The game-theory tag has no wiki summary.

**9**

votes

**1**answer

421 views

### M-matrix plus S-matrix is P-matrix?

I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would ...

**9**

votes

**3**answers

862 views

### Nim game for odd number of stones

Consider the classical Nim game with total number of stones being odd. Then the first players wins, of course, what follows from the general description of winning positions. But is there some shorter ...

**12**

votes

**2**answers

708 views

### An unfair game involving an odd number of pieces of chocolate

Two greedy chocolate eaters play the following game involving $n$ pieces of chocolate
and an additional parameter $\alpha$ with initial value $1$: Each player eats either $\alpha$
pieces of chocolate ...

**4**

votes

**0**answers

147 views

### Examples of functions from matrices to real numbers with certain properties

Let $M(\mathbb{R})$ be the set of all matrices (of any size) over $\mathbb{R}$. Let $v : M(\mathbb{R}) \rightarrow \mathbb{R}$ be a function which satisfies the following 5 properties:
If ...

**15**

votes

**1**answer

481 views

### The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...

**15**

votes

**2**answers

3k views

### Simple proof of the existence of Nash equilibria for 2-person games?

Is there a nice elementary proof of the existence of Nash equilibria for 2-person games?
Here's the theorem I have in mind. Suppose $A$ and $B$ are $m \times n$ matrices of real numbers. Say a ...

**2**

votes

**5**answers

987 views

### Algorithm to solve Sokoban-like game on graphs - move chips from one set of vertices to another

The question is close to the Sokoban game (thanks to Dima Pasechnik !), but a little different in details.
Consider a directed graph (multi-graph). Consider some set of marked chips (chip1, ...

**10**

votes

**12**answers

4k views

### Are there any interesting connections between Game Theory and Algebraic Topology?

I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? ...

**2**

votes

**1**answer

486 views

### game theory - coin flipping question

Lets say 2 players A and B try to have the most money at the end after playing a casino game in which they have a $49\%$ chance to double a wager.
Here are the rules to the bet between A and B:
...

**48**

votes

**9**answers

7k views

### Does War have infinite expected length?

My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers.
The ...

**6**

votes

**1**answer

516 views

### Determinantal formula for the nullspace of a singular matrix

In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should ...

**0**

votes

**2**answers

213 views

### Mysterious sentence in a paper: what's the ultimate distribution of pure strategies?

Does anybody know how to interpret the sentence: For any set $T$ of mixed strategies, let $D[T]$ denote the set of probability distributions over the elements of $T$, each expressed as vector, ...

**7**

votes

**1**answer

221 views

### Optimum Tournament Strategy

Consider a symmetric N-player game in which all players partition one total unit of
energy among individual games. The probability of winning each game is simply proportional to the spent energy ...

**0**

votes

**0**answers

466 views

### Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...

**5**

votes

**2**answers

514 views

### Gandhi's quote formalized [closed]

Hello,
I hope this question is appropriate for Mathoverflow. Gandhi said, "Be the change that you wish to see in the world". I don't understand anything in Game/optimization theory (I don't know ...

**0**

votes

**0**answers

70 views

### Computing maximum point for minimal function of a family of linear functions

Let $x \in S^n $ where $S^n = ${$ [x_1,x_2,...,x_{n+1}]\in \mathbb{R}^{n+1} \mid x \ge 0 , \sum x_i = 1 $} and let $f_i : I^n \to \mathbb{R}$ be a finite $m$-sized family of LINEAR functions such ...

**3**

votes

**2**answers

317 views

### Uniqueness of equilibrium from infinite strategies

I took the following game from the Peter Winkler collection (chapter "Games"):
Two numbers are chosen independently at random from the uniform distribution on [0,1]. Player A then looks at the ...

**1**

vote

**0**answers

193 views

### Allocation game optimal strategy

There are two players, Alice and Bob. There is an initial pool of 100 dollars. Alice proposes an allocation of the dollars (real numbers, not necessarily integers), and Bob can either accept or ...

