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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Pareto optimality - Game theory

Good morning, I have this game theory problem. Let's consider 5 farmers, each of them has 2 cows to put into the field. So every farmers can put 0,1 or 2 cows. I denote the three stategies by ...
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9 views

Single bid auction: calculating bid as function of winning probability

I'm simulating a auction game with sealed single bid, where each of the $n$ players has winning probability $p_i,i=1,...,n$, and their bids $b_i$ have to be calculated to meet the $p_i$. Supposing ...
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11 views

Reducing an I-optimal problem to a Pareto-optimal problem

Given a set $\textbf y\subset\mathbb R^2$, let $y = (y_1,y_2), y'=(y'_1,y'_2)\in\textbf y$ be elements of that set, let $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
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1answer
31 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 ...
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1answer
24 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 $y$, ...
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1answer
19 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 ...
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8 views

Chance of Meeting [closed]

Two persons A&B decided to meet during lunch hours but due to some reasons A will arrive randomly from 1:20 pm to 4:45pm while B will arrive randomly from 2:05pm to 5:10pm (of the same day). If A ...
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1answer
50 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 ...
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29 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, ...
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64 views

If $X$ is finite and $R$ is a complete and reflexive binary relation on $X$, then $M(R, S) \neq \emptyset$ on any $S \subset X$ iff $R$ is acyclic.

Could you help me to verify my proof and my writing? Definition 1: A binary relation $R$ on $X$ is complete if, for all $x, y \in X$ such that $x \neq y$,$xRy$ or $yRx$ or both and reflexive if, for ...
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39 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 ...
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2answers
31 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: $$ ...
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2answers
57 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, ...
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1answer
67 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 ...
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1answer
15 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 ...
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1answer
19 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 ...
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2answers
36 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 = ...
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1answer
34 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 $v(S)=4$ ...
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16 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 ...
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1answer
21 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 ...
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1answer
77 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 ...
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1answer
45 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 ...
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16 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 ...
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1answer
31 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 ...
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2answers
92 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 ...
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1answer
49 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
174 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
43 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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28 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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2answers
26 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
84 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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38 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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39 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
41 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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227 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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37 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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75 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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67 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
36 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
34 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
48 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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32 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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17 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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1answer
30 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
37 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
69 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
30 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
89 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
46 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
55 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 ...