tl;dr: all that’s needed to observe objective mathematical differences between crowd-goal and patron-goal is (A) pledges are not all the same amount and (B) we have payouts before reaching 100% of the goals.
The reason the games behave differently is: In crowd-based mechanism, players can have only binary 0 or 1 direct impact toward reaching the goal. In dollar-based mechanism, players can have a variable impact toward reaching the goal.
For the two games to be the same, we would have to provide both variable dollar pledge-size option and let each pledge choose whether to count as a partial patron or multiple patrons etc.
After our phone chat this evening, we worked to find some falsifiable situation that would show how the two mechanism variations are different games, not just different framings/labels that would affect real people with emotions and imagination etc.
Well, @mray here’s my example to falsify your assertions of utter sameness in the core math:
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I actually got out 3 dice and rolled them 24 times to get a pretty random pool of 24 patrons who have different size pledges. It came out to 241 total, so I reduced a random pledge by 1 to get an easier round 240 total, average of 10.
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We can consider for this game that this is all the money that exists (but patrons get a fresh allotment of it each month). The patrons have no choices. They just do pledge all their money to the project.
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Here’s the 24 pledge sizes: 13, 12, 10, 10, 10, 8, 7, 11, 7, 12, 10, 6, 13, 10, 9, 9, 8, 12, 9, 9, 14, 16, 9, 8.
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In order to compare at all, we set the project goals to 100% of the available crowd and money from this limited universe.
- Crowd goal is 24 patrons
- Dollar goal is $240
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I rolled a single die to decide how many patrons would join before each of a series of payout points.
The following charts do drop some partial cents, but the rounding issues don’t matter
1st payout: initial 2 patrons
higher than $10 average, so dollar goal gives higher output
Pledges |
crowd-goal % |
crowd-goal donations |
dollar-goal % |
dollar-goal donations |
$13 |
1/12 |
$1.08 |
25/240 |
$1.35 |
$12 |
1/12 |
$1 |
25/240 |
$1.25 |
|
crowd-goal total: |
$2.08 |
dollar-goal total: |
$2.60 |
2nd payout: up to 7 patrons
new pledges bring average back to $10, so outputs are the same
Pledges |
crowd-goal progress |
crowd-goal donations |
dollar-goal progress |
dollar-goal donations |
$13 |
7/24 |
$3.79 |
70/240 = 7/24 |
$3.79 |
$12 |
7/24 |
$3.50 |
70/240 = 7/24 |
$3.50 |
$10 |
7/24 |
$2.92 |
70/240 = 7/24 |
$2.92 |
$10 |
7/24 |
$2.92 |
70/240 = 7/24 |
$2.92 |
$10 |
7/24 |
$2.92 |
70/240 = 7/24 |
$2.92 |
$8 |
7/24 |
$2.33 |
70/240 = 7/24 |
$2.33 |
$7 |
7/24 |
$2.04 |
70/240 = 7/24 |
$2.04 |
|
crowd-goal total: |
$20.42 |
dollar-goal total: |
$20.42 |
3rd payout: up to 11 patrons
continues the $10 average, so still equal
Pledges |
crowd-goal progress |
crowd-goal donations |
dollar-goal progress |
dollar-goal donations |
$13 |
11/24 |
$5.96 |
110/240 = 11/24 |
$5.96 |
$12 |
11/24 |
$5.50 |
110/240 = 11/24 |
$5.50 |
$10 |
11/24 |
$4.58 |
110/240 = 11/24 |
$4.58 |
$10 |
11/24 |
$4.58 |
110/240 = 11/24 |
$4.58 |
$10 |
11/24 |
$4.58 |
110/240 = 11/24 |
$4.58 |
$8 |
11/24 |
$3.67 |
110/240 = 11/24 |
$3.67 |
$7 |
11/24 |
$3.21 |
110/240 = 11/24 |
$3.21 |
$11 |
11/24 |
$5.04 |
110/240 = 11/24 |
$5.04 |
$7 |
11/24 |
$3.21 |
110/240 = 11/24 |
$3.21 |
$12 |
11/24 |
$5.50 |
110/240 = 11/24 |
$5.50 |
$10 |
11/24 |
$4.58 |
110/240 = 11/24 |
$4.58 |
|
crowd-goal total: |
$50.42 |
dollar-goal total: |
$50.42 |
4th payout: up to 14 patrons
average drops below $10, so the crowd goal gets slightly more money
Pledges |
crowd-goal progress |
crowd-goal donations |
dollar-goal progress |
dollar-goal donations |
$13 |
14/24 |
$7.58 |
139/240 |
$7.53 |
$12 |
14/24 |
$7.00 |
139/240 |
$6.95 |
$10 |
14/24 |
$5.83 |
139/240 |
$5.79 |
$10 |
14/24 |
$5.83 |
139/240 |
$5.79 |
$10 |
14/24 |
$5.83 |
139/240 |
$5.79 |
$8 |
14/24 |
$4.67 |
139/240 |
$4.63 |
$7 |
14/24 |
$4.08 |
139/240 |
$4.05 |
$11 |
14/24 |
$6.42 |
139/240 |
$6.37 |
$7 |
14/24 |
$4.08 |
139/240 |
$4.05 |
$12 |
14/24 |
$7.00 |
139/240 |
$6.95 |
$10 |
14/24 |
$5.83 |
139/240 |
$5.79 |
$6 |
14/24 |
$3.50 |
139/240 |
$3.48 |
$13 |
14/24 |
$7.58 |
139/240 |
$7.53 |
$10 |
14/24 |
$5.83 |
139/240 |
$5.79 |
|
crowd-goal total: |
$81.08 |
dollar-goal total: |
$80.50 |
5th payout: up to 20 patrons
The average goes down further below $10, resulting in a greater reduction to the dollar-based total compared to the crowd-based total.
