Strategic interaction models every growth plan should know

Your rivals are doing the maths too

Most growth plans assume the market holds still while you move. You cut a price or order more stock, and the spreadsheet assumes everyone else carries on as before. They do not. A rival reads the same market, notices your move and answers it. What you earn depends on your choice and on theirs.

Game theory is the study of exactly that: situations where each player's result depends on what everyone chooses.¹ In my experience a handful of its classic models cover almost every competitive question a growing shop or software business meets.

Each model below gets a worked example. The businesses and numbers are invented so that I can show every step. Nobody real, no client. If payoff matrices and the Nash equilibrium are new to you, my article on game theory in pricing builds them from scratch with a Black Friday example, and I lean on it here rather than repeat it.

Two tools carry every model: the payoff matrix and the equilibrium

A payoff matrix is a table of what each player earns for every combination of moves. Your options are the rows, the rival's options are the columns, and by convention the first number in each cell is the row player's payoff and the second is the column player's.¹ Writing it down forces every assumption into the open.

A Nash equilibrium is a combination of moves, one per player, where nobody can do better by changing only their own move.¹ John Nash showed in 1950 that every finite game has at least one, once players are allowed to mix their moves at random.² The equilibrium is stable, which is not the same as good. That gap is what every model below is about. A payoff matrix is just a grid of numbers (Foundations: matrices).

Worked example: reading a cell

Suppose the cell "you hold, rival cuts" reads €3,000, €8,000: you earn €3,000 and the rival €8,000. To test a cell, ask each player in turn: if I switched and the other did not, would I earn more? If either says yes, the cell is not stable. If both say no, it is an equilibrium.

Bertrand: with identical products, a price war ends at cost

The first model describes far too many online markets. In the Bertrand model two firms sell the same product and each chooses a price. Customers buy from whoever charges less and split evenly when prices match.³

Each firm's profit follows three cases:

$$\pi_i = \begin{cases} (p_i - c)\,D(p_i) & \text{if } p_i < p_j \\ \tfrac{1}{2}\,(p_i - c)\,D(p_i) & \text{if } p_i = p_j \\ 0 & \text{if } p_i > p_j \end{cases}$$

Here $\pi_i$ is the profit of firm $i$, $p_i$ is the price firm $i$ sets, $p_j$ is the rival's price, $c$ is the cost of one more unit (the marginal cost, the same for both), and $D(p_i)$ is how many units the whole market buys at price $p_i$. The fraction $\tfrac{1}{2}$ is the even split when prices are equal. The cost of one more unit is explained in Foundations: marginal versus average.

The logic is brutal. At any common price above cost, either firm can shave a cent and take the whole market, which roughly doubles its sales for a tiny loss of margin. So no common price above cost is stable. The only equilibrium is both firms pricing at marginal cost and earning nothing on each sale.³ Two firms are enough to wipe out the margin.

Worked example: two Cork shops and the same filter cartridge

Two online shops in Cork sell the identical replacement cartridge for a popular water filter jug. Call them Lee Supplies and Douglas Home. Each cartridge costs €6 landed. The market buys about 2,000 a month at prices in this range, and I keep that number fixed to keep the arithmetic clean.

Step one, both at €10. They split the market: 1,000 each. Each earns $1{,}000 \times (€10 - €6) = €4{,}000$ a month.

Step two, Lee cuts to €9.50. It takes all 2,000 buyers and earns $2{,}000 \times (€9.50 - €6) = €7{,}000$. Douglas earns nothing. The cut was worth €3,000 to Lee.

Step three, Douglas answers at €9. Now it earns $2{,}000 \times €3 = €6{,}000$ and Lee earns nothing.

Step four, the logic repeats. At €7 the winner earns $2{,}000 \times €1 = €2{,}000$. At €6.50 it is €1,000. At €6 it is zero.

Every single move was rational. The destination was zero.

How to get out of the Bertrand trap

The model needs identical products, customers who see every price and switch instantly, and sellers with unlimited stock. Break any one and the price stops falling to cost. For a shop that means: make the product less identical (a bundle, a refill subscription, faster delivery, better advice), make switching less automatic (an account, saved orders, a loyalty scheme), or accept that a commodity line is a footfall product and earn the margin elsewhere. I go through those options in detail in beating price wars in homogeneous markets.

