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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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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29 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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31 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 ...
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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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39 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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31 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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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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119 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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1answer
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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1answer
49 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 ...
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58 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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1answer
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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30 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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1answer
40 views

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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64 views

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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4answers
121 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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47 views

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 ...
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33 views

does the max-min inequality hold for constrained functions?

I would like to know if the max-min inequality holds for constrained functions (e.g. MILPs). my problem looks like: $$\min_{w \in W} \max_{q \in Q(w)} F(w,q) \ge ? \max_{q \in Q} \min_{w \in W} ...
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27 views

Game Theory Rulerette, Sprague-Grundy Theorem

Here is the question: Rulerette. Suppose in the game Ruler, we are not allowed to turn over just one coin. The rules are: Turn over any consecutive set of coins with at least two coins being ...
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33 views

Why is a weakly dominated strategy not played in a mixed strategy equilibrium mixture?

This question has been bugging me a lot. If you take this game, for example 1/2 | L | R U | 12,1 | 8,8 D | 15,1 | 8,-1 My instructor said the equilibrium to this is (D,L), even though (U,R) is a ...
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1answer
55 views

Some zero-sum game

Consider a game with the following payoff-matrix $\textbf{M}=\begin{array}{c|c c c c} \! & A & B & C & D \\ \hline A & 0 & 1 & -1 & 0 \\ B & -1 & 0 & 1 ...
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1answer
63 views

Maxmin and minmax strategies

I was solving for a stable equilibrium in the following 2 player zero sum game. I need to calculate the equilibrium using maxmin and minmax strategies. In this game they should come out to be ...
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2answers
17 views

A question about Game Theory Notation in symmetric Games

In Martin Osbourne and Ariel Rubenstien's book A Course In Game Theory (page 20) the authors say that a 2 person game is symmetric the A1 = A2 (both players have access to the same action set) and ...
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1answer
30 views

Verifying minimum of a function when second derivative is 0

I was solving for a stable equilibrium in a game. I calculated expected payoff of one of the players which came out to be the function $$E= -4pq + 0.1p + 0.2q + 0.7$$ where p is probability of an ...
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39 views

Blackjack Dice Probability - Odds of winning and where to stand

Hi I'm looking for help/advice in finding the solution to the following problem. Suppose I have a fair $101$ sided dice labelled $0$ to $100$, Your objective is to roll the dice any number of times ...
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1answer
40 views

Winner in this game with divisors

I was solving this SPOJ question, which is as below: N wooden pieces (marked with numbers 1 to N) are placed in a transparent bottle. On his turn the first player takes out some piece (numbered x) ...
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48 views

Name and related theory for game: “n-th Player to respond wins”

What is the classic name of the game "The N-th player to respond wins"? What are general strategies for the case when the number of total players is not known (or maybe there's only a prior on the ...
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1answer
67 views

Nash equilibrium for n players game

There is a question that I am trying to solve but I am not sure about my approach and is hoping I could get some help. Here is the question: There are $n$ companies sharing a water reservoir, let's ...
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1answer
27 views

Gaining intuition as to why maximal lotteries use randomness to break general ties

The maximal lottery is a voting system based on choosing an optimal candidate game-theoretically. If a winner isn't clear (there is no condorcet winner), then it will return probabilities as to which ...
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118 views

A coin tossing game with random probabilities

Let $p$ a random variable, uniformed distributed in $[0,1]$. Two player $A$ and $B$ play the following game: Starting from A, a player gets a random value $p(\omega)\in[0,1]$, and he has two choices: ...
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1answer
35 views

How to graph a rational reaction set?

In my game theory class, I am asked to graph the rational reaction set for the follow matrix: $\begin{bmatrix} (3,-1) &(-1,-4) \\ (-1,2) & (1,2) \end{bmatrix}$ I am not sure how to graph ...
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33 views

Uniqueness of saddle point solution to zero-sum game

Considering a two player zero-sum game, is a found saddle point solution unique? And if not, are there any conditions under which the saddle point solution is unique?
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12 views

Game theoretic expected utility function with two unknown parameters

I am working on a game theoretic model for my Master's thesis in Political Science, and I have calculated the following expected utility function for player $i$: ${EU}_{i} = 3-12{V}_{i}S$. Both ...
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1answer
43 views

Mechanism design with known utilities (game theory)

I'm trying to prove that in an n-party setting, where each party has a private value, the dominant strategy is always to reveal it. I'm assuming that parties only care about monetary payoffs and ...
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1answer
25 views

Does exchanging the rows or columns of a matrix game affect the outcome?

This is part of an assignment. I feel like this is a really trivial question or I'm missing some key idea. I'm asked whether exchanging the rows affects a matrix game. I believe it doesn't because the ...
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37 views

Optimizing generalized ternary search

There are $N$ socks numbered $1$ to $N$, one containing a gift. Dave needs to find the sock with the gift. He can ask some questions in order to find that sock: in each question, Dave chooses $2$ ...
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1answer
114 views

Game theory - Finding Nash Equilibria

i have to solve these questions. I already came up with a solution. Can anyone look over it and state their opinion? Many thanks in advance :) CHALLENGE I 1. Consider the following game between ...
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1answer
74 views

How to prove that the zero sum game's optimal (security) strategy do not change when payoff matrix increase by a constant factor

I am not sure how to construct a proof for something so obvious, can someone give me some pointers on proving the following Suppose a constant K is added to each element of a pay-off matrix ...
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2answers
46 views

Best Strategy in a Non-Cooperative Game of Perfect Information (Pure Strategies only)

Suppose there are two players in this game, and each player has $4$ dollars prior to making his move. Each player has as his strategy space the ability to submit an integer number of these $4$ initial ...
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34 views

Game Theory rationalizable strategies that are all the same

Suppose player 1 has strategies u, m, and d, and player 2 has strategies A, B, and C. The strategies A, B and C are all identical (payoff equivalent). Can any of them be eliminated using the ...
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140 views

Playing rock paper scissors over online chat.

Is there a way to play rock, paper and scissors fairly over internet chat? By this, I mean that both players cannot play their hands simultaneously, one of them has to go first and the second player ...
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69 views

What is the optimal losing move?

I had a hard time trying to find the best-suited stackexchange site to ask this question. I'm still not sure whether this is the right place, so please guide me to the right one if you think this is ...
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46 views

Optimal strategy to escape spotlight

Here is the setup. A prisoner is being held in the center a circular yard with radius $r$ and can run in any direction at some velocity $v$, there is a spotlight which illuminates a line on the circle ...