What a Semester of Graduate Game Theory Taught Me (and What It Didn't)

The setup

I took Econ 2048 expecting to learn "how to find Nash equilibria." I left with something different: a sense for when equilibrium concepts are the right question, and when they're covering up the wrong one. The course moved fast — normal form games, Bayesian games, extensive form, refinements, monotone comparative statics, supermodular games, potential games — all in one semester, taught by Shengwu Li with the kind of precision that makes you realize how much hand-waving you've been doing your whole life.

What follows is my attempt to reconstruct the intellectual arc. Not a textbook summary — more like "here's what actually stuck, and why."

The normal form

The course opens with a quote from Von Neumann and Morgenstern that reframes what a "strategy" is. In everyday language, a strategy is a vague plan. In game theory, a strategy is a complete contingent plan: a function that specifies what you'll do in every possible situation you might encounter, given every possible piece of information you might have.

This is an extreme definition, and the course is upfront about that. Savage's famous critique is right there in the notes: requiring someone to "envisage every conceivable policy for the government of his whole life" is "utterly ridiculous" — not because you might regret it later, but because the computational task is beyond human possibility. You can't even plan a picnic this way.

But the power of the normal form is exactly this abstraction. A game $(N, S, u)$ consists of players $N$, strategy sets $S_n$ for each player, and utility functions $u_n : S \to \mathbb{R}$. That's it. Every sequential game, every dynamic interaction, every elaborate mechanism — all reduce to choosing from a set of complete plans and getting a payoff.

"The idea that this mathematically simple normal form is a general model for all games was the key simplifying insight that allowed the first great advances of modern game theory." — Myerson (2004)

Dominance and rationalizability

Before asking "what will players do?", the course asks a gentler question: "what won't they do?" A strictly dominated strategy is one that is always worse than some alternative, no matter what opponents do. Rational players won't play them.

But the real insight comes from iterating. If I know you're rational, I know you won't play dominated strategies. So I can eliminate those from your strategy set. But now, with your strategy set trimmed, some of my strategies might become dominated. We can keep going — this is iterated deletion of strictly dominated strategies.

What survives is the rationalizable set. A beautiful theorem connects the two ideas: a strategy is not strictly dominated if and only if it's a best response to some belief about opponents. The proof uses the separating hyperplane theorem — you show that the set of achievable payoff vectors and the set of payoffs that would strictly dominate your strategy are two disjoint convex sets, and the separating hyperplane gives you the belief.

This is the first of many moments in the course where a seemingly semantic distinction — "not dominated" vs. "best response to something" — turns out to be a real mathematical equivalence with a nontrivial proof.

Nash equilibrium

A Nash equilibrium is a strategy profile where everyone is best-responding to everyone else simultaneously. It's the most famous concept in game theory, and it exists in every finite game.

The existence proof uses Kakutani's fixed point theorem. Define the best-response correspondence $B : \Sigma \rightrightarrows \Sigma$, where $B_n(\sigma) = \arg\max_{\sigma'_n} u_n(\sigma'_n, \sigma_{-n})$. A Nash equilibrium is exactly a fixed point of $B$ — a profile that maps to itself under best-responding. Kakutani guarantees this fixed point exists because the strategy simplex is compact and convex, best-response sets are nonempty and convex, and the correspondence is upper hemi-continuous.

But here's the uncomfortable fact the course doesn't shy away from: finding a Nash equilibrium is computationally hard. It's PPAD-complete (Daskalakis et al., 2009). You might hope that the extra structure of games makes it easier than generic fixed-point problems. It doesn't.

This matters more than it sounds. If Nash equilibrium is the "right" prediction for rational play, but neither humans nor computers can efficiently find it, then in what sense is it a prediction at all?

The refinement ladder

Nash equilibrium has a problem: there are usually too many of them, and some are clearly unreasonable. The refinement program tries to narrow down which equilibria are "plausible" by adding extra requirements.

The idea behind perfect equilibrium (Selten, 1975) is: what if players occasionally tremble — accidentally play the wrong action with tiny probability? A perfect equilibrium survives this test. It's the limit of $\varepsilon$-perfect equilibria as trembles vanish.

Proper equilibrium (Myerson, 1978) goes further: not just that you rarely tremble, but that worse mistakes are less likely than better ones. If action $a$ gives lower payoff than $a'$, then $\sigma_n(a) \le \varepsilon \cdot \sigma_n(a')$. This is a much more structured trembling story.

The course presents a game that illustrates the difference cleanly. Three Nash equilibria — but only two are perfect, and only one is proper. The proper equilibrium is the one where players' trembles are "ordered by rationality" — a genuinely elegant constraint.

Correlated equilibrium

This was one of the most eye-opening parts of the course. Correlated equilibrium (Aumann, 1974) allows a mediator to recommend actions to players using a shared random signal. Each player sees only their own recommendation and has no incentive to deviate.

