The study of competitive and non-competitive games, equilibrium concepts such as Nash equilibrium, and related subjects. Combinatorial games such as Nim are under (combinatorial-game-theory), and algorithmic aspects (e.g. auctions) are under (algorithmic-game-theory).

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386
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25answers
52k views

Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...
63
votes
0answers
2k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
55
votes
3answers
5k views

Mathematical research of Pokémon

In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some ...
45
votes
8answers
11k views

Why do people lose in chess?

Zermelo's Theorem, when applied to chess, states: "either white can force a win, or black can force a win, or both sides can force at least a draw [1]" I do not get this. How can it be proven? ...
42
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2answers
3k views

A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe

Consider a 9 by 9 matrix that consists of 9 block matrices of 3 by 3. Let each 3 by 3 block be a game of tic-tac-toe. For each game, label the 9 cells of the game from 1-9 with order from left to ...
32
votes
1answer
1k views

Is War necessarily finite?

War is an cardgame played by children and drunk college students which involves no strategic choices on either side. The outcome is determined by the dealing of the cards. These are the rules. A ...
27
votes
3answers
684 views

Three against the devil: a combinatorial game

A team of three sinners plays a game against the devil. They confer on strategy beforehand; then they go into three separate rooms, and there is no more communication between them. The play in each ...
24
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4answers
1k views

rock, paper, scissors, well

Everyone knows rock, paper, scissors. Now a long time ago, when I was a child, someone claimed to me that there was not only those three, but also as fourth option the well. The well wins against rock ...
22
votes
6answers
2k views

A beautiful game of gold and silver coins

A stack of silver coins is on the table. For each step we can either remove a silver coin and write the number of gold coins on a piece of paper, or we can add a gold coin and write the number of ...
21
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7answers
6k views

Game theory - self study

I want to self study game theory. Which math-related qualifications should I have? And can you recommend any books? Where do I have to begin?
19
votes
3answers
7k views

How is prisoner's dilemma different from chicken?

Chicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash. So ...
19
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3answers
978 views

Optimal strategy for the Rope Climbing Game

Define a two-player, turn-based climbing game as follows. Each turn, players have the option to climb or tie a knot at his current position. If the player chooses to climb, there is a 50% chance ...
18
votes
2answers
352 views

Game involving points on $[0,1]$

You're given a list of $22$ points in $[0,1]$ (not necessarily distinct), and you're asked to select, at every iteration, $2$ points to be substituted by their midpoint. After $20$ iteration, you ...
18
votes
1answer
1k views

Number of moves to solve a flood-it/sock-dye game

[ Question based on the sock dye game ] [ Update: It appears that this game is better known as "Flood it" and is NP-hard. Also, "the number of moves required to flood the whole board is $\Omega(n)$ ...
17
votes
4answers
471 views

A Category Theoretical view of the Kalman Filter

Some basic background The Kalman filter is a (linear) state estimation algorithm that presumes that there is some sort of uncertainty (optimally Gaussian) in the state observations of the dynamical ...
17
votes
1answer
265 views

A game played on a rectangle

Suppose two players play the following game on a $m$ by $n$ rectangle. Alternatingly they have to make a cross in some empty $1\times 1$ square. They are not allowed to make a cross next to another ...
17
votes
3answers
2k views

Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff ...
17
votes
4answers
380 views

Salvaging a damaged cable

Let's say we have a cable of unit length, which is damaged at one unknown point, the location of which is uniformly distributed. You are allowed to cut the cable at any point, and after a cut, you'd ...
15
votes
7answers
1k views

Game theory textbooks/lectures/etc

I looking for good books/lecture notes/etc to learn game theory. I do not fear the math, so I'm not looking for a "non-mathematical intro" or something like that. Any suggestions are welcome. Just put ...
15
votes
9answers
486 views

Understanding the solution of a riddle about lions and sheep.

I heard a riddle once, which goes like this: There are N lions and 1 sheep in a field. All the lions really want to eat the sheep, but the problem is that if a lion eats a sheep, it becomes a sheep. ...
14
votes
5answers
1k views

Probability of dice sum just greater than 100

Can someone please guide me to a way by which I can solve the following problem. There is a die and 2 players. Rolling stops as soon as some exceeds 100(not including 100 itself). Hence you have the ...
14
votes
1answer
472 views

How many possible board states in 2048?

I recently found out about the famous 2048 game. For those of you who don't know how it works, it consists on a 4x4 board on where tiles which are powers of 2 are placed. On every turn, you "swipe" ...
13
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5answers
1k views

A lady and a monster

A famous problem: a lady is in the center of the circular lake and a monster is on the boundary of the lake. The speed of the monster is $v_m$, and the speed of the swimming lady is $v_l$. The goal of ...
13
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3answers
3k views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ ...
12
votes
3answers
361 views

A number guessing game

Alice chose a positive integer $n$ and Bob tries to guess it. In every turn, Bob will guess an integer $x$ $(x>0)$: If $x$ equals $n$, then Alice tells Bob that he found it, and the game ends. ...
12
votes
3answers
603 views

Best Strategy for a die game

You are allowed to roll a die up to six times. Anytime you stop, you get the dollar amount of the face value of your last roll. Question: What is the best strategy? According to my calculation, for ...
12
votes
3answers
571 views

Formula for picking time closest to (but after) target

Let's say you have an arbitrary length of time. You are playing a game in which you want to push a button during this time span after a light comes on. If you do so, you win ($+1$), if not, you lose ...
12
votes
2answers
357 views

