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].

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6
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4answers
69 views

Guess the number despite false answer

This is the Guess-The-Number game with a twist! Variant 1 Take any positive integer $n$. The game-master chooses an $n$-bit integer $x$. The player makes queries one by one, each of the ...
1
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0answers
9 views

While finding an optimal strategy for a mixed nash equilibrium, why do we not consider strategies which are never a best response?

"A strategy cannot be plausibly chosen by a rational player if and only if it is never a best response." I understand the logic behind neglecting the strategies that are strictly dominated. But why ...
6
votes
1answer
235 views

Coordination game

Consider the following game in normal form: $$\begin{bmatrix} & x_B=0& x_B=1 \\ x_A=0& 1-\theta_A,1-\theta_B & 0,0 \\ x_A=1 & 0,0 & 1+ \theta_A,1+\theta_B \end{bmatrix}$$ ...
0
votes
0answers
56 views

Probabilities in this blackjack variation

Let's say I play blackjack (52 cards, figures count for 10, aces count for 1 or 11) and alone (no dealer). The cards I use for one particular game are always removed at the end of that game and won't ...
1
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0answers
74 views

A variation of Nim game

There are two players X and Y . They write N integers on paper ( A_1 , A_2 , A_3 , .... A_N ). They have also p integers (b_1 , b_2 , b_3 , .... b_p ) . Now , Player X always takes turn first . He ...
3
votes
0answers
77 views

Mathematical game with numbers

We invented a mathematical game, which i am going to explain here. The first player choose a natural number, lets call it $n$ (if you play it for real, you must choose a sufficiently big number so ...
3
votes
0answers
142 views

Alice and Bob make all numbers to zero game

Alice and Bob are playing a number game in which they write $N$ positive integers. Then the players take turns, Alice took first turn. In a turn : A player selects one of the integers, divides it ...
0
votes
1answer
31 views

Three games of two-players each being played by three players simultaneously

Has the game theory literature considered situations wherein there are three two-player games being played by three players concurrently with each other; and the outcomes of those games may impact the ...
0
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1answer
28 views

Strategy optimisation

This is a question from the Singapore Invitational Mathematics Challenge 2016. The question paper can be found here. (Part C:Question 2) ...
0
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1answer
769 views

I can't find the Nash equilibrium of this 3x2 game.

Sorry for my English, I am French but i couldn't find help on the French website (so I am here). I have a question about this two-player game: $$ \begin{array}{c|cc} & y_1 & y_2 \\ \hline ...
3
votes
2answers
77 views

How can the policeman catch the gangster?

I try to solve the following problem (Moscow Mathematical Olympiad, 1978) There is a town with six streets: four sides of a square and two its middle lines. Policeman tries to catch a gangster. If ...
2
votes
1answer
606 views

Reduce the payoff matrix using (weakly) dominated strategies

Below is the payoff matrix of a game. Use the principle of elimination of (weakly) dominated strategies to simplify the payoff matrix. What is the optimal solution of the game for the row player? ...
1
vote
1answer
92 views

Game of chocolates

Two players $A$ and $B$ play a game alternatively and $A$ starts the game. Their are $2$ boxes of chocolates and we are Given the number of chocolates in both the boxes, let them be $c_1$ and $c_2$, ...
0
votes
0answers
9 views

Looking for info on game theory for a scenario where participants earn points, and the top K earners receive a reward determined by rank

Say there's some competition that lasts for a week and takes place in a community. Participants receive points for collecting littered cans in the streets. Each can collected is a point. At the end ...
0
votes
1answer
16 views

Finding extreme solutions to zero-sum games

For the matrix B above, I'm not able to understand how they have extended to solution (1,0) to (0,1,0). I understand why this extension is necessary ( because A is a 2 by 3 matrix and so y must have ...
3
votes
2answers
540 views

Game involving tiling a 1 by n board with 1 x 2 tiles?

