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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110 views

Nash equilibrium: Can I delete weakly dominated strategies in this case?

As far as I know, an equilibrium can involve a weakly dominated strategy, but cannot involve a strictly dominated strategy. Is there a general rule for when/if you can safely delete a weakly dominated ...
0
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1answer
34 views

Biobjective optimisation, pareto non-domination

Ok, so, I have a function $f_I(y_1, y_2) = \max\{\alpha y_1 + (1-\alpha)y_2:\alpha\in[\alpha_{min},\alpha_{max}]\}$ that I'm trying to minimise, and I'm asked to find, amongst a set of vectors $y$, ...
0
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1answer
91 views

Calculating Nash equilibrium in mixed strategy in a game where a Nash equilibrium in pure strategy exists

Let's say I want to calculate Nash equilibrium with mixed strategies for a two-players game, in which there is no Nash equilibrium with pure strategies (no dominant strategy for any of the two ...
3
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2answers
81 views

Street Fighter: is the game balanced?

Suppose that $A$ is a matrix that describes the matchup information of any pair of Street Fighter characters e.g., considering $3$ characters, assume that the first row/collumn is associated with a ...
1
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0answers
64 views

optimal strategies for 2-player zero-sum games of perfect information

Do finite-state 2-player zero-sum games of perfect information with only win-draw-loss outcomes always have deterministic-and-memoryless optimal strategies for both players? In other words, ...
2
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0answers
46 views

Book or paper recommendation about “Rube Goldberg Mathematics” // e.g. Longest path problems

First: My question is not be very specific, since I lack a concrete overview, but my idea/thoughts in a nutshell: I would like to have a recommendation of a good book, paper or article about processes ...
1
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1answer
73 views

Cournot Duopoly Game - Nash equilibrium

I have this problem about Cournot Duopoly game. Actually I don't know if I have understood the "real sense" of the problem. I consider CD game described by the following payoff fucntions: $$ ...
0
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2answers
115 views

Math theories in Game Theory

What are all the mathematical theories in Game Theory? I have taken Mathematical Modelling, including: application of linear systems, matrix operations, inverse of matrix, leontif input-output model, ...
2
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1answer
176 views

Finding the Nucleolus

Given the following table of values and excesses of coalitions S and imputation $\vec{x} = (9,6,9)$: How do I find the Nucleolus? My book wasnt clear on the method of calculating it, so Id like to ...
0
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1answer
28 views

Understanding the proof of Gibbard-Satterthwaite theorem

Let $n$ be the number of voters and $A$ be the set of alternatives. For voter $i$, we denote by $a \succ_i b$, if $i$ prefers $a$ to $b$, where $a,b \in A$. Let $L(A)$ denote the set of all strict ...
0
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1answer
95 views

Finding support of optimal mixed strategy

Considering a finite zero-sum game represented by a matrix $\mathbf{A}$, I understand one method to solve for the game is to use the principle of indifference. If the optimal strategy of Player 1 (who ...
1
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2answers
103 views

Implicit function theorem in comparative static problem

The individual lives for two periods. He has a utility function $u(c_{1} )+ bu(c_2)$. His budget constraint requires that his period I consumption be his period I endowment minus any savings, $c_1 = ...
0
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1answer
45 views

Determining the core of a permutation game

For my game theory class I have to determine the core of the following cooperative game: $\mathrm{N}={1,2,3}$ $S=\{1\}$ gives $v(S)=2$ $S=\{2\}$ gives $v(S)=5$ $S=\{3\}$ gives $v(S)=4$ ...
0
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0answers
32 views

Translating these “payoff matrices” to normal form.

I'm used to looking at payoffs to given players in normal form but here I'm given the following matrices and asked to interpret them as payoff matrices. Can someone explain how I'd translate the given ...
0
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1answer
39 views

Optimum type auction for seller's profit maximisation

I am trying to formulate an auction, in which sellers will create a cartel and ask for the highest possible price that buyers pay. Practically, I would have a multi-part game in which in each part I ...
0
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1answer
141 views

Finding the nucleolus of a jury case

I am a bit stuck on this problem: (a) $\quad $ Consider a jury system with $12$ jurors in which a defendant is found guilty if voted guilty by $10$ or more of the jurors. We represent this jury ...
1
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1answer
93 views

Game theory on selling shoes

I am stuck in this question: A pair of shoes consists of a left shoe and a right shoe, and can be sold together for $ \$10 $. Consider a coalitional game with $a+b$ players: $a$ of the players have ...
0
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0answers
19 views

every simple, monotone 3 player TU game is a weighted majority game.

I'm trying to prove that every 3 player TU game $G=(\{1,2,3\},v)$ which satisfies: $G$ is simple: if $ T\subseteq N$ then $ v(T)\in \{0,1 \}$ $G$ is monotone: if $T\subseteq S $ then $ v(T)=1 ...
0
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1answer
44 views

What is the optimal strategy for a peace war game with unequal power and varying peace agreements?

Consider a variant of the peace war game in which nation A can "harm" nation B much more than B can harm A if they both go to war, but each nation can also give the other nation tribute. To formalize ...
0
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2answers
169 views

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 ...
1
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1answer
83 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
268 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 ...
0
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1answer
70 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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0answers
38 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 ...
1
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2answers
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 ...
6
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1answer
95 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 ...
2
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0answers
64 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 ...
2
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0answers
81 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
89 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 ...
14
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2answers
248 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 ...
1
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0answers
60 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 ...
2
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0answers
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 ...
3
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0answers
95 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
votes
2answers
332 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 ...
0
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1answer
65 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 ...
2
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1answer
69 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
67 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 ...
2
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1answer
40 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 ...
1
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1answer
113 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 ...
0
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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 ...
1
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1answer
133 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 ...
0
votes
1answer
121 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 ...
2
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2answers
80 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 ...
1
vote
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 ...
5
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2answers
128 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 ...
5
votes
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 ...
0
votes
1answer
41 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 ...
7
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0answers
62 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 ...
1
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1answer
91 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 ...
0
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0answers
85 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 ...