Courses / Game theory for business
In most business decisions the outcome depends on more than you. You cut the price, and the rival answers. You trust a supplier, and he decides whether keeping his word pays. You sign a contract up to a set date, and the other person counts how much is left. Game theory studies exactly these situations: when the others think too. The course follows Marta. She roasts coffee and sells it online, and every chapter brings her a new game: a price war, Black Friday discounts, a supplier, a remote team, a fight for a hire, negotiations and a new "fast road" that makes things worse for everyone. Each game gets one rule, and a simulation on the page checks it.
≈ 45–60 minutes, in 5-minute parts if you like; your progress is saved. The maths is school fractions and simple probability.
Four situations where the decision is made once: a price, positioning, a Black Friday discount and guessing what the others will do. In each, your result depends not only on you but on what a rival who also does the maths decides. Each chapter starts with a question: guess first, then check.
"The shop next door cut its price by a dollar. I'll go a dollar below them and the customers come back. What's there to think about?"
Marta roasts coffee and sells it online. The example is made up, but its numbers are honest. A pack costs Marta $8, and she sells it for $14. Another online shop sells the same arabica from the same farm, also for $14. Every week 100 customers look for this coffee and buy where it's cheaper, and at equal prices they split in half. So each shop sells 50 packs and makes $300 a week.
This week the rival went to $13 and took every customer. Either can cut by a dollar at any time, and customers see both prices in a second.
Both shops answer every cut as long as it pays them. At what price does it stop?
You set Marta's price for the week. Next week the rival picks the price that earns him the most against yours.
Every step here is sensible: a dollar off buys a week of the whole market. But the rival does the same maths, and within a few weeks both sell at $10. The first to cut won one week, and both lost. Back in 1883 the French mathematician Joseph Bertrand noticed: if the product is the same and the customer goes where it's cheaper, two sellers are already enough to push the price down to cost. Raising it back doesn't work either: whoever raises first is left without customers.
This builds on what you haven't done yet: chapter 1, “A dollar cheaper”
"Why do all coffee shops look alike: the same medium roast, the same landing pages, newsletters at 10 in the morning?"
After the price war (chapter 1) Marta decided to differ in taste. The roast can be chosen on a scale from 0 (very light) to 100 (very dark). Customers' tastes are spread evenly along the scale, and each buys the coffee nearest to their taste. Prices are equal. Right now Marta is at 25 and the rival at 75: each gets half the customers.
Why do rivals' products so often drift towards the same thing?
This logic works wherever a customer can be "bitten off" from the neighbour: roast, pricing plans, the tone of a landing page, the send time of a newsletter. Each step towards the neighbour pays the one who takes it, and so the market pulls together into sameness. Customers at the edges lose: their taste is further from what's on offer.
But in this model prices are equal. If two coffees sit next to each other on the scale, they're almost the same to the customer, and the price war from chapter 1 begins. So when they compete on price too, sellers do better to move apart: d'Aspremont, Gabszewicz and Thisse showed this in 1979. Whoever differs can hold the price.
A kilometre of beach and two ice cream sellers who can move their kiosks. Every customer walks to the nearest one. Where will the sellers stand?
The end of this game is both in the middle, at 50, each with half the customers, as at the start. But the average distance from taste to coffee doubles: 25 points instead of 12.5 when Marta and the rival stood at 25 and 75. The sellers gained nothing, and the customers lost. Harold Hotelling used this in 1929 to explain why shops, parties and churches resemble each other.
"Everyone discounts on Black Friday. Maybe skip it this time? If only the neighbour skips it too."
Black Friday is coming. Marta and the rival shop each decide on their own: a 20% discount or none. A pack costs $20 and $8 to make: without a discount Marta makes $12 a pack, with one (a $16 price) $8. Each shop has 100 regulars who buy at their own shop at any price. Another 300 bargain hunters go where it's cheaper, and at equal prices split in half. Agreeing isn't an option: that's price fixing, and nobody trusts anybody anyway.
Both shops count the money and decide once. What do they do?