Cournot: when you commit to stock first, the war is softer

Now change one thing. Instead of choosing prices, the firms choose quantities first, and the price then settles wherever the market clears. This is the Cournot model, dating from 1838, and it fits any business that has to commit to stock or capacity before the selling season.⁴

Market price falls as total supply rises. With a straight line demand curve:

$$P = a - b\,(q_1 + q_2)$$

Here $P$ is the market price, $a$ is the price at which nobody would buy at all, $b$ is how much the price drops for each extra unit on the market, and $q_1$ and $q_2$ are the quantities the two firms bring. How strongly quantity answers price is the subject of Foundations: elasticity. Each firm then picks its own quantity to maximise profit, given what it expects the rival to bring. Setting the extra revenue from one more unit equal to its cost gives each firm's best response, called its reaction function:⁴

$$q_1 = \frac{a - c - b\,q_2}{2b}$$

Here $q_1$ is firm one's best quantity, $a$ and $b$ are the demand numbers above, $c$ is the cost of each unit, and $q_2$ is what firm one expects the rival to bring. Read it in plain words: the more the rival brings, the less I should bring, by half a unit for each of theirs. The step "extra revenue equals extra cost" is the marginal reasoning from Foundations: marginal versus average.

When both firms have the same cost, solving the two reaction functions together gives each firm the same quantity:

$$q^{*} = \frac{a - c}{3b}$$

Here $q^{*}$ is each firm's equilibrium quantity, and $a$, $b$ and $c$ are as before. It is the point where both reaction functions hold at once, still the marginal reasoning of Foundations: marginal versus average.

Worked example: two Galway Christmas tree sellers

Two online Christmas tree sellers in Galway, Salthill Trees and Oranmore Pines, must order their trees from growers in October. Once the trees are cut, they are sold for whatever the market will pay in December. Both pay €20 a tree, delivered. From past seasons, demand is roughly $P = 80 - 0.02\,Q$, where $Q$ is the total number of trees both bring. So at 2,000 trees in total the price is €40, and every extra 100 trees knocks €2 off.

Step one, the reaction function. With $a = 80$, $b = 0.02$ and $c = 20$:

$$q_1 = \frac{80 - 20 - 0.02\,q_2}{0.04} = 1{,}500 - 0.5\,q_2$$

Here $q_1$ is Salthill's order and $q_2$ is Oranmore's (Foundations: marginal versus average). If Oranmore orders nothing, Salthill should order 1,500. Every tree Oranmore orders cuts Salthill's best order by half a tree.

Step two, the equilibrium. Each orders $q^{*} = \frac{80 - 20}{3 \times 0.02} = \frac{60}{0.06} = 1{,}000$ trees.

Step three, the price. Total supply is 2,000, so $P = 80 - 0.02 \times 2{,}000 = €40$.

Step four, the profit. Each earns $1{,}000 \times (€40 - €20) = €20{,}000$.

Step five, check it is stable. Suppose Salthill orders 1,200 while Oranmore stays at 1,000. Supply is 2,200, the price is $80 - 44 = €36$, and Salthill earns $1{,}200 \times €16 = €19{,}200$. Less. Suppose it orders 800 instead. Supply is 1,800, the price is €44, and it earns $800 \times €24 = €19{,}200$. Less again. Neither firm gains by moving alone, so 1,000 each is the equilibrium.

Under Bertrand rules the same two sellers would fight the price down to €20 and earn nothing. Committing to quantity first softens the fight, because neither can flood the market later. In my experience businesses with real stock commitments are rarely in the vicious price wars that pure resellers fight.

Stackelberg: moving first pays, if the commitment is real

The Stackelberg model, from 1934, keeps the Cournot setting but lets one firm move first. The leader chooses its quantity, the follower sees it and then chooses.⁵ The leader knows the follower will play its reaction function, so it chooses its own quantity with that answer already built in. Solving backwards like this gives:

$$q_L = \frac{a - c}{2b}, \qquad q_F = \frac{a - c - b\,q_L}{2b}$$

Here $q_L$ is the leader's quantity, $q_F$ is the follower's, and $a$, $b$ and $c$ are the demand and cost numbers from the Cournot section. The second formula is simply the follower's reaction function with the leader's quantity put in. Choosing the best quantity once you know how the other side reacts is the same marginal reasoning as before, in Foundations: marginal versus average.

