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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102 views

The name of a game from the 2013 Putnam

Does the following game from the 2013 Putnam have an official name? Based on my searches, it seems to have been created just for the exam. Let $n\geq 1$ be an odd integer. Alice and Bob play the ...
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1answer
398 views

What is the probability of a $4$ appearing in the game $2048$? [closed]

I'm not sure if this is the appropriate SE, so please suggest a more appropriate website if not. I'm making a facsimile of $2048$, and I've just one question I've been unable to work out: what is the ...
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0answers
28 views

n-player zero-sum rating

I am trying to make a rating system like Elo-rating for n-players in SET with rules here. In each game 1 player plays against n-1 other players. The player collects a number of SETs, $s$. Based on ...
4
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1answer
90 views

A game problem- double or increment by 1

Its a two player game. Initially $P=1$, and there is some fixed integer $Q>1$. A valid move consists of either increasing $P$ by $1$ or doubling it iff on doing so $P$ does NOT exceed $Q$.The ...
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0answers
52 views

Game theory Zero Sum Game Proof

Zerosum games. A coailitional game with transferable payoff is zerosum if $v(S) + v(N - S) = v(N)$ for every coalition $S$; it is additive if $v(S) + v(T) = v(S \cup T)$ for all disjoint $S$ and $T$. ...
8
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2answers
337 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 ...
12
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273 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 ...
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1answer
106 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
5
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1answer
177 views

Which mathematical game or puzzle did you invent?

A couple of weeks ago, a friend of mine showed me a extension of a game we are all familiar with that he was working on. The game we know is called Tic-Tac-Toe, and he was working on his own version ...
3
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1answer
350 views

Game Theory - A joint project problem

I'm new to this forum and as such not sure if this is the correct place to ask for help on Game Theory. As such I am currently working out of the Introduction to Game theory book by Martin J Osborne ...
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1answer
22 views

A game of lines and points

Consider the following scenario: $\mathcal{A}$ and $\mathcal{B}$ play a game inside the unit disc $\mathcal{D}: $ $\mathcal{A}$ chooses a point $p_0\in \mathcal{D}$. At step $n, ...
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58 views

Dilemma at the dining table

I created this problem while I was having my supper a few days back. So there maybe flaw in the formulation. Please point them out as you see one. Suppose, there is a circular dining table with ...
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2answers
53 views

Existence of a winning strategy against the probability of winning

Edit: I've made the question clearer. Suppose a game is played between $A$ and $B$, in which there exists a winning strategy for $A$. Suppose $A$ and $B$ play their moves at random, do we have ...
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2answers
222 views

Real world application of dominating set?

can anyone tell me about the application of vertex coloring problem and algorithm for vertex color problem in graph or networks.
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1answer
34 views

What number of robbers, under the model of the prisoner's dilemma, would be optimal?

The prisoner's dilemma is defined as "Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with ...
13
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3answers
5k 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 $4\times 4$ ...
0
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1answer
77 views

Finding Nash Equilibria of a finte game of 2 players.

In a finite game, suppose player 1 has strategies $\{\alpha_1,\alpha_2\}$ and player 2 $\{\beta_1,\beta_2\}$ with payoffs as below. \begin{array}{c|c|c} &\beta_1&\beta_2\\ \hline\\ ...
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1answer
69 views

How to 'show your work' with game theoretic notation

Everything I've read on game theory seems to describe the game in notation and solves it in natural language. How do you work with notation in game theory? Could you recommend a straight-forward ...
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0answers
75 views

Linear Programming with Matrix Game

It seems from an easy google of "learning linear programming" that a common way of learning it is to work with Matrices that represent "games" for two players. Here is one I have stumbled across. We ...
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1answer
146 views

Game Theory - trying to find game name by description

My hobby AI research have led me to a thorethical game of particular design. As design is pretty simple, I'm almost sure that such game has well-known name and tons of research already done around it. ...
4
votes
3answers
348 views

How to denote this in game theoretic notation

I'm writing a paper that demonstrates that linguists can use the concepts in game theory to infer what interlocutors naturally infer when the literal meaning of their utterances doesn't ostensibly ...
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1answer
134 views

Game Theory - Nash equilibrium question

Consider a game in which 2 players transmit packets in a network with a selected power $x ∈ [1, A]$ and $y∈ [1, A]$, respectively. The utility of the players can be expressed as: $$u_{i} (x,y) = ...
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2answers
168 views

Winning a restricted game of Nim?

Given the following piles, find the Grundy number of the initial position and make the first move in a winning strategy given that no more than two sticks may be removed from a pile at any time. Pile ...
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1answer
73 views

Determining Grundy Numbers for an inverted takeaway game

Given the following game, I need to determine a winning strategy and find the set of positions in the kernel. I figure the best way to do so would be with Grundy numbers. Rules: The game consists ...
4
votes
1answer
307 views

Game Theory/Bayesian approach to a bluffing game

Two players play the following card game with a deck consisting of (A,2,3,4,5). A dollar is placed in the pot by some third party, and player 1 is dealt a card. If it is an A, he has a winning card, ...
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2answers
88 views

Solution to $n$ by $n$ game of lights out

How can I solve an $n$ by $n$ game of Lights Out?
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1answer
61 views

Subscript before a function symbol?

