Table of Contents
Picture a lawyer and a mathematician walking into a conference room to draft a contract. The mathematician pulls out a flowchart mapping every possible future scenario. The lawyer rolls their eyes so hard you can hear it. This scene plays out in boardrooms and courtrooms more often than anyone admits, and at the heart of this professional cold war lies a deceptively simple concept: the complete contract.
Game theorists dream of complete contracts the way architects dream of buildings that never need repairs. A complete contract, in theory, specifies exactly what happens in every possible state of the world. Your business partner steals your clients? The contract covers it. A pandemic shuts down global supply chains? Already in clause 47b. Aliens land and disrupt the semiconductor market? Check appendix Q.
Lawyers hate this idea. Not because it’s wrong, but because it’s impossibly right.
The Mathematician’s Fantasy
Game theory treats contracts like elegant equations. Two rational parties sit down, calculate all possible futures, assign probabilities, and write down the optimal response to each scenario. The result is a document so complete that neither party ever needs to return to the negotiating table. Disputes evaporate because every contingency has a predetermined solution. Courts become obsolete. Lawyers become museum pieces next to typewriters and rotary phones.
The beauty of this framework lies in its clarity. When economists model contracts, they assume both parties have perfect information about their preferences, capabilities, and constraints. They know every possible outcome and can calculate the expected value of each choice. The contract becomes a comprehensive map of the future, leaving no room for interpretation or disagreement.
This is not mere academic fantasy. The logic is sound. If you could write a truly complete contract, you would eliminate enormous costs. No litigation. No renegotiation. No ambiguity eating away at business relationships. The efficiency gains would be staggering.
There’s just one problem. The world doesn’t work this way.
The Reality Tax
Consider what would actually go into a complete contract for something as mundane as selling a car. You would need to specify what happens if the engine fails in month three versus month four. What if it fails because of manufacturing defects versus buyer negligence versus an undetectable flaw in the metal alloy? What if the failure occurs during a heatwave versus winter? What if the buyer used premium fuel versus regular, drove mostly highway versus city, exceeded the speed limit by five percent of the time versus ten percent?
Now multiply this by every component of the car. Then account for changes in emissions regulations, fuel prices, insurance costs, and resale values. Factor in the possibility of recalls, natural disasters affecting the supply of spare parts, and shifts in consumer preferences that might make your vehicle unsellable.
You would need a document longer than the car’s operating manual. And you would need to update it constantly as new information emerged. The transaction costs of writing, negotiating, and enforcing such a contract would exceed the value of the car itself.
This is not a flaw in the math. Game theorists understand transaction costs perfectly well. But here’s where lawyers start grinding their teeth: the mathematical models often treat these costs as unfortunate friction rather than fundamental limits on what contracts can achieve.
The Incompleteness Theorem of Business
Lawyers know something that game theorists model but don’t always internalize. Contracts are incomplete not because people are lazy or stupid, but because completeness is economically irrational.
Take the example of employment contracts. A truly complete employment contract would specify the employee’s required effort level for every possible task, market condition, and personal circumstance. It would define exactly what constitutes satisfactory performance when the employee has a sick child, when the company is under financial pressure, when new technology makes old skills obsolete, or when a global pandemic forces everyone to work from home.
Writing this down is impossible. But even if you could write it down, you couldn’t verify it. How do you prove in court that someone gave 87 percent effort instead of the contractually required 94 percent? How do you measure the creativity demanded by clause 23 or the judgment called for in section 15?
So we use incomplete contracts with vague terms like “reasonable effort” and “good faith.” Lawyers don’t do this because they’re imprecise thinkers. They do it because precision would be more expensive than the problem it solves.
Where Money Disappears
Game theory reveals something counterintuitive about incomplete contracts. They don’t just save on writing costs. They can actually create value.
Imagine two companies forming a joint venture. If they tried to write a complete contract, they would spend months negotiating every detail. This negotiation would reveal their private information, bargaining positions, and future strategies. Some of this information is competitively sensitive. Some of it they don’t want to share because it would weaken their position.
An incomplete contract allows them to agree on the broad strokes and leave details for later. This isn’t kicking the can down the road. It’s a rational response to the fact that neither party knows yet which details will matter. Maybe the venture will pivot within six months and half the negotiated clauses will be irrelevant. Why pay lawyers hundreds of billable hours to argue about scenarios that will never occur?
The math here is brutal for the complete contract ideal. The cost of specifying a contract term rises exponentially as the likelihood of that contingency decreases. You pay the full cost upfront but only get value if that specific scenario occurs. For low probability events, you’re essentially burning money.