Pledges |
crowd-goal progress |
crowd-goal donations |
dollar-goal progress |
dollar-goal donations |
$13 |
20/24 |
$10.83 |
195/240 |
$10.56 |
$12 |
20/24 |
$10.00 |
195/240 |
$9.75 |
$10 |
20/24 |
$8.33 |
195/240 |
$8.13 |
$10 |
20/24 |
$8.33 |
195/240 |
$8.13 |
$10 |
20/24 |
$8.33 |
195/240 |
$8.13 |
$8 |
20/24 |
$6.67 |
195/240 |
$6.50 |
$7 |
20/24 |
$5.83 |
195/240 |
$5.69 |
$11 |
20/24 |
$9.17 |
195/240 |
$8.94 |
$7 |
20/24 |
$5.83 |
195/240 |
$5.69 |
$12 |
20/24 |
$10.00 |
195/240 |
$9.75 |
$10 |
20/24 |
$8.33 |
195/240 |
$8.13 |
$6 |
20/24 |
$5.00 |
195/240 |
$4.88 |
$13 |
20/24 |
$10.83 |
195/240 |
$10.56 |
$10 |
20/24 |
$8.33 |
195/240 |
$8.13 |
$9 |
20/24 |
$7.50 |
195/240 |
$7.31 |
$9 |
20/24 |
$7.50 |
195/240 |
$7.31 |
$8 |
20/24 |
$6.67 |
195/240 |
$6.50 |
$12 |
20/24 |
$10.00 |
195/240 |
$9.75 |
$9 |
20/24 |
$7.50 |
195/240 |
$7.31 |
$9 |
20/24 |
$7.50 |
195/240 |
$7.31 |
|
crowd-goal total: |
$162.50 |
dollar-goal total: |
$158.44 |
6th payout: the whole crowd has now joined
I don’t need a chart here, both goals are reached, all patrons give their full pledges for $240 total.
The two highest pledgers in the whole list at $14 and $16 come in here to raise the average back to $10 after a couple rounds of being below $10.
Totals over the whole time
- Crowd-based goal: $556.50
- Dollar-based goal: $552.38
Because it randomly happened that more of the payouts had crowds below the total crowd average, the dollar-based mechanism produced slightly reduced total income.
If it worked out the other way, that more of the payout periods were higher than average (i.e. if the most above-average patrons happened to pledge early instead of late), then the difference would be reversed.
Conclusions
These two mechanisms behave differently given identical setup and identical inputs and identical crowds. The only change is which mechanism we use. The only thing necessary to observe a mathematical difference is to have payouts happen before reaching the goal point.
The subtle difference could have been more significant if the variations and clustering of the pledges happened to be more extreme. I used this random dice-roll approach to help illustrate that I was not choosing a particular contrived example just to illustrate some contrived point.
We cannot conclude that one mechanism or the other will increase the total funding. That depends on the particular pattern of pledges over time. If I ran this random-dice experiment a thousand times, the difference would average out so that both mechanisms would get the same totals over all thousand trials combined. But the fact that the mechanisms behave differently is enough to recognize that the behavior can influence patron motivations toward higher or lower pledges without any appeal to framing effects.
How to study the difference via game theory modeling
Once we recognize that the two mechanisms behave differently, we can ask whether this motivates different patron behavior if we allow it. In other words, if we give the patrons an option: “you may pledge all your money, but you don’t have to”, then if as assign them certain motivations, they can make their choices to optimize their goals.
Here is a game-theory instance (far from the only instance possible) which would be analogous to the sorts of game-theory setups used to study prisoner’s dilemma and snowdrift dilemma games:
- Players are specified as having a goal of maximizing funds for the project at the lowest cost to themselves
- We could specify how many points they get for money to the project and how many points they lose for money they spend
Given such defined players, they do indeed have a mathematically factual (not emotional, this could be computer programs, and this is completely unrelated to the meaning of any labels) strategy of minimizing their pledge in crowd-based goals and maximizing their pledge in dollar-based goals.
The strategies for other players would of course be different. Some other games we could set up:
- The player’s goal is to maximize the crowd size
- The player’s goal is to freeride (such players will never pledge ever)
- The player’s goal is to get as much money as possible to the project and has no concern for personal cost (such players will always pledge all their possible money)
The key distinction I made before still holds. If the players know that the crowd will hit the goal no matter what, then their behavior will be the same for either crowd-goal or dollar-goal games. If they do not know when or if the goal will ever be hit and can assume payouts will happen along the way, then their calculations will be different for crowd-goal and dollar-goal games because the games are not the same mathematically.
None of the differences are conclusive in terms of which model is better, primarily because that is a judgment about our mission (do we want larger crowds or more dollars?) and because the behavior of real people and real projects in how they set goals and make pledges will probably be the far more decisive difference in practice.
But I hope we can finally drop the argument about whether the models are exactly the same and it’s only framing. They are not the same. We can accept that and still recognize that the way forward is based on the practical matter of real projects and people and indeed about how we frame things. The mathematical differences aren’t enough to use directly to favor one or the other mechanism.