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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107
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4k 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 ...
16
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0answers
333 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 ...
12
votes
0answers
397 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
7
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0answers
52 views

Diagonal-free Sudoku grid

I have a Sudoku grid with the property that diagonally adjacent elements are distinct (it is also a torus under the same property). The grid offers new and exciting logical possibilites. My question ...
7
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0answers
149 views

Is there a totally asymmetric 2P $0$-sum game with all payoffs $\pm1$, with a unique Nash eq. which assigns positive probability to each strategy?

Is there a totally asymmetric 2-player zero-sum game with all payoffs $\pm1$, with a unique Nash equilibrium which assigns positive probability to each strategy? By totally asymmetric, I mean that ...
6
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0answers
57 views

What is the most effective strategy to win at this game?

The game is as follows. Alice secretly selects three real numbers $a_{1},a_{2},a_3$ such that $1\geq a_1\geq a_2\geq a_3\geq 0$ and $a_1+a_2+a_3=1$. Bob secretly selects three real numbers ...
5
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0answers
80 views

Fastest way to meet, without communication, in a toroidal palace?

I was interested by a similar question asked here, but wanted to pose a slightly different variant that avoids some of the pitfalls and ambiguities in the first question in order to ask something more ...
5
votes
0answers
108 views

Guess the number of a liar

Sue picks a number from 0 to 3. Tom asks questions about the number, with yes/no answers. For example, "Is it odd" or "Is it 3?" If Sue picked X, she is allowed to lie at most X times. For ...
5
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0answers
100 views

Apple game question

Player A and Player B play a game. On the middle of the table there is a pot full of $N$ apples of different weights. Player A starts first and chooses an apple and starts eating it. Losing no time ...
4
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62 views

Wizard against two dwarfs: guess the whole function

An evil wizard plays the following game with two dwarfs $A$ and $B$: he thinks of a function $f:\mathbb{R}\to\mathbb{R}$ (which is not required to have any regularity properties, such as ...
4
votes
0answers
91 views

StackEgg optimal algorithm

What is the minimum number of days that is needed to complete the StackEgg game? (It's on the right if anyone didn't notice.) There are four markers (Questions, Answers, Users, Quality) I believe each ...
4
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0answers
29 views

Nash Equilibria for the Effort Level Game

Player $i$ chooses an effort level, $e_i \in [0, 10]$. Let player $i$ have the following payoff function: $90 -e_i$ if $e_i > e_j$ and $80 -e_i$ if $e_i \leq e_j$. What is the Nash Equilibrium (NE) ...
4
votes
0answers
65 views

Who has a winning strategy in the hamilton-circle-game?

The game starts with a graph with $n$ vertices and no edges. The players alternately add edges until the graph contains a hamilton-circle. The player who made the last move loses. Who has a winning ...
4
votes
0answers
143 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
4
votes
0answers
86 views

Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
4
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0answers
132 views

Unexpected hanging paradox maxmin strategies

I have a question about strategies of the players of Unexpected hanging paradox (I am very sorry for a long topic, topic exist already for a while, during this time I try to develop idea how to solve ...
4
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0answers
255 views

$n$ players of paper scissor rock

Suppose there are $n$ players $(3\leq{n})$ showing Paper, Scissor or Rock simultaneously. If there is no winner then there is no payoff to any player. If there are winners and losers (e.g. $k$ ...
4
votes
0answers
141 views

Understanding Blackwell's Approachability Theorem

I'm not super solid on my linear algebra, so I am getting lost in the discussions of halfspaces. Can someone give me an intuitive explanation (possibly with a concrete toy problem) of Blackwell's ...
3
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0answers
49 views

Conway's Game OF Life maximum periods on a set x by x game board.

I have taken interest in Conway's Game of Life and want to know if you guys can help me with a mathematical problem :) That is what this website is for right? You need to be familiar with the rules ...
3
votes
0answers
82 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
3
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0answers
48 views

Simple undetermined games

We know that, under AC, there exists a game in which two players play finite numbers and neither one has winning strategy. There are also such undetermined games when we consider players playing ...
3
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0answers
106 views

Maximizing dot-product score by asking queries

Let $a>b>0$, and let $T=\{a,b\}^n$ be the set of all $n$-tuples each entry of which is $a$ or $b$. Let $X\subseteq\{0,1\}^n$ with $|X|>1$, and let $f:T\rightarrow X$ be a function. For each ...
3
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0answers
68 views

News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
3
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0answers
46 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
3
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0answers
65 views

Terminology questions about a game where one may “save his progress” at the cost of a turn.

The game is for $p$ players who each start at square $1$. Each turn, a player can either roll an $m$-sided dice or place a marker on his current square. If he rolls $x\in\{2,\ldots, m\}$, he ...
3
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0answers
115 views

mixed vs behavior strategies for zero-sum game with infinite extensive form

edit: No responses to this post after a week, so I'm cross-posting it to cstheory.stackexchange here. I'm looking for a known theorem stating that, for appropriate kinds of two-player zero-sum games ...
3
votes
0answers
471 views

Simple game with coins - strategy

Let's play a game: There are $n$ stacks of coins in a row. $i$-th stack consists of $d_i$ coins. Two players: $\text{Player1},\text{Player2}$ make moves alternately. Player in his turn can only take ...
3
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0answers
139 views

Optimal strategy in a VCG auction with partial collusion?