The result is that the leader brings more, the follower brings less, the total rises and the price falls.⁵ The leader earns more than in Cournot, the follower less.

Worked example: Salthill orders in September, in public

Same market, same costs. This year Salthill signs its grower contract in September and says so: a press note, a preorder page, a visible commitment Oranmore can see before it orders.

Step one, the leader's order. $q_L = \frac{80 - 20}{2 \times 0.02} = \frac{60}{0.04} = 1{,}500$ trees.

Step two, the follower's answer. Oranmore plays its reaction function: $q_F = 1{,}500 - 0.5 \times 1{,}500 = 750$ trees.

Step three, the price. Total supply is 2,250, so $P = 80 - 0.02 \times 2{,}250 = €35$.

Step four, the profits. Salthill earns $1{,}500 \times (€35 - €20) = €22{,}500$. Oranmore earns $750 \times €15 = €11{,}250$.

Step five, a check. Had Salthill ordered 1,200, Oranmore would answer with 900, the price would be €38 and Salthill would earn $1{,}200 \times €18 = €21{,}600$, less than €22,500.

The horizontal axis shows the model and which seller, the vertical axis the profit for the season in euro. Only the rule of the game changes. Price competition with instant restocking wipes the profit out, committing to stock protects it, and committing first and visibly shifts €2,500 to the leader and costs the follower €8,750.

The catch: a commitment the rival cannot see, or can see you undo, is worth nothing

The leader's advantage comes entirely from commitment, and it needs the follower to observe the leader's quantity. If the follower cannot see it, the game is strategically the same as Cournot.⁵ The same goes for a commitment you could quietly reverse. An order you could cancel next week moves nobody. A signed contract or stock in the warehouse does.

That is the first mover advantage I believe in: not being first to an idea, but first to a visible commitment that changes the rival's best answer.

The prisoner's dilemma lives inside your ad account

A prisoner's dilemma is any game with four payoffs in a particular order. The temptation to defect while the other cooperates, $T$, beats the reward for both cooperating, $R$, which beats the punishment when both defect, $P$, which beats the sucker's payoff for cooperating alone, $S$.⁶

$$T > R > P > S$$

Here $T$ is the temptation, $R$ the reward, $P$ the punishment and $S$ the sucker's payoff, all for the same player. With that ordering, defecting is the better move whatever the other side does, so both defect and both end up with $P$, although both would prefer $R$. The four payoffs sit in a two by two grid, Foundations: matrices. The price version is in my game theory in pricing article. Here is a version most shops meet without noticing: bidding on each other's brand names in search.

First, how the auction actually charges you

It is still common to read that Google Ads is a second price auction, where the winner pays the bid of the runner up. That is not how Google describes it. Ads are ordered by Ad Rank, which Google says is based on your bid, the quality of your ads and landing page, the Ad Rank thresholds, the competitiveness of the auction, the context of the search (location, device, time) and the expected impact of assets.⁷ What you pay per click is, in Google's words, what is minimally required to clear the Ad Rank thresholds and to beat the Ad Rank of the competitor immediately below you. If no one is below you, you pay only the reserve price.⁸

So on your own brand name, a rival appearing below you raises what you pay to stay on top. I explain the whole auction in auction theory for paid media.

Google's trademark policy does not restrict a competitor's trademark as a keyword, though it can restrict one in the ad text.⁹ Whether bidding on it is wise is a game.

Worked example: two Limerick bike shops and each other's names

Two online bike shops in Limerick, Castletroy Cycles and Treaty Wheels, each get steady searches for their own name. Each earns about €12,000 a month in contribution from people who search its name and buy.

Castletroy's owner estimates three effects.

1. Bidding on the rival's name costs €1,500 a month in clicks and wins sales worth €3,500 in contribution. Net gain: €2,000.
2. Being bid on loses €3,500 of contribution to the rival.
3. Being bid on also raises the cost of the shop's own brand ad, because there is now an advertiser below it to beat. Estimated extra cost: €1,000 a month.

