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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2
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2answers
41 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 ...
1
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0answers
117 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
104 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
195 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
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1answer
35 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 ...
3
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2answers
95 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 ...
0
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0answers
11 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
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1answer
35 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) http://www.nushigh.edu.sg/qql/slot/u90/file/simc/...
0
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2answers
29 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 ...
2
votes
1answer
38 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 ...
0
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1answer
28 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
130 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 ...
2
votes
1answer
156 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
43 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
vote
1answer
40 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 $$...
3
votes
1answer
67 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 ...
1
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0answers
115 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 ...
8
votes
4answers
116 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
16 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 ...
3
votes
1answer
24 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 ...
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0answers
40 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 ...
6
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0answers
88 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: ...
1
vote
1answer
29 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
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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
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1answer
27 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)
0
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0answers
35 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
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0answers
31 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
18 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 ...
0
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0answers
52 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: https://www.youtube.com/...
1
vote
1answer
22 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 $...
0
votes
1answer
45 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 ...
5
votes
1answer
176 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
1answer
30 views

Second price auction when number of items and bidders are the same

For an advertising slot bidding problem, let's say there are two slots and two bidders. If bidder A bids \$10 and bidder B bids \$8, bidder A will win the first slot and pays \$8. How about bidder B? ...
6
votes
1answer
241 views

Coordination game

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

Game theory probability

I have a variant of two-finger morra game, where the winner is determined by the parity of the sum of the two numbers thrown, but the amount won or lost is the product of the two numbers. There are ...
1
vote
1answer
27 views

How to compute a mixed Nash Equilibrium where only one payoff is given.

Let's say I have this: $$ \begin{matrix} & A & B \\ X & 1 & 2 \\ Y & 2 & 1 \\ \end{matrix} $$ That is the payoff for me if I make ...
5
votes
3answers
44 views

Game is winnable if and only if $n \neq k$

Integers $n$ and $k$ are given, with $n \ge k \ge 2$. You play the following game against an evil wizard. The wizard has $2n$ cards; for each $i = 1, \ldots, n$, there are two cards labelled $i$. ...
0
votes
0answers
13 views

Confidence interval / incertainty for Shapley Value

I want to know if there is a way to compute a confidence interval (or some measure of incertainty) for Shapley values in a cooperative game. Since the calculation of the Shapley value requires ...
0
votes
1answer
20 views

Is the Shapely value of this voting game in the core?

Given a voting game where $v(1),v(2),v(3) = 0, v(1,2)= \frac{1} {3}, v(2,3) = \frac{5} {6}$, $v(1,3)= \frac{1} {6}$ and $v(1,2,3) = 1$ I know the Shapely coefficients for a 3 player game, for $|s|=1,...
0
votes
1answer
33 views

Prove that the core of this game is empty

A game (N, v) is simple if for every coalition S is a proper subset of N, either v(S) = 0 or v(S) = 1, In a simple game, a player, i, is said to be a veto player, if v(N \ {i}) = 0. Suppose (N,v) is ...
2
votes
1answer
34 views

Find Bayesian Nash Equilibria

A rich, honest, but mischievous father told his two sons that he had placed $10^n$ and $10^{n-1}$ in two envelopes respectively, where n ∈ $\{1,2,3,\ldots,10\}$. The father then randomly handed each ...
0
votes
0answers
9 views

Predicting Nash equilibrium after one player enters or leaves

Suppose I have a game with $N$ players, and that the Nash equilibrium can be calculated. If one player enters or leaves the game, is it possible to predict or quickly calculate the resulting Nash ...
0
votes
1answer
30 views

Problem with proof of the upper hemicontinuity of correspondence

I have a problem with a proof I found here of the upper hemicontinuity of the best-reply correspondence in the Nash Theorem. Below there is the proof, and here my problems: Problems: Is here ...
0
votes
1answer
45 views

2 finger morra game probability

Consider the 2-finger. Bob pays Alice $\$(a + b)$ if $a + b$ is even Alice pays Bob $\$(a + b)$ if $a + b$ is odd Suppose Alice plays $one\; finger$ with probability $\frac 12$ and $two\; fingers$ ...
1
vote
1answer
51 views

calculate price based on demands and maximize revenue

I believe I have a simple question which I am struggling to answer. It is as follows: We have 400 items, each item costs £100. Retailer bought these items before the season started. The forecasted ...
5
votes
1answer
127 views

Solving the Gobblet game

In 1995 the Connect-4 Game was solved with a brute force approach. Using the standard 6 high / 7 wide grid, first player can force a win in 41 moves. Complexity of the Connect-4 game could be ...
0
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0answers
30 views

Software or code to solve a congestion game for n players

First of all, I am pretty new to game theory so if I say something wrong, please correct me. I have a congestion game (similar to the bookcase of machine job scheduling problem). The jobs are the ...
2
votes
0answers
23 views

Solve a hypothetical mass-level extinction scenario [closed]

Summary: A single, immortal alien life form that kills by touch seeks to eradicate all life on Earth as efficiently as possible. After killing a victim, he is able to sense the next human nearest to ...
0
votes
0answers
25 views

Mixed strategy, find an equilibrium pair

In this example I need to find equilibrium pair X = ($x_1$, $x_2$, $x_3$), Y = ($y_1$, $y_2$, $y_3$) The matrix looks like this $[0, -1, 2]$ $[3, 1, 0]$ $[-2, 2, 1]$ P(X, $B_1$) = 3$...