov will, with the same amount of work, be better off delegating his vote.

All the above has been rather theoretical, of course, but the argument itself is not: vote splitting will not happen when it is more profitable (and just as easy) to delegate your votes.

In the specific context of TCRs, delegating stakes can also lower the barrier to participation by obviating the need to go through the commit and reveal process for the average player.

Note 1: Illustrating the "bounded rationality" of human kind, Oblomov may just as likely do nothing at all.

Note 2: Risk-averse agents may choose the splitting vote strategy anyway, even if it pays out less.