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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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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0answers
56 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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24 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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4answers
2k views

Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff ...
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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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2answers
300 views

Existence of Saddle Point of a Matrix (Shapley's Theorem)

A $m\times n$ matrix $M=(a_{ij})_{m\times n}$ with real entries is said to have a pure saddle point at $(i_0,j_0)$ if $\min_j \max_i (a_{ij}) = \max_j \min_i (a_{ij}) = a_{i_0j_0}$. Here the notation ...
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1answer
469 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 ...
406
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25answers
52k views

Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...
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1answer
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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2answers
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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0answers
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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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
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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1answer
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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1answer
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
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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1answer
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
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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1answer
465 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 $k$ ...
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2answers
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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
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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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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0answers
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
169 views

Is convergence to a Nash Equilibrium dependent on turn order?

Is convergence to a Nash Equilibrium dependent on turn order? Namely, if you change the turn order or switch between synchronous (all players move at once) and asynchronous turns can the outcome ...
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1answer
437 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 ...
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1answer
864 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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362 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
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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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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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1answer
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
134 views

Dice roll game fair price

You and I each roll 1 die each at the same time. I win if I roll a six on one roll, and then a five on the next. You win if you roll two sixes in a row. Who would you bet your money on? Note that You: ...
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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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10answers
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Mathematician vs. Computer: A Game

A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same ...
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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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1answer
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
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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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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1answer
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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1answer
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 ...
2
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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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3answers
79 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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1answer
291 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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2answers
124 views

Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
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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
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 ...