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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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61 votes
5 answers
5k 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 ...
Trevor Powell's user avatar
49 votes
4 answers
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 ...
Jafe's user avatar
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33 votes
3 answers
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 ...
IQAndreas's user avatar
  • 714
32 votes
6 answers
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 ...
Rand al'Thor's user avatar
28 votes
4 answers
5k views

Can you beat random?

It is known that one cannot beat a truly random player in Rock Paper Scissors. Here you will play a variant of the game. In this variant you are not allowed to play the same shape twice in a row. So ...
Dmitry Kamenetsky's user avatar
26 votes
5 answers
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 ...
Dmitry Kamenetsky's user avatar
26 votes
9 answers
3k views

N prisoners locked in cells

N prisoners are locked in N separate cells, numbered 1 to N. The warden visits each cell in order with a bowl. Each prisoner can decide whether to put a single coin in the bowl, depending on its ...
Eric's user avatar
  • 6,536
25 votes
3 answers
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 ...
durron597's user avatar
  • 2,374
24 votes
3 answers
3k 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 ...
Nellington's user avatar
20 votes
8 answers
3k views

Which parent should you start playing against?

Let $k < n$, $k$ even, $n$ odd. Mary is to play $n$ chess games against her parents, alternating between her father and mother. To receive her allowance she must win $k$ games in a row. Given the ...
Simd's user avatar
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19 votes
4 answers
1k 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 ...
Eric's user avatar
  • 6,536
18 votes
1 answer
654 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). ...
happystar's user avatar
  • 7,664
17 votes
6 answers
1k views

Can you stop copy Alice?

At time 0, Alice and you both move freely at speed 1 on the plane. As each hour passes, all copies of Alice will subdivide into 2, each moving at half the speed of the parent. So if left alone, at ...
Eric's user avatar
  • 6,536
17 votes
1 answer
593 views

Game of the glasses on the windowsill

The windowsill above the sink is where my wife and I place our dirty wine glasses. And while both of us love each other, neither of us love loading the dishwasher. As a result, these dirty glasses ...
Feryll's user avatar
  • 2,379
16 votes
2 answers
734 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 ...
Feryll's user avatar
  • 2,379
15 votes
2 answers
894 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 ...
Jafe's user avatar
  • 77.4k
14 votes
4 answers
668 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 ...
Jafe's user avatar
  • 77.4k
14 votes
5 answers
573 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 ...
Milo Brandt's user avatar
  • 7,881
14 votes
2 answers
689 views

Who is most likely to win all the chips?

Alice, Bob and Charlie each have 5 chips. Starting from Alice, they take turns to act in the order of Alice, Bob and Charlie. The $n$-th person to act must take a total of $n$ chips from their ...
Eric's user avatar
  • 6,536
13 votes
2 answers
704 views

2019 gold coins to share [duplicate]

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 ...
Oray's user avatar
  • 30.3k
13 votes
3 answers
898 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 ...
Eric's user avatar
  • 6,536
12 votes
3 answers
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 ...
xnor's user avatar
  • 27k
12 votes
3 answers
862 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 ...
Dmitry Kamenetsky's user avatar
12 votes
2 answers
2k 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 ...
Herr K.'s user avatar
  • 223
12 votes
4 answers
792 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 ...
Culver Kwan's user avatar
  • 5,600
12 votes
1 answer
752 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 (...
Omega Krypton's user avatar
11 votes
3 answers
504 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 ...
Tyler Seacrest's user avatar
11 votes
1 answer
649 views

The median game

Five friends play a simple game with the following rules: Players play consecutively one after the other. Each player must call out a whole number between 1 and 10 (inclusive), such that it hasn't ...
Dmitry Kamenetsky's user avatar
11 votes
1 answer
890 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 ...
P...'s user avatar
  • 517
11 votes
1 answer
417 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 ...
Jafe's user avatar
  • 77.4k
11 votes
1 answer
425 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 ...
Jafe's user avatar
  • 77.4k
10 votes
3 answers
675 views

A game of 101 stones—win at (not quite) all costs

In the land of Humilia, a tournament game is played between two players with 101 stones in a pot between them. On each turn, a player may take up to five stones from the pot (and must take at least ...
Feryll's user avatar
  • 2,379
10 votes
5 answers
2k 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 \...
JimSi's user avatar
  • 309
10 votes
2 answers
669 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 ...
happystar's user avatar
  • 7,664
10 votes
2 answers
328 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 ...
Culver Kwan's user avatar
  • 5,600
10 votes
1 answer
380 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 ...
Hugh Meyers's user avatar
  • 23.7k
10 votes
3 answers
431 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 ...
Eric's user avatar
  • 6,536
10 votes
1 answer
474 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 ...
JMP's user avatar
  • 35.6k
9 votes
3 answers
1k views

Alternative solutions to a Michael Rabin puzzle

I first heard the puzzle below around 1990 while working at Carnegie Mellon University; the puzzle was attributed to Michael Rabin. After solving it, I realized that alternative solutions are ...
Timothy Chow's user avatar
9 votes
2 answers
325 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 ...
Dmitry Kamenetsky's user avatar
9 votes
2 answers
480 views

Moving tokens game

I've been thinking about the following two player game played on a 3x3 grid: Red goes first and players alternate in taking turns. Red moves tokens horizontally right. Blue moves tokens vertically ...
Dmitry Kamenetsky's user avatar
9 votes
1 answer
445 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 ...
Jafe's user avatar
  • 77.4k
9 votes
1 answer
672 views

Toroidal tic-tac-toe

Part of the Monthly Topic Challenge #10: Möbius Strips, Klein Bottles, and other unusual topological surfaces Warm-up for topological thinking. You are playing tic-tac-toe with the usual 3x3 grid,...
quarague's user avatar
  • 1,853
9 votes
2 answers
562 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: ...
Eric's user avatar
  • 6,536
9 votes
0 answers
355 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 ...
Feryll's user avatar
  • 2,379
8 votes
6 answers
2k 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: ...
Bass's user avatar
  • 77.6k
8 votes
12 answers
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,...
rapt's user avatar
  • 197
8 votes
3 answers
516 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 ...
athin's user avatar
  • 34.2k
8 votes
6 answers
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 ...
ghosts_in_the_code's user avatar
8 votes
3 answers
1k 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 ...
MetaInformation's user avatar