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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The Shapley value and the core [on hold]

I had one task on exam, which confused me, can you give me some ideas ? The task was: We know that $(1,1,1,1,1)$ belong to the core. What can we tell about the Shapley value? I think the only thing we ...
5
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1answer
48 views

Construct a game with only pure strategy nash equilibrium.

I'm trying to construct a normal-form game with $2$ players such that it satisfies the following three properties: $1)$ Each player has exactly $4$ strategies. $2)$ The game has exactly $4$ Nash ...
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4 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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12 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
17 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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26 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
25 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
17 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 ...
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2answers
42 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? ...
0
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0answers
19 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 ...
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18 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
22 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
57 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
23 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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42 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
28 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
17 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
21 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: ...
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0answers
32 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
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0answers
43 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 ...
2
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0answers
23 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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21 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 ...
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0answers
15 views

Finding Bayesian Nash equilibria in specific centipede game

Please, could you help me with following game of incomplete information ? I have never solved game in this form. I can only solve the basic centipede game.
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1answer
78 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
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1answer
29 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: ...
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0answers
22 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
131 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
21 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
22 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 ...
4
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1answer
59 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} ...
2
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1answer
33 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, ...
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1answer
62 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 ...
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1answer
36 views

Mixed Strategy Equilibrium of a Game

I am having some problems with an exercise that showed up in my game theory course. Consider the two player game where each player bids a non-negative integer multiple of five cents. The highest ...
2
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1answer
302 views

What is the probability that the loser has won exactly $k$ games when the match is over?

Adam and Eve play a series of games of tennis, stopping as soon as one as them has won $n$ games. Suppose that they are evenly matched and that Adam wins each game with probability $1/2$, ...
2
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0answers
22 views

A Generalized Mechanism for Gale-Shapley

I am working on some problems in my applied graph theory course, and we have just gotten to matching problems. We are currently working on a graph problem where instead of there being two types of ...
2
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3answers
40 views

Terminology for a game in which Black and White have the same “probability” to win

Consider a game between two players, Black and White. The game is sequential and ends after finitely many moves. White moves first. The game ends either in the victory of one of the players or in a ...
0
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1answer
25 views

Calculations for grid based games

For a grid game to match items in chains of 3 or more, how can a difficulty be calculated? I have a number of moves, and a grid with a cell count and an element type count n, e.g. 6 different ...
0
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2answers
32 views

Does a game need below-average players

I am no mathematician, just a programmer and gamer who thought about this problem. I reckoned it's more relevant here than one of the gaming-SEs, as it's not really about any particular game, but ...
0
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1answer
46 views

Who will win the game?

There are two players $A$ and $B$. There are two bags with $n_1$ and $n_2$ things in it. $A$ will start the game and can take out $x$ where $1\leq x \leq \min(n_1,n_2)$ number of things from either ...
1
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1answer
18 views

SPNE of a finitely repeated game

So I have this little game between two players, played T times without any discouting factor with the following payoff table: now if T is 2, is there a SPNE where (B,L) is being played first round? ...
2
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1answer
63 views

A game involving a Poisson process

Let T > 1. We observe a Poisson process of rate 1 on the time interval (0, T ). Each time a point occurs, we may decide to stop. Our goal is to stop at the last point which occurs before time T; if ...
3
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2answers
53 views

Nash equilibrium: comparing different definitions

A Nash equilibrium seems to be defined in different ways in different books. Sometimes a Nash equilibrium refers to a single strategy (Definition $1$ below) and sometimes a Nash equilibrium is defined ...
2
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0answers
120 views

How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which ...
3
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2answers
70 views

What is the axiom of quasideterminacy?

This is something mentioned in the "See also" of the wikipedia page for the axiom of determinacy, but when you click on it it takes you to the page for "Determinacy" and the section for ...
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1answer
21 views

Verifying Correlated Equilibria for a game of Chicken

I'm reading Computing Correlated Equilibria in Multi-Player Games (C.H. Papadimitriou, T. Roughgarden). Consider the following extracts While the following extract is from here, it is ...
2
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2answers
78 views

First mover advantage in a Stackelberg game

I am considering a simple game with two firms. Each firm faces the following demand function \begin{equation*} q_i(p_i,p_j)= a- b p_i + cp_j, \end{equation*} where $i,j\in \{1,2 \}$ and $i\neq j.$ ...
2
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1answer
46 views

7 hats, 6 prisoners

I came across this variant of the prisoner-hat problem the other day and couldn't seem to muster a proper solution: $6$ prisoners are on an island and are each assigned a hat numbered $1$ through ...
2
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1answer
72 views

Game theory: Finding Nash equilibrium in $3\times 3\times 3$ matrices

I tried to find how to solve $3\times 3\times 3$ matrix to find Nash equilibrium but I could not find anything on the web. Maybe I am searching with wrong keywords... I understand how to solve Nash ...
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0answers
16 views

Difference between Sequential and Weak Sequential Equilbria

This is in reference to the Game theoretic concepts as Nash equilibrium refinements. Sequential equilibrium are often defined as satisfying two conditions: consistency and sequential rationality. ...
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1answer
25 views

Recursive utilities in a repeated game

I am trying to set up utilities for an infinitely repeated game and I am having some trouble figuring out how to write the correct functional form. This game has a stochastic component where a ...