The study of competitive and non-competitive games, equilibrium concepts such as Nash equilibrium, and related subjects. Combinatorial games such as Nim are under (combinatorial-game-theory), and algorithmic aspects (e.g. auctions) are under (algorithmic-game-theory).

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if $G$ has no Nash equilibrium in pure strategies then $G$ has single Nash equilibrium in mixed strategies.

Let $G=(S,T,\pi _1 ,\pi_2)$ be a 2 player game with strategies $T$ for player 1 and $S$ for player 2 such that $|T|=|S|=2$, and payoff functions $\pi _1 ,\pi_2$. prove that if $G$ has no Nash ...
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57 views

Game to maintain distinct number of balls in glasses

There are $n$ glasses, containing $n+1,n+2,\ldots,2n$ balls, respectively. Two players $A$ and $B$ play a game, alternately taking turns with $A$ going first. In each move, the player must choose some ...
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26 views

Supply and demand law from game theory

I am trying to retrive the law of supply and demand from game theory. I don't understand the result. Suppose we have a probability $p$ to sell a good at price $q$. I can calculate $p$ as the fraction ...
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44 views

A combinatorial game theory problem

In details, Let, there are four bishops on a chessboard where every two bishops are in pair ( as there are 4 bishops that means 2 pairs and in each pair they sit in vicinal squares). How many ...
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59 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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2answers
24 views

Mixed strategies in 3x3 game - can strategies be negative?

Heres the payoff for player one. I'm searching for mixed strategies of player two. However I do the algebra, i get:p=4/5, q=3/10 and z=1-p-q=-0,1. Could anybody please explain, how negative ...
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1answer
26 views

Expected earning when Player B randomly guesses a number player A picked

(Introduction to Probability, Blitzstein and Nwang) Player A chooses a random integer between 1 and 100, with probability pj of choosing j (for j = 1, 2, . . . , 100). Player B guesses the ...
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42 views

Second price auction, page 82-84 of Osborne's An Introduction to Game Theory

Consider the second price auction defined and discussed on pages 82-84 of Osborne's An Introduction to Game Theory $($pages 80-82 here in this online draft version of the textbook: Martin J. Osborne, ...
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27 views

Solving genereal zero sum games

Suppose I have this payoff matrix for a zero sum game \begin{array}{ccc} 8 & 3 & 4 & 1 \\ 4 & 7 & 1 & 6 \\ 0 & 3 & 8 & 5 \end{array} Since it has no saddle point ...
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16 views

Doubt regarding optimizing leader's problem in Stackelberg model

Suppose there is a leader with net profit function $EC(K_{1},v_{2},A_{1},A)$ where $K_{1},v_{2},A_{1}$ are decision variables of leader. The follower's problem is given as follows: ...
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24 views

Modified parcheesi game

A "modified Parcheesi" game starts with the following position: First $x$ flips a fair coin. If heads he can move two spaces or pass. If tails he can move one space or pass. If he occupies the ...
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29 views

Pure and mixed strategy in Nash Equilibria with n player

I got confused when I see the following problem: There are n staffs and they want to raise their salary, if any one or more than one of these staffs suggest their boss they want raise salary, all ...
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64 views

Towers game strategy

Given the following game, what is the strategy to win? Given $N$ towers of different heights. Two players play against each other. Each player (in his turn) divides each of the towers which are ...
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1answer
23 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 ...
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1answer
84 views

What is the pure strategy Nash Equilibria of asking your professor to cancel class?

Each student in a class has the option to remain silent or ask the professor to cancel class. If any students asks to cancel class, all students get a payoff of $r$. However, the student that asks ...
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24 views

game theory atomic selfish routing

An asymmetric scheduling instance di ers from an atomic selfi sh routing instance in the following two respects. First, the underlying network is restricted to a common source vertex s, a common sink ...
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2answers
41 views

What exactly is a strategy stealing game and is it bad?

Some time ago, I asked myself if infinite gomoku is a first player win, which seems not proven yet, and while searching for an answer I often heard the term "strategy stealing game". I just thought ...
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1answer
53 views

Can a symmetric equilibrium yield superior social welfare in a symmetric game?

