Game Theory Basics

Most probability puzzles have one decision-maker facing a coin or a die. Game theory starts where that ends: when the thing you're up against is other people who are also thinking, and your best move depends on theirs while theirs depends on yours. This is the world of competing market makers, bidders in an auction, and a surprising number of trading-interview games. Game theory is the language for reasoning about it clearly.

Every game has three ingredients. Players — the decision-makers. Strategies — the choices available to each. Payoffs — the reward each player gets for every combination of choices. Lay the payoffs out in a grid and most small games become solvable by inspection.

Two ideas do nearly all the work. A best response is your optimal move given what the others do. A dominant strategy is even stronger: a move that's your best response no matter what anyone else does — when you have one, you should just play it and stop thinking about the opponent.

Nash equilibrium

The central solution concept is the Nash equilibrium: a combination of strategies where each player is already playing a best response to the others, so no one can do better by changing their move alone. Formally, a strategy profile $s^\star$ is a Nash equilibrium if for every player $i$,

$$u_i(s_i^\star, s_{-i}^\star) \;\ge\; u_i(s_i, s_{-i}^\star) \quad \text{for all } s_i.$$

Reading it in words: $u_i$ is player $i$'s payoff, $s_i^\star$ is their equilibrium move, and $s_{-i}^\star$ is everyone else's moves. The inequality says player $i$'s equilibrium move earns at least as much as any other move they could make while everyone else stays put. It's a no-regret, no-unilateral-deviation resting point.

Worked example: the prisoner's dilemma

Two traders can either cooperate (keep a wide, profitable spread) or defect (undercut the other to grab all the flow). The payoff grid, higher numbers better:

	Other cooperates	Other defects
You cooperate	you 3, other 3	you 0, other 5
You defect	you 5, other 0	you 1, other 1

Solve it by looking for a dominant strategy. If the other player cooperates, you get 5 by defecting versus 3 by cooperating — defect is better. If the other player defects, you get 1 by defecting versus 0 by cooperating — defect is still better. Defect beats cooperate no matter what they do, so defect dominates. The same is true for them by symmetry. Both defect, landing on payoffs (1, 1).

Here's the sting: both would be better off at (3, 3) if they'd only cooperated — but that outcome isn't stable, because each has a private incentive to break it. The Nash equilibrium is individually rational and collectively lousy.

Why this is a trading concept

The prisoner's dilemma is exactly the logic of two market makers competing for order flow. Each is tempted to shave a tick off the other's quote to win the trade, and once both do it repeatedly, the spread is competed down toward the bare cost of providing liquidity — great for customers, thin for the quoters. That collapse to a competitive spread is a market Nash equilibrium, worked through in Nash Equilibrium in Markets. The same machinery prices bids in Auction Theory for Markets and underlies why prices in a crowded market drift toward efficiency: if a profit is sitting there for the taking, someone's best response is to take it until it's gone.

Pitfalls and refinements

• Rationality is assumed. Equilibrium analysis presumes everyone reasons perfectly and knows the others do too. Real people don't, which is why games like guess-two-thirds-of-the-average settle away from their equilibrium — see Nash Equilibrium in Markets.
• One-shot vs repeated. The prisoner's dilemma flips when it's played over and over: cooperation can become sustainable because defection today invites retaliation tomorrow. Time horizon changes everything.
• Multiple equilibria. Some games have several Nash equilibria, and the theory alone doesn't say which one gets played; coordination and focal points matter.
• Mixed strategies. When no single move is a stable best response (think rock-paper-scissors), the equilibrium is a randomized mix. Every finite game has at least one such equilibrium — that's Nash's theorem.

For interviews, the move is to set up the payoff grid, check for dominant strategies first, then find the cell where each player's best response agrees. If someone frames a competition between traders, quoters, or bidders, they're almost always asking you to find the Nash equilibrium — and often to notice it's worse for the players than cooperating.