The game-theory tag has no usage guidance.

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29 views

### How to model how players affect each others in a cooperative game?

The Shapley value is a very useful concept to evaluate the importance/contribution of a player based on how he affects different possible coalitions. Now based on this information, is it possible to ...

**1**

vote

**0**answers

47 views

### Game Theory Cake Cutting [closed]

I'm familiar with Game Theory concepts (I took one course at College, but it was rather superficial), but my mathematical skills aren't at the best level though. However, I'd like to hear from more ...

**4**

votes

**1**answer

259 views

### Time-efficient way of calculating the least number of 1s in a representation of $n$ using only the operations $+,!$

This was inspired by the following paper:
J. Arias de Reyna, J. van de Lune, "How many $1$s are needed?" revisited, arXiv link.
It might help explain my question better, because my question is ...

**1**

vote

**1**answer

149 views

### Averaged geometric series with floor function

Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression:
$$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor ...

**-5**

votes

**1**answer

103 views

### What means the expression “#{…}”? [closed]

I was reading the definition of a congestion game in the theory of games. Then I read the following expression:
"For each element $e$ and a vector of strategies $(P_1,P_2,...,P_n)$, a load $x_e = ...

**7**

votes

**1**answer

274 views

### What does “game theory” cover and how should it be called?

There seems to be a huge discrepancy in what people refer to when they speak of "game theory". I tend to think of it as including, among other things:
Combinatorial game theory dealing with certain ...

**3**

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**2**answers

325 views

### Determinacy of (infinite, possibly loopy) combinatorial games

I am looking for references and hopefully enlightening proofs of the following statement(s) concerning the determinacy of not-necessarily-well-founded (i.e., possibly infinite, possibly loopy) ...

**3**

votes

**1**answer

166 views

### What's the best betting strategy to double money if we have $\delta$ advantage?

Suppose that I am very skilled in a gambling game, and any day that I bet $x$, I get back $2x$ with probability $\frac 12+\delta$ (and nothing with probability $\frac 12-\delta$). My goal is to double ...

**-2**

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**1**answer

33 views

### how to resolve the infinite nesting of interactive POMDP

I am reading papers about I-POMDP. I cant understand the finitely nested I-POMDPs given in these papers.
The belief update of the algorithm has a problem that agents' belief updates mutually depend ...

**1**

vote

**1**answer

90 views

### A bound on the number of bilinear functions needed in order to obtain the minmax

For $n\in\mathbb N$, let $\Delta(n)=\{x\in\mathbb R^n:x_i\geq 0, \sum_ix_i=1\}$ be the set of probability vectors in $\mathbb R^n$.
Is there a function $m:\mathbb N\to\mathbb N$ such that for any ...

**4**

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**1**answer

116 views

### standard auction model

I'm not familiar enough with the auction theory to know where to look, but this seems close to what seems to be known as the "standard auction model".
Say an asset is up for auction.The true value of ...

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**0**answers

42 views

### Projecting on a convex compact polytope with special form

Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...

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**0**answers

378 views

### On the Theory of Infinite Step Processes of Sequential Decision Making

On the border of Set Theory and Game Theory there is a broad class of Infinite Step Processes of Sequential Decision Making that can be characterized by the main following property: a Future ...

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**2**answers

378 views

### Pursuit-Evasion type game on graph (“Flyswatter game”)

An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the ...

**12**

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**0**answers

182 views

### Is the game Hanabi NEXPTIME-complete?

The game Hanabi is a cooperative, hidden-information game. You can read the rules elsewhere, but broadly speaking the players are attempting to cooperatively build a fireworks display by playing cards ...

**3**

votes

**1**answer

196 views

### Minimal Birthdays

In combinatorial game theory: The birthday of a game is defined recursively as 1 plus the maximal birthday of its options, with the zero game having birthday 0.
Suppose we define the quasi-birthday ...

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**0**answers

237 views

### Nimbers and Surreal Numbers [closed]

I have been researching Combinatorial Game Theory. One common theme is the assignment of values to games in order to classify the game as a win for a specific player. One such way is class of surreal ...

**1**

vote

**1**answer

135 views

### Convergence to equilibrium via gradient descent

J. B. Rosen proved that in concave games of n players (which assumes that Cartesian product of strategy profiles is convex) if the game satisfies the condition of diagonally strictly concave then ...

**8**

votes

**1**answer

145 views

### Optimal strategy for game of 'online sorting' into a poset

Consider a single-player game played with an arbitrary finite poset, and a random number generator with a known distribution:
Each turn, the RNG produces a number, and the player must assign that ...

**4**

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**0**answers

143 views

### Generalization of Sprague-Grundy Theorem

In my research on Combinatorial Game Theory, I used a certain theorem that is essentially a generalization of the Sprague-Grundy theorem. Because the result hinges too much on the work of others to be ...

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**0**answers

251 views

### Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...

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**0**answers

123 views

### Maximizing Expected Utility

I am currently trying to solve a maximization problem given by
$\max_{f(x)} \int_0^1 \int_\mathbb{R} (c-y\cdot f(x)-d\cdot (x+f(x)-b)^2) \ h(x) \ dx \ dy$.
Or in other words, I have a utility ...

**1**

vote

**1**answer

179 views

### Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...

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**9**answers

17k views

### John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends.
Maybe this is an appropriate time to ask a ...

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votes

**1**answer

165 views

### How can we solve the TSP problem using game theory? [closed]

Is there a known way to model the traveling salesman problem (TSP) using non-cooperative game theory?
I only found in the internet cooperative game theory. Why there is no work that solves the TSP ...

