Game Theory Essentials: Dominance, Nash, and Auctions

The pirate game, market-making against informed flow, guessing games at interviews — all run on a small set of strategic concepts. This lesson is that set, with the interview problems each one unlocks.

Dominance and iterated deletion

A strategy is dominated if another does at least as well against everything. Rational players never use dominated strategies — and if everyone knows that, strategies dominated after that deletion die too, and so on.

The famous demonstration is the 2/3-of-the-average game: everyone names a number in $[0,100]$; closest to 2/3 of the average wins. Numbers above 66.7 are dominated; deleting them makes numbers above 44.4 dominated; iterating collapses everything to 0 — the unique equilibrium. Reality check interviewers love: against actual humans, the winning guess is usually 15–25, because real opponents run only a few deletion rounds. The lesson quoted back to you at trading firms: play the equilibrium against theorists, play the population against markets.

Nash equilibrium

A strategy profile where no player gains by deviating unilaterally. Existence is guaranteed (Nash, allowing randomization); uniqueness isn't. The two benchmark games:

• Prisoner's dilemma: defection dominates, so (defect, defect) is the equilibrium despite (cooperate, cooperate) being better for both — individual rationality ≠ collective optimality. Market analog: two market makers tightening spreads to compete away the edge.
• Matching pennies / mixed strategies: no pure equilibrium; you must randomize. The equilibrium mix makes your opponent indifferent — the key computational trick. In markets: predictable order-timing gets exploited, which is why execution algorithms randomize — a mixed strategy in production.

Backward induction

Solve sequential games from the last move backward — the pirate game's engine, and the chess-like logic of any take-it-or-leave-it negotiation chain. Its known fragility (the centipede game unravels to immediate defection in theory, but humans cooperate for many rounds) mirrors the 2/3-game gap: perfect logic assumes perfectly logical opponents, and pricing the opponent's actual sophistication is itself the skill.

Auctions and the winner's curse

Trading interviews adore auction questions because markets are auctions:

• Second-price (Vickrey) auction: bidding your true value is dominant — you never affect what you pay, only whether you win. Elegant, and the standard "prove it" prompt.
• First-price auction: shade your bid below value; equilibrium shading with $n$ symmetric bidders on uniform values is bidding $\frac{n-1}{n}$ of value.
• Common-value auctions and the winner's curse: when the item has one true value everyone estimates noisily (an oil field, a block of shares), winning is bad news — you won because your estimate was the most optimistic. Rational bidders shade for this; naive bidders systematically overpay. The direct market translation: getting filled on your quote is winning an auction against someone who chose to trade with you — the winner's curse is adverse selection, connecting this lesson straight to the microstructure course.

The interview version

"I'll auction this jar of coins to the table." — Estimate the value, then bid meaningfully below it, and say why: "whoever wins overestimated most, so I shade for the winner's curse." "Why do you randomize in rock-paper-scissors against a good player, and with what mix?" — Uniform thirds; any tilt is exploitable, and the equilibrium logic (make the opponent indifferent) is the answer. Game theory questions rarely need computation — they need the right concept named at the right moment.