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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133 views

Is this Game Theory

A coffee shop wants to bring back a group of customers that are socially connected to each other. It wants to offer them a great discount as an incentive to return. The coffee shop wants to empower an ...
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1answer
39 views

Move of Nature - Ultimatum game

Let $N=\left \{1,2 \right \}$, $A_1=\left \{(x,y): x+y= \omega \right \}$. Player 2, upon seeing a proposal $(x,y)$ can either accept or reject the offer. Player 2 (the responder) has an outside ...
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34 views

Stochastic game matrix

I am trying to solve this exercise and I can't find all Nash Equilibrium. I posted as picture because I couldn't make this matrix!.
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18 views

The variation of Nim game [duplicate]

Can you throw some tips to solve this one variation of Nim game? Two players. The first one throw some stones in heap, the second one throw another natural number in the second heap. Both players ...
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2answers
37 views

The Nash Demand Game

I am having troubles to figure out how to find all pure strategies and two mixed strategies (one in which the shares received by players are always $0$, and one in which $\omega$ is often - but may be ...
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6 views

Game theoretic generalization of decision graphs?

Decision trees can be generalized into game trees (to model interacting decisions amongst 2 or more players) and further into game trees with information sets (to recognize imperfect knowledge of game ...
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29 views

Two person zero-sum game question

$A$ and $B$ have each $\$100000$ in cash and they are equal partners in business worth $\$60000$. They wish to dissolve partnership and they agree that each of them will make a sealed bid between $\$0$...
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48 views

The game of coins. [duplicate]

Two players play the game: There are two bowls, each of which can be fitted by some number of coins. In the beginning the first player puts in the first bowl some natural number of coins of his ...
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44 views

Maximum of minimums

Suppose $v_1,\ldots, v_k \in \mathbb{R}^n$ are vector with all coordinates non-negative. How to explicitly calculate: $$ \max_{x\geqslant 0, ||x||_1=1} \min_{1\leqslant i \leqslant k} <x,v_i>$$ ...
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118 views

Game: two pots with coins

Rules of the game with two players. First player puts any number of coins in the first pot. Then second player, knowing that number, puts any amount of coins in the second pot. Then they in turns (...
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40 views

Winning strategy for a certain graph-drawing game

Given $G$ a finite simple undirected graph and $n \geq 2$ a natural number, let $\mathcal{G}(G,n)$ denote the following game between players $A,B$: Fix a countable set of vertices $\lbrace v_0, v_1,\...
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1answer
26 views

Making sense of weak dominance exercise

I am working on the exercise $4.4$ of Game Theory: an Introduction by Steven Tadelis. The Chapter $4$ is about Beliefs, Best-Response Correspondences and Rationalizability in Game Theory. I am ...
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23 views

How do you seed a “non-binary” tournament?

In a typical tournament, seeding is arranged to provide proportionate advantage to competitors based on their perceived relative ability. (Good teams get to play bad teams, bad teams have to play good ...
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14 views

$N=\{1,2,3,4\}$ abd $v(124)=v(134)=v(234),v(1234)=1$ and $v(3)=0$

Let $N=\{1,2,3,4\}$ abd $v(124)=v(134)=v(234),v(1234)=1$ and $v(3)=0$ otherwise $(a)$ Is $C(v)\neq \phi$ ? $(b)$ Write down a stable set for the game Can anyone please tell how to solve above two ...
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55 views

How to calculate the payoff in Battleship Game Theory

Consider a 3 by 3 board and suppose that Player I hides a destroyer(length 2 squares) vertically or horizontally on this board. Then Player II shoots by calling out squares of the board, one at a time....
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37 views

Finding all mixed Nash equilibria in a $3\times 3$ game

I was looking at the exercise $2$ in this file http://isites.harvard.edu/fs/docs/icb.topic1531493.files/Practice%20Problem%20Solutions%20on%20Nash%20Equilibrium.pdf pages 4 to 7. I do not understand ...
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69 views

Prisoner's Dilemma and Centipede Game - what's wrong with this analysis?

