Questions tagged [game-theory]

Puzzles that involve strategic decision making, usually involving either competitive or cooperative choices between a number of participants.

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57
votes
5answers
4k views

Who shall come to the tea party?

All of your stuffed animals are assembled, anxiously waiting to learn who will be be invited as the Guest of Honor to your next tea party. This is to be an exclusive party, after all, as there is ...
47
votes
4answers
6k views

Rock, Paper, Scissors and Trump

Alicia and Robert are two high-stakes gamblers who are playing a match of rock-paper-scissors. Each winning throw is worth 10 €, paid by the other player. These players have deep pockets, so matches ...
33
votes
2answers
2k views

The Nuclear Missile Standoff

Three world powers have just entered a nuclear war! They have been waiting for this day for decades; everyone has their nuclear missiles ready to fire, and only need to choose a target. The Chinese ...
32
votes
6answers
6k views

Will a greedy algorithm solve Tatham's Flood?

I was just investigating some of the puzzles on Simon Tatham's website, and came across Flood, in which we start with an $n\times n$ grid of cells each of which is filled with one of $k$ predetermined ...
26
votes
5answers
6k views

Shortest game of Monopoly

This video claims that the fastest game of monopoly involves 4 turns and 9 rolls of the dice. I am wondering if this is actually the shortest game or can we do better in theory? What is the ...
24
votes
3answers
2k views

Angels & Demons (Open question)

This puzzle was given to me by a friend of mine, who in turn heard it from someone else who neglected to provide the answer. The text of the puzzle as it was given to my friend is below: Angels ...
23
votes
3answers
2k views

In the iterated Prisoner's Dilemma, how would a change in the payoff matrix affect strategy?

Quick summary of the Prisoner's Dilemma: Two criminals are charged with a crime. Each is given an opportunity by the police to confess. If they both stay silent, they don't have enough evidence to ...
18
votes
1answer
565 views

Lap Theory Optimal Betting Game

Three prisoners are seated at a table. Each of them has a mobile phone on their lap, and they are not allowed to look at anyone else’s phone (and obviously no other form of communication is allowed). ...
18
votes
2answers
610 views

How long can you survive at the devil's playground?

The devil has trapped you in his playground. The devil knows that you can't cross over the burning boundary of his circle, so he allows you to choose a position within the circle before he starts to ...
16
votes
2answers
680 views

Topological Nim

You and your friend's new favorite game is "topological nim": You take your favorite compact metric space $X$ and a radius $r$. Each player removes an open disk of radius $r$ from the space ...
14
votes
2answers
778 views

Reverse dots and boxes

Alice and Bob are playing a game of reverse dots and boxes. The rules are simple: The players take turns adding one horizontal or vertical line in one free spot on the grid (marked with light ...
14
votes
4answers
605 views

Racetrack designers, assemble!

Note: Turns out the intended answer to this doesn't work after all (see comments to Sconibulus's answer). The rules have been changed slightly after the fact to ensure there can be valid answers (see ...
14
votes
5answers
547 views

Frugality 101: How to Replace Your Police Force with Cardboard Cutouts

In the far away land of Frugalia - a land where the conservation of money has been elevated to the status of any of Newton's laws - a conundrum is being had. There has been an upsurge in citizens ...
12
votes
3answers
3k views

Tic-tac-toe 1000 in a row

Consider a variant of Tic-Tac-Toe and Gomoku on a board that's infinite in every direction. X wins by getting a line of 1000 X's ...
12
votes
2answers
1k views

Winning strategy in a calendar game

Consider a two-player game. The players move in turns to pick either a month or a day. The winner is the player who picks the date December 31 ($12/31$). The following rules apply: The game starts ...
12
votes
2answers
662 views

2019 gold coins to share

You and your friend has found in total 2019 gold coins in a treasure hunt. Since your friend found the place to dig up, he deserves a bit more gold than you so you offer a game to share the coins to ...
12
votes
4answers
758 views

A packing game!