This is the prisoner's dilemma, the best-known game in game theory. A discount here is a dominant strategy: it's better whatever the rival decides. The pair "both discount" is a Nash equilibrium: neither gains by changing their decision alone. And at the same time both are worse off than if both had held back. Each acted sensibly, and together it turned out badly.
A set of strategies where nobody gains by changing their own while the others keep theirs.
An option that gives a player more than any other option of theirs, whatever the others choose.
Two broken vases; each owner writes a price from $2 to $100. Both get the lower one, with a $2 bonus for writing it. What would you write?
The traveler's dilemma is the same trap stretched over a hundred steps: each "a dollar less" is sensible, and the chain leads to $2, the worst pair for both. But real people write 95–100 and earn more than the "rational" ones. Kaushik Basu described it in 1994 as exactly a paradox of rationality: logic that both know to the end hurts both.
"I'm running a contest for subscribers. I want to know in advance which number wins, so the prize doesn't go by chance."
Marta announces a contest in her newsletter: everyone names a whole number from 0 to 100, and whoever is closest to 2/3 of the average of all numbers wins. The prize is a kilo of coffee. The number depends only on how the others think. Whoever assumes the others pick at random (average 50) will name 33: that's one step. Whoever thinks everyone takes that step will name 22: two steps. Three steps give 15, and so on all the way down to 0.
How many steps of this chain should you take to win among ordinary subscribers?
The mix of players here is illustrative: those "at random" name a random number from 0 to 100, the rest 33, 22 or 0. Real contests are noisier still, but the picture is the same.
The most logical answer, 0, wins only among people who also go to the end. Among ordinary people it's far from the target. The winner isn't whoever thinks deepest but whoever guessed correctly how deep the others think and took one step more. In 1936 John Maynard Keynes compared the stock market to this: you should pick not what you like, but what others think others will like.
Business works the same way. A sale should start a little before most rivals, but not so early that customers aren't thinking about gifts yet. A price is set a little below what rivals will actually charge, not what they'd charge if they reasoned to the end.
Everyone names a whole number from 0 to 100. Whoever is closest to two thirds of the average of all the numbers wins. Which number do you name?
The real numbers. In Rosemarie Nagel's experiment (1995), students gave a first-round average of 36.73, and none named 0. In 1997 Richard Thaler ran the game in the Financial Times: 1,476 readers gave an average of 18.91, and 13 won. The most popular guesses were 33 and 22, that is one and two steps, though there were plenty of zeros too.
In the first part every game was played once, and logic pushed everyone to a bad outcome. But Marta plays with her supplier, her contractor and her team every month. Four chapters on what repetition changes: how to answer a trick, what a known end date does, how many ties trust can bear, and when repetition becomes a trap.
This builds on what you haven't done yet: chapter 3, “The discount dilemma”
"My supplier slipped me cheaper beans once already. Forgive or punish?"
Every month Marta buys green coffee from one supplier. Each of the two can play fair or cut corners: Marta can pay on time or drag out the payment, the supplier can ship the agreed grade or mix in a cheaper one. The monthly gain works like the discount table in chapter 3, so this is a prisoner's dilemma again: both fair, $300 each; both cutting corners, $100 each; whoever cheats a fair partner gets $500, and the fair one gets $0.
Unlike Black Friday, this game repeats, and each sees what the other did last month. The simplest rule for such a game is tit for tat: play fair in the first month, then do whatever the partner did last month.
Marta and the supplier both play tit for tat and work fairly. In month 4 the supplier's warehouse slips up, and a lower-grade batch arrives, though he didn't mean to cheat. What happens next if nobody explains anything?
The grudger plays fair until cheated, then cuts corners to the end. The forgiving rule punishes only after two tricks in a row. The bad batch happens with every supplier.
faircuts cornersMonths 1 to 12, the supplier plays tit for tat.
With no accidents, tit for tat and the grudger earn the same: both play fair with the fair and don't let themselves be cheated twice. One bad batch shows the difference. After it the grudger cuts corners to the end of the year, even with a fair supplier. Tit for tat falls into an echo: each answer becomes the reason for the next. And the forgiving rule lets one trick pass and loses almost nothing.
