Game Theory

Three Blue-Red Hackenbush stalks stand side by side. Find the value of the whole position as a fraction.

Under instant-runoff voting, how few ballots must change to hand the win to the candidate currently eliminated first?

A pile, three allowed removal sizes, and a thousand starting sizes to test. For how many does the second player win?

Two bidders with known, unequal valuations run an all-pay auction in mixed strategies. What does the seller expect to collect?

Sixteen coins in a row, five showing heads. A move flips a block of consecutive coins ending in a head. How many opening moves win?

Four shareholders vote by weight, and 51 wins. The smallest holder is far from powerless. How much power does the Shapley-Shubik index assign?

Two players haggle over a pile of coins that loses 15 coins each time an offer is rejected, and the game ends after six rounds. What does the first proposer keep?

A zero-sum 3×3 matrix with no saddle point and no dominated strategy. Find what the row player can guarantee on average.

Three shooters take turns firing, and any of them may deliberately miss. With these accuracies, what are the first shooter's chances of being the last one standing?

A 3×3 non-zero-sum game with no pure equilibrium. Its single Nash equilibrium uses every strategy of both players.

Two firms wait out a standoff over a prize. In the mixed equilibrium, the firm that values the prize less ends up winning more often.

Two bargainers alternate offers over a dollar, but each round of waiting costs each of them a fixed amount rather than a fraction. Who benefits from being cheaper to wait?

Two players alternate offers over one dollar, and each round of delay shrinks the value of money differently for each of them. What share does the first proposer get?

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?

Colonel Blotto on three battlefields with 5 soldiers against 4. How much of an edge does the extra soldier really give?

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

On a 4×7 Chomp bar exactly one opening bite wins. After it, how many squares of chocolate are left?

Players name divisors of 360, deleting every multiple of the name. Naming 1 loses. What is the first player's only winning move?

A chair can choose any of the 24 orders in which four proposals are voted on, against sophisticated voters. How many of those orders hand the win to Alpha?

Sixteen coins, each heads or tails, and a game of flipping short blocks. For how many of the 65536 arrangements does the second player win?

A weighted committee of five, quota 8. Count how often each member can flip the outcome, then normalise. What share does the 4-vote member hold?

In a four-pile game of Nim the first player is winning. How many different first moves keep that win?

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

Two players place dominoes on a 4×4 board, covering two adjacent squares each time. The player who cannot place loses. Who wins?

A cutter who knows the chooser's tastes can do better than half. How much of the cake, by the cutter's own measure, can they secure?

No pure strategy is dominated by another pure strategy, yet iterated elimination still singles out one outcome.

Employers cannot see ability, only schooling. How much schooling can both types be pushed into when everyone chooses the same level and wages follow the average?

In Wythoff's game with piles of 15 and 23, how many different first moves hand the opponent a lost position?

A pile of 76 stones, a cap on each move that depends on the opponent's last move. What is the only winning first move?

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

Five men and five women rank each other. A matching is stable if no pair would both rather elope. How many stable matchings does this instance have?

A 4×4 zero-sum game that looks like a linear program but collapses to a 2×2 after a few strict dominance deletions.

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?

Firms enter a Cournot market until profit no longer covers a fixed cost. Because firms come in whole numbers, the exact count takes a little care.

One firm commits to a quantity first; two rivals then play Cournot against each other. How much does the leader earn?

Players hold single gloves, and only matching pairs are worth anything. What does the Shapley value say a right-glove owner deserves?

In a small beauty contest your own guess counts toward the average. Chase the chain of best replies three steps from a naive 50.

Three Nim piles, each between 1 and 15 stones. How many of the 3375 starting positions are lost for the first player?

Two firms with zero cost split loyal buyers but fight over price-comparing shoppers. In the mixed equilibrium, how likely is a deep discount?

Maker wants an entire row or column of a 4×4 grid. Breaker wants to stop that. With Maker moving first, who wins?

A 2×2 game with no pure equilibrium. Row's equilibrium payoff is pinned down by what Column is willing to randomize over.

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

In tic-tac-toe where completing three in a row loses, most opening squares lose against perfect play. How many?

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?

Four piles; a move either takes stones from one pile or cuts a pile into two smaller ones. Exactly one first move wins: how many stones does it remove?

Nim, except a move may reduce one pile or two piles at the same time. Piles 5, 8, 11, 13: how many stones remain after the only winning move?

Both players write X on a 3×3 grid, and whoever completes three in a row loses. Which opening square wins?

Only one player has a dominated strategy at the start, yet repeated rounds of elimination pin down a single outcome of this 3×3 game.

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 take turns putting a queen on a 7×7 board so that no two queens attack each other. Who wins?

A pile, and each turn you remove a prime power or exactly one stone. How many starting sizes up to 1000 lose for the mover?

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

Heaps of 14, 20 and 29. From a heap you may take 1, 2, 4, 8, ... stones. How many opening moves win?

Rock-paper-scissors where each winning hand pays a different amount. How often should you throw rock?

Two firms set prices for products that are close, but not perfect, substitutes. What price do they settle on?

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

Five voters decide among four proposals in a fixed sequence of pairwise votes, and everyone sees where each vote leads. Which proposal survives?

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.

On each turn you remove 1, 4, 9, 16, ... stones. Among piles of size 1 to 100, how many are lost for the player to move?

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

A chain store faces twenty would-be entrants, one per town, and fighting looks like a good way to scare off the rest. Does rational play agree? Compute its total profit.

Three voters rank three candidates at random. How likely is it that majority preference chases its own tail?

Two Cournot firms face a demand curve, but one has costs so high that the usual formulas go wrong. What does the other earn?

In a second-price auction, the winner keeps the gap between the two highest values. How big is that gap on average with three bidders?

Three players can share a total of 100, but pairs can walk away with their own sums. What is the most the middle player can claim without anyone defecting?

Two bidders have exponentially distributed values. A seller who chooses the best possible auction and reserve earns how much in expectation?

One bargainer can walk away to a sure outside payoff, but only at certain moments. Does the threat to walk change what the first proposer gets?

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?

A candidate leads in first-place votes, yet a different one wins when every ranking is scored. Who takes the Borda count?

A king walks toward the corner of a chessboard, but whoever steps onto the corner loses. On how many squares does the player to move lose?

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?

A reserve price in a first-price auction shifts every bid upward, even for bidders far above it. How much does the top type bid?

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 of three identical Cournot firms merge. By how much does their combined profit change?

Three bystanders each wish someone would act. In the symmetric mixed equilibrium, how often does exactly the right number volunteer?

A token walks over a 3×3 grid of squares and may never revisit a square. From how many starting squares does the first mover win?

In an all-pay auction everyone pays, win or lose. With three bidders, how much should a bidder with value 3/4 put in?

Heaps of 38, 53 and 71, but you may only remove 2, 5 or 6 stones at a time. Count the first moves that preserve the win.

An incumbent can build capacity before a rival decides whether to pay a fixed cost to enter. Is flooding the market to keep the rival out actually worth it?

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?

Two bidders draw private values from a distribution tilted toward high numbers. What is the expected revenue of a first-price auction?

Two partners choose effort levels, but output depends only on the lower one. Count the pure-strategy equilibria.

Run the men-proposing deferred acceptance procedure on four men and four women. Which man is left holding Z at the end?

Three citizens report how much a bridge is worth to them, and the Clarke pivot rule charges each for the harm they cause others. What is the total collected?

How many starting positions in Wythoff's game, with both piles between 1 and 50, are lost for the player to move?