Amy and Ben are playing a game which is suggested by a genie. Amy first chooses $a,b,c\in\mathbb{R}^+$. Then a empty cuboid box with internal measurements $a+b,b+c,c+a$, and infinite supply of cuboid ...
12
votes
3answers
788 views

Outmaneuver your opponents in the duel

A, B and C are in a three way duel. Starting with A, they rotate in the order of A-B-C, each firing one shot at a time. They stand close to one another, so that each can kill one of the others or ...
12
votes
1answer
699 views

Reverse dots and... cubes? - Snowflake edition

A Reverse dots-and-boxes puzzle inspired by @jafe Albert and Robinson are playing a game of reverse dots and boxes. The players take turns adding one plane in one free spot on the grid (...
11
votes
3answers
414 views

Dollar Auction with Perfect Logicians

The dollar auction is a famous game that shows how seemingly logical choices can end up with very illogical results. Wikipedia says the best strategy is not to play, which is probably true in most ...
11
votes
3answers
773 views

A knight chased by three knights

Two players are playing a variant of chess on a 8x8 grid. The first player controls a white knight that starts in the top-left corner. The second player controls three black knights that start in the ...
11
votes
1answer
355 views

Birthday girl's casino game

My daughter's birthday is coming up. Here's a fun party game we can play to celebrate. (Bring your wallet!) I have a pair of 6-sided dice, and a standard deck of 52 cards. You place your ...
10
votes
2answers
302 views

Add a divisor! A game

Let $k$ be a positive integer. Amy and Ben are playing a game, with the number $1$ written on the whiteboard initially. Amy and Ben do the following in order, starting with Amy: Suppose the number on ...
10
votes
1answer
778 views

Pirate Puzzle Double Twist

Five pirates, fresh from their first bout of pillaging, return to their island base to celebrate and divide up the treasure. While this was their first time working together, the start of what they ...
10
votes
1answer
328 views

Who Wins At Horsie-Push?

Allan, Brian, and Charlie had a chess set and an afternoon with nothing to do. A chess tournament was suggested but to three towering intellects such as these, regular chess is a frivolous, juvenile ...
10
votes
3answers
363 views

Find a good strategy to survive the multiplayer duel

Alice is involved in a duel with $N$ other opponents. She can choose a gun for herself with any hit probability as she wishes. She also prepares guns for her opponents, whose hit probabilities she can ...
10
votes
1answer
430 views

Is Oh Tetris Oh! always a draw?

Alice and Bob were playing Oh-Tetris-Oh!, when they decided that every game must be a draw (played intelligently). Can you prove this? (Oh yes, the rules: A $10\times10$ grid, you can pick any ...
10
votes
1answer
360 views

Reverse dots and boxes – Guess the last move

Alice and Bob are playing a game of reverse dots and boxes. The players take turns adding one horizontal or vertical line of size 1 in one free spot on the grid (marked with light gray lines ...
9
votes
2answers
613 views

Buzzword Bingo Boss Battle!

Two bosses of different companies take turns delivering long speeches. Every speech much “claim” at least one unused square by uttering the corresponding buzzword(s) in the table below. Victory is ...
9
votes
2answers
260 views

A knight chased by four knights

This is a follow up to A knight chased by three knights Two players are playing a variant of chess on a 11x11 grid. The first player controls a white knight that starts in the centre square. The ...
9
votes
5answers
967 views

Two person game, choosing an integer between 1 and 30?

Suppose two players are playing a game. The first player chose an integer, n st. $n \in \left\{ 1,\ldots ,30\right\}$. The second player picks a different one. After that, a random integer x st. $x \...
9
votes
1answer
378 views

Paper, pencil and a bunch of bars

An entry for the Fortnightly Topic Challenge #40. Alice and Bob are two rational players who are playing a simple pencil-and-paper game. In the beginning, there are fifteen small vertical bars on ...
9
votes
2answers
429 views

Three way duel, which guns to choose?

