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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908 views

game theory - coin flipping game

Lets say 2 players $A$ and $B$ make a bet, who can have more money at the end after playing the following game: a coin is flipped: with 51% probability it lands tails, with 49% probability it lands ...
4
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1answer
3k views

Does Black have a winning strategy in Gomoku(freestyle)?

Gomoku is actually a finite two-person game of perfect information. Moreover, if we consider draw as victory of White, then by Zermelo's theorem, exactly one of the two has a winning strategy, either ...
3
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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 ...
4
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1answer
813 views

Analytically solving (calculating Nash equilibrium for) 3-player extensive form games

Let's say we extend the popular half-street Kuhn poker variant to 3 players. The rules would be as follows: ...
0
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1answer
64 views

Which distributions should be used to model the winning & 2nd bids in second price auctions?

With second price auction which distributions should I use to model the winning bids and 2nd bids (separately)? I'm thinking of using Gaussian. However for the winning bids r.v, it has to satisfy: $$ ...
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0answers
62 views

Parameterized convex optimization

I'm trying to formulate a game so that at Nash equilibrium I achieve supply equales demand. Then I ran into this problem. For all $i,$ $v_{i}\left(x_{i}\right)$ is concave in $x_{i}$. The value ...
1
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1answer
288 views

Does this multiplayer generalization of the Prisoner's Dilemma exist?

While thinking some things over in my head, I came across the idea of trying to generalize the Prisoner's Dilemma to multiple participants, and trying to do it in the simplest way possible. In ...
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2answers
380 views

Game theory: Nash equilibrium in asymetric payoff matrix

I have a utility function describing the desirability of an outcome state. I weigh the expected utility with the probability of the outcome state occuring. I find the expected utility of an action, a, ...
2
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0answers
231 views

Explanation of Mixed Strategy Definition in Game Theory

Definition: Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$. ...
8
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1answer
344 views

Why is the best position for LCR not the last person?

For the uninitiated, LCR is a game in which each player starts with three "tokens" and rolls up to three dice (at most as many as tokens they have). Each die has three sides which indicate that ...
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0answers
92 views

Can the Nash bargaining solution be applied in repeated game?

I am trying to develop a model involving two agents who interact strategically to set an optimal time for a joint work. These agents will have to meet repeatedly. I want to derive the optimal time for ...
3
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0answers
105 views

mixed vs behavior strategies for zero-sum game with infinite extensive form

edit: No responses to this post after a week, so I'm cross-posting it to cstheory.stackexchange here. I'm looking for a known theorem stating that, for appropriate kinds of two-player zero-sum games ...
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3answers
279 views

How to formally model the “hesitation” in the hat-guessing puzzle?

Hua Luogeng (in Chinese, 华罗庚) took a hat-guessing puzzle as an illustration in a booklet focusing on mathematical induction. The following description is a literal translation from Chinese. ...
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2answers
134 views

Some Questions About Chess

I have to questions about the chess game: please help me to understand it. 1- How can a computer program know if this move or that move is better? It calculates all possbile continuation and examine? ...
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1answer
137 views

Mathematics behind Incentive Design

I was working on an Applied Math project on allocation and I had an interesting idea about extending it to providing incentives to different "players" in the allocation process. But I am clueless ...
0
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2answers
632 views

unanimity game, calculate core/shapley value

Another (quick) question; Let $T \subset N$ be a coalition. The unanimity game on $T$ is the game $(N, u_T)$ where $u_T(S)=1$ if $T \subset S$ and $u_T(S)=0$ if $T\S$. In other words, a ...
1
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1answer
100 views

Nucleolus core element?

I dont know how to get started with the following question: How do I show that in a game with a nonempty core the nucleolus always is a core element? I mean, if the core is nonempty its quite obvious ...
4
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2answers
3k views

Nash Equilibria for zero-sum games (Rock Paper Scissors)

I'm trying to figure out a nash equilibria strategy for rock paper scissors and when the strategy would not be optimal. I know it's a zero sum game and I must use a mixed strategy but the practice ...
3
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2answers
1k views

Does chess have more Nash equilibria than you can find through backwards induction?

