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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5
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1answer
45 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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0answers
20 views

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 ...
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0answers
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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3answers
12k views

How is prisoner's dilemma different from chicken?

Chicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash. So ...
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1answer
56 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 ...
1
vote
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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0answers
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 $\preceq$ ...
2
votes
1answer
520 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? ...
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0answers
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
votes
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 ...
1
vote
1answer
80 views

Is the expected utility function linear?

Given the definition of the mixed extension of a finite game as in the link below (only first 7 lines): How to find perfect equilibria in a finite game? We define the expected utility function in the ...
0
votes
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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5answers
6k views

Deal or No Deal: Monty Hall?

This question was inspired by another question posted today: Monty Hall Problem Extended. So I thought that the comments an answers brought up a great point about increasing the doors to 100 or ...
1
vote
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 ...
0
votes
1answer
473 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 ...
0
votes
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
votes
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 ...
4
votes
1answer
618 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 ...
0
votes
0answers
17 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 ...
1
vote
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 ...
2
votes
2answers
912 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 ...
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votes
1answer
203 views

Calculating the value of a bi-matrix game.

So I know this question is very simple, however in my text and from what I can find online, the solution tends to simply be given (such as in this example) Example: Let the following bimatrix game ...
3
votes
1answer
1k 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 ...
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0answers
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 ...
0
votes
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. ...
2
votes
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 ...
1
vote
1answer
765 views

Osgood Box (Doctor Who) Dilemma

Today's episode of Doctor Who featured an interesting dilemma, centered around an object called the "Osgood Box". I'm not well-versed enough in game theory to recognize it as an existing problem, but ...
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2answers
577 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 ...
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vote
1answer
496 views

Optimal strategy for dominoes game

Here is the basic principle of the game I'm trying to find an optimal strategy for: Two players (say P and Q) are given a 2x3 grid and a domino. Then P chooses one way of positioning the domino on ...
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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: ...
1
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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 ...
28
votes
7answers
13k views

Game theory - self study

I want to self study game theory. Which math-related qualifications should I have? And can you recommend any books? Where do I have to begin?
6
votes
1answer
497 views

Bounded sequence with divergent Cesaro means

Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)? More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that ...
4
votes
1answer
155 views

Can't EF game theory be applied to finite languages WITH function symbols?

Let $\mathcal{M}$ and $\mathcal{N}$ be two structures in a language $\mathcal{L}$. We define the finite determined game $G_n(\mathcal{M},\mathcal{N})$ as a game with $n$ rounds where in each round ...
2
votes
1answer
592 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 ...
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4answers
4k views

Game Theory - Unsure how to proceed with this question

A company has a competition to win a car. Each contestant needs to pick a positive integer. If there’s at least one unique choice, the person who made the smallest unique choice wins the car. If there ...
4
votes
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 ...
4
votes
1answer
1k views

The Notion Of Degenerate Two Player Game

I try to get the intuitive understanding of the notion "degenerate two player game". Definition. A two-player game is called non degenerate if no mixed strategy of support size $k$ has more than $k$ ...
2
votes
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 ...
0
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0answers
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 ...
10
votes
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 ...
1
vote
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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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.
0
votes
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: ...
1
vote
1answer
22 views

Distribution of coalition cost among coalition members (game theory) on the basis of contribution in coalition

Does any one know or point out the method or technique used for the distribution of the coalition cost among the coalition members depending upon their contributions in the coalition. In other words, ...
2
votes
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 ...
0
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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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vote
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
votes
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} ...
1
vote
1answer
27 views

Disprove that the given strategy pair is a solution to the game.

Problem: For the following matrix game, prove or disprove that the given strategy pair is a solution to the game. \begin{align} A &= \begin{bmatrix} -1 & 2 & -3 \\ 3 & -4 & ...