Alice, Bob and Carole are involved in a game of three way duel. They take turns to shoot in the order of Alice-Bob-Carole-Alice-... until only one surviver is left standing. The rule is very simple: ...
8
votes
12answers
3k views

Time to call the bluff [closed]

You are located outside of two rooms which have a door between them. Each room has one or more computers whose total power is unknown, so they cannot be reliably used as proof-of-work to compare times,...
8
votes
6answers
1k views

Single-pile Nim with Three Players

Three perfectly rational logicians (Alice, Bob and a Quetzalcoatlus) are playing a game. (Charlie couldn't make it, so he asked the Quetzalcoatlus to fill in for him.) The rules are very simple: ...
8
votes
3answers
454 views

Four-in-a-line Puzzle

Disclaimer: This is an open-problem, I don't have a complete solution for this puzzle yet. We are playing a $2$-player game: you as a challenger and me as a judge. Initially, there is an empty ...
8
votes
6answers
1k views

Devious auction game

Saw this on TV today: A man is auctioning a real $20\$$ bill. There are a vast number of bidders. A person may make as many bids as he wants. The starting bid is $5\$$. No $2$ bids must be ...
8
votes
1answer
792 views

Reverse Puzzling 3

This is a continuation of Reverse Puzzling 2 "Okay George, let's try this one. Pretty simple." I gave George the puzzle. He picked up his marker and quickly sketched out the following. &...
8
votes
2answers
797 views

Reverse dots and boxes, swastika edition

Alistair and Roberta are playing a game of reverse dots and boxes. The players take turns adding one horizontal or vertical line in one free spot on the grid (marked with light gray lines in ...
8
votes
1answer
344 views

No point shuffling, just pick your cards

Alice and Bob are playing a card game with the following rules: Alice selects two piles of 3 cards from a standard 52-card deck and places the piles in front of her, face up. Bob can see these ...
8
votes
5answers
562 views

Which gun is Alice's best choice?

Alice, Bob and Charles are involved in a deadly three way duel game. Before the game starts, the referee prepares three guns for the players to choose from. She randomly picks three guns from a very ...
8
votes
3answers
399 views

Rock, Paper, Scissors, Anti-Rock

Alice and Bob are both well-known perfect logicians who have the ability to internally randomize and often pass their time by playing rock-paper-scissors. One day, tired of the usual RPS game, Alice ...
8
votes
0answers
263 views

A Glutton and a Split Check

N, T, A, B are arbitrary constants in this problem. I have only definitively solved the N=1 subproblem, so, if the general problem cannot be solved, I will be satisfied to mark as accepted a solution ...
7
votes
4answers
1k views

Higher or lower?

The game “Higher or Lower” is a simple one in which playing cards are drawn randomly from a deck, and the player has to guess whether each subsequent card will be higher or lower in value than the ...
7
votes
3answers
653 views

The 15 Pebbles Game

This is a game for 2 players - Each player uses a different coloured marker or pencil, there are 15 pebbles in total. Players take turns to colour 1, 2 or 3 pebbles (player chooses how many). When all ...
7
votes
1answer
291 views

Domino Tiling Game

Two players are presented an $n\times n$ grid. They take turns placing a piece on the board until no more moves are possible. The last player able to make a move is the winner. The pieces are $2\times ...
7
votes
4answers
659 views

'Companion Planting' Logic Puzzle - Is there an elegant solution?

Problem: You're a vegetable gardener planning to plant out seedlings in a new garden bed, however your farmer's almanac has outlined a system of rules about what plants can and cannot be placed next ...
7
votes
1answer
274 views

Given an $8×8$ square and a set, which contains the pentominoes and four $1×1$ squares

I have a game. Given an $ 8 × 8 $ square and a set, which contains the pentominoes and four $ 1 × 1 $ squares. Players alternately pick one item from the set. Then players (starting with the player ...
7
votes
2answers
329 views

Can the fugitive escape?

A fugitive is surrounded by N police officers, with the nearest one at distance 1 away. The fugitive and the officers move alternatively. In a fugitive move, the fugitive can travel no more than a ...
6
votes
1answer
510 views

Playing blindfolded 2048

Suppose you are playing many 2048 games, all blindfolded. As usual you can press left, right, down, or up to move the tiles. When there are no more moves the game terminates and you are told your ...