All equilibria found with backwards induction on a tree of a perfect information game are Nash equilibria, but in general the reverse is not true: ...
2
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1answer
166 views

vickery auction question(second-price auction)

The question is as follow, Alice and Bob would both like to own the same manuscript. The manuscript is worth 5 million to Alice and worth 3 million to Bob. The present owner of the manuscript ...
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1answer
71 views

Correlated Equilibrium - Transforming a non-linear objective function into a linear one

I am trying to transform a non-linear objective function into a linear one, in order to create a LP. How might I go about to do this (I have never taken a course in linear programming). I have that I ...
0
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1answer
275 views

subgame perfect nash equilibrium for war of attrition

the question is as follow: suppose that two players are playing war of attrition, that means both of them could choose either to fight or quit, if either one of them quit, the game ends, and if ...
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2answers
4k views

cournot competition with N-firms

The question is as follow: Here is how we can think of N-firm Cournot competition. Assume all the firms have the same marginal cost C > 0. Firm 1 chooses Q1, Firm 2 chooses Q2, and so on. The market ...
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1answer
2k views

cournot equilibrium and stackelberg equilibrium question

Question is as follow: there are 2 firms that want to enter the apple juice market in country A. There are no existing firms in the market or potential entrants. They need to decide on yearly ...
4
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1answer
171 views

Always win without a winning strategy

On Page 141, Axiom of Choice, Herrlich(2006) Show that if in a game of the form $G(1, X_1, Y_1, A)$, the first player has no winning strategy, then the second player can always win, even ...
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0answers
399 views

Simple game with coins - strategy

Let's play a game: There are $n$ stacks of coins in a row. $i$-th stack consists of $d_i$ coins. Two players: $\text{Player1},\text{Player2}$ make moves alternately. Player in his turn can only take ...
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0answers
114 views

Card game-ordering a deck [duplicate]

Possible Duplicate: Game Theory Matching a Deck of Cards Suppose we take a blank deck of $52$ cards, write the number $1$ on the first card, $2$ on the second card, and so on until we write ...
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1answer
160 views

Cournot-game problem

I'm so stuck with an exercise about Cournot game and was hoping if someone could help me out here. Would appreciate all the help. This is the exercise: Consider the market for Blue Turtle (a new ...
2
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2answers
134 views

Poker, number of three of a kind, multiple formulaes

I wanted to calculate some poker hands, for a three of a kind I infered, 1) every card rank can form a 'three of a kind' and there are 13 card ranks, 2) there are $\binom{4}{3}$ ways to choose three ...
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0answers
311 views

Determine market price and quantities produced; non-cooperative cournot game

$P(Q)$ represents a market where demand $Q$ is related to price $P$ by $$P(Q) = Q^{-\frac{1}{2}}$$ In this market there are $m$ identical producers, say firm 1, 2, up to $m$ which can produce any ...
0
votes
1answer
107 views

Tree search on game of Imperfect Information

Ok... I have a game of imperfect information and I want to compute its Minimax Value.From the lectures I know that in order to find the value for a Perfect information game I can use tree search ...
17
votes
1answer
281 views

A game played on a rectangle

Suppose two players play the following game on a $m$ by $n$ rectangle. Alternatingly they have to make a cross in some empty $1\times 1$ square. They are not allowed to make a cross next to another ...
3
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1answer
640 views

Game Theory - Centipede Game

I am working on a game theory question. The question is as follows: Characterize the unique subgame-perfect equilibrium in the following game. Can you find any Nash equilibrium which is not ...
0
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2answers
145 views

Why does menace work?

Consider the repeated Prisoner's Dilemma. Every day, for many days, two players play this game: $\left(\begin{array}{ccc} \left(3,3\right) & \left(0,10\right) & (-2,-2)\\ (10,0) & (1,1) ...
3
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0answers
129 views

Optimal strategy in a VCG auction with partial collusion?