This is an analysis which to me seems trivial, but which I very rarely see brought up in any discussion of games like The Prisoner's Dilemma or The Centipede Game which are well known for having '...
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38 views

Jumping on the Coordinate lattice grid

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show ...
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91 views

What is the optimal strategy when playing `head or tail` per team

Introduction Once a week, we are playing head or tail in my favorite bar. There are $N$ people in the room and each person is guessing whether ...
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313 views

Winning strategy of a game

The following is a game on monomials. Let $M(X,Y)$ denote the set of all monomials in $X$ and $Y$, i.e., $$ M(X,Y)=\{X^aY^b\mid(a,b)\in\mathbb{N}^2\}, $$ where $\mathbb{N}=\{0,1,\dots\}$. ...
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34 views

Pursuers (a game)

From pure entertainment I am interested in the following questions: Let $N$ (particularly $N=1$ case is special) pursuers pursue running at max speed $u$ a runner running at max speed $v$. ...
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30 views

Bayesian formulation of the Shapley Value

I wanted to know if there is a Bayesian formulation of the Shapley value in cooperative Games. I'm not sure if my problem really fits the definition of Bayesian games so here is the problem : For a ...
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1answer
55 views

Does it pay to know what you know?

Let's play a game. I ask you question a yes/no question, and you answer. You don't answer with a yes or no though, you answer with a probability of it being yes ($P \in (0,1)$). For example, I might ...
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52 views

Choosing optimal strategy for real number assignment game

My opponent is dealt a real number $r\in [0, 1]$ uniformly at random and I know that with probability $1-r$ she chooses to discard that $r$ and be dealt a new number in $[0, 1]$ uniformly at random ...
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35 views

How to simplify decision tree for sequential game

I am working on the exercise 2.8 of Game Theory: an Introduction by Steven Tadelis. I thought to solve this exercise by undertaking the steps sequentially, in order to minimize the loss in case of ...
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217 views

Finding p positions with 2 subtraction sets in the take-away game [closed]

Find the set of P-positions for the takeaway game with the subtraction sets: $S = {1,3,5,7}$ $S = {1,2,4,8,16,32}$ Who wins each game when there are 100 tokens on the table to start, the first or the ...
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74 views

Game, stealing edges in a graph.

I was inventing a problem for a math contest, I was really pleased with it, but then I found a mistake in my solution and have not been able to solve it. It is as follows: Alice and Bob play a game. ...
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1answer
71 views

What's the difference between a Nash, Correlated, and Extreme equilibrium?

As the title states, what's the difference? As I understand it: The Nash Equilbirum (NE) is a solution concept in non-cooperative games where no player has incentive to unilaterally deviate from a ...
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1answer
31 views

Find the Nash equilibria

A law is passed requiring a monopolistic soft-drink manufacturer to separate the production department and the marketing department. The marketing department chooses the price $P > 0$ to charge for ...
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1answer
11 views

Find all Nash equilibria in pure strategies

We consider the following public good provision game. There are 2 players, each choosing the amount of money $x_i$ ($i$ denotes 1 or 2) they will give to build a public good. We assume that each ...
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26 views

Find subgame Nash equilibrium

Two players, A and B play the following game. First A must choose IN or OUT. If A chooses OUT, then the game ends, and the payoffs are: A gets 2 and B gets 0. If A chooses IN, then B observes this and ...
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87 views

value of a matrix game

Suppose I have a matrix $$ \begin{bmatrix} 1&4&2\\ 3&2&1 \end{bmatrix} $$ How do I find the minimax value of the matrix? ( It will be considered as a matrix of a matrix game where ...
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77 views

Placing circles inside of a regular polygon.