Consider a $1$ by $n$ tiled rectangle. You want to play a game with one opponent in which you place $1$ by $2$ "dominoes" on this rectangle. The player who places the last domino wins. Which player ...
0
votes
1answer
26 views

Enunciating utility maximization problem using set theory

I need to enunciate a problem using set theory and I am not sure how to start. The problem goes like this: You are a car manufacturer and need to decide how many colours to use in your next bash of ...
4
votes
2answers
105 views

Variation of Nim, where one has to divide a pile into any number of piles.

I am learning the basics of combinatorial game theory (impartial games). After learning about decompose a game into the sum of games, I feel comfortable with games that can divided into the sum of 1 ...
0
votes
1answer
218 views

Gamblers ruin formula

Hello , I have been reading about gamblers ruin and I found this formula can anyone confirm its accuracy ? I assume they only bet one chip a time
3
votes
1answer
55 views

Tic Tac Toe: What is the probability that a random player draws against an infallible player?

I have simulated a tournament between an infallible Tic Tac Toe player and one that chooses its moves randomly. Even after 5 million games, the infallible player wins every single game. I know that ...
0
votes
2answers
26 views

Dice role: What is the probability to observe 2 times 1 and 2 times 5 with the outcome of a fifth die role being unknown?

I tried to solve the following exercise: Given a dice with $P(X=2) = P(X=4) = P(X=5) = \frac{2}{15}$ and $P(X=1) = P(X=6) = P(X=3) = \frac{2}{10}$. What is the probability to observe 2 times 1 and 2 ...
0
votes
0answers
15 views

A bin-assignment infinite 2-player zero-sum game

What is known about the following infinite 2-player zero-sum game? There are $k$ bins. Each player has 1 unit of mass and, simultaneously, divides it arbitrarily among the $k$ bins. The player wins ...
2
votes
1answer
140 views

He who has the largest real number in $[0,1]$ wins

Let's play a game: Let $X,Y \sim U (0,1)$ be random variables uniformly distributed over $[0,1]$. The game is as follows: I obtain a realization of $X$. You obtain a realization of ...
0
votes
1answer
38 views

Stackelberg problem?

Suppose that two firms have different production costs: Player I's cost of producing x is x+2, while Player II's cost to produce y is 3y+1. Suppose that the price function is p(x,y)=17−x−y, where x ...
1
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1answer
35 views

Bertrand Duopoly

Consider the following version of the Bertrand model with differentiated products. Specifically, if player I sets price $p_1$ and player II sets price $p_2$ for goods, then the demand is given by ...
1
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0answers
42 views

Position games: how to fill a matrix with dominos? [duplicate]

Dominos of size $2 × 1$ can be placed on a $m × n$ board so as to cover two squares exactly. Two players alternate placing dominos. The first one who is unable to place a domino is the loser. I can ...
3
votes
2answers
1k views

Splitting the dollar Nash equilibrium

I'm working on a game theory problem I can't seem to figure out. Players 1 and 2 are bargaining over how to split $\$10$. Each player names an amount $s_i$, between 0 and 10 for herself. These ...
6
votes
0answers
69 views

Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
3
votes
1answer
23 views

Two person cooperative non-zero sum game

This a two person cooperative non-zero sum game. The shaded area is the negotiation set. $s_a$ and $s_b$ are the security levels for $A$ and $B$ respectively. I do not understand the part I have ...
0
votes
1answer
14 views

Upper and Lower Value of a two person zero sum game

I understand that if a game's lower value V$_{L}$ is equal to it's upper value V$_{U}$ then the game has a value V$=$V$_{U}$=V$_L$. Just to be sure it is also the case that if a game has a value V ...
0
votes
1answer
16 views

Explanation of Nash Equilibrium using Gambit software

I am trying to understand how Nash Equilibrium works in the Gambit software but I can't figure it out. I have created a simple game shown below and I have calculated just one Nash equilibrium by ...
1
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0answers
34 views

Big Two's classification according to game theory

Is the game Big Two, as described in https://www.pagat.com/climbing/bigtwo.html, classified as: a game of perfect or imperfect information? deterministic or stochastic? EDIT: I am fairly certain ...
1
vote
1answer
27 views