Programs play the prisoner's dilemma for 200 rounds, each against each. Who scores most: an eternal defector, a sneak, Tit for Tat or a coin toss?
In Axelrod's first tournament in 1980, programs played the prisoner's dilemma for 200 rounds, each against each. Both the first tournament (14 programs) and the second (62) were won by Anatol Rapoport's tit for tat, the shortest program of all. It didn't beat a single opponent in a match, since it never scores more than its partner, but it collected many points with those who played fair.
This builds on what you haven't done yet: chapter 5, “Tit for tat with a supplier”
"A one-year contract, everything in writing. Why does the work slow down near the end?"
Marta hires a contractor for 12 months to run her ads and newsletters. Every month each decides how to play: the contractor can try hard or do a sloppy job, Marta can approve and pay the bonus on time or drag her feet. The gains are the same as with the supplier (chapter 5): $300 each for a fair month, $500 to whoever cheats a fair partner, $0 to the fair one and $100 each if both cut corners. Both know that month 12 is the last.
Both are purely rational and think ahead. By strict logic, from which month does the slacking start?
faircuts corners
When there is no end date, every month there's a chance of another one. A trick gains $200 at once (500 instead of 300), but then the partner stops trusting, and each following month brings $100 instead of $300, that is, $200 less. If the work goes on with chance p each month, on average p/(1 − p) months lie ahead. A trick doesn't pay when that is at least one, that is, when the chance of a next month is at least 1/2. In general the threshold is (500 − 300)/(500 − 100), and with these numbers that's exactly 1/2.
So it's not about how long the work lasts but about whether the end is in sight. By strict logic a 12-month contract with a known date unravels completely, while work with no date that lasts only 2 months on average (a 1/2 chance of going on) already holds, just at the threshold.
The theory explains such a slide, though in a real contract ordinary money motives play a part too: near the end a person is already thinking about the next job.
This builds on what you haven't done yet: chapter 5, “Tit for tat with a supplier”
"We're going remote. If I trust people in the office, I'll trust them online too, right?"
Marta has a team now: roasting, support, marketing. Helping a colleague means spending an hour of your own work to save them several. Let an hour of help save a colleague 5 hours: benefit b = 5, cost c = 1. Whoever helps, helps everyone they work with. Whoever doesn't, pays nothing but enjoys the help of others. People watch their neighbours and pick up the habit of whoever is doing better.
In the office everyone works with 4 people every day. After the move to remote, everyone has 24 contacts in chats and calls, a little with each.
In which team does the habit of helping survive, if 5 hours of benefit per 1 hour of cost is the same for both?
helpsdoesn't help
At each step someone leaves the team, and the newcomer in that place takes the habit of one of its neighbours; the more that neighbour earns, the better its chance. Half help at the start, and the simulation runs 150 steps per person.
The rule b/c > k is an approximation: it was derived for a weak effect of earnings on choices and for large networks. On a grid your neighbours' neighbours are often your neighbours too, so in the simulation the threshold is a bit below k, especially for 24 ties. But the direction is the same: the more ties, the bigger the benefit of help must be for helping to survive. Helpers survive in clusters: inside a cluster they get help from their own, and at its edge they lose to those who only take.
"We've been fighting for this roaster for four weeks. One more week and he's ours."
The best roaster in town is looking for a new job, and two want him: Marta and a big coffee chain. To each he's worth $10,000 of extra profit. While the fight goes on, every week costs each side $500: the recruiter, counter-offers, an empty position. Whoever gives up gets nothing and doesn't get the money back. Whoever holds out longer makes the hire.
Nobody knows in advance how long the rival will hold out. Game theory has an equilibrium for such a fight: each side may give up at any moment with a small chance, set so that one more week of fighting is neither a gain nor a loss for the rival.
Both sides play this equilibrium. How much do they spend on the fight together, on average?
You are Marta. The chain plays the equilibrium: it gives up when its turn comes, after 20 weeks on average, and each week with the same chance, however long it's been.