Now fill the four cells. Castletroy chooses the row, Treaty the column.

Neither bids: each keeps €12,000.

Castletroy bids, Treaty does not: Castletroy earns $12{,}000 + 2{,}000 = €14{,}000$. Treaty earns $12{,}000 - 3{,}500 - 1{,}000 = €7{,}500$.

Both bid: each earns $12{,}000 + 2{,}000 - 3{,}500 - 1{,}000 = €9{,}500$.

	Treaty stays off	Treaty bids on Castletroy
Castletroy stays off	€12,000, €12,000	€7,500, €14,000
Castletroy bids on Treaty	€14,000, €7,500	€9,500, €9,500

Check the order: $T = €14{,}000 > R = €12{,}000 > P = €9{,}500 > S = €7{,}500$. It is a prisoner's dilemma. Read Castletroy's options column by column. If Treaty stays off, bidding pays €14,000 against €12,000. If Treaty bids, bidding pays €9,500 against €7,500. Bidding wins either way, the same holds for Treaty, and both end at €9,500, each €2,500 a month worse off than if neither had started. The one party better off is the auction, which now sells both shops €5,000 a month more in clicks.

Repeated games: the future is what keeps a truce alive

The brand bidding game is not played once. It is played every day, for years. That changes the answer, because today's move shapes tomorrow's.

Two strategies are worth knowing. GRIM cooperates until the other side defects once and then defects for ever; Nash himself noted that it is an equilibrium of the indefinitely repeated prisoner's dilemma.⁶ Tit for tat starts by cooperating and then copies whatever the other player did last round. In Robert Axelrod's two computer tournaments it scored highest in the first and won the second, against 63 entries. Axelrod put that down to four traits: it is nice (never the first to defect), retaliatory, forgiving and clear.⁶

Whether restraint is stable depends on how much next month matters. Game theorists put that into a discount factor, which can equally be read as the chance that the game continues.⁶ Under GRIM, cooperating for ever beats one month of temptation followed by punishment for ever when:

$$\frac{R}{1 - \delta} \geq T + \frac{\delta\,P}{1 - \delta} \quad \Longleftrightarrow \quad \delta \geq \frac{T - R}{T - P}$$

Here $\delta$ is the discount factor, a number between zero and one that says what a euro next month is worth to you today, including the chance there is a next month. $R$, $T$ and $P$ are the reward, temptation and punishment from the matrix. The left side, $\frac{R}{1 - \delta}$, is the value of earning $R$ every month for ever. The right side is $T$ once and then $P$ every month after, discounted. Valuing a stream of future payments is explained in Foundations: discounting, and the reading of $\delta$ as a chance of continuing rests on Foundations: probability.

Worked example: how much the future must matter in Limerick

Put in the bike shop numbers: $T = 14{,}000$, $R = 12{,}000$ and $P = 9{,}500$.

Step one, the gain from defecting once: $T - R = 14{,}000 - 12{,}000 = 2{,}000$.

Step two, the loss each month once punishment starts: $T - P = 14{,}000 - 9{,}500 = 4{,}500$.

Step three, the threshold: $\delta \geq \frac{2{,}000}{4{,}500} = 0.44$.

Here $\delta$ is still the discount factor and the numbers are euro a month. If each owner puts more than about a 44% weight on next month, staying off the other's name is rational, and two established shops are far above that. The truce breaks when someone's future shrinks: a shop up for sale, a new owner under pressure to hit this quarter's number, an agency paid on clicks. In my experience that is where most brand bidding wars start.

The line you must not cross

Holding back on your own, and answering like tit for tat when the rival starts, is ordinary competition. Ringing the other owner to agree a truce is something else. Section 4 of the Competition Act 2002 prohibits agreements between businesses that restrict competition in the State, and among its examples are agreements that fix selling prices or limit or control markets.¹⁰ Article 101 of the Treaty on the Functioning of the European Union says the same wherever trade between member states may be affected.¹¹ The Competition and Consumer Protection Commission (CCPC) counts agreeing on discounts, margins or minimum prices as price fixing, and its guidance lists what follows: fines of up to €50 million or 20% of turnover, director disqualification for five years, and up to 10 years in prison.¹² A deal between rivals not to bid on each other's names is an agreement about how hard you compete, and I would not sign one without a competition solicitor. The repeated game is played through what you visibly do, never over a pint with the other owner.