Does anyone know what the subscript before the function means? $$ _pf_p $$ It's part of a definition for selfish routing in networks: Let $N = (V,E)$ be the network, which is a directed graph. ...
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1answer
103 views

Game theory, gambling odds

I'm looking for mathematics around such game: Two games: 50%, max bet: 155000, price x1.96 0.0015%, max bet: 5, price x64224.3 Let, I will place 155,000 for 50% ...
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1answer
95 views

Topological games

I have seen in a few abstracts, as this for instance: A survey of topological games the remark that the subject Topological games has applications in other fields of mathematics. I am familier with ...
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1answer
82 views

Game Of Strings

There are two strings A and B. Initially, some strings A’ and B’ are written on the sheet of paper. A’ is always a substring of A and B’ is always a substring of B. A move consists of appending a ...
3
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2answers
190 views

Finding optimal thresholds for “guess if number is highest” game

Consider the following game: five numbers are chosen randomly in the interval [0..1] with uniform distribution. The player is shown each number in turn and asked if it is the highest. The game ...
5
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0answers
90 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 ...
2
votes
2answers
113 views

Game Theory: players' gender convention?

What is the Game Theory convention of using gender terms (male/female) for the players? I found only one reference suggesting that odd-numbered players are male and even-numbered players are female. ...
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1answer
67 views

Fair three-way sandwich division

This question discusses fair three-way sandwich division. Mentoined solutions include the Selfridge–Conway discrete procedure and the moving-knife procedure. I posed the question to the guys at the ...
3
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2answers
319 views

Existence of Saddle Point of a Matrix (Shapley's Theorem)

A $m\times n$ matrix $M=(a_{ij})_{m\times n}$ with real entries is said to have a pure saddle point at $(i_0,j_0)$ if $\min_j \max_i (a_{ij}) = \max_j \min_i (a_{ij}) = a_{i_0j_0}$. Here the notation ...
1
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1answer
44 views

Prove $f(x,y) > g(x,y)$ for all $x,y \in [0,1]$

I'm trying to prove the following: $$ 4xy + 4(1-x)(1-y) < \max\{8xy,8(1-x)(1-y),3\} \qquad \forall x,y \in [0,1] $$ In the language from the class, I'm trying to show that: $m_2 < ...
0
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1answer
127 views

Special Counterexample to Kakutani's Fixed-Point Theorem

For reference, here is the statement of Kakutani's fixed point theorem. Let $X$ be a compact, convex subset of $\mathbb{R}^n$ and let $f:X\to \mathcal{P}(X)$ be a set-valued function such that $f(x)$ ...
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1answer
90 views

Card game: How much will you pay to gamble?

You turn over the cards 2 at a time, if they are both red, you keep the cards, if they are both black I keep the cards. If one is red and the other is black then neither you nor I get a card. If you ...
2
votes
1answer
130 views

Brouwer's fixed point theorem

Theorem: If $f:D^n\rightarrow D^n$ is continuous then there is $x \in D^n$ such that $f(x)=x$. To prove the theorem we assume that $f$ is cts but has no fixed point, that is $f(x)\neq x$ for all ...
1
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1answer
61 views

Coin based subtraction game

I'm having a problem in Game Theory where I am trying to understand how a subtraction game can be interpreted by a coin based game. From my book: The problem I'm having is if I have 9 coins and the ...
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1answer
94 views

Explanation of basic definitions in game theory.

In the article entitled Non-Cooperative Game written by Nash in 1951, he discussed about the symmetries of games. Due to my lack of basic knowledge in permutations and symmetries, I looked up some ...
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0answers
46 views

On a certain type of card game

Suppose two players are playing a card game, which is described as follow. Each player is allowed to construct their own decks of exactly $n$ cards with an additional repeatable card, where each ...
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2answers
97 views

Why should a GE fail to exist in non-convex sets?

In an exchange economy with $2$ goods and $m$ identical Households where each household has utility function $u(x_1, x_2)$, together with positive endowments. If preferences are not convex, then why a ...
4
votes
1answer
231 views

A game of Chess - Ideal Solution

I am a student of physics. I have learnt some basic group theory, and I am wondering if there is any ideal solution for a given Chess game (like solving Rubik's cube). I know the no. of permutations ...
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0answers
32 views

Prove that the partial derivatives of $(y-g_i+a\sum^n_{j=1} g_j)$ are positive

I have a function: $$\pi_i^1=y-g_i+a\sum^n_{j=1}g_j,$$ where 0 < a<1< na, and I need to prove this: $$\frac{\partial(\sum^n_{i=1}\pi^1_i)}{\partial g_i}=-1+na>0.$$ I am not very ...
6
votes
1answer
213 views

Monkey typing ABRACADABRA and gamblers

Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
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1answer
46 views

A little question about the existence theorem of Nash equilibrium in game theory

Recently when I started reading Nash's paper, I found a little question about the linearity of payoff functions. Is it an assumption? Or did I miss some idea about the payoff function and its ...
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1answer
54 views

Questions about Auctions

I am having a hard time figuring out a problem. In a first price auction with a reserve price R and values of the bidders are U[0,1], how do we find expected revenue given the strategy of both of them ...
2
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1answer
78 views

Deriving statistical distributions from games

The normal distribution can be derived from basic principles and calculus The Normal Distribution: A derivation from basic principles. Are there other distributions that can be derived like this from ...
2
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2answers
89 views

A Special Type Of Guess The Number Game

There is Guess Number game like this: In this game, the player must find a hidden positive number by at most $T$ guesses (or turns). The parameter $T$ together with a health parameter $H$ is ...