16 Building the Game
Nobody hands you a payoff matrix, and the payoffs are the whole difficulty
The last chapter ended one step short. Hogi Yogi could buy the advertising, and so could Smart Cookie, and each of them has to decide without knowing what the other will do.
That is a game, and this chapter is about how to build one.
The distinction matters more than it sounds. Almost everything written about game theory teaches you to solve games: here is a matrix of numbers, find the equilibrium. It is a reasonable thing to teach and it prepares you for a situation that never happens. Nobody will walk into your office holding a payoff matrix and ask you to solve it. What you will have is a decision, a competitor, and no numbers at all.
Building the game is the work. It is also where every mistake lives.
Solving Is the Easy Part
The conventional five steps of a game are worth stating, mostly so the weighting can be corrected.
Identify the players. Identify the actions available to each. Determine the payoffs from every combination of actions. Identify each player’s strategies. Solve for the equilibrium.
The last step is the one textbooks spend their pages on and it is genuinely mechanical. Given a completed matrix, finding dominant strategies and equilibria is a procedure you can carry out on paper in a minute, and an AI will do it instantly and correctly every time. Hand a machine four numbered cells and you will get the right answer.
Step three is where the difficulty is, and it is not close. What are the payoffs? is a question about what would actually happen to your business under four different futures, only one of which you will observe. Answering it requires knowing how customers respond to price, what serving them costs, how many of them exist, and how the other firm would price in each case.
Which is to say: it requires everything in the previous fifteen chapters. The engine that turns parameters into equilibrium profit is the payoff function. That is why game theory belongs at the end of this book rather than in a chapter of its own somewhere near the front. Every other chapter was building the thing that fills in the cells.
Payoffs Are Profit
There are two ways to fill a matrix, and they are not equally useful.
Ordinal payoffs rank outcomes. Best, second best, third, worst, or 4, 3, 2, 1 with no claim that the gaps mean anything. They are quick, they need no data, and for some games they are enough: if you only need to know which outcome each player prefers, ranking gets you to an equilibrium.
They also smuggle in the answer. To rank the outcomes you must already know which you prefer, and in most real competitive decisions that is precisely what you do not know. Is it better to advertise while your rival does, or to sit still while they advertise alone? The last chapter showed that the intuitive ranking of exactly that question is wrong.
Cardinal payoffs are numbers with meaning — profit, or the net present value of profit over the life of the commitment. They require the analysis. They also answer questions ranking cannot: how much better, whether the gap is worth the risk, and at what cost the decision would flip.
Ordinal and cardinal payoffs — a ranking of outcomes versus a measurement of them. Ordinal is faster and assumes you already know the order. Cardinal has to be computed and tells you how much is at stake.
For this book, payoffs are profit. You have a demand system, a cost structure, and a procedure for finding where prices settle. Every cell of a matrix is one run of that procedure under one combination of decisions.
Building the Game, One Cell at a Time
Back to the ice cream sandwiches, with the agency now selling to both shops. Each firm can buy the campaign or not. The campaign costs $0.50 per person per month and adds one unit of consumption per person per month to whoever buys it.
The players are Smart Cookie and Hogi Yogi. The actions are advertise or do not. That gives four combinations, and each one is a different market with different parameters, so each one needs its own equilibrium.
That is the whole construction. Change the parameters to match the combination, solve the pricing game, record the profits, subtract the cost of the campaign from whoever bought it.
| Smart Cookie | Hogi Yogi | p sc | p hy | π sc | π hy |
|---|---|---|---|---|---|
| does not | does not | $1.53 | $1.02 | $1.33 | $0.96 |
| does not | advertises | $1.74 | $1.18 | $2.11 | $1.14 |
| advertises | does not | $1.79 | $1.05 | $1.84 | $1.04 |
| advertises | advertises | $2.00 | $1.20 | $2.86 | $1.24 |
Four rows, four equilibria, and every number in them computed rather than assumed. Notice that the prices differ in every row. The firms are not choosing between four versions of the same market, they are choosing which market to be in.
Arranged as a matrix, that is the game:
| Hogi Yogi does not advertise | Hogi Yogi advertises | |
|---|---|---|
| Smart Cookie does not advertise | $1.33, $0.96 | $2.11, $1.14 |
| Smart Cookie advertises | $1.84, $1.04 | $2.86, $1.24 |
Solving It
This part takes a paragraph, which is the point.
Look down each column for Smart Cookie. If Hogi Yogi sits still, Smart Cookie earns $1.33 by doing nothing and $1.84 by advertising. If Hogi Yogi advertises, Smart Cookie earns $2.11 by doing nothing and $2.86 by advertising. Advertising is better in both cases, so it is a dominant strategy — Smart Cookie does not need to predict anything.
Now across each row for Hogi Yogi. Against a passive Smart Cookie, $0.96 or $1.14. Against an advertising Smart Cookie, $1.04 or $1.24. Also dominant.
Both firms advertise. The equilibrium is $2.86 and $1.24, and neither firm has any reason to deviate.
Dominant strategy — an action that is best whatever the other firm does. When one exists you do not have to forecast your rival, which is the easiest kind of decision competition ever offers.
Two things about that outcome deserve notice, and neither was visible before the game was built.