Suppose you control the bid prices in a multiple-item VCG auction for a partial coalition of bidders. Each bidder is only allowed to win one item out of the set of multiple items, which are all ...
0
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2answers
288 views

Determine Nash equilibrium

How can I determine the nash equilibria in the following matrix? $$\begin{pmatrix}-\pi,-\pi & e,0 \\ 0,e & -\pi,-\pi \end{pmatrix}$$ I know the definition of a Nash equilibrium, but because ...
2
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1answer
1k views

Value of a zero sum game

I've been looking some around on the net for some info on zero-sum games, But I don't think I fully understand the principle; If we consider the (simple) matrix: $$\begin{pmatrix}\pi&0 \\ 0&e ...
2
votes
2answers
183 views

The name of the game (Hawk-Dove variant?)

I stumbled upon the following symmetric two-person game. We have two objects $X,Y$ with positive value $x$ and $y$, and two persons that have to pick independently form each other simultaneous one of ...
2
votes
1answer
207 views

mixed strategy nash equilibrium question!

Suppose the game consists of only $2$ players, player $1$ and player $2$, and each of them has only $2$ strategies to choose between. This gives a $2$ by $2$ payoff matrix. Player $2$ has no ...
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1answer
237 views

Question on mixed nash equilibrium!

The question is as follows: Think of the Golden Ball game. Now player 1 is money-minded and jealous, and player 2 is very good-hearted, so the payoff matrix is follows: ...
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0answers
212 views

On a zero-sum game betting market

I'm trying to come up with the rules for a betting game. My problem is to figure out under what constraints that game has zero-sum game properties: in other words, I want to make sure that no money is ...
32
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1answer
1k views

Is War necessarily finite?

War is an cardgame played by children and drunk college students which involves no strategic choices on either side. The outcome is determined by the dealing of the cards. These are the rules. A ...
2
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0answers
143 views

Game Theory - Extensive Zero-Sum Game Property Proof

How I might I go about to prove (or disprove, but I believe that this is true) the following: We call a 2-player extensive game $\Gamma$ a zero-sum game if the sum of the 2 payoffs for an terminal ...
3
votes
1answer
357 views

Game Theory Matching a Deck of Cards

Moderator Note: This question is from a contest which ended 1 Dec 2012. Suppose we have a deck of cards labeled from $1$ to $52$. Let them be shuffled in a random configuration, then made ...
2
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1answer
557 views

Saddle points in zero sum game

We only had one lecture about the subject and already have quite difficult questions, could someone please help me? The matrix looks something like this: \begin{matrix} 3 & 2 & 1 & 4 ...
4
votes
1answer
392 views

Which side has winning strategy in Go?

Go is actually a finite two-person game of perfect information and cannot end in a draw. Then by Zermelo's theorem, it is exactly one of the two has winning strategy, either Black or White. So my ...
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2answers
76 views

graph-theory combinatorics

Here is a combinatorics problem having to do with graph-theory Ten players participate at a chess tournament. Eleven games have already been played. Prove that there is a player who has played at ...
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1answer
365 views

Relevant Topics for Presentations

I have to make an presentation as a part of our math course. We are provided relavent topics but I am looking for somethat that is challenging, related to economics something like game theory or ...
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0answers
252 views

Carrier of a mixed strategy in a 3x3 bimatrix game

We are given a bimatrix game (A,B) = $\begin{bmatrix}0,4 & 4,0 & 5,3\\4,0 & 0,4 & 5,3\\ 3,5 & 3,5 & 6,6 \end{bmatrix}$. Suppose (p, q) is a Nash equilibrium in (A,B). Prove ...
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1answer
52 views

Game theory multiple cooperative adversaries

Are there any papers talking about games with multiple cooperative adversaries? I do research in computer science, and I am interested in this type of game. I am really not that knowledgeable in game ...