Representation of a game with simultaneous movements

In Game Theory, can you use a tree structure to represent a game with simultaneous movements or you have to use a matrix form? In a sequential game it is logical to use a tree, as every node ...
1
vote
1answer
28 views

Randomised strategies

I don't understand what is meant my assigning probabilities to randomised strategies. Randomised strategies are themselves probability distributions over the pure strategies.
0
votes
1answer
25 views

Value of Zero sum game

In part iii) I am unsure as to why we subtract 1 from the value of the game (underlined in green)
4
votes
1answer
642 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 ...
9
votes
3answers
856 views

Optimal Strategy for Rock Paper Scissors with different rewards

Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward. If you win with Rock, you get \$9. Your opponent loses the \$9. If you win with Paper, you get \$3. ...
1
vote
1answer
698 views

Relationship between regular Nim and Lasker's Nim

So I'm trying to do qn $6$ (on pg I-13) about staircase Nim in Game Theory by Ferguson Game Theory, Ferguson and it's asking to prove that $(x_1, x_2, \ldots, x_k) \in P $ only if $(x_1, x_3, x_5, ...
5
votes
1answer
165 views

An envelope game

Each of two individuals receive a ticket on which there is an integer from one to five indicating an amount of money he may receive. The individuals' tickets are assigned randomly and independently. ...
0
votes
0answers
28 views

Visualizing Nash Equilibria of a 4 dimensional matrix

Are there any good ways to visualize Nash equilibria of a 4-d matrix? I have created an game theory model which consists of of four players (P1; P2; P3; P4) who can all choose between a set of 27 ...
1
vote
0answers
28 views

Set of equilibrium points generically finite and odd

Let $\Sigma$ denote a finite product of unit simplices and $E:\mathbb{R}^k \rightrightarrows \Sigma$ an upper-hemicontinuous and compact valued correspondence with graph $\Gamma$. By $\pi: \Gamma \to ...
0
votes
0answers
17 views

How you can model the other players replies in a game theoretic model?

In a game theory field, the payoff function of a player n is basically a function of the other players responses which are considered as constants. I'm trying to solve the maximization of the payoff ...
1
vote
2answers
771 views

Nash equilibrium in first price auction

I'm trying to understand Exercise 18.2 from Martin J. Osborne and Ariel Rubinstein A Course in Game Theory about finding pure Nash equilibria in a first-price auction. There are $n$ players, named ...
0
votes
0answers
43 views

An Interesting Variation to the “Pebbling a Checkerboard” Puzzle

Pebbling a Checkerboard (or chess board) was a puzzle proposed by Maxim Kontsevich in 1985, which was very interesting and fun to try, and you can find a great video on it at: ...
1
vote
1answer
12 views

Average discounted payoffs after T periods formula.

I am being told that; The average discounted payoffs after T periods is given by; $$\pi_i = \frac{1 - \delta }{1 - \delta^T}\sum_{t=0}^{T-1} \delta^tg_i(a^t) $$ $\delta$ is the discount rate ...
-2
votes
1answer
90 views

Puzzle: Players $A,B,C,D$ are in a line

Players $A,B,C,D$ stands in a line. Players $A, D$ do not move. round $1:$ player $B$ moves one distance closer to the midpoint of $A$ and $C$ round $2:$ player $C$ moves one distance closer to ...
4
votes
1answer
2k views

Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question: Is ...
4
votes
2answers
3k views

Does chess have more Nash equilibria than you can find through backwards induction?

All equilibria found with backwards induction on a tree of a perfect information game are Nash equilibria, but in general the reverse is not true: ...
0
votes
1answer
40 views

Does the first player have a winning strategy?

Two players play a game where they alternatively cross out a number from the numbers written on the board ($1-21$). They stop when two numbers are remaining. If thie sum of these two numbers is ...
3
votes
1answer
609 views

Game Theory. Repeated Games. Strategy set.

I'm reading the book "Strategic games" by Krzysztof R. Apt. I have a question about the strategies in Prisoner Dilemma repeated game. On page 63 there is expression: "In the first round each player ...