Markets: the worked examples

These examples explore the ideas in the short essay on incentives through explicit market models. Read whichever question interests you.

You run a coffee shop. The shop across the street sells a similar coffee, and you are deciding whether to lower your price. A lower price earns less per cup, but might bring in more customers. How many depends partly on what the other shop charges.

Your decision depends on someone else’s decision. That is the kind of problem game theory studies. In a market economy, it appears in pricing, wage negotiations, supplier relationships, investment, and the rules governing trade.

Two businesses choosing a price

Suppose each shop can choose a higher or lower price. For this example, assume demand and costs produce these weekly profits:

Your shop ↓ / Other shop →	Higher price	Lower price
Higher price	(3, 3)	(1, 4)
Lower price	(4, 1)	(2, 2)

The numbers are invented profit units, not the prices of the coffee. Each cell lists (your profit, their profit). If you charge less while they charge more, you gain enough customers to make 4 while they make 1. If both cut prices, you each make 2.

Hold their choice still. Against their higher price, your lower price gives 4 instead of 3. Against their lower price, your lower price gives 2 instead of 1. Lower is your best response in either case. It is also theirs, for the same reason.

When both charge less, neither can improve by changing price alone. That is a Nash equilibrium. A best response answers one person’s question, given the other’s choice; an equilibrium requires those answers to fit together. Yale’s business-partnership lecture develops this way of comparing incentives.

Select a price combination to compare each shop’s possible switch. The slider changes the assumed profit from being the only shop to cut its price; the other payoffs stay fixed.

Both shops would earn more at the higher-price outcome. But that does not establish that higher prices are better for everyone: customers are absent from this profit table. Their costs and benefits matter too. “Best for the two businesses” and “best for society” are different questions.

The result also depends on our invented demand and cost assumptions. A different product, loyal customers, limited capacity, or another available price could change the comparisons. Some games have several equilibria. Finding one does not prove that people will reach it.

Game theory calls the outcome numbers payoffs, or utilities. We used profit because this example concerns profit-seeking firms. A different model could include working conditions, fairness, environmental concerns, or care for other people. The mathematics does not require everyone to be selfish; the model has to say what each decision-maker values.

Bargaining depends on your alternatives

Suppose a buyer values an item at 12, and supplying it costs the seller 5. At a price of 8, the buyer gains 12 − 8 = 4, and the seller gains 8 − 5 = 3. Trade creates a total gain of 7; the price determines how they divide it.

That does not tell us which price they will agree on. Consider their alternatives.

The buyer can get an equivalent item elsewhere for 9, gaining 3. The seller can sell this item to someone else for 7, gaining 2. These are their outside options: what each can obtain without this particular agreement.

Now the buyer will not prefer paying more than 9, and the seller will not prefer receiving less than 7. Prices between 7 and 9 leave neither worse off than their alternative. At 8, both are strictly better off.

That range still does not select a unique bargain. Who can make an offer? Can the other person counteroffer? Who can afford to wait? How much does each know about the other’s alternatives? Those details belong in the game. Yale’s bargaining lecture shows how conclusions change with the bargaining procedure, patience, and information.

The same reasoning helps frame wage negotiations: another credible job offer changes a worker’s alternative to accepting the current offer. An employer’s ability to hire someone else matters on its side. Saying both parties are free to reject an offer does not tell us whether their alternatives are equally attractive.

A promise can change after an investment

A buyer wants a custom part. Making the necessary setup costs a supplier 2, and the buyer promises to pay 3 for the finished part. Assume there are no further production costs. The buyer values it at 6. If the deal happens as promised, the supplier earns 1 and the buyer gains 3.

But suppose the setup has no other use and the promised price is not binding. After the supplier has spent the 2, the buyer can either honor the price or replace it with an offer of 1. The supplier now compares accepting 1 with receiving nothing. The setup cost has already been paid either way. Accepting leaves a loss of 1; refusing leaves a loss of 2.

If the supplier accepts the reduced payment, the buyer gains 6 − 1 = 5, instead of 3 under the original promise. The buyer therefore has an incentive to change the terms after the investment.

Knowing this in advance, the supplier may refuse to invest: an expected loss of 1 is worse than staying out and earning 0. A project that could have benefited both sides never starts.

This reasoning works backward from the later decision. It exposes why a statement made beforehand may not be credible when the time comes to carry it out. Yale’s discussion of commitment distinguishes an announcement from a decision that actually changes later choices.

A payment made in advance, an enforceable agreement, or equipment that can serve other customers would change this example. Each changes a concrete feature of the game: what can be withheld, which choices remain available, or what happens if the relationship ends.

Repeated trade gives the future a role

Now consider a supplier who expects regular orders. Providing the agreed quality earns 3 per month. Quietly cutting quality would earn 5 today. Suppose the buyer detects the change, switches to an equally good alternative, and never returns; the supplier then earns only 1 per month elsewhere.

Cutting quality gains an extra 2 today, but loses 2 in each later month. Whether it pays depends on how much those future earnings matter. This gives reliable service an economic explanation without requiring us to assume the supplier cares only about money.

The future consequence has to be believable. The buyer needs to detect the problem and have a usable alternative. If quality cannot be observed, switching is prohibitively costly, or the supplier expects the relationship to end anyway, this particular incentive becomes weaker. Yale’s repeated-game lecture analyzes the gain from breaking an agreement against the value of continuing a trading relationship.

The calculation, if you want it

Assume infinitely many months and a discount factor $0\le\delta<1$: next month’s payoff counts by $\delta$, the following month’s by $\delta^2$, and so on.

Keeping quality gives

$$3+3\delta+3\delta^2+\cdots=\frac3{1-\delta}.$$

Cutting quality now, followed by the specified loss of business, gives

$$5+\delta+\delta^2+\cdots=5+\frac\delta{1-\delta}.$$

The first is at least as large when $\delta\ge1/2$. At $\delta=0.75$, the values are 12 versus 8. This checks the supplier’s incentive under our stated assumptions; it is not a universal threshold for trust or proof that an actual trading relationship will behave this way.

Reputation can matter because past behavior affects later choices. Mistakes complicate that mechanism: a late delivery might reveal unreliability, or it might be an exceptional disruption. A rule that ends every relationship after one bad outcome can destroy valuable trade too.

Who is missing from the calculation?

Return to the coffee shops. So far, every payoff table has included only the people making the immediate deal. A decision can also affect people outside it.

Imagine a firm saves 2 by using a disposal method that imposes 5 in cleanup costs on nearby residents. If it does not bear that cost, its own accounts favor the cheaper method even though the combined cost is higher. This is a negative externality: a cost falling on others that is not included in the decision-maker’s private comparison. External effects can also be beneficial; Yale’s partnership example examines effort that benefits someone besides the person supplying it.

Rules can change that comparison. In this invented example, a charge of 3 for the damaging method would outweigh the firm’s saving of 2. That illustrates an incentive change, not an argument that 3 is the right charge. Measuring harm, enforcing a rule, and accounting for its other consequences are additional problems.

This is where game theory connects to questions about capitalism: competition, bargaining power, ownership, contracts, and regulation affect the choices people face. The method can help explain what a particular arrangement encourages. It cannot choose the goals we should value by itself.

Actual people also have incomplete information, habits, loyalties, and conflicting priorities. Before treating a predicted outcome as an explanation of the world, check whose interests entered the table, which alternatives were available, and whether the promised consequences could really happen.