**5**

votes

**1**answer

360 views

### Resources-Aware Combinatorial Game Theory

First of all, I preemptively apologize if my question happens to be naive, I am no expert of CGT (or general game theory, for that matter).
Now the question:
**is there such a thing as the study of ...

**3**

votes

**1**answer

223 views

### Indeterminacy of long games

Hello, all,
Several months ago I sat in on a seminar on AD+, which was incredibly wonderful even though I could barely follow it at all. AD+ is a technical variant of AD, the axiom of determinacy, ...

**4**

votes

**1**answer

1k views

### Finding the Nash Equilibrium of $0-1$ poker with one betting round

While working on a hobby project I encountered a difficult math problem. Or at least, difficult for me. Here is the problem:
Given an $a > 0$, find all pairs of a value $λ \in [0,1]$ and a ...

**14**

votes

**0**answers

437 views

### The multiplication game on the free group

Fix $W\subseteq\mathbb F_2$ and consider the following two-person game: Player 1 and Player 2 simultaneously choose $x$ and $y$ in $\mathbb F_2$. The first player wins, say one dollar, iff $xy\in W$. ...

**9**

votes

**3**answers

438 views

### Representing a real number as the value of a countably infinite game

Is it true that for any real number $p$ between 0 and 1, there exist finite or infinite sequences $x_m$ and $y_n$ of positive real numbers, and a finite or infinite matrix of numbers $\varphi_{mn}$ ...

**2**

votes

**3**answers

936 views

### Motivation for the Sprague-Grundy theorem

The Sprague-Grundy theorem states that every impartial combinatorial game under the normal play convention is equivalent to a (unique) nimber.
What does the equivalence relation thus defined tell us ...

**5**

votes

**2**answers

571 views

### Pursuit-Evasion on a Manifold

I know pursuit-evasion has been studied in many contexts, including
on a manifold (e.g., Melikyan,
"Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"),
but I have not seen this version:
...

**6**

votes

**1**answer

412 views

### Cake-cutting and amenable groups

I recently heard Alan Taylor speak about envy-free fair division and started wondering if questions like these make sense if we replace finitely additive measures with invariant means on amenable ...

**6**

votes

**1**answer

757 views

### Drawing lines and removing squares - an Alice and Bob game

Thought about the following while in a Complex Analysis lecture:
Let there be a $N \times N$ grid of squares and two players $A$ and $B$. First, $A$ needs to draw a line $l$ that needs to intersect ...

**3**

votes

**0**answers

188 views

### References for this game

Hello everybody,
I would like to know how the following game is known in the literature and, possibly, to have references for related papers.
Description of the game: Fix a space $X$ and two Borel ...

**1**

vote

**0**answers

698 views

### What would be nice open problem in evolutionary game theory ? [closed]

Hello,
i was trained as a biologist, but have taught myself mathematics to a level that is roughly equivalent to that of a masters degree in math. I decided to try do some phd-research in ...

**2**

votes

**3**answers

383 views

### Why are Nash-Equilibria inside the Simplex S_n unique ?

Hello,
i came across a remark that states that nash equilibria inside the simplex S_n are unique. Or stated differently: If there is more then one such equilibrium, then they have to lie on the ...

**5**

votes

**1**answer

511 views

### Decay of Relative Growth in Conway's Game of Life

Intro
The question is about Game of Life.
Let us denote the set of points obtained from initial configuration $A$ after $m$ steps as $A(m)$ (we are only interested in finite initial configuration, ...

**3**

votes

**2**answers

580 views

### Set Cardinality Game - Can a player with numbers in R win over a player with numbers in N as each of them in one turn has to present a new number?

Let there be 2 players, p$\mathbb{N}$ and p$\mathbb{R}$. They are playing the Set Cardinality Game where p$\mathbb{N}$ has to present some number n that has not been used in the game so far by either ...

**8**

votes

**0**answers

2k views

### Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog

I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao ...

**4**

votes

**1**answer

676 views

### Conditioning on one term of a sum of random variables

Let $\theta$ be normally distributed with mean $\bar \theta$ and variance $s^2$. Let $Z$ be normally distributed with mean $0$ and variance $\sigma^2$, and chosen independently of $\theta$. Define ...