Both firms end up better off than if neither had advertised: $2.86 against $1.33, and $1.24 against $0.96. This is not a trap. The spending genuinely creates value here, because the campaign brings new consumption into the market rather than shuffling existing consumption between two shops. Not every competitive escalation works this way, and the next chapter is about telling the difference.
And the gap widens again. Smart Cookie’s lead goes from thirty-six cents to a dollar sixty-two. Hogi Yogi advertises, correctly, and finishes further behind than when it started. The advertising was worth buying and it was never going to fix the problem.
What Would the Ad Have to Cost?
Here is the return on having built the game rather than been handed it.
A payoff matrix is four numbers. A payoff function is the machinery that produced them, and it will answer questions the numbers cannot. The most useful one: the agency is charging $0.50, so at what price would this stop being worth it?
That question has an answer, because the gross gains are computable and the cost is the only thing that changes.
| Situation | Worth buying while the campaign costs less than |
|---|---|
| Hogi Yogi, if Smart Cookie sits still | $0.67 |
| Hogi Yogi, if Smart Cookie advertises | $0.70 |
| Smart Cookie, if Hogi Yogi sits still | $1.02 |
| Smart Cookie, if Hogi Yogi advertises | $1.25 |
At $0.50 the agency is selling below every one of those thresholds, which is why advertising is dominant for both firms. Raise the price and the game changes character in stages. Above about $0.70, Hogi Yogi drops out whatever Smart Cookie does, and the equilibrium becomes Smart Cookie advertising alone. Above about $1.02, nobody buys.
So the same two firms, the same customers, the same campaign effectiveness, produce three different games depending only on what the agency charges. If you had been handed the matrix you could have solved it. Having built it, you can negotiate, and you know that Hogi Yogi’s willingness to pay tops out at about seventy cents while Smart Cookie’s runs past a dollar, which is worth knowing before either of them walks into the agency.
What a two-by-two hides
Real decisions are rarely binary. Advertising comes in amounts, not in a switch, and the campaign that adds one unit per person is one point on a curve that probably bends.
A two-by-two is a deliberate simplification that buys clarity about the structure: who has a dominant strategy, whether the outcome is a trap, which firm the spillover favors. Use it for that. When the question is how much rather than whether, the same machinery still works, but you are optimizing over a continuous choice rather than reading a cell, and the answer stops being a game in the textbook sense.
The Same Logic at Ten Billion Dollars
The construction does not change with the size of the decision, only the horizon over which profit is counted.
Consider two aircraft manufacturers deciding whether to develop a new, technologically advanced aircraft, which is the situation Airbus and Boeing faced. Each can innovate or not. The payoffs are the net present value of Bertrand competition profits over twenty years, computed under each of the four combinations, with the development cost carried as a fixed commitment.
| Boeing does not innovate | Boeing innovates | |
|---|---|---|
| Airbus does not innovate | $30B, $40B | $20B, $30B |
| Airbus innovates | $40B, $20B | -$10B, -$20B |
Solve it the same way. Boeing earns $40B rather than $30B if Airbus stays out, and $20B rather than losing $20B if Airbus goes ahead. Not innovating is dominant for Boeing. Knowing that, Airbus compares $30B with $40B and develops the aircraft.
The structure is identical to the ice cream sandwiches. What differs is that the payoffs took months of demand analysis and cost forecasting to produce, and that the mistaken cell, the one where both firms innovate, would have cost thirty billion dollars.
That is the argument for building games carefully. It is also, as it happens, where the research this part closes on began.
Before you trust a game you have built
- Is every cell a computed equilibrium? If any payoff was estimated by intuition while the others were calculated, the comparison between them is meaningless.
- Did you let prices move in every cell? The most common error is holding price fixed at today’s level in the cells where the structure changed. The last chapter showed what that costs.
- Are the payoffs on one clock? Per person per month, or NPV over twenty years, but not one of each. Mixing horizons across cells produces a matrix that looks solvable and is not.
- Have you included the cost of the action in the cells where it was taken? Only there. This is the easiest arithmetic slip in the whole construction.
- Would a small change in one parameter flip the equilibrium? If so, say so out loud. A game whose answer depends on the third decimal place of an estimated coefficient is telling you the decision is close, not that it is resolved.
Ask yourself — what are the four futures?
Name one non-price decision in front of you and one competitor who could make the same decision.
Write the four combinations. Both do it, you alone, they alone, neither. Do not put numbers in yet. Just say, in a sentence each, what the market looks like in that future.
Now ask which of the four you have actually thought about. Most people have considered two: the world where they act and the world where they do not, both assuming the rival stands still. The two cells that involve the competitor moving are usually blank, and one of them is where the trouble is.
Then, for the cell that worries you most, ask what you would have to know to put a number in it. That list is your next piece of analysis, and it is shorter than it looks. It is demand, cost, and scale under one set of assumptions.
The move: Build the game before you solve it. Solving is arithmetic and a machine will do it; the payoffs are judgment plus every chapter that came before, and a matrix filled in by intuition gives a confident answer to a question nobody asked.
Both games in this chapter had a comfortable property: the equilibrium left everyone better off than doing nothing, or at least left the winner clearly ahead. That is not guaranteed. There are competitive moves where both firms act rationally, both end up worse than if neither had moved, and no amount of skill at solving the game gets you out of it. Telling those apart in advance is the difference between a competitive investment and an expensive reflex.