Game Theory

A couple prefers being together but disagrees on the venue. In the mixed equilibrium, how often does each attend their own favorite?

Two identical firms set prices and the cheaper one takes the whole market. Where do prices end up?

Players eat chunks of a chocolate bar whose bottom-left square is poisoned. Who wins with optimal play, and why is the proof famous?

Two firms simultaneously choose quantities with price P = 120 − Q and zero costs. What does each produce in equilibrium?

Two bidders with uniform private values bid in a first-price auction. What fraction of their value does each bid in equilibrium?

In an infinitely repeated Prisoner's Dilemma, how patient must players be for grim-trigger cooperation to hold?

Hawks fight over a resource at a cost; Doves share and flee. With what probability is Hawk played in the evolutionarily stable strategy?

Match pennies: Player 1 wins $1 on a match, Player 2 wins $1 on a mismatch. What is the value of this game?

Nim with piles of 3, 4, and 5: the first player wins with optimal play. How many stones does the unique winning first move remove?

Two players pick sequences of three coin flips. The player whose sequence appears first wins. How many distinct sequences can Player 2 choose to guarantee a winning edge over any sequence Player 1 picks?

Players alternately add 1–10 to a running total; whoever says exactly 100 wins. What is the winning first move?

A zero-sum game with payoff matrix [[3, −1], [−2, 4]]. What is the value of the game?

A casino flips a coin repeatedly. You win 2^n dollars where n is the flip on which heads first appears. What is the expected value of this game?

Same market as Cournot, but one firm commits to its quantity first. What does the leader produce?

Hunt the stag together for the big payoff, or play it safe with a hare? Find the mixed-equilibrium probability of hunting stag.

Two players could grow a pot over 100 rounds by passing. On which round does backward induction say the game ends?

Five perfectly rational pirates divide 100 gold coins. The most senior proposes a split; a strict majority rejection means he dies. How many coins does the senior pirate keep?

Two suspects are interrogated separately with the classic payoffs. How many years does each serve in the unique Nash equilibrium?

A proposer splits $100; the responder can accept or leave both with nothing. What does a purely rational proposer offer?

The highest bidder wins but pays the second-highest bid. What is the optimal bid, no matter what anyone else does?

Three shooters with hit rates 1/3, 2/3, and 1 duel in turns, weakest first. What should the weakest shooter do first?

Two travelers claim a value from $2 to $100 for lost luggage, with a $2 bonus for the lower claim. What do rational players claim?

Everyone picks a number in [0, 100]; closest to two-thirds of the average wins. What do perfectly rational players pick?