Consider a 2-player symmetric game given by a payoff matrix $A\in [0,1]^{n,n}$ for the row player (i.e. the column player matrix is $A^t$). Define the social welfare as the sum of payoffs for both ...
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74 views

Game between Alice and Bob involving extremal numbers

Alice generates $4$ numbers in $(0,1)$ independently and uniformly at random. She discloses one of the numbers to Bob, who is requested to guess whether the disclosed number is extremal (i.e. the ...
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122 views

What does $\overline{z}\mathbb{1}$ and $\underline{z}\mathbb{1}$ mean?

I'm working on some paper concerning auction analysis. I have trouble with understanding what is the meaning of symbols: $\overline{z}\mathbb{1}$ and $\underline{z}\mathbb{1}$ Do you have any ...
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1answer
29 views

Two methods for the Nash equilibrium give different answers; which is correct?

Suppose we have a game, played in which Alice and Bob play mixed strategies: (Sorry about the spacing, but I don't know how to put a table or tab spacing in this text box.) Alice plays Dove with ...
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What is the most effective strategy to win at this game?

The game is as follows. Alice secretly selects three real numbers $a_{1},a_{2},a_3$ such that $1\geq a_1\geq a_2\geq a_3\geq 0$ and $a_1+a_2+a_3=1$. Bob secretly selects three real numbers ...
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21 views

Deciphering game formula

Hello I'm trying to find a Formula of a certain system(Game) and would like some help. I will try not to get into the context of the game too much, but some times it will be necessary for better ...
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1answer
47 views

Multiplying game strategy

Given the following game, what is the strategy to win? Given $X,N\in \mathbb{N}$ such that $N>X$ and $N>1000$, two players play against each other. Each player multiply $X$ by $2$ or by $3$ by ...
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28 views

Calculating mixed strategy Nash equilibria: using the derivative?

From roaming around and looking for ways to calculate the mixed strategy Nash equilibrium, I learned that a general way to do it is by determining the probability of choosing a strategy in such a ...
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21 views

Static game with complete and incomplete information

I am currently trying to learn game theory on my own. I have a question regarding the solution methods for static games with complete information vs that of incomplete information. The textbook ...
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1answer
27 views

What is the optimal reserve price in a second price sealed bid auction?

Consider a seller who must sell a single private value good. There are two potential buyers, each with a valuation that can take on one of three values,θi∈{0,1,2}, each value occurring with an equal ...
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28 views

Revelation Principle

Would someone be so kind as to explain me the Revelation Principle with a simple example with two agents bidding for one good where one agent would lie about his perceived value of the good?
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1answer
43 views

Games on betting from a set

Two players each chooses a number from the set $\{1,2,4\}$ and correspondingly bets an amount of \$$1$, \$$2$, or \$$4$. There is no collaboration between players. Rules: $1.$ If the two chosen ...
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31 views

A 2 Player Pure Strategy Game

There are two players each has $n$ balls. At the same time they distribute their balls among $m$ boxes. For each box 1 point is given to the player with more balls and zero points to other one (When a ...
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1answer
51 views

How can I tell if a two-person game is non-degenerate, given its payoff matrices?

Consider a two-person game with payoff matrices defined by \begin{equation} P= \left( \begin{array}{ccc} 0 & 4 & 1 \\ 2 & 2 & 4 \\ 3 & 2 & 2 \end{array} \right) \quad ...
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53 views

Payoff matrix with a specific form

I am very stuck on this question: Suppose that $b \in \mathbb{R}^m$, $c \in \mathbb{R}^n$, $A$ is a $m \times n$ real matrix, and all components of $A$, $b$ and $c$ are positive. Consider the ...
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Is there a game theory for doing the dishes in a shared living situation?

