Questions tagged [game-theory]

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

Game theory is the study of mathematical models of strategic interaction between rational decision-makers. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.

3688 questions
697
votes
26 answers

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 hour to an hour, and then just grins at me, and…
VividD
  • 16,306
144
votes
1 answer

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 $p_0 > 0$ and $p_{k+1} > p_k$ for all $k$ . In…
Yuval Filmus
  • 58,333
120
votes
18 answers

Fastest way to meet, without communication, on a sphere?

I was puzzled by a question my colleague asked me, and now seeking your help. Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and the sphere is way bigger than you two. There are no…
Rob Audenaerde
  • 1,245
103
votes
11 answers

Mathematician vs. Computer: A Game

A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same range. If the numbers chosen share a prime factor,…
user139000
94
votes
1 answer

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 : ($6 \times 6$ example) Rules Start with a grid made…
user139000
73
votes
3 answers

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 online servers let the players see the opponents…
Andrew Thompson
  • 4,529
58
votes
3 answers

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

Consider a $9 \times 9$ matrix that consists of $9$ block matrices of $3 \times 3$. Let each $3 \times 3$ block be a game of tic-tac-toe. For each game, label the $9$ cells of the game from $1$ to $9$ with order from left to right, from above to…
mez
  • 10,957
  • 5
  • 59
  • 112
56
votes
8 answers

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?
user12358
  • 289
44
votes
3 answers

Staff icebreaker - is stasis ever attained?

Yesterday at work we had a staff day, where we were asked to play an interesting game as an icebreaker. We (50 or so people) were told to stand in a circle and choose 2 people at random out of the group. We were then asked to walk to a point so that…
martin
  • 9,226
39
votes
3 answers

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 and scissors (because both fall into it) but…
celtschk
  • 44,837
39
votes
1 answer

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 standard $52$ card deck is shuffled and dealt evenly…
Alexander Gruber
  • 28,405
37
votes
3 answers

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 room follows the same rules, as described below. A…
user75900
35
votes
5 answers

Help: rules of a game whose details I don't remember!

In a probability course, a game was introduced which a logical approach won't yield a strategy for winning, but a probabilistic one will. My problem is that I don't remember the details (the rules of the game)! I would be thankful if anyone can…
Behzad
  • 2,375
35
votes
1 answer

Can the lion protect the sheep from three wolves?

Generally, in pursuit-evasion games, there's one prey and one or many pursuers. I'd like to know how extending the food chain would change the dynamics of such games. Specifically, let's consider a closed, circular shape arena in $\mathbb{R}^2$ .…
Eric
  • 1,939
  • 14
  • 18
33
votes
3 answers

A coin flipping game

I've been thinking about the following game for a while and am curious if anyone has any ideas of how to analyze it. Problem description Say I have two biased coins: coin 1 that shows heads with probability $p$ and coin 2 that shows heads with…
Erik M
  • 2,461
1
2 3
…
99 100