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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Nash equilibrium indifference principle

In the Hebrew wiki page on Nash equilibrium there is a reference to an indifference principle which means that once we know the other player uses the equilibrium strategy then the first player can use ...
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57 views

What are the optimal mixed strategies for this game?

Fix $k < n$ positive integers, and two players play the following game: each player picks a positive integer between 1 and $n$. If the two numbers picked are within $k$ of each other, the larger ...
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38 views

Playing roulette using martingale

A player with unlimited money decides to play roulette. He bets $1$ on red, if he loses, he bets $2$, if he loses again he bets $4$ and so on till he wins. Prove that he is guaranteed to make a profit....
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Differential equation, Cournot competition, Game theory

I would like to solve differential equation derived from differentiating $u_i(q_1,q_2)=q_ip(q_1+q_2)-c_i(q_i)$ by $q_1$ and $q_2$, resp. and putting them equal to zero,and taking $q_2$ to be $r_2(q_1)$...
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55 views

Calculate the expected value of this game

Every night, different meteorologist gives the probability of rain for the next day. To judge their predictions, we use the following scoring system: if a meteorologist predicts rain with the ...
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50 views

Queen moves — The Squared Chain Puzzle

Karl Scherer made the interesting Squared Chain Puzzle. Start with a $7\times7$ board, with a queen somewhere. Make a legal move with the queen, placing coins over all squares visited. For subsequent ...
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52 views

Probability of winning a certain amount of money in this roulette martingale

Given this classic roulette system: A gambler starts by betting $1$ unit on red. Every time he loses his bet he will double his wager until he wins. Afterwards he will start this scheme all over ...
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40 views

Game theory problem: the Man and the Lion

I have been reading this solution to the somewhat famous problem about a man and a lion closed in a circular arena: they both move at the same speed, and the lion is trying to eat the man. The ...
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44 views

Matchstick game problem

I'm going through past exam papers and this question is proving to be tricky. I can't seem to solve it and have no clue how to, I tried checking numerous ways like odds,evens,primes etc but i'm sure ...
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34 views

Symmetric game with selecting numbers

Two players select numbers from the set 1,2,...,N. Denote x the number selected by Player 1 and y the number selected by Player 2. Here is the payoff: If $x < y$ => Player 1 wins and gets 1 dollar....
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52 views

Game theory: power and mod

Given two non negative integers $a, b$. Two players alternate turns. If at any state of the game the two integers are $a\le b$ then the player with the turn can either replace $b$ with $b\bmod a$ or $...
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874 views

Osgood Box (Doctor Who) Dilemma

Today's episode of Doctor Who featured an interesting dilemma, centered around an object called the "Osgood Box". I'm not well-versed enough in game theory to recognize it as an existing problem, but ...
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28 views

Best voting strategy

Let us assume there are two parties in a country with population $p$. There are $c$ constituencies. Each party knows exactly who are their supporters and can choose where each supporter votes. Note ...
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39 views

Object and number trick

You are telling your friends Adam and Steve a trick and here is how it goes: Tell Adam to take out $3$ objects from his pocket which all the objects have a different amount of letters. Lets ...
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15 views

Class-participation problem modeled with game theory

I'm taking a class and the teacher has set up a system of class-participation to encourage us, the students, to, well, participate more actively. The system is as follows: each student is given 20 ...
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50 views

Is this Nash-Equilibrium valid?

The game is as follows: $$\begin{array}{c|c|c|} &A&B\\ \hline A&2;3&2;3\\ \hline B&-1;2&1;2\\ \hline C&-1;3&4;2\\ \hline \end{array}$$ I've written a program that ...
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1answer
80 views

Winning strategy - nim variation

i was reading about different variations of nim game and i'm trying to find winning strategy to one of them: There are n empty places on the circle. Two players are placing their "coins: on empty ...
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47 views

Expected Value in multi-round game

Say I have a game structured as follows: 1) if you decide not to play, I reward you $16 Otherwise: 2) you can flip a coin OR you can throw a ball into a basket (numbered 1 thru 10 with the ...
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45 views

Can anyone explain this more clearly?

I'm new to CGT so i might need help but could anyone simplify this and explain it to me please- "set f ⊕ f = 0 for any f. (A nice correspondence can be made if we think back to the original game of ...
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1answer
50 views

Formulating a game in an economic setting

I'm trying to teach myself Game Theory, and have come across the following question: Suppose that a company, $L$, produces left shoes only, and a company $R$ produces right-shoes. If $L$ charges $...
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61 views

The application of Nimbers to Nim strategy

I've been reading about combinatorial game theory, and some works start with the game of Nim. After that, they introduce Nimbers, which are numbers that represent Nim games. So far so good. I get ...
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53 views

Prove that the set of $(m \times n)$-matrix games is dense and open

Show that the set of $(m \times n)$-matrix games with unique optimal strategies is dense and open. Let $\mathbb{R}^{nm}$ be a $nm$-vector space of all matrix games and let $M \subset \mathbb{R}^{...
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31 views

How to construct potential function in game theory

Can someone explain me how can I find a potential function for a potential strategic game? I know the theory and definitions, but can't figure out the reasoning process. Potential function that I ...
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31 views

Partitioning integers into sets

Try to find out if it is possible to partition 6 consecutive positive integers into two sets, $A_1$ and $A_2$ such that the product of the elements in $A_1$ is equal to the product of the elements in $...
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31 views

Finding the numbers on cards

This question is a bit confusing. I decided to start by showing that all cards don't contain the same number. Lets say I pick a subset of $10$ cards then they have $10-19$ $10+11+12+13+..+18+19=145$ ...
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48 views

