Game theory is the mathematical study of how rational agents make decisions when outcomes depend on each other's choices.
It's not about board games. It's about any situation where your best move depends on what others do — negotiations, pricing wars, arms races, elections, even nature.
The decision-makers in the game. Could be people, firms, countries, or even genes.
The complete set of actions available to each player.
The outcomes — rewards or costs — each player receives for every combination of choices.
Use the navigation above to explore the key concepts interactively. Start with the Prisoner's Dilemma →
Two suspects are arrested. They can't communicate. Each must decide: Cooperate (stay silent) or Defect (betray).
| B: Cooperate | B: Defect | |
|---|---|---|
| A: Cooperate | A gets 1yr, B gets 1yr | A gets 3yr, B gets 0yr |
| A: Defect | A gets 0yr, B gets 3yr | A gets 2yr, B gets 2yr |
You're Prisoner A. The AI plays Prisoner B. What do you choose?
Defecting is always individually rational (it's the dominant strategy). But if both defect, both are worse off than if both cooperated. Individual rationality ≠ collective rationality.
A Nash Equilibrium is a set of strategies where no player can benefit by unilaterally changing their strategy, given what everyone else is doing.
Named after mathematician John Nash (of A Beautiful Mind). Every finite game has at least one Nash Equilibrium.
It's a stable state — like a ball resting at the bottom of a bowl. No player has an incentive to deviate. It doesn't mean it's the best outcome, just a stable one.
Two firms choose price: High or Low. Select the Nash Equilibrium from the matrix below.
| Firm B: High Price | Firm B: Low Price | |
|---|---|---|
| Firm A: High Price | A:$8M, B:$8M | A:$2M, B:$11M |
| Firm A: Low Price | A:$11M, B:$2M | A:$5M, B:$5M |
Use the best response method: for each strategy of B, what is A's best response? Mark it. Do the same for B. A Nash Equilibrium is where both best responses overlap.
In the Prisoner's Dilemma, (Defect, Defect) is the Nash Equilibrium — even though (Cooperate, Cooperate) is better for everyone.
When a game is played repeatedly, cooperation can emerge. Players can reward or punish based on past behavior. Let's simulate strategies competing over multiple rounds.
Both Cooperate → Both get 3pts | Both Defect → Both get 1pt
One defects, one cooperates → Defector gets 5pts, Cooperator gets 0
Tit-for-Tat: Start cooperative, then copy opponent's last move. Simple, forgiving, retaliatory.
Grim Trigger: Cooperate until opponent defects once — then defect forever. Maximum punishment.
Insight: In Robert Axelrod's famous tournaments, Tit-for-Tat won consistently — showing that being nice, retaliatory, forgiving, and clear is the optimal long-run strategy.
Auctions are games too. How you bid depends on what others might bid, what the item is worth to you, and the auction format.
In a common-value auction (e.g., oil rights), the winner often overpays — because winning means everyone else thought it was worth less than you bid.
Each bidder has a private value for the item. They submit one secret bid. Highest bid wins — and pays their bid (First-Price) or the second-highest (Vickrey/Second-Price).
In a second-price auction, your dominant strategy is to bid your true value. Why? Bidding higher can't help you (you'd overpay), and bidding lower risks losing. It's a mechanism that makes truthfulness individually rational — a beautiful application of game theory to market design.