It occurred to me this morning (when I was intentionally not tidying up my flatmate's dishes) that doing the dishes in a shared living situation, such as at an office, or living with housemates, might ...
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31 views

Optimization problem with variables in the subscript

I want to solve a optimization problem, which mimics the actions between a seller and several buyers. A seller has several goods, 1, 2, ... J, with prices $p_j$ and quantity $q_j$. A buyer can only ...
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1answer
21 views

Need help with finding pure strategy nash equilbria

In the following game, how can I find the pure strategy Nash equilibria? The answers are apparently (b,d) and (b,g) but I'm not sure why. I have realised the following: Player one (rows) has no ...
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1answer
39 views

Should you choose highly owned or little owned players in fantasy sport

Here's the situation: It's a fantasy soccer game where players score points for my team based on their actual performances on the pitch. I have a team of 11 players and their is no limit to the ...
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1answer
45 views

Multiplayer finitely ultimatum game

Imagine a 3 member legislature that must decide how to allocate an asset of unit value. There are three rounds to the game and in each round a randomly assigned proposer must make an offer to each of ...
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1answer
31 views

Could there be multiple symmetric equilibriums in a symmetric games?

Given a finite symmetric 2 player game with a strategy space $S$, a (mixed-strategy) symmetric equilibrium is a distribution $d\in \Delta(S)$ such that $(d,d)$ is a Nash equilibrium. A known result ...
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Is there a name for the ratio between the optimal social-welfare equilibrium and the worst social-welfare equilibrium of a strategic game?

Suppose you have a $n$ players strategic game, and assume that the "social-welfare"(SW) of the game is defined as the sum of payoffs to the players. Two well known measures about the "efficiency" of ...
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2answers
39 views

Strategic form: Nash equilbrium

I am currently working through a question where I have to find any Nash equilibrium not in pure strategies, together with the associated payoffs. I have managed to identify the pure strategy Nash ...
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23 views

What is the Hamel or Schauder basis for functions from the subsets of the natural numbers to the reals

I'm trying to prove that some linear operator (Shapley value) $\varphi:\mathbb{R}^{P(\mathbb{N})}\rightarrow\mathbb{R}^{\aleph_0}$ is unique, where I'm using $\mathbb{R}^{P(\mathbb{N})}$ to denote all ...
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1answer
39 views

convex for nash equilibrium

I have trouble understanding this question, the first question to my understanding is asking me that for a fixed p , (p,q) is nash equilibrium, prove that all (p,q) are convex. and for the ...
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1answer
29 views

What is the pareto optimal payoff vectors for war of attrition game?

The game works as follows: two player are involved in a dispute over an item. the value of the object to player i is vi>0. time is modeled as a continuous variable that starts at 0 and runs ...
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1answer
30 views

how to find mixed Nash equilibria for 3x3?

A (3,2)(3,0)(2,2) B (1,0)(3,3)(0,3) C (0,2)(0,0)(3,2) p q 1-p-q So what I have done is : 3p+3q+2(1-p-q)=p+3q q=1 this is when A=B p+3q=3(1-p-q) p=-3/4 this is when B=c I don't know ...
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prove that if $n=k$ then white has a winning strategy in $S_{n,k}$.

Black and white play sequentially the game $S_{n,k}$ with $k,n\in \mathbb N \space 0\leq k\leq n$ the game board consists of all subsets $A\subseteq\{1,2,...,n\}$ such that $1\leq |A|\leq k$. every ...
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Can “tit for tat” strategy be defined in monadic second-order logic?

Prisoner's dilema game can be represented as a game tree, which could be infinite game with corresponding infinite game (binary) tree in common case. There is well-known tit for tat strategy, which ...
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Optimal Strategy Game of Communicating without Overlap

Today, I had a conversation which proceeded very poorly; indeed, I had this conversation with $n$ people, including myself, and everyone had something to say. Problem was that, for an unreasonably ...
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1answer
31 views

Probability of coin flip betting

Imagine a situation where you and a friend both have 5 dollars, and you play him in a 50/50 coin flip "duel" where if it flips heads you receive a dollar from them otherwise you lose a dollar to the ...
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1answer
15 views

Difference in the function (Game Theory)

I was hoping to know the difference between equation (1) and equation (2). Would they be considered equal or is equation (2) less strict compared with equation (1)? Let the correspondence ...
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1answer
23 views

What is the difference of pure strategic equilibria and nash equilibria?

Are they the same thing just named differently or with minor differences?