Lawyers understand this in their bones. They’ve seen clients waste fortunes trying to button up every hypothetical. Game theorists understand it in their models, but the models often skip past this insight too quickly, treating incomplete contracts as a temporary deviation from the ideal rather than the ideal itself.
The Exploitation Gap
Here’s where the story gets interesting. Incomplete contracts create gaps. Those gaps are where lawyers make their money.
When a contract doesn’t specify what happens in a particular situation, someone has to decide. That someone might be a judge, an arbitrator, or the parties themselves in renegotiation. Whoever makes that decision has power. Lawyers help clients position themselves to either make that decision or influence the person who does.
Game theorists see this and get uncomfortable. The gaps look like inefficiencies, like places where value leaks out of the economic system. If only the parties had written a complete contract, they wouldn’t need to fight over interpretation later.
But look closer at what happens in those gaps. Often, the parties find creative solutions they couldn’t have anticipated when writing the original contract. The world changes. New technologies emerge. Markets shift. The gaps in the contract give the parties flexibility to adapt.
A complete contract locks you into yesterday’s understanding of tomorrow. An incomplete contract admits that tomorrow might look different than anyone expects.
The Courtroom as Laboratory
Legal disputes over contract interpretation generate something valuable: information about how courts will handle ambiguity. Each case creates precedent that fills in the gaps for future contracts.
This is a distributed problem solving system. Instead of every party trying to anticipate every contingency, the legal system gradually builds up a database of how to handle specific situations. When a court decides that “reasonable notice” means fourteen days in a particular industry context, every future contract in that industry can rely on that interpretation without spelling it out.
Game theorists might model this as a learning process or an evolutionary system. The law evolves through case law the way species evolve through natural selection. Successful interpretations survive and propagate. Failed interpretations disappear.
Lawyers don’t usually think in these terms, but they operate within this system expertly. They know which terms are well defined by precedent and which remain ambiguous. They know when to be precise and when to be deliberately vague. This isn’t mathematical optimization, but it’s a different kind of intelligence.
The Nash Equilibrium Nobody Wants
Game theory proves something disturbing about contracts. Even when both parties would benefit from a complete contract, they might rationally choose to leave it incomplete.
Suppose you’re negotiating with someone you don’t entirely trust. A complete contract would eliminate their ability to exploit you later. But writing that complete contract requires revealing information about your business, your constraints, and your alternatives. That information makes you vulnerable during the negotiation itself.
You face a dilemma. Stay vague and preserve your bargaining power, or be specific and risk exploitation before the ink dries. Both parties face the same dilemma. The result is an equilibrium where everyone writes incomplete contracts even though everyone would prefer the certainty of completeness.
Lawyers navigate this terrain constantly. They know how to signal credibility without revealing too much, how to be specific about important terms while staying vague on others, how to build in flexibility without inviting abuse. These skills don’t fit neatly into mathematical models, but they’re essential to making contracts work.
Why Math Still Wins
After all this praise for legal realism over mathematical idealism, here’s the turn. Game theory still matters, and the complete contract remains a valuable benchmark even if it’s unachievable.
Consider what happens when you try to draft a contract without thinking about incentives, probabilities, and strategic behavior. You get documents that look good on paper but fall apart under pressure. You create perverse incentives where both parties benefit from failure rather than success. You leave gaps that invite exploitation rather than enabling flexibility.
The mathematical approach forces clear thinking. What are we trying to achieve? What could go wrong? How will each party behave given the incentives we’re creating? These questions don’t guarantee a complete contract, but they produce much better incomplete contracts.
Take performance bonuses. A lawyer might write: “Employee receives a bonus based on satisfactory performance as determined by management.” This is incomplete by design. It gives management flexibility to respond to circumstances the parties can’t anticipate.
But a lawyer who understands game theory will recognize the problem. Management has an incentive to set an impossibly high bar or move the goalposts after the employee has already committed effort. The employee, anticipating this, has an incentive to withhold effort or focus on easily measured metrics rather than what actually creates value.
A better incomplete contract might specify: “Employee receives a bonus equal to 10 percent of department revenue growth above 5 percent, with revenue calculated using the same methodology as the previous year unless both parties agree to changes.” This leaves room for interpretation and adjustment, but it constrains the ways either party can exploit the ambiguity.
The math doesn’t write the contract. But it illuminates the strategic landscape in which the contract operates.
The Evolution of Trust
Repeated interactions change everything. Game theory proves that cooperation can emerge even without complete contracts if the relationship continues long enough.
