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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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
40 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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0answers
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 ...
2
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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 ...
3
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2answers
123 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 ...
9
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2answers
335 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^...
7
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1answer
90 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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0answers
36 views

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
39 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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2answers
44 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
35 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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1answer
29 views

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 ...
0
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0answers
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 ...
2
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0answers
49 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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0answers
24 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 ...
6
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1answer
271 views

Game theory, olympiad question

I've seen the following question in the brazilian olympiad for university students, and I couldn't solve it. Thor and Loki play the game: Thor chooses an integer $n_1 \ge 1$ , Loki chooses $n_2 \gt ...
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0answers
66 views

Are there different forms of the Mexican standoff?

The Mexican standoff is a stalemate situation where three mutually-hostile gunmen have each trained their gun on the next gunman. If any gunman shoots first, the remaining gunman is now free to shoot ...
5
votes
1answer
122 views

Construct a game with only pure strategy nash equilibrium.

I'm trying to construct a normal-form game with $2$ players such that the game has exactly $4$ Nash Equilibria From the above properties, I know the game has to be a $4 \times 4$ matrix game, and it ...
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0answers
9 views

Continuity of utility function in normal form games

I want to characterize the utility functions of normal form games. Let $G$ be a game with a finite number of players $k$ given by the action sets $S_1,\ldots,S_k$ and utility function $u:S_1\times \...
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0answers
15 views

Preference relation

Let $A=\{a,b,c\}$ and $\preceq$ is quasi-linear order on $\mathcal{L}(A)$. We also know that $a\prec b\prec c$ and for every lotery $L \notin \{[a],[c]\} $ we have $L\approx [b]$. Is $\preceq$ ...
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1answer
25 views

The core -symmetric players

We have $n$-persons ($n\ge 3$) cooperative game. And we know that player $1$ and $2$ are symmetric. So for each element $(x_1,x_2,...,x_n)$ from the core we have $x_1=x_2$ ? Is that true ? Never seen ...
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0answers
91 views

Game Theory Projects for Undergrads

I am looking for some project ideas for beginning math students in the topic of game theory. I am not very knowledgeable in the topic so it would be great if I could get an good introductory source to ...
2
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1answer
41 views

Win/Lose ratios and selection strategies

Imagine the following scenario: You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...
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1answer
31 views

Security level and equilibrium payoffs in $3-$person zero sum game.

Let the security level ($p-$payoff, $M-$set of all strategies) $$B_i:=\sup_{\sigma_i \in M_i} \inf_{\sigma_{-i}\in M_{-i}}p_i(\sigma_{-i},\sigma_i)$$ Now I consider $3-$person zero sum game. The ...
0
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2answers
43 views

When solving linear equations what does ${0x_n = 0}$ mean? What if the system is used to find Nash equilibrium?

When solving systems of linear equations one sometimes gets result like ${0x_n = 0}$ what does it mean for solving the system? Is it error on part of the solver or just feature of the assignment? ...
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0answers
26 views

How are lottery winnings calculated?

I'm pretty familiar how most chance games payouts are calculated - the ratio shoul be inversely proportional to the probability of winning, minus house edge. If we bet the same amount on the same game,...
0
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0answers
20 views

Summation Equivalence In Game Theory

Let $N =$ {$1, ... ,n$}, $i \in N$, $A_i \subset N$ such that $i \notin A_i$, $x \in \mathbb R^n$ such that $x = (x_1, ... , x_n)$ and $\sum_{i=1}^n x_i = 1$ Knowing that $\sum_{j≠i} x_j ≥ \sum_{j ...
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1answer
43 views

Take away games

Takeaway Game Consider the takeaway game with the subtraction set $S = {1,4,5}$. Assuming there are two players and Player 1 moves first, if there are 87 tokens on the table, who wins with smart ...
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1answer
265 views

References on a game with white and black stones

I'm looking for references on this game (name, strategies analysis, ...) : It's a two player game with two players (Black and White) A position of the game is a single line (sequence) of black and ...
0
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1answer
24 views

Let $f(x,y) = y^2-x^2, C=D=[-1,1]$

Part a) Find $v^{+}=\min_{y \in D} \max_{x \in C}$ $f(x,y)$ and $v^{-}=\max_{x \in C}\min_{y \in D} $ $f(x,y)$ I'm sure I got this: $v^{+} = (-1)^2 - 1^2 = 0$ and $v^{-} = 1^2-(-1)^2 = 0$ Part b ...
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0answers
51 views

Proof of existence of optimal strategy in $2\times 2$ zero-sum game.