Suppose you control the bid prices in a multiple-item VCG auction for a partial coalition of bidders. Each bidder is only allowed to win one item out of the set of multiple items, which are all ...
2
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0answers
50 views

Cournot competition: profit maximizer vs. market share maximizer

Today during an informal conversation with an established business researcher, I learned such a fact: In the classical Cournot competition model, if one player is a profit-maximizer, the other ...
2
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0answers
24 views

Finding the core of a coalition game

I need to find the core of a 3-player coalition game graphically, given that $v(\phi)=0$, $v(1) = 9, v(2)=8, v(3) = 9, v({1,2}) = 14, v({1,3})=15, v({2,3}) = 13, v({1,2,3}) = 21$ So I'm following the ...
2
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0answers
30 views

Cournot Oligopoly in Bayesian Game Theory

I have this Cournot game in which $n$ firms produce quantities $q_1, \ldots, q_n$ with respective marginal costs $c_1, \ldots, c_n$. They all sell at price $P=1-(q_1 + \cdots + q_n)$. For any $i$ ...
2
votes
0answers
71 views

Should I maximize points in the beginning of a long match?

I have a question that may really be about mathematical modelling as much as math itself, but I will try to give it a formulation suited for this site. Suppose that I am going to play some game, ...
2
votes
0answers
52 views

How to find optimal strategies for infinite betting games?

Suppose we have a game structure of the following form: Game: You have an $n-$sided die. At any point in the game, there is a "value" associated to the rolls that have already occurred. You can ...
2
votes
0answers
97 views

Is it possible to calculate the balanced cost of parameters' increase in card game? How?

I wonder if it's possible to calculate the balanced cost of parameters' increase for the card game. Game rules: Each player draw 7 cards at the beginning of the game and then one card each turn ...
2
votes
0answers
173 views

stable marriage algorithm problem

Better of the two Suppose that in the stable marriage problem with $n$ men and $n$ women, we have found two (possibly different) stable matchings $S$ and $T$. We will show how to combine $S$ and $T$ ...
2
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0answers
50 views

Showing $\frac{1+c}{a+b}\leq \varphi$, when $\frac{1}{1+b}\leq a\leq 1$, $\frac{c^2}{a+c}\leq b\leq 1$ and $0\leq c\leq 1$

The title pretty much says it all. Let $\varphi\triangleq\frac{1+\sqrt 5}{2}$ be the golden ratio. Let $a,b,c$ be some non-negative numbers such that: $\frac{1}{1+b}\leq a\leq 1$ ...
2
votes
0answers
43 views

Book or paper recommendation about “Rube Goldberg Mathematics” // e.g. Longest path problems

First: My question is not be very specific, since I lack a concrete overview, but my idea/thoughts in a nutshell: I would like to have a recommendation of a good book, paper or article about processes ...
2
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0answers
51 views

Game Theory: What are Best Strategies for High-Low game (game details are below)?

High Low game is one where one person picks a number between a range (say 1-100) and another person have to guess it. With each guess, s/he is told whether the guess was high, low or correct. If the ...
2
votes
0answers
59 views

Correlation of belief distributions from distinct signals

Anne and Bob are two Bayesians who initially share a non-degenerate prior about a binary state of the world. Anne observes some signal (i.e., an experiment in Blackwell's terminology) about the state ...
2
votes
0answers
80 views

A combinatorial game theory problem

In details, there are four bishops on a chessboard in two pairs. In each pair they sit in orthogonally adjacent squares. How many positions can there be to place the two pairs on the chessboard ...
2
votes
0answers
86 views

Can “tit for tat” strategy be defined in monadic second-order logic?

Prisoner's dilema game can be represented as a game tree, which could be infinite game with corresponding infinite game (binary) tree in common case. There is well-known tit for tat strategy, which ...
2
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0answers
30 views

Optimal Strategy Game of Communicating without Overlap

Today, I had a conversation which proceeded very poorly; indeed, I had this conversation with $n$ people, including myself, and everyone had something to say. Problem was that, for an unreasonably ...
2
votes
0answers
15 views

Bingo based on number of hits on card?

I was wondering how many possible combinations there were to win based on the number of hits on a bingo card (25 spots, 1-75 etc.)? I know that if you get 1,2,3 hits on the card, there are no chances ...
2
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0answers
18 views

dimension of Weber set and selectope (as a operator)

Let $\Omega$ be a finite set of players. For a selector $\alpha:(2^{\Omega}-\{\emptyset\})\rightarrow\Omega$, we define a marginal value operator as a linear operator $m^{\alpha}$ ...
2
votes
0answers
108 views

Game theory: connect four?

Through Allis' solution etc... P1 can force a win if he places the first stone in the highlighted region. Assuming both players have perfect information, it will take 41 turns maximum (if I recall ...
2
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0answers
113 views

Game theoretical approach to other branches of mathematics

Are there some methods and ideas derived from game theory that are successfully applied to better (or more intuitively) understand theorems and proofs or tackling problems from other areas of ...
2
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0answers
87 views

What is the largest value one can get in game 2048 without new tiles appear

This is a simplified version of the famous game 2048. Given a 4x4 grids with some values chosen from {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048}. A value of 0 indicates that the position in ...
2
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0answers
28 views

VCG - plynomial time algorithm when bidders are unit demand

Is there a polynomial time algorithm to run a VCG when bidders are unit-demand? I though to look at the Bipartite graph when the left side is the bidders the right is the items and the edges are the ...
2
votes
0answers
413 views

What is the highest possible score in 2048 hard?

There is a variant of the popular game 2048, called 2048 hard or 2048 impossible, which automatically places each new tile in the hardest possible location. Is this variation possible to solve, and if ...
2
votes
0answers
137 views

What is the optimal strategy for this 2 player game?

Let some finite array of integers is given initially. There is a number k which is initially '0'. In a move, a player will select a number from the array arr[i] and change k to gcd(k,arr[i]). Also, ...