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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Problem in game theory related to traffic networks

I have learnt game theory for a short period of time and I am not familiar with multi-player non-zero sum games. Here is a problem from my book which I am stuck: In this road network below each of ...
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1answer
80 views

Winning strategy for solitaire?

I'm talking about the Klondike solitaire, turning three cards at once to the waste and placing no limit on passes through the deck. I know there isn't always a winning strategy, a counterexample can ...
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2answers
262 views

Alice and Bob number sum game

Alice and Bob play a game with first $N$ positive numbers. Out of these $N$ integers some $K$ integers are missing. So both decided to play with remaining $N-K$ integers and in this game Alice wants ...
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1answer
67 views

Extensive and strategic form in game with uncertainty

I have to solve the following problem for my gametheory course: Software Inc. and Hardware Inc. are in a joint venture together. The parts used can be defective or not; the probability of defective ...
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37 views

Balancing a game, or proving the imbalance of a game

Consider a real-time game with two teams $T_1$ and $T_2$ fighting each other, each team composed of $n$ players $p_1^t,\dots,p_n^t$, where $p_i^t$ denote the $i$th player of the team $T_t$. Each ...
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39 views

Nash Bargaining Equilibrium with exponential utilities

I'm trying to derive the answer to the following question: Two players play the classic divide-the-dollar game, which is an imperfect information version of the ultimatum class of games. Utility ...
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1answer
94 views

The game with countable amount of steps

Here is a cute problem. The angel and the devil play a game. Firstly the angel has an empty box and the devil has a box which contains all numbers from $\mathbb{N}$ (one copy of every natural ...
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0answers
60 views

Game Theory: What are Best Strategies for High-Low game (game details are below)?

High Low game is one where one person picks a number between a range (say 1-100) and another person have to guess it. With each guess, s/he is told whether the guess was high, low or correct. If the ...
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77 views

Correlation of belief distributions from distinct signals

Anne and Bob are two Bayesians who initially share a non-degenerate prior about a binary state of the world. Anne observes some signal (i.e., an experiment in Blackwell's terminology) about the state ...
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2answers
86 views

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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247 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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58 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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81 views

A combinatorial game theory problem

In details, there are four bishops on a chessboard in two pairs. In each pair they sit in orthogonally adjacent squares. How many positions can there be to place the two pairs on the chessboard ...
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91 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 ...
2
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2answers
309 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
59 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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1answer
68 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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2answers
61 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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1answer
39 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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1answer
106 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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1answer
73 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
124 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
118 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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2answers
76 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
60 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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126 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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1answer
134 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
39 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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60 views

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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1answer
88 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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0answers
83 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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2answers
48 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
134 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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1answer
124 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?
2
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1answer
58 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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1answer
41 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
84 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 ...
2
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1answer
93 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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3answers
115 views

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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0answers
37 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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2answers
51 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 ...
0
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1answer
44 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
67 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
36 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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0answers
15 views

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
67 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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30 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
57 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
43 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
69 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 ...