The kicker: unlike Nash equilibria, correlated equilibria can be computed in polynomial time. The CE constraints are linear in the probabilities, so you can find one via linear programming. Nash constraints become nonlinear the moment you require the joint distribution to be a product of independent marginals — that's what makes Nash hard.

Three interpretations of CE that the course highlights: (1) explicit coordination by a trusted mediator, (2) informational robustness — a CE captures what you can predict about play without assuming independence, and (3) learning — players who meet repeatedly and play myopic best-responses to calibrated forecasts converge to CE.

Extensive-form games

The normal form collapses all sequential structure. The extensive form brings it back — who moves when, what they know at the time, what they've observed. The notation is heavier (histories, information sets, active player functions, behavioral strategies) but the payoff is that we can talk about dynamic rationality: not just "is this a good plan?" but "would you actually follow through with it?"

A key result: Kuhn's theorem — under perfect recall, mixed strategies and behavioral strategies are equivalent. This sounds technical but it's what lets us work with decision-by-decision strategies instead of exponentially large complete plans.

Subgame-perfect equilibrium (SPE) is the natural first refinement: require Nash equilibrium in every subgame. Computable via backward induction. But SPE is fragile — "seemingly-irrelevant transformations of the game tree can eliminate subgames," and with them all the rationality requirements that SPE imposed.

Sequential equilibrium

The course spends significant time on the problem of off-path beliefs. In extensive games, some information sets are never reached in equilibrium. But players still need beliefs about what's happening there — and those beliefs can be arbitrary, sometimes supporting unreasonable equilibria.

Weak Perfect Bayesian Equilibrium (WPBE) requires sequential rationality and Bayesian updating on the path of play. But it doesn't restrict off-path beliefs, which leads to three bugs the course illustrates with examples:

1. Players can "signal what they don't know" — off-path beliefs let them convey information they shouldn't have.
2. Players can make "different inferences from the same information" — two players who've seen identical histories can hold contradictory beliefs.
3. WPBE doesn't even imply SPE in some games.

The early literature added ad hoc patches (Bayesian updating "wherever possible," no signaling what you don't know, agreement across players). Shengwu's comment in the notes: "This was a mess."

Sequential equilibrium (Kreps & Wilson, 1982) cleans this up with a single elegant idea: consistency. Beliefs are the limit of what Bayes' rule would give under fully mixed strategies (where every action at every information set gets positive probability). As the trembles vanish, the limiting beliefs pin down what you "should" believe even off-path.

Sequential equilibrium implies both WPBE and SPE, and every finite extensive game has one.

Monotone comparative statics

The traditional approach to comparative statics: maximize $b(x, t) - c(x)$, take the first-order condition, apply the implicit function theorem:

If the cross-partial $b_{xt} > 0$, $c$ is convex, and $b$ is concave in $x$, then $x^*$ is increasing in $t$. This works — but the course immediately shows it's overkill. The conclusion holds even if $b$ and $c$ aren't differentiable, $c$ isn't convex, $b$ isn't concave, and $x$ isn't even real-valued. The differentiability was only a servant to the method.

The intuitive idea: if doing more of $x$ becomes more attractive when the parameter $t$ is higher, then optimal $x$ goes up. The modern approach strips this down to what actually matters: lattice structure and increasing differences.

Topkis's monotonicity theorem

The centerpiece result. If $f(x, t)$ is supermodular in $x$ and has increasing differences in $(x, t)$, and the feasible set $S(t)$ is increasing in the strong set order, then the set of optimizers $X^*(t) = \arg\max_{x \in S(t)} f(x, t)$ is increasing in $t$ (in the strong set order).

The beauty is that in many applications, some conditions are trivially satisfied. If $X \subseteq \mathbb{R}$, it's automatically a lattice and every function is supermodular in $x$. If constraints don't vary with $t$, condition 4 is free. You only need to check increasing differences — which often reduces to checking a cross-partial is nonneg.

Supermodular games

Topkis's theorem is about single-agent optimization. Supermodular games extend it to strategic settings. A supermodular game (Milgrom & Roberts, 1990) is one where players' strategies are strategic complements: when one player increases their action, others want to increase theirs.

The headline result: supermodular games have extremal pure-strategy Nash equilibria — a smallest and a largest. And these bound the entire rationalizable set: every rationalizable strategy is between the smallest and largest equilibrium.

This is remarkable. In a generic game, Nash, correlated equilibrium, and rationalizability can make very different predictions. In supermodular games, they all agree on the upper and lower bounds. And a corollary: a supermodular game is dominance solvable if and only if it has a unique pure-strategy Nash equilibrium.