**4**

votes

**1**answer

641 views

### Are there any interesting connections between game theory and engineering?

I am doing a senior project and it must be based off game theory, but I am having trouble finding any connections to engineering, possibly structural, or architectural, maybe even civil or mechanical. ...

**14**

votes

**5**answers

2k views

### Irreversible chess

Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ...

**8**

votes

**2**answers

389 views

### A competitive root finding game

Inspired by a question about bisection I wondered about the following: The are two players X and Y and a moderator Z who knows two (random,independent, uniformly chosen) hidden reals $x$ and $y$ from ...

**9**

votes

**2**answers

1k views

### The duel problem

The following duel problem is due to Ben Polak (maybe there's earlier origin, which I'll be glad to be informed about). The rule is as follows:
Two players 1 and 2 start a duel $N$ steps away from ...

**3**

votes

**2**answers

367 views

### Truel extended to n persons

n players numbered 1~n play a shooting game. Their accuracy rates p1~pn are strictly between 0 and 1, and strictly increases from p1 to pn. This is common knowledge.
Before the game starts, the ...

**0**

votes

**1**answer

283 views

### Zermelo's stone game in 3 dimensional space

Well, first let me make this clear: I'm actually not sure about the background of the game, whether it was really posed (and solved) by Zermelo. But I'll state the game anyway (perhaps someone can ...

**7**

votes

**4**answers

1k views

### Infinite games: are they well defined?

It is just my curiosity about this question where we have an infinite game and (according to the answers) winning strategies for both players. I am familiar with terminating games only, and I am ...

**2**

votes

**2**answers

530 views

### Non-measurable sets and Determinacy…

Assume AC. Suppose $X$ is a subset of the irrationals (Baire Space) for which neither player has a winning strategy (i.e. the game $G(\omega, X)$ is not determined). Is $X$ non-measurable in the ...

**3**

votes

**1**answer

312 views

### Approximating the maximin value of a zero-sum game

For square matrix $P$, define
$$V(P) = \sup_x \inf_y x^T P y^T$$
where $x$ and $y$ lie on the unit $n-1$-simplex.
($P$ is a payoff matrix for a symmetric game, $x$ and $y$ are mixed strategies, ...

**5**

votes

**0**answers

759 views

### A Fun Game with Coins [closed]

Assume you have a pair number of coins $2n$ with possibly different values, ordered in a line. Let us enumerate the coins as $x_1,x_2,\ldots,x_{2n}$. The coins are not ordered in any particular way.
...

**6**

votes

**2**answers

253 views

### Truthful multi-unit auctions that guarantee selling all items

Suppose an auctioneer has $k$ units for sale. There are $n$ bidders, each of whom are interested in a single good, and have value $v_i$ for it. If bidder $i$ has to pay $p_i$ and gets the good, he ...

**2**

votes

**2**answers

648 views

### Games of imperfect information (e.g. Blackwell's games) in Set Theory?

Hello,
Intro about standard two player games
Gale-Stewart games are the well-known games played by two Players $I$ and $II$, which in turn play natural numbers for infinitely many ($\omega$) steps. ...

**1**

vote

**1**answer

522 views

### Simple(?) game theory

3 players are playing a game where they get to pick independently without knowing the other players picks one of 2 prizes (A,B) and the payout is (a,b) for the two prizes, divided by how many people ...

**3**

votes

**2**answers

606 views

### What is the optimal strategy for participants in this situation?

Consider a simplified version of eBay where everyone bids once on an item, nobody sees each-other's bid, and the highest bid wins. This is called a "First-price sealed-bid auction".
One day you find ...

**2**

votes

**1**answer

795 views

### Applications of Algebraic Geometry in Evolutionary Game Theory

Hello,
do you know any papers or books that use algebraic geometry in evolutionary game theory ?

**15**

votes

**3**answers

977 views

### Fairest way to choose gifts

Suppose that a parent brings home from a trip $2n$ gifts of roughly
equal value for his/her two children. The children get to choose one
at a time which gifts they want. What is the fairest way to do ...