Weighted participation and participation credits

[mensch72]

A larger community might decide to allow all their members to take part in all polls that might affect more than one member, even in those where just part of the community is known to be affected in a significant way, because otherwise it might be difficult to decide who is allowed to take part in what polls, and that decision might not be possible in an objective way and thus might lead to conflict.

To still make sure that in each single poll of such a community, those members of the community that have higher stakes in this poll or have stronger preferences in this poll also have more power in this poll, the community might want to establish a system of weighted participation with participation credits.

In such a system, every member of the community would have a stock of participation credits that they can spend on participation in any poll of the community. When a member spends x participation credits on some poll, they control a share of that poll's total budget that is proportional to x. More precisely, they control a share that equals x divided by the total amount of credits spent by any members on this poll. Once the poll ends, these x participation credits are subtracted from the member's participation credit stock.

At any time, the member may allocate a share of their current participation credit stock in any way to the currently running polls, saving the rest for future polls. in vodle, this would be done in the Running section on the My Polls page via sliders. (These sliders should have a different design than the wap sliders on the poll page, to avoid confusion.) To this end, the Running section would contain a subsection containing all polls of the community, headlined by the name of the community. At the top of that section the current participation credit stock and the current amount of unallocated credits would be listed, and below that all polls belonging to that community, with a slider for controlling how many credits to allocate to this poll. As long as the poll has not ended, this allocation can be changed. Only once the poll ends, the credits are actually "spent" on the poll.

To allow members to participate regularly in polls, each member's participation credit stock grows by one credit per day. To prevent members from saving credits indefinitely and thus gaining potentially too much power in any single poll, the existing participation credit stock will also decay at a rate of 1/100 days. This means that as long as no poll ends to which the member has allocated some credits, their credit stock will grow continuously, but ever more slowly, and will not exceed a limit of 100 credits but will rather approach this limit ever more slowly. If a member spends, for example, roughly one credit every two days, their stock will stabilize at about 50. If they spend roughly five credits a week, their stock will stabilize at about 28, if they spent six credits a week, it will stabilize at about 14, and so on. Obviously, no member can continuously spend more than one credit per day on average.

A newly entering member of the community will initially be given a stock of zero. If the member does not participate for the first month, their stock will have grown to 26 by then, or to 45 after two months, or 60 after three months. This will allow new members to take part fast, without giving them too much power too soon.

[mensch72]

A game-theoretic analysis of this mechanism shows that rational members should allocate more credits to polls in which they have a larger interest and a larger prospective influence per credit, and saving credits depending on what polls they expect to come up in the future.

More precisely, assume a rational member, i, evaluates their stakes in any poll, j, with the value v(i,j), and expects there to be W(j) credits being spent on poll j in total by the members of the community. If poll j still runs for r(j) days, a credit allocated to j now will be decayed to exp(–r(j)) many credits when the poll closes. Thus their expected utility of allocating one credit on poll j now would be u(i,j) := v(i,j) exp(–r(j)) / W(j). The optimal strategy is thus to

• allocate some credits to that poll j which maximizes u(i,j) among the currently running polls,
• allocate no credits at all to any other running poll, and
• save the remaining credits for future polls.

(To optimize the ratio of spent to saved credits is more complicated and depends strongly on their belief about future polls.)

As this optimal allocation of member i depends on the other members' allocations via the numbers W(j), the next step in the analysis is to see whether there exists a strategic (Nash) equilibrium in all members' allocations, and whether this equilibrium is unique. It appears that under almost all circumstances (more precisely: when all values v(i,j) are different), there indeed exists a unique Nash equilibrium. In this Nash equilibrium, the members with the largest share in a poll will typically be those who care the most about this poll, as we hoped when designing this credit mechanism in the first place.

[mensch72]

In our scheme, each credit spent on a poll translates into an equal share in that poll. In other words, to get twice as large a share in the poll, one has to spend twice as many credits. This means the "cost" of participating in a poll in credits is a linear function of the share one wants to get in that poll.

An alternative "cost" function that is discussed much in current theoretical literature is quadratic voting. In that scheme, the cost is a quadratic function. This means, to get twice as large a share, one needs to spend four times as many credits. The rationale for choosing such a cost scheme is that this way it gets harder to acquire very large power in a poll. The downside of quadratic voting is that under that pricing scheme, a rational member would spend at least a few credits on every poll and would have a harder time optimizing the exact amounts to spend on each poll. As a consequence, the members would have to get informed in detail about more different issues to be able to vote effectively in all polls.

Our proposed scheme already contains a different mechanism that limits a single member's power, namely the decaying of credits. I therefore see no reason at the moment to use the more complicated quadratic cost function instead of the proposed linear cost function.

Another problem with nonlinear cost schemes may arise when it is not completely clear whether some issue makes one or two polls. E.g., two yes-no questions could be considered two binary polls or one poll with four options. This decision should not make too much of a difference in terms of optimal credit allocation decisions. So investing one credit into each of the two binary decisions should be equivalent to investing two credits into the alternative four-option poll. A linear cost scheme might seem more natural in this respect. Maybe this can even be formally shown in some way.

[mensch72]

Further analysis of the two-poll, two-faction situation suggests a general issue with participation credits:

• The resulting credit-allocation game might be very difficult to play:
  • Optimal allocation might depend strongly on beliefs about others' allocations
  • These beliefs might be incorrect due to:
    • ignorance of others' stakes
    • ignorance of others' level of rationality / cognitive abilities
    • existence of multiple Nash equilibria
• As a consequence, the resulting behaviour might not conform to a Nash equilibrium, might be hard to predict, and might not realize the theoretical welfare benefits of a participation credit system
• In addition, participants' additional efforts for deciding where to spend their credits might outweigh their spared efforts for deciding how to vote in those polls they do invest in.

[mensch72]

However, those theoretical analyses might also miss important features from the real world:

• In a situation with many polls, if members know that if they were to fully optimize their credit allocation, they would likely spend only few credits on most polls, while that optimization might incur fixed costs for researching others stakes that increase at least linearly with the number of polls considered. As a consequence they will likely first scan all polls very roughly and decide which to spend their cognitive efforts on at all, thereby reducing the number of polls voted in considerably. It seems plausible that this first filtering decision will be mainly based on the member's estimate of their own stakes, and that this estimate correlates strongly with the member's utility difference between their favourite option in that poll and the group-wide benchmark lottery for that poll. Hence it should be expected that regardless of the actual Nash equilibria, members will generally only spend credits on a few polls in which their perceived stakes are highest.
• If this is the main thing we want, one could then use an even more nonlinear vote pricing scheme than the quadratic one: A member can only buy zero or one votes, but no more and no fractional votes. In other words, members only decide which polls to vote in, likely based on their perceived own stakes, and then have one vote in those polls, as now.