Focal points: when everyone must guess the same answer

Some games have several equilibria and the players cannot talk. Then the question is not what is stable but which stable outcome everyone will guess. Thomas Schelling's answer was the focal point: the option that stands out, so each player expects the others to pick it.¹ Round numbers, obvious dates, the default everyone already uses.

Worked example: a Kilkenny homeware shop picks its sale week

A Kilkenny homeware shop, Nore Linen, can run its big autumn sale in Black Friday week or in a quiet week in mid October. Its customers can look for deals in either week. The shop's payoff is extra contribution. The customers' payoff is the total they save, in euro, across the shop's mailing list.

	Shoppers look in Black Friday week	Shoppers look in mid October
Sale in Black Friday week	€9,000, €4,000	€2,000, €0
Sale in mid October	€1,000, €0	€11,000, €4,000

Step one, test "Black Friday, Black Friday". The shop moving alone drops from €9,000 to €1,000, shoppers from €4,000 to nothing. Stable.

Step two, test "October, October". The shop moving alone drops from €11,000 to €2,000, shoppers from €4,000 to nothing. Also stable.

Step three, which one happens? October is better for the shop, because the competition for attention is lower. But shoppers have no reason to expect it, and Black Friday is the date everyone already knows. It is the focal point. Without something that moves the expectation, the shop lands on €9,000.

Step four, what it costs to move the focal point. The shop needs shoppers to know about October in advance: an announced date, a countdown in the newsletter, an early access list. It is worth spending up to $€11{,}000 - €9{,}000 = €2{,}000$ on that. The lesson: when you need people to coordinate with you, choose the obvious option, or pay to make your alternative the obvious one.

Signalling: a promise only counts if faking it would hurt

The last model is about hidden information: the seller knows its quality, the buyer does not. Michael Spence showed in 1973 how a worker who knows more about their own ability than an employer can signal it through education.¹³ The trick is not that the signal proves anything directly. The trick is that it costs the high quality type less to send than the low quality type, so only the high type sends it. Economists call this the single crossing condition.¹⁴ Later work showed the same logic for products: a firm that produces high quality goods can signal it by offering a warranty.¹⁴

For a signal to separate the two types, the premium buyers pay for it must cover what it costs the high type, and must not cover what it would cost the low type:

$$c_H \leq \Delta p < c_L$$

Here $c_H$ is the expected cost of sending the signal for a high quality seller, $c_L$ the expected cost for a low quality seller, and $\Delta p$ the extra price buyers pay when they see the signal. The costs are expected values, a probability times a cost, as in Foundations: expected value.

Worked example: a 12 month warranty on refurbished laptops in Waterford

Two kinds of seller offer refurbished laptops online from Waterford. A careful one, which tests every machine properly, sees 4% of laptops fail within a year. A careless one sees 20% fail. Buyers cannot tell them apart from the listings. A warranty claim costs either seller about €60 to handle and repair.

Step one, the careful seller's expected cost per laptop: $0.04 \times €60 = €2.40$.

Step two, the careless seller's expected cost per laptop: $0.20 \times €60 = €12$.

Step three, the premium. Buyers who see a 12 month warranty pay €8 more, because they read it as a sign of quality.

Step four, does each seller offer it? The careful seller gains $€8 - €2.40 = €5.60$ a laptop, so it offers the warranty. The careless seller would gain $€8 - €12 = -€4$ a laptop, so it does not. The condition holds: $€2.40 \leq €8 < €12$. The warranty separates the two, and buyers are right to trust it.

Step five, where it breaks. At a €15 premium the careless seller gains $€15 - €12 = €3$ and offers one too, and the warranty tells buyers nothing.

The lesson is uncomfortable for a lot of marketing. A signal that costs nothing to fake tells the buyer nothing. A polished design or a badge anyone can paste on a page is not a signal. A long warranty, free returns or a named person who answers for the work is, because a weaker business could not afford to copy it.