Once each side has spent $2,000 over four weeks, the next week seems a trifle: "another $500 and the hire is ours". But that $2,000 won't come back whatever happens, and the next week, like every week before it, is worth zero on average. The trap is thinking about what's spent, when the only decision is the next step. The same trap awaits buyers at an auction; chapter 2 of the auctions course is about it.
The price of a week only changes the length. The cheaper the week, the longer the fight drags on, and together the sides still spend $10,000: in equilibrium the war of attrition burns the whole value of the prize.
A one-dollar bill is up for auction. Both top bidders pay, but only the winner gets the dollar. How much does the seller usually collect?
In Martin Shubik's dollar auction (1971) both top bidders pay, and the bidding usually goes past a dollar: each step seems worth it, because giving up loses your bid. Shubik wrote that total payments of 3 to 5 dollars for one dollar are not uncommon. The war of attrition is the same game with weeks instead of bids: in equilibrium the sides together burn the whole prize, and people who can't stop in time burn more.
Three rivals instead of two, yes-or-no negotiations, a promise that can't be taken back, and a new option that makes everyone worse off. Here what matters is not only what you do, but in what order the players move and who sets the rules of the game.
"Mega is taking my customers. Should I strike it first, before it's too late?"
Marta's coffee is now sold in three places in the city, and her niche has two more players: a mid-sized roastery, Brew, and a chain, Mega. Each of the three may go on the attack once a month against one rival: steep discounts aimed at its range, poaching its café clients. Marta's attack pushes a rival out of the niche with probability 1/3, Brew's with 2/3, and Mega, with its budget, never misses. They take turns: Marta, Brew, Mega and round again, until one is left in the niche. A month may also be skipped, attacking nobody.
It's a simplified model, but it catches the main thing: each knows the others' strength and asks who is the biggest danger to them.
Whom should Marta attack in the first month?
After Marta's first move everyone plays their best: Brew attacks Mega, Mega attacks Brew, and Marta skips while both are in, then attacks whoever is left.
Attacking the strongest seems logical, but Mega is under fire anyway. Brew strikes Mega, because while Mega is in the game its next attack is on Brew. And Mega strikes Brew, because Brew is more dangerous than Marta. Marta's weakness becomes her shield: nobody deals with her first. And a successful attack only hurts Marta: then she is left one-on-one with the survivor, who moves first.
Three shoot in turn: Ann hits 1 shot in 3, Bob 2 in 3, Sam always. Who has the best chance of being the last one standing?
The truel has the same numbers: Ann hits with 1 shot in 3, Bob with 2 in 3, Sam always. And the worst shot has the best chance of surviving if she fires her first shot into the air: 39.7% for Ann, 38.1% for Bob and only 22.2% for Sam the sharpshooter. Marta is the same: the weakest of the three, yet the one most often left in the niche.
This builds on what you haven't done yet: chapter 6, “When the end is in sight”
"The recipe is mine, the brand is mine. Ten dollars should do for the partner: it beats nothing, right?"
A café chain wants to sell Marta's signature blend under both logos. The joint blend will bring $1,000 of extra profit a month, and nothing without both agreeing. The chain said it plainly: "Name the split once. If we agree, we launch; if not, we part." There's no haggling: this game is called the ultimatum.
Look from the end, as in chapter 6. The last move is the chain's: accept or refuse. Any amount beats zero, so a sensible partner will agree even to $10. Knowing that, Marta can offer $10 and keep $990. That's what logic says. And what do people say?
How much of the $1,000 should Marta offer the chain to earn the most on average?
The model is illustrative: the chance of a yes rises by a quarter with every $100 offered and is 100% from $400. It follows the shape of real data, not its exact numbers.
Reasoning from the end isn't wrong, it's just incomplete: the partner's payoff holds more than money. Turning down $10 costs the partner $10, but it gives the feeling of not being cheated and shows Marta she can't treat them that way. So solve the game from the end, but put a real person at the end: what they get if they say no, and what "fair" means to them.
Five logical pirates split 100 coins, and a captain whose split is not backed by half the crew goes overboard. How many coins does he keep?
