Demonstration 1 of 4
The participation threshold
How many members must an alert network expect before joining pays?
Utility rises by v for each extra member. The threshold is where the network value covers the part of the fee that stand-alone value does not.
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n is the number of organizations expected to join, out of 80. Each gets stand-alone value 12, pays the fee and gains v for every other member.
Predict first. Cutting the fee from 30 to 24 is a 20 percent cut. Does the threshold of 31 fall by 20 percent too?
Choose an example
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Constructed example: the chapter's hypothetical alert network (80 organizations, stand-alone 12, fee 30, 0.60 per member); fees of 24 and 36 and v = 0.40 are added for comparison.
Calculated values
- Smallest viable membership n
- 31
- Utility at n = 30
- -0.6
- Utility at n = 31
- 0.0
- Utility at n = 70
- 23.4
- Viable within 80 organizations
- yes
Joining needs 12 + 0.60(n - 1) - 30 >= 0, so n - 1 >= (30 - 12) / 0.60 = 30 and n = 31. At 30 members utility is 12 + 0.60(29) - 30 = -0.6, so a marginal organization stays out; at 31 it is 0.0. At 70 members it is 12 + 0.60(69) - 30 = 23.4.
Worked steps
- n - 1 >= (30 - 12) / 0.60 = 30
- Smallest integer n = 31
- U(30) = 12 + 0.60 x 29 - 30 = -0.6
- U(31) = 12 + 0.60 x 30 - 30 = 0.0
- U(70) = 12 + 0.60 x 69 - 30 = 23.4
Use the idea
Size the guaranteed launch cohort from (fee - stand-alone value) / value per member, plus one.
Where the conclusion applies
Every member values every other member equally, expectations are fulfilled and organizations are identical. Content that adds no reachable members does not move the threshold.
Check your understanding: With a fee of 36 and v = 0.60, what is the smallest viable membership?
Chapter 41 source: section "Direct network effects".
Demonstration 2 of 4
Compatibility can reverse a choice
How good must the interface be before the better stand-alone tool wins?
Compatibility adds the other installed base to each tool, which helps the small network more. Once the bases are close, stand-alone quality decides.
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Tool A has 480 users and stand-alone value 16; tool B has 120 users and value 22. Each reachable user adds 0.025. Theta is the share of interactions with the other tool's users that the interface carries, applied to both tools.
Predict first. At theta = 0.30, which tool wins?
Choose an example
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Constructed example: the chapter's hypothetical engineering tools (480 and 120 users, values 16 and 22, 0.025 per user) at theta 0, 0.60 and 1; theta 0.30 and A's symmetric partial access are added.
Calculated values
- Effective base, A
- 480
- Effective base, B
- 120
- U_A
- 28.0
- U_B
- 25.0
- Choice
- Tool A
With 0.00 of cross-system interactions carried, A reaches 480 + 0.00(120) = 480 users and B reaches 120 + 0.00(480) = 120. U_A = 16 + 0.025(480) = 28.0 and U_B = 22 + 0.025(120) = 25.0, so the buyer picks Tool A. B overtakes A once 9 theta > 3, that is theta above 1/3.
Worked steps
- Base A = 480 + 0.00 x 120 = 480
- Base B = 120 + 0.00 x 480 = 120
- U_A = 16 + 0.025 x 480 = 28.0
- U_B = 22 + 0.025 x 120 = 25.0
- Choice: Tool A
Use the idea
Measure how much useful interaction actually crosses an interface before counting the rival's users as reachable.
Where the conclusion applies
Linear value per reachable user and the same interface share in both directions. The book's 0.60 case compares B with incompatible A at 28; giving A the same partial access (29.8) is added here.
Check your understanding: At theta = 0.30, which tool wins, and by how much?
Chapter 41 source: section "Compatibility".
Demonstration 3 of 4
Early leads and tipping
How large an early lead lets the lower-quality format take over?
Each format's value rises with its own adopters, so the side above the crossing attracts the next adopter and the lead grows.
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120 firms choose format A (stand-alone 3) or B (3.3). Each expected adopter of the same format adds k; portability lowers k from 0.012 to 0.004.
Predict first. With portability (k = 0.004), does any split shown here let A lead?
Choose an example
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Constructed example: the chapter's hypothetical accounting formats (120 firms, values 3 and 3.3, k 0.012 and 0.004, splits 75 and 70); the split at 73 is the book's integer tipping point.
Calculated values
- U_A
- 3.90
- U_B
- 3.84
- Crossing n_A
- 72.5
- Adoption moves toward
- A
U_A = 3 + 0.012(75) = 3.90 and U_B = 3.3 + 0.012(45) = 3.84, so adoption moves toward A. Setting 3 + 0.012 n_A = 3.3 + 0.012(120 - n_A) gives n_A = 72.5, so A needs at least 73 expected adopters to lead.
Worked steps
- U_A = 3 + 0.012 x 75 = 3.90
- U_B = 3.3 + 0.012 x 45 = 3.84
- Crossing: 0.024 n_A = 0.3 + 1.44, so n_A = 72.5
- Adoption moves toward A
Use the idea
Compare an early lead with the crossing point before assuming the better product will win.
Where the conclusion applies
Linear same-format benefits, myopic adopters and incompatible formats. Coordination by large adopters can move expectations past the crossing at once.
Check your understanding: Where is the crossing with k = 0.004?
Chapter 41 source: section "Market tipping".
Demonstration 4 of 4
Same receipts, different participation
Do fee pairs with the same projected receipts earn the same profit?
Receipts projected at fixed participation hide the fact that each side's decision depends on the fee it pays and on the other side's presence.
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150 buyers and 30 sellers. A buyer values 3 plus 0.40 per active seller; a seller values 0.20 per active buyer. Serving a buyer costs 2 and a seller 8. A negative seller fee is a payment to sellers.
Predict first. Does the fee pair (16, -17) earn the 1,890 it projects?
Choose an example
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Constructed example: the chapter's hypothetical procurement platform with fee pairs (9, 18), (11, 8) and (16, -17); the pair (13, -2) is added.
Calculated values
- Buyer fee
- 9
- Seller fee
- 18
- Buyer utility
- 6
- Seller utility
- 12
- Active buyers
- 150
- Active sellers
- 30
- Projected receipts
- 1,890
- Profit
- 1,350
Buyer utility is 3 + 0.40(30) - 9 = 6 and seller utility is 0.20(150) - 18 = 12. Both sides join. Projected receipts are 150(9) + 30(18) = 1,890, and profit is 150(9 - 2) + 30(18 - 8) = 1,050 + 300 = 1,350.
Worked steps
- Buyer: 3 + 0.40 x 30 - 9 = 6
- Seller: 0.20 x 150 - 18 = 12
- Projected receipts: 150(9) + 30(18) = 1,890
- Profit: 150(9 - 2) + 30(18 - 8) = 1,050 + 300 = 1,350
Use the idea
Check each side's participation at the proposed fees before comparing revenue plans.
Where the conclusion applies
All-or-nothing participation on each side, linear values and fixed serving costs.
Check your understanding: What is the largest buyer fee that keeps buyers in if projected receipts stay at 1,890?
Chapter 41 source: section "Platform Economics and Two-Sided Markets".