Determining the number of valid TicTacToe board states in terms of board dimension

I am attempting to find a closed form equation in terms of $n$, for the number of valid Tic-Tac-Toe board states (ignoring symmetry), where the board has dimension $n \times n ,\; 0 \lt n,\;n \in \Bbb ...
12
votes
1answer
248 views

Optimal strategy for Jackpot Rock Paper Scissors

Jackpot Rock Paper Scissors is a gambling variant of Rock Paper Scissors, wherein ties result in the wager being carried forward into a jackpot. If a player plays the same hand (rock, paper or ...
12
votes
0answers
223 views

The Right Triangle Game

I am looking for a deeper understanding, particularly the optimum strategy and the maximum score as a function of grid size, of the following (single-player) game played with an $n$ by $m$ grid: ($6 ...
11
votes
5answers
438 views

The Price is Right optimal play

The following situation happened on the Price is Right and I was curious about the optimal response. The rules are: A contestant rolls a wheel with 5 cent increments from 5 - 100 (20 numbers total). A ...
11
votes
2answers
313 views

Optimal Strategy for Bar Dice

This game is played in bars in Wisconsin, USA, but I'm sure variations are played many places around the world. The game has practical value, since once mathematicians figure out the best strategy, ...
11
votes
4answers
170 views

How do you create a nonlinear game that the player can always win?

I thought a lot about this question — and initially, I intended to ask this on gamedev.stackexchange.com — but due to its rather theoretical aspects, I think it might be more appropriate to address a ...
11
votes
2answers
139 views

Fastest way to get \$1 million of two different currencies in a video game

This question actually relates to a video game, I came across the scenario and I realized I had no idea how to go about solving something like this or even what branch of mathematics it falls under. ...
10
votes
1answer
172 views

Can we qualitatively predict the strategy of the German and US teams in today's World Cup soccer match?

In today's World Cup soccer match between Germany and the US, both teams only need a draw to advance to the next round. There's been speculation about possible collusion, especially given the friendly ...
10
votes
5answers
1k views

What is the probability of getting yahtzee?

What is the probability of getting yahtzee using N dices with X sides in Y throws in a single round? Which side of the dice one gets yahtzee with doesn't matter (e.g. it doesn't matter if it is ones ...
10
votes
1answer
158 views

What are the relevant properties of cardinal utility functions for defining a notion of expected utility for mixed strategies in games?

For (finite) normal form games we can develop everything for pure strategies by considering ordinal utility functions to capture only a players preferences over the pure strategy space of the game. ...
10
votes
2answers
254 views

Strategy for a game of breaking sticks

Two persons have 2 uniform sticks with equal length which can be cut at any point. Each person will cut the stick into $n$ parts ($n$ is an odd number). And each person's $n$ parts will be permuted ...
9
votes
3answers
685 views

Optimal Strategy for Rock Paper Scissors with different rewards

Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward. If you win with Rock, you get \$9. Your opponent loses the \$9. If you win with Paper, you get \$3. ...
9
votes
5answers
281 views

How can the observed strategies* in this actual auction be explained?

This is a "real world" question. Recently I witnessed the separate auctions of about 30 houses. The place where I went uses the following rules. The following describes the procedure for the ...
9
votes
3answers
366 views

Is there experimental evidence that people ever play mixed Nash equilibrium in real games?

Have any studies been done that demonstrate people (not game theorists) actually using mixed Nash equilibrium as their strategy in a game?
9
votes
2answers
620 views

Determine the winner of a tic tac toe board with a single matrix expression?

Assume a tic-tac-toe board's state is stored in a matrix. $$ S=\begin{bmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ \end{bmatrix} $$ Here, $X$ is mapped to $1$, $O$ is ...
9
votes
1answer
171 views

The Goblin Game

Goblin Game is a Magic: the Gathering card. The full text of the spell is: Each player hides at least one item, then all players reveal them simultaneously. Each player loses life equal to the ...
8
votes
4answers
389 views

Probability question: optimal strategy

I am really confused about how to think about this question. It was presented as a challenge by a peer. Two people seek to kill a duck at a location $Y$ meters from their origin. They walk from ...
8
votes
3answers
144 views

Seemingly similar but different probability games

Burger King is currently running its "family food" game in which each piece can be modeled as a scratch off game where exactly one of three slots is a winner and you may only scratch one slot as your ...
8
votes
1answer
342 views

Common knowledge as a fixed point

I read on a wikipedia page that from the modal logic formalization CK can be formulated as a fixed point. If it also holds for the set theory formalization? If it does, where I can find about it? ...
8
votes
3answers
266 views

number choose game

Three people A,B,C attend the following game: from 0~100, the Host will come up a number with Uniform, but he doesn't tell them the number, the attendee will guess a number and the closes one will ...
8
votes
2answers
281 views

A modified NIM game

Let's play a game of NIM, but with a catch! We have exactly three piles of stones with sizes $a$, $b$ and $c$, all of which are different. We move in turns. In every move, we can select a pile and ...
8
votes
3answers
254 views

Markov Perfect Equilibrium with Incomplete Information

Since the pathbreaking paper Stochastic Games (1953) by Shapley, people have analyzed stochastic games and their deterministic counterpart, dynamic games, by examining Markov Perfect Equilibria, ...
8
votes
1answer
147 views

Feedback loop in real-time voting of TV show?

Today, a German TV casting show ("Unser Star Fur Baku") introduced a new "real-time" voting system that works as follows: 10 contestants take part in a song competition. Viewers can call in and vote ...