Game Theory: Basic to Advanced

The mathematics, not the anecdotes

Game theory is unusually prone to being taught as a collection of stories: the prisoner's dilemma recounted, Nash equilibrium invoked loosely, a conclusion about human nature drawn. That version is entertaining and teaches almost nothing.

Here it is a formal subject — strategies, payoffs, equilibrium concepts, theorems and proofs — running from payoff matrices and dominance to Sprague-Grundy, repeated games and auction design.

What Nash equilibrium does not promise

A Nash equilibrium is a strategy profile in which no single player can improve their own payoff by unilaterally deviating. That is the entire content, and four things follow that popular accounts routinely obscure.

It does not mean the outcome is good — the prisoner's dilemma equilibrium is worse for both players than the cooperative outcome, which is precisely why the example is famous. It does not mean the equilibrium is unique; many games have several. It does not mean players will find it. And it says nothing about fairness.

A student who holds those caveats understands the concept better than most people who cite it.

Representation first

How a game is written down determines what you can say about it.

Normal form — the payoff matrix — suits simultaneous play. Extensive form — the game tree, with information sets — is required when moves are sequential or when a player does not know what has already happened. Converting between the two is routine and instructive, and getting the representation wrong makes the rest of the analysis answer a different question than the one asked.

Backward induction and non-credible threats

Finite games of perfect information are solved from the terminal nodes upward, and the resulting subgame perfect equilibrium is sharper than Nash equilibrium alone.

The reason is worth stating. Nash equilibrium permits threats a player would never actually carry out — a strategy that says "if you do that, I will do something that hurts us both" can support an equilibrium even though nobody believes it. Subgame perfection rules those out by requiring the threat to be optimal at the point it would have to be executed. That distinction is one of the genuinely useful ideas in the subject, and it has obvious analogues outside mathematics.

Impartial games and the theorem that unifies them

A separate strand of the subject deals with combinatorial games: two players alternate, the moves available depend only on the position rather than on whose turn it is, and the last player to move wins.

Nim has a complete solution. The Sprague-Grundy theorem then does something remarkable: it shows that every impartial game is equivalent to a single Nim heap of some size. So once a student can compute nim-values, a whole class of competition problems — chips in boxes, stones in piles, tokens moved on a board — collapses into one technique.

This is the unit that makes the Integral Cup game theory track approachable, and it is the material students most often report enjoying.

Why cooperation can be rational

The prisoner's dilemma played once has a bleak and unambiguous answer. Played repeatedly, with the end date unknown, it does not.

Repeated games, the folk theorem, trigger strategies and tit-for-tat show that cooperation can be sustained as an equilibrium when the shadow of future interaction is long enough. Discounting makes "long enough" precise. Evolutionary game theory then removes the rationality assumption entirely and asks which strategies survive in a population — a different question with a strikingly similar answer.

Designing the game instead of playing it

The final unit inverts the subject. Instead of analysing a game, you design the rules so that the behaviour you want is optimal for the participants.

The second-price sealed-bid auction is the clean example: bidding your true valuation is a dominant strategy, which is an engineered property rather than a happy accident. Revenue equivalence, incentive compatibility and the basics of mechanism design follow. This is the branch of game theory with the clearest commercial footprint — advertising auctions and spectrum sales are built on it.

Who this suits

Students from Class 9 upward through undergraduate study. This is the most self-contained course in the series: it needs school algebra and basic probability for the mixed-strategy material and little else, where the other courses assume more.

For that reason it is often the right third track for an Integral Cup entrant, and a good first course for a student who wants to see what mathematical reasoning looks like before committing to a heavier syllabus. Classes are live and online in groups of at most eight.