When parties expect to work together again, they have an incentive to build reputation rather than exploit every short term advantage. An employer who cheats on bonuses will struggle to attract good employees. An employee who shirks will get worse assignments or no contract renewal. These future consequences discipline behavior more effectively than any contract clause.
Lawyers see this clearly. Many business relationships rely more on reputation and repeat dealing than on contract terms. The contract exists mainly to handle extreme cases where the relationship breaks down completely.
But here’s where game theory adds something valuable. It shows exactly when reputation mechanisms work and when they don’t. If the relationship will end soon, if the stakes are high enough, or if one party can profit more from cheating than from preserving the relationship, reputation won’t save you. You need contract terms with teeth.
The math tells you which risks you can manage through relationship building and which require legal protection. Lawyers provide the protection. Game theorists calculate whether you need it.
The Efficient Breach
Here’s where lawyers and game theorists occasionally unite against common sense. Both groups recognize that sometimes breaking a contract is economically efficient.
Suppose you contract to deliver widgets for ten dollars each. Later, someone offers you twenty dollars for the same widgets. If the penalty for breach is twelve dollars per widget, you should breach the original contract, pay the penalty, and sell to the higher bidder. You’re better off, the original buyer gets compensated, and the higher bidder gets the widgets they value more. Total wealth increases.
This offends moral intuitions about keeping promises. But the math is clear, and the law reflects this logic through the expectation damages remedy. You don’t have to perform. You just have to make the other party whole.
Complete contracts would specify exactly when breach is efficient and structure the damages accordingly. Incomplete contracts rely on courts to calculate the right damages after the fact. Both approaches recognize the same underlying truth: contracts are economic instruments, not moral absolutes.
Lawyers monetize this gap between moral and economic reasoning. Clients who think contracts are sacred pay more in legal fees than clients who view them as financial tools with built in exit clauses.
The Real Divide
The tension between lawyers and game theorists isn’t really about complete versus incomplete contracts. It’s about different types of expertise and different claims to authority.
Game theorists claim authority through mathematical rigor. Their models prove theorems. The logic is airtight even if the assumptions are debatable. This gives them credibility with engineers, economists, and anyone who values quantitative precision.
Lawyers claim authority through institutional knowledge and practical judgment. They know how courts actually rule, how negotiations actually unfold, and how contracts actually get enforced. This gives them credibility with business people and anyone who values experience over abstraction.
The conflict arises when each group dismisses what the other brings to the table. Game theorists sometimes treat legal reasoning as muddy thinking that could be replaced with clearer mathematics. Lawyers sometimes treat game theory as ivory tower nonsense disconnected from how the world works.
Both are wrong. The best contracts combine mathematical clarity about incentives and outcomes with legal sophistication about interpretation and enforcement. The best negotiators understand both the theory of optimal bargaining and the practice of reading the room.
Why You Should Side With the Math
Despite everything said in defense of lawyers and incomplete contracts, the mathematical perspective deserves the final word.
Game theory doesn’t just model contracts. It reveals fundamental truths about strategic interaction, information asymmetry, and collective action problems. These truths don’t change based on industry norms or legal traditions. They’re as reliable as physics.
When a lawyer tells you that a vague contract term is standard in your industry, game theory lets you ask: standard for whom? Does this term benefit the stronger party at the expense of the weaker? Does it create incentives for the behavior you actually want?
When a lawyer tells you not to worry about a particular contingency because it’s unlikely, game theory lets you calculate whether unlikely means one in a hundred or one in a million, and whether you can afford to be wrong.
When a lawyer tells you to trust the relationship and not over specify, game theory lets you model exactly how much trust is rational given the incentives and the expected duration of the relationship.
The math doesn’t replace legal judgment. But it keeps legal judgment honest.
Lawyers hate game theorists because mathematics exposes the gaps in legal reasoning. It shows when traditional practices are inefficient. It reveals when vague language masks exploitation. It proves that many things lawyers claim are necessary are actually choices that benefit some parties at the expense of others.
You should side with the math not because it gives you complete contracts, which remain impossible, but because it gives you clear thinking. In a world where contracts must be incomplete, clarity about what you’re leaving out and why matters more than ever.
The lawyer and the mathematician walk out of that conference room with a contract that will inevitably leave some contingencies unspecified. But if they’ve done their job well, informed by both legal experience and mathematical rigor, they’ve specified the right things and left the right gaps.
That’s not a complete contract. It’s something better: an intelligent incomplete contract, written by people who understand why completeness is impossible and what to do about it.
The game never ends. But knowing the rules helps you play better.