Which model fits your question

This is the sequence of questions I run through before I plan a move against a rival.

Read it from the top: each yes points to the model in the box below it. Several can apply at once, as the Limerick bike shops show.

Model	The question it answers	The one line lesson
Bertrand	What happens when identical products compete on price?	The price falls to cost, so differentiate first
Cournot	What happens when stock or capacity is chosen first?	Commitment softens the fight
Stackelberg	Does moving first pay?	Only if the commitment is visible and irreversible
Prisoner's dilemma	Why do sensible rivals end up worse off?	The stable cell is not the good one
Repeated game	When does restraint survive?	When the future weighs more than $\frac{T - R}{T - P}$
Focal point	Which of several outcomes will people guess?	The obvious one, unless you pay to move it
Signalling	How do I prove what a buyer cannot see?	Only with a promise a weaker rival cannot afford

How I use these models without fooling myself

The models are only as good as the payoffs I put in, so I write the matrix with real numbers from the business, even rough ones, and show it to the owner. A cell they disagree with is the most useful thing in the meeting. I look for the commitment, because in every example above the decisive fact was something hard to reverse. And I remember the rival is reading the same market. If my plan only works when the rival does nothing, it is not a plan. It is a hope with a spreadsheet attached.

There is a further branch of the field, evolutionary game theory, which studies which strategies spread when players copy whatever is doing well.¹ It is a topic of its own and I leave it for another article.

Footnotes

1. Don Ross, Game Theory, The Stanford Encyclopedia of Philosophy, substantive revision 3 September 2023, https://plato.stanford.edu/entries/game-theory/

2. John F. Nash Jr., Equilibrium Points in N Person Games, Proceedings of the National Academy of Sciences, volume 36, issue 1, pages 48 to 49, 1950, https://pmc.ncbi.nlm.nih.gov/articles/PMC1063129/

3. Stefano DellaVigna, Economics 101A Lecture 22: Oligopoly, Cournot and Bertrand, University of California Berkeley, spring 2008, https://eml.berkeley.edu/~webfac/dellavigna/e101a_sp08/lec22.pdf

4. Shachar Kariv, Game Theory in the Social Sciences, Lecture V: Applications of Nash equilibrium, Cournot's oligopoly, University of California Berkeley, 2026, https://eml.berkeley.edu/~kariv/110_lecture_5.pdf

5. Shachar Kariv, Game Theory in the Social Sciences, Lecture VII: Applications of extensive games, Stackelberg's duopoly model, University of California Berkeley, 2026, https://eml.berkeley.edu/~kariv/110_lecture_7.pdf

6. Steven T. Kuhn, Prisoner's Dilemma, The Stanford Encyclopedia of Philosophy, substantive revision 24 October 2025, https://plato.stanford.edu/entries/prisoner-dilemma/

7. Google, Ad Rank: Definition, Google Ads Help, accessed November 2026, https://support.google.com/google-ads/answer/1752122

8. Google, Actual cost per click (CPC): Definition, Google Ads Help, accessed November 2026, https://support.google.com/google-ads/answer/6297

9. Google, Trademarks, Google Advertising Policies Help, accessed November 2026, https://support.google.com/adspolicy/answer/6118

10. Law Reform Commission, Competition Act 2002, revised, section 4, Revised Acts, updated to 1 July 2026, https://revisedacts.lawreform.ie/eli/2002/act/14/revised/en/html

11. European Union, Consolidated version of the Treaty on the Functioning of the European Union, Article 101, EUR-Lex, 2008, https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:12008E101

12. Competition and Consumer Protection Commission, Price fixing, competition law guidance for businesses, CCPC, accessed November 2026, https://www.ccpc.ie/information-for-businesses/guidance-for-businesses/competition-law-guidance/competition-law-breaches/cartels/price-fixing

13. Michael Spence, Job Market Signaling, The Quarterly Journal of Economics, volume 87, issue 3, pages 355 to 374, 1973, https://ideas.repec.org/a/oup/qjecon/v87y1973i3p355-374..html

14. Joel Sobel, Signaling Games, University of California San Diego, 2007, https://econweb.ucsd.edu/~jsobel/Paris_Lectures/20070527_Signal_encyc_Sobel.pdf