Games and moats

A game is any situation where what you get depends on what somebody else chooses as well as on what you choose. A strategy is a full plan of what you will do. A strategy is dominant when it is your best plan no matter which plan the other player picks. Most games have no dominant strategy, which is why they are hard. The prisoner's dilemma is famous because each player has one, and following it makes both worse off.

Read the table by columns. Suppose Duarte holds: Beira earns 100 by holding and 130 by cutting, so cut. Suppose Duarte cuts: Beira earns 60 by holding and 80 by cutting, so cut again. Cut wins in both columns; it is dominant. Duarte, reading the rows, reaches the same conclusion. Both cut, both earn 80, and the 100 each would have earned by holding is gone. Nobody made a mistake. Each played the best reply to what the other would do, and the outcome is worse for both than a pair of moves neither can hold on their own.

The data habit this gives you is the one Foundations session 2's number cannot: an elasticity says how buyers answer, and only buyers. Before any price move, write the rival's reply in the next cell, and ask whether the number you computed still describes the world after they make it.

The table is played once. Real sellers face each other every month, and that changes the arithmetic without changing the table. If Beira cuts this month, Duarte can match next month, and then both sit at 80 for as long as they both remember. A seller who expects to be on the shelf next year weighs 130 now against 80 forever, and holding starts to look better than the single-round table says. Repetition is what lets two sellers hold a price without ever agreeing to, and it is why the standoff you can see on many shelves is quieter than the dilemma predicts.

Two conditions have to be true for the quiet to last. The rival must be able to see the cut, which on a marketplace with public prices they can, and the rival must be expected to stay, which on a marketplace with 3,095 sellers is not always true for any one of them. When either condition fails the game collapses back to the single round, and the cut comes.

Metcalfe's law counts pairs. With n members there are about n squared over two possible connections, so if every connection is worth the same, value grows as n squared. Doubling the members quadruples the value. It is a clean story and it is why the phrase "network effects" turns up in pitch decks as if it were arithmetic.

The rebuttal attacks the assumption that every connection is worth the same. Briscoe, Odlyzko and Tilly argued that a member's connections fall off in value the way most things do: the most useful link is worth a lot, the tenth is worth a tenth of that, the hundredth a hundredth. Sum a series like that and each member gets about log n of value, so the network as a whole is worth about n log n. Doubling from a thousand members then gives 2.2 times the value, not four. That is pure arithmetic on the two claims, and it is the only number on this page about network size.

The honest position is that both are claims. The mechanism, that a bigger network is worth more to each member, is real and is why marketplaces exist. The exponent is not a law of nature and has never been measured on the marketplace whose data runs through these sessions. When somebody quotes a square, ask them which of the two they mean, and whether they can show a single connection that is worth the same as the first one.

A network effect is direct when a member gains from more members of the same kind: a messaging app is worth more to you when your friends are on it. It is indirect when a member gains from more members of the other kind: a buyer gains from more sellers, a seller from more buyers, and a seller gains nothing, or less than nothing, from another seller. A marketplace runs almost entirely on the indirect kind, which is why it is called two-sided, and why it has a chicken-and-egg problem at birth: nobody sells where nobody buys, and nobody buys where nobody sells.

The distinction matters for the claim in Part 2. Pairs of sellers are not connections anyone values, so counting all pairs among all members overstates a marketplace's network even before the log n argument arrives. The connections that carry value cross the platform from one side to the other. Deep-dive session 5 draws both sides with the real counts and does the arithmetic at ten times the members rather than two.

Write a switching cost as a list and it stops being a slogan. For a seller on a marketplace: the hours to re-upload a catalogue, the reviews that stay behind, the search rank earned over months of sales, a contract's notice period, the team's habits. Each brick has a price, and the sum is the wall. The incumbent builds the wall; the rival builds the door: an importer for the catalogue, a launch promotion that stands in for the missing rank, a buyout of the notice period. The wall's height is therefore not a property of the market. It is an ongoing contest.

Winner-take-most is what network effects and switching costs together produce when they work: the biggest platform is worth the most to join and the most expensive to leave, so it keeps growing. The limit on it is multi-homing. A seller who lists on two marketplaces at once has no wall to climb; they are already on both sides. A buyer with two apps on their phone is the same. Where multi-homing is cheap, the biggest platform still wins the most, but the second and third survive on the same sellers and the same buyers, and "all" never happens.

The test for any moat claim is one question: where is the door, and who is building it? If nobody can name the door, the claim is probably true for now. If the door is already open, as it is wherever multi-homing is common, the moat is a wall with a queue walking through it.

Part 1's word, dominant, has one famous application beyond the dilemma. In a sealed-bid auction where the highest bidder wins but pays the second highest bid, the best thing to do is bid exactly what the item is worth to you, whatever anybody else bids. Bid more, and the only time it changes anything is when you win at a price above your value and lose money. Bid less, and the only time it changes anything is when you lose an item you would have paid for. Truthful bidding is dominant. That result is Vickrey, in the Journal of Finance, 1961, volume 16, issue 1, pages 8 to 37.

Why it belongs in a session on games: it is the cleanest case of a mechanism designed so that the game has a dominant strategy, which removes the guessing. Most auctions people actually meet, including the ones that price ads, are not designed that way, and deep-dive session 6 shows what goes wrong when they are not.

Commitment. In the dilemma, Beira would love to promise "I will hold if you hold". The promise is worthless because holding is not Beira's best reply once Duarte holds. A commitment only changes a game if it is credible: if breaking it costs more than keeping it. A public price-match guarantee is a credible commitment to cut, which oddly makes cutting less attractive for the rival, because they know the reply arrives automatically. The lesson is that words move nothing; what moves a game is a change in the payoffs behind the words.

Data as a moat. The most common moat claim in a data business is "we have more data, so our model is better, so we win more customers, so we get more data". The loop is real at the start. It stops being real for a mechanical reason: learning curves flatten. Early rows teach a model the common cases and accuracy climbs fast; later rows are mostly common cases again, and what is left to learn is rare. Each doubling of data buys less than the doubling before it. Once both you and a rival are on the flat part, a rival with a fraction of your data is a few points behind, not a generation behind. Where the loop does keep compounding is where the data itself is hard to get, or where it is tied to a switching cost that keeps the customer producing it for you. That is a wall again, and the door question applies. Deep-dive session 5 fits the curve to constructed data and reads where it flattens; no published exponent is quoted here because none is on file for these pages.