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vote

**1**answer

183 views

### Random graphs with boundary in a game (Tsuro)

Suppose we have an $n \times n$ board and we have $n^2 - 1$ square tiles. These tiles consist of a 8 vertices, two on each edge, and every vertex is connected to precisely one other vertex. These ...

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**0**answers

121 views

### Analysis of Nim-Like Game? [closed]

There are a finite number of heaps, each with a finite number of counters. Two players take turns; on each move, they may remove exactly one counter from any heap, and also, if the heap is of size ...

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votes

**4**answers

566 views

### Fair cake-cutting between groups

The cake-cutting game is usually played between individuals. What if we try to play it between groups?
A certain land has to be divided between two states. There are $n$ citizens in each state. ...

**5**

votes

**1**answer

122 views

### Anything known about the Grundy Ordinal of Sylver's Coinage

Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia:
The two players take turns naming positive integers that are not the
...

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**0**answers

128 views

### Convergence proof for fictitious play!

In "Fictitious play property for games with identical interests" by D. Monderer and L.S. Shapley, the convergence of fictitious play to a Nash equilibrium is proved for a potential game with players ...

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**0**answers

211 views

### When does a “stable” assignment of buyers into goods exist?

Consider a setting of $n$ buyers and $m$ goods.
We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...

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**0**answers

88 views

### On subset of Deterministic games

Denote strings $u,v$ from $\{0,1\}^n$.
Denote concatenated pair $[uv]$.
Denote
$$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$
collection of pairs with Hamming distance $1$ from $[uv]$ string ...

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**0**answers

103 views

### optimal strategies for 2-player zero-sum games of perfect information

I asked essentially this on math.SE slightly more than
3 days ago, and it hasn't received any answer there.
Do finite-state 2-player zero-sum games of perfect information with only win-draw-loss ...

**3**

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**0**answers

165 views

### What mathematical models can analyze and optimize systems based on gossip?

I look for a mathematical model that can accommodate, analyze and suggest optimizations for a system that can be humanly described as people gossiping about stuff.
System description:
We have a ...

**3**

votes

**2**answers

301 views

### (linear algebra) - Can a symmetric equilibrium achive higher social-welfare than some equilibrium with the same support?

EDIT: rewritting the question to linear algebra to make it more accessible.
Denote by $\Delta([n])$ the set of all probability distributions over $\{1,2,\ldots,n\}$, that is:
...

**2**

votes

**1**answer

64 views

### Is there always a symmetric “subset equilibrium” for an equilibrium in a symmetric game?

Consider a 2-player symmetric game given by a payoff matrix $A\in [0,1]^{n,n}$ for the row player (i.e. the column player matrix is $A^t$).
Let $s=<s_1,s_2>$ be a (possibly mixed-strategies, ...

**1**

vote

**0**answers

288 views

### What is known about multiplayer poker with flop?

I am interested in the following simplified version of poker.
Each player gets a card (for example, either A or B).
Then they bet knowing their own cards (for example, the pot initially has 1 euro, ...

**2**

votes

**1**answer

116 views

### How to find equilibrium of the following game?

I'm looking at the following game:
2 symmetric players $\{A,B\}$, each choose a number $x_a,x_b\in [0,1]$.
The utility of player $A$ is:
$$U_A = \
\begin{cases} p\cdot x_a &\mbox{if } x_a> ...

**16**

votes

**1**answer

794 views

### Removing pawns - the game

Here is a simple game I've invented (if the idea is not fresh, then please let me know):
The game is played on a board.
The board has some (finite) number of lines drawn on it.
A pawn is placed on ...

**3**

votes

**2**answers

366 views

### Graph game minimum vertex degree

Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ \gg \log(n)$. Players are BR and MA (BR moves first):
BR claims an unclaimed edge from $E$, adds it ...

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votes

**2**answers

195 views

### Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - ...

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votes

**4**answers

923 views

### Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game

The game
Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is ...

**9**

votes

**1**answer

380 views

### Guessing the larger integer: A game-theoretic twist

The starting point for this question is the old chestnut, already discussed on MO, about a game show on which the host has chosen two distinct integers and the contestant gets to reveal one of them at ...

**7**

votes

**1**answer

497 views

### Explicit examples of undetermined games

Suppose we have a game between two players in which they take alternating turns. The game can have finite length, length $\omega$ or any transfinite number of steps (however, I'm not concerning games ...

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**3**answers

322 views

### Limiting probabilities for two-player game drawing random uniform numbers

Consider this simple 2-person game I just made up:
Player A goes gets to draw a uniform U[0,1] number up to X times. At any time, he may either keep his number, or draw a brand new uniform number. ...

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votes

**1**answer

123 views

### What is the (mixed strategies) equilibrium of this game?

Given a weight vector $w\in [0,1]^d$ such that $\sum w_i=1$, the game goes as follows:
Two players, $X,Y$ choose strategies $x,y\in [0,1]^d$ such that $\sum x_i = \sum y_i = 1$.
The utility (profit) ...

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votes

**1**answer

412 views

### Stromquist's 3 knives procedure

(copied from math.SE)
BACKGROUND: A cake has to be divided among 3 people with possibly different tastes, such that each person receives a single connected piece, and no person prefers another ...

**10**

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**1**answer

300 views

### Game on the tree [closed]

There's a problem from programming competition which already finished:
http://codeforces.com/contest/458/problem/F
Two weeks already passed but still nobody solved it yet - in fact you can see here ...

**0**

votes

**1**answer

850 views

### Maximal score for the 2048 game [duplicate]

t's been weeks (months?) since the 2048 game--by Gabriele Cirulli--took Internet by storm. I have an explicit integer $X$ which is greater or equal than any score of this game. Possibly my $X$ is the ...

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votes

**5**answers

3k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...