Comparative statics for supermodular games fall out cleanly: if you parameterize the game and payoffs have increasing differences in the parameter, then the extremal equilibria are monotone in the parameter. You get predictions about how equilibria move without solving for them.

Potential games

A potential game is one where a single function $F$ captures every player's incentives: any player's gain from deviating equals the change in $F$. If $F$ has a maximum, it's a Nash equilibrium.

This immediately gives existence of pure-strategy NE (just maximize $F$) and the finite improvement property: start anywhere, let players make beneficial deviations one at a time, and you must reach an equilibrium because $F$ strictly increases at each step. No cycles possible.

The canonical example is congestion games — think traffic routing. Each player picks a set of roads; each road has a cost that increases with congestion. The potential function sums up "accumulated costs": $F(a) = -\sum_{x} d_x(n_x(a))$ where $d_x(m) = c_x(1) + c_x(2) + \cdots + c_x(m)$.

A beautiful application: VCG auctions with endogenous investment. Players can invest to change their values before a VCG mechanism runs. The investment game turns out to be a potential game with potential function equal to social welfare. So at any pure-strategy Nash equilibrium, no single player can improve social welfare by changing their investment. The mechanism aligns individual incentives with collective good — at least locally.

What the course didn't teach

Every course makes choices, and the omissions are as revealing as the inclusions.

No repeated games. The folk theorem — that almost anything can be sustained as an equilibrium in infinitely repeated games — is one of the most important (and unsettling) results in game theory. The course skipped it entirely. I think this is defensible: the folk theorem is about what's possible in equilibrium, not what's likely, and the course prioritized tools that make sharp predictions.

No mechanism design. How to design rules of a game to achieve a desired outcome — auctions, voting systems, matching markets. The VCG example at the end was a taste, but full mechanism design is a separate course (and one I want to take). This is where game theory becomes engineering rather than analysis.

No evolutionary game theory. What happens when "players" don't optimize but instead follow simple rules, and selection pressure does the work? This connects to learning dynamics, replicator equations, and — I suspect — to how neural networks learn strategic behavior during training.

Limited learning and dynamics. The CE-as-learning-outcome result (Foster & Vohra) was mentioned but not developed. How players actually converge to equilibria — or fail to — is arguably more important than the equilibria themselves.

The alignment connection

Principal-agent as the core framing. The alignment problem is, at its root, a game between a principal (humans) and an agent (AI) with misaligned information and possibly misaligned objectives. The extensive-form game structure — sequential moves, imperfect information, beliefs about opponent types — maps directly. What makes alignment hard is that we're playing a Bayesian game where we don't know the agent's type (its learned objectives) and can only partially observe its actions (its internal reasoning).

Refinements as a theory of "what kind of rationality to assume." The refinement ladder — from rationalizability to Nash to perfect to proper to sequential — is really about how much intelligence to attribute to the players. For AI alignment, this is directly relevant: how capable is the model at strategic reasoning? A model that only avoids dominated strategies behaves very differently from one that plays sequential equilibria. As models become more capable, we may need to assume stronger rationality concepts — and design our oversight mechanisms accordingly.

Correlated equilibrium and the mediator problem. In alignment, we want to be the mediator — the entity that coordinates the game by sending private recommendations that players voluntarily follow. Constitutional AI, RLHF, and system prompts are all attempts at mediation. The CE framework suggests a precise question: under what conditions will the AI voluntarily follow the mediator's recommendation? The answer depends on whether the AI's incentive constraints are satisfied — which requires knowing its utility function. And that's exactly what mechanistic interpretability is trying to figure out.

Supermodular games and capability cascades. If AI capabilities are strategic complements — one lab advancing makes other labs want to advance faster — then the AI development landscape is a supermodular game. The extremal equilibrium result says the rationalizable strategies are bounded by the extremal equilibria. In a race with strategic complements, this means the "rational" range of behavior is pinned between "everyone goes slow" and "everyone goes fast." Comparative statics tells you how these bounds shift when you change parameters like compute costs or regulation.

Potential games and alignment by design. If we could design the "game" between AI systems and humans so that it has a potential function aligned with social welfare, then individual incentive-compatible behavior would automatically promote collective good. The VCG-with-investment result is a prototype: mechanism design can align local incentives with global welfare, at least at equilibrium. Whether this extends to the AI setting — where "players" are learned systems with opaque objectives — is a research question I'm very interested in.

Where the analogy breaks down. Classical game theory assumes players know the rules of the game, have well-defined utility functions, and can compute best responses. AI systems violate all three: they learn the "rules" from data, their "utility functions" are emergent properties of training, and their "best responses" are approximated by gradient descent over billions of parameters. The gap between the theory and the application is large. But I think the structural analogies — the information asymmetries, the strategic complementarities, the monitoring problems — are real, and the formalism gives us a language to make them precise.