In the pirates' puzzle perfectly logical pirates reason from the end, so the captain keeps 98 coins and gives one each to pirates 3 and 5. Real people would hardly agree to such a split: one coin in a hundred is 1%, and that gets rejected even in the lab.
This builds on what you haven't done yet: chapter 1, “A dollar cheaper”chapter 3, “The discount dilemma”
"The rival's site now says: 'Find it cheaper and we'll match the price.' Will he lure away all the customers now?"
Remember the two shops from chapter 1: cost $8, price $14, and customers go wherever it's cheaper? And the discount dilemma from chapter 3: a discount pays for each shop alone, but both are worse off together. Now both shops print the same promise: "Find it cheaper and we'll match the price." The match is automatic, no screenshots needed.
What happens to prices once both shops make this promise?
In each cell: Marta's profit / the rival's profit. The frame marks an equilibrium: a cell neither shop gains by leaving alone.
One promise isn't enough: against the shop that didn't promise to match, a discount still wins customers, and the dilemma returns. When both promise, "no discount" is never worse than a discount. Notice what the promise did: it gave the shop nothing now, it took away a choice. Thomas Schelling wrote about this as burning bridges: an army with nowhere to retreat convinces its opponent better than any words, because its threat is believable.
To a customer a price-match promise sounds caring. Economists see something else: in 1986 Steven Salop described such guarantees as a way for rivals to hold prices up without any collusion. But there's a limit: if a match means finding a receipt, sending a screenshot and waiting, some customers won't bother, a discount wins some of them again, and the promise "leaks".
Transparent box A holds $1,000. A predictor that is right 99% of the time put a million in box B only if it foresaw you taking B alone. What will you take: only B or both?
In Newcomb's paradox the predictor puts the million in for those who predictably take one box: those who take only B leave with $990,000 on average, and those who take both with $11,000. It's the same mechanism as commitment: when others can predict your choice, it pays to be the one who has tied their own hands in advance.
"I'll add free express delivery. Whoever wants it gets their parcel sooner, and the rest are no worse off, right?"
Marta now ships 4,000 orders a week across the country. Customers choose the delivery themselves and see a live estimate of how many hours it will take. There are two options. "Courier and post": Marta's courier takes the parcel to the post depot, and the more parcels, the longer it takes (number / 100 hours: 2,000 parcels, 20 hours), then the post, always 45 hours. "Post and courier": the post picks the parcel up at the roastery and carries it to the customer's city in 45 hours, where a local partner courier delivers it, also number / 100 hours. Customers have split evenly, 2,000 each, and every parcel travels 20 + 45 = 65 hours.
Now Marta arranges for her courier to hand parcels straight to the local partner at one depot, with no delay at all. The new option, "Courier to courier", is free, and every customer still picks the fastest one for themselves.
How long will a parcel travel on average once customers get used to the new option?
Each customer compares only their own hours and doesn't see that their parcel adds hours for everyone using the same courier. The best for all would be to let only 500 parcels take the new option: then 64.7 hours on average. Free choice gives 80. The ratio between what free choice gives and the best possible split is called the price of anarchy. In 2002 Tim Roughgarden and Éva Tardos proved that when the time on each leg grows linearly with the load, this ratio never exceeds 4/3. Here it is 80 / 64.7 ≈ 1.24.
4,000 drivers take 65 minutes to get to work. The city opens a free, instant shortcut between the routes. Will the trip get shorter or longer?
These are the same numbers as in Braess's classic example: 4,000 drivers, narrow streets at T/100 minutes, highways at 45, and a free shortcut turns 65 minutes into 80. Close the shortcut, and everyone gains 15 minutes. Delivery works the same way: removing "Courier to courier" gives every parcel back 15 hours.
One question from each chapter. Go through them before you change a price, sign a contract or sit down to negotiate.
If the game is an auction or a tender, the course "How to win an auction without losing" takes it from there: the bid limit, the winner's curse and the formats. The classic puzzles themselves are in the catalogue's "Games" topic: Axelrod's tournament, the truel and Braess's paradox.