Game theory, Duopoly problem

. Suppose two firms compete in micro-chip industry. Each period firm 1 produces q1 chips and firm two produces q2 chips and the firms face a demand curve of P = 1000−20Q, where Q = q1 +q2. Both firms ...
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105 views

Cournot Duopoly Question [closed]

(Repeated Games). Suppose two firms compete in micro-chip industry. Each period firm 1 produces q1 chips and firm two produces q2 chips and the firms face a demand curve of P = 1000−20Q, where Q = q1 +...
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32 views

Game theory participation constraint

What are the implications of a participation constraint that is unsatisfied? I am confused by a problem I am working on.
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34 views

Perfect Bayesian Equilibria of the following game

Consider the following game between a monopolist firm and a consumer. Consumer's income is $1$, and he needs to allocate it between period 1 and period 2 consumption to maximize his utility $u(c_1,c_2)...
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30 views

Randomized Mechanism - Price of Anarchy

We are given a mechanism where randomization is introduced by the designer and not the players. For example, a player gets an item with probability equal to $f(x)$, where $x$ is the input vector of ...
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3answers
135 views

3 contestants choosing a smallest number to win a car

Each of 3 contestants chooses a positive integer. The contestant who chooses the unique smallest positive integer number wins a car. If all of them chose the same number, then no body wins car. What ...
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44 views

Find the set of undominated strategies in Cournot duopoly

Consider a version of the Cournot doupoly game, where firms 1 and 2 simultaneously and independently select quantities to produce in a market. The quantity selected by firm $i$ is denoted $q_i$ and ...
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32 views

Ideal Card Game

I have invented a very interesting card game. All the cards from 2 to 10 (in four colours) are divided evenly between the two players (the deck is shuffled before dealing the cards, of course). Now ...
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49 views

Numbers written on a board

The numbers $1,2,...,n$ are written on a board ($n\in\mathbb N$). In each step we take any two numbers $a,b$, remove them, and write either $a-b$ or $a+b$ on the board. After $n-1$ steps there will be ...
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20 views

Generating a sequence of random numbers as a random seed for gambling

We have a simple game where a player wins if a random number is greater than some threshold (say 0.5 on 0-1 scale). The player commits to a game first, and then we generate proceed to generate the ...
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2answers
43 views

stacks with odd number of cards

Given $n$ stacks of cards, stack $i$ contains $a_i$ cards ($1\le i\le n$) such that each $a_i$ is odd. Two players $A$ and $B$ play a game. Players alternate turns. In a move, a player takes an ...
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17 views

Finding a number with certain properties

I am having a hard time finding this number. Here is my work: $RLPNM$ $3(RLPNM)=RLPNM-1$ However, how am I supposed to solve this equation of $5$ unknowns? Are there any different approaches to ...
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1answer
30 views

Perfect information game with heap of objects

I have to find the winning strategy for the following game. There is a heap with $N$ objects. Two players take objects in turn, but there is a limitation: if there were taken $K$ objects on previous ...
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2answers
51 views

A probabilistic approach of Prisioner's Dilemma

In a prison there were three prisoners, $A_1$, $A_2$ and $A_3$. A draw had been made to give two of them a pardon. $A_1$ asked a guard (who knew the draw) the name of another prisoner who had been ...
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120 views

Very fascinating probability game about maximising greed?

Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number ...
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24 views

Understanding pure nash strategy for general sum game

Given a general sum game with the cost function $A = \begin{bmatrix} 2000 & 0 & 2000 \\ 1000 & 100 & 1000 \end{bmatrix}$ $B = \begin{bmatrix} 400 & 0 & 0 \\ 1 & 1 & ...
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1answer
25 views

How does this textbook compute the Nash Equilibrium of the two person zero sum game?

In Tamer Basar Noncooperative Game theory pg $33$ there is a $2 \times 3$ game (zero sum game) $A = \begin{bmatrix} 1 & 3 & 0 \\ 6 & 2 & 7 \end{bmatrix}$ (each element is a cost, ...
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51 views

What is the intuition for two player games, mixed strategies are computed with respect to pure strategies instead of mixed strategies?

Let $x$ be the mixed strategy of player $1$ Then the mixed strategy for player $1$ is calculated with respect to $[1, 0], [0, 1]$, the pure strategies of player $2$. i.e. $x^*$ = $\max \min x^TA([1,...
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59 views

Why view mixed strategies as points in a simplex?

In some game theory books (e.g., Evolutionary Games and Population Dynamics), mixed strategies are described as points in a simplex. More specifically, for a game with $N$ strategies, from $R_1$ to $...
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51 views

Dynamic game with removable players

As I know, with game theory we can compute the equilibrium of a game (i.e. the best strategy that each player uses according to other players best strategy) and in dynamic games, the utility function ...
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31 views

'Unhappy with my proof' problem: Connectactoe

Connectactoe is tictactoe played in a $3\times3$ Connect4 grid, so gravity plays a part. Player 1 wins if they go in a corner and Player 2 doesn't go in the other corner (see EDIT) when it is a draw: ...
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Moves to P-positions in Nim

Let $A$ be an N-position in Nim such that all moves to P-positions reqire exactly $k$ tokens to be removed. What can we say about $A$?
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Probabilistic Game Theory

I would appreciate help on the following problem: Problem. You just bought a new a card printer which continuously prints cards in red or blue, chosen independently and uniformly at random. You play ...
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142 views

In a game of drawing straws, why are all turns equally good?

For example, there are 3 straws in a pile - 2 long and 1 short. The person who draws the shortest straw loses. When a straw is drawn, it is removed from the pile. Drawing first, second, or last all ...
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Solution of $Connect4^{TM}$

It says here that Connect4 can be won by Player $1$ if their first counter goes in the middle column $4$, a draw if they play in columns $3$ or $5$, and Player $1$ loses everywhere else. As far as I ...