I am trying to solve the following task and don't know where to start from: Given that in a $2\times 2$ matrix zero-sum game the first player has optimal pure strategy, prove that the second ...
2
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1answer
33 views

Switching balls among 3 piles

There are 3 piles of balls. Each hour, I take a ball from one pile and move it to another. The amount of points I earn from this move is the amount of balls in the pile I took the ball from minus the ...
0
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1answer
21 views

How to solve mixed strategy Nash equilibrium.

Lets say I have following problem: Zero sum game. Payoff matrix for player one: -1 4 4 2 6 -2 I start by writing equations for each strategy. ...
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0answers
31 views

Dealing with an infinitely repeated game

I have been playing around with problems related to game theory, and I ran into this issue related to an infinitely repeated game. Consider this game repeated an infinite number of times: $$\begin{...
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0answers
36 views

Game theory and sharing

What sort of mathematical knowledge is required for applying game theory specifically in relation to evaluating fairness in a shared resource environment? One of the things I'd like to explore in my ...
4
votes
0answers
59 views

A Game Between a Panda and a Polar Bear

I've been working on some problems related to Bayesian games, and I reached this dynamic game that I have been having some problems with. Consider a game where a polar bear and panda bear are choosing ...
3
votes
1answer
39 views

Optimal Strategies in a Quantum Game

I've been playing around with problems involved in introductory quantum game theory, but I am having problems figuring out strategies in this one game. For background, consider the 2x2 Pauli spin ...
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0answers
27 views

A Simultaneous Game Played $N$ Times

I'm working on problems related to repeated games in my game theory course, and I came across a problem related to finitely repeated games. Consider the two-player simultaneous game $$\begin{array}{|c|...
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1answer
148 views

How to interpret negative probability for a strategy in mixed nash equilibrium?

I am trying to get the mixed strategy in Nash equilibrium for the following matrix. $$\begin{pmatrix} 0 & 3 & 4 & 5 & 6 \\ 3 & 0 & 5 & 6 & 7 \\ 4 & 5 & 0 &...
0
votes
1answer
49 views

Meaning of cost allocation in a coalition

I want to know about the meaning of cost allocation in a coalition. I know we have some solutions for this(Shapley or Nucleolus value). Consider following interpretation from cost allocation: ...
2
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0answers
26 views

Are there any types of tournaments that allow for absences?

I'm trying to organize an online tournament with about 50 people that will span across 1 or 2 months, and inevitably some people won't be able to play their match every week. Is there a tournament ...
10
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3answers
144 views

Optimal Strategy for this schoolyard game - (Charge, block, shoot)

I encountered this game when I was a kid (we called it Street Fighter back when it was all the rage) and recently saw it again with my nephews playing the same game with a different name and slightly ...
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0answers
33 views

A Dual Matching Market In the Roommate Problem

I am working on developing some new mechanism related to the Roommate Problem, which is a problem where given a set of $n$ agents, each agent can establish preferences on the other $n-1$ agents in the ...
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1answer
23 views

Re-implementing Matching Pennies

I'm a little confused on a problem from my game theory course. I am reviewing the standard ``matching pennies'' game where player $1$ wins $1$ and player $2$ loses $1$ if the their two pennies match ...
5
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1answer
72 views

How can I find the Nash-equilibrium of the following zero sum game?

I want to find the Nash-equilibrium of the following zero sum game. $$A=\begin{bmatrix}0&2&-1\\-2&0&3\\1&-3&0\end{bmatrix}$$ I used the Minimax Theorem. $$min_{x \in X} max_{...
2
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1answer
37 views

Strategies on a symmetric chess play

The idea is that the difficulty of the game of chess is derived primarily from the asymmetry between the king and queen. all other chess pieces are arranged symmetrically and can move symmetrically, ...
0
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1answer
67 views

Paradox of Random Natural Numbers

I've got a question about a game taken from a book called Rachunek prawdopodobieństwa dla (prawie) każdego by Jacek Jakubowski and Rafał Sztencel. Adam and Bolek have a machine that generates a pair ...