Alice and Bob play the following game: on a table there is a regular $n$-gon. On each person's turn, they are required to place a circle of radius $r$ fully in the interior of the $n$-gon such that it ...
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1answer
37 views

Nash Equilibria of P-Beauty

I'm a little confused with the work I am currently doing in Game Theory. Here is the questions I'm working on: ...
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22 views

Mixed Strategy Probability Distribution

Problem Two firms with equal capacity constraints k but different marginal costs $c_i$ compete in a pay-as-bid auction for a fixed demand of (balancing) energy. The information on capacity ...
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1answer
24 views

Proving a nash equilibria does not exist

At a certain warehouse, the price of tobacco per pound in dollars, $p$, is related to the supply of tobacco in pounds, $q$, by the formula $p=10−(q/100000)$ Thus the more tobacco farmers bring to ...
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114 views

Game-winning strategy

Player A and Player B are playing a turn-based game. At the beginning of the game there are $N(N \ge 3)$ points in a plane. In each turn one of the players chooses exactly $3$ different points and he ...
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307 views

Factorization game, can we find winning strategy?

I'm thinking about a game theory problem related to factorization. Here it is, Q: two players A and B are playing this factorization game. At very first, we have a natural number $270000=2^4\times 3^...
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84 views

Polynomial game problem: do we have winning strategy for this game?

I'm thinking about some game theory problem. Here it is, Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and "Girl"...
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Can't solve matrix for Nash Equilibrium?

So, I have the following 9 by 9 probability matrix. I want to solve it for a nash equilibrium. https://docs.google.com/spreadsheets/d/16Y1FqxRIAHsHpgEz1ckxDt2sEOInOG3zz_wU8kBHvB4/edit?usp=sharing For ...
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2answers
36 views

Definitional question: difference between a correspondence and a function

Is there a difference between a correspondence and a function? For example, in game theory I am told that for a given strategy set, $\Sigma_i$, the best response given by $BR_i(\sigma_{-i})=\text{...
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1answer
27 views

Check that a Nash equilibrium point is given by $\left(0,\frac {1} {2}, \frac {1} {2}\right)$ $\left(0,\frac {1} {2}, \frac {1} {2}\right)$

Given the game matrix \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 2 \end{bmatrix} I already see a Nash equilibrium in pure strategies, which is $a_{11}$, ...
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1answer
35 views

Iterated Best Response to find Pure Nash Equilibria

The context of this question is Game Theory. I've been trying to apply a simplified (?) version of the Iterated Best Response (IBR) technique to find Pure Nash Equilibria (PNE) in games generated by ...
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1answer
34 views

Game Theory - $3\times3 $Matrix - Mixed Strategy

I am trying to solve the following $3\times4$ game: \begin{array}{c|rrrr} & A & B & C & D \\\hline X & -3 & 5 & 2 & -1 \\ Y & 4 & -1 & 1 & -3 \\ Z & ...
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Card Game Probability, the 15th card

Playing a card game in which 2 52 card decks are combined to create the pile and from this pile each player is dealt 14 cards. One rule of the game is if any player is dealt 3 doubles (a double being ...
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18 views

How to find convergence with a learning rule that depends on the outcome of a game?

my first post here and really excited about the community. In a game theory set in which agents choose from a finite set of actions with a probability distribution, how can I look for convergence ...
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48 views

Find a Mixed-strategy Nash equilibrium in an all-pay auction

This is an all-pay auction (Highest bidder wins the object, all players pay what they bid, player 1 wins all ties): Player 1 has $300$ dollars, Player 2 has $500$ dollars, the object being auctioned ...
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1answer
59 views

Factor Game Calculation

Game: There are two players. The game starts off with the numbers 1 - 30 and player1 chooses a number and gets that number added to their score; however, the sum of all the available factors of that ...
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23 views

Find a Nash Equilibrium point in mixed strategies using the simplex algorithm

Here is the game matrix A. First I note that $C_4<C_1$ and $C_4 <C_3$ So I can remove $C_1 $ and $C_3$ since they are dominated strategies. Then considering only $C_2,C_4$, I note that $R_3$ ...
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1answer
58 views

Math for game theory

I have read a bunch of book on Game Theory, and I find that one of the best is the the book of Osborne and Rubinstein. Nevertheless, all books which I found are not made for mathematician, meaning ...