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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74 views

Where's the Nash Equilibrium here? $ \begin{pmatrix} (2,-2) & (2,-2)\\ (1,-1) & (3, -3) \\ \end{pmatrix} $

I just opened a book on Game Theory, so I'm totally new to this. My book says that the only Nash Equilibrium in the example below is (2, -2) -first row, first column-, and I really don't see why... ...
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1answer
65 views

Finding Mixed Strategy Nash Equilibria

Okay, so I was working through this problem: Now, I understand the computations. What I don't understand is why the solution says that each player will play H with probability p=2/3. I would have ...
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0answers
43 views

Reference to mean field game theory

Can someone suggest a good reference to introductory mean field game theory? It would be good if it explains with examples. Thank you.
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1answer
94 views

Consider a game in which two players take turns removing any positive number of pebbles they want from one of two piles of pebbles.

Consider a game in which two players take turns removing any positive number of pebbles they want from one of two piles of pebbles. The player who removes the last pebble wins the game. Show that if ...
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1answer
212 views

Expected payment in second price seal-bid auction

Environment Suppose $n$ bidders participate in a second price sealed-bid auction, in which one object is being sold. Each bidder $i$ values the object at $v_i$, and each $v_i$ is independently and ...
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1answer
80 views

Game of Stones - Count the ways

We are given a number of piles of stones. and we can remove two stones , where both stones come from different piles. We do this until all the piles are finished or only one pile is left as we cannot ...
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2answers
118 views

Simple game-theoretical problem

I have a confusion regarding the following problem: Suppose there are three players, and each of them has to pick a number out of $1,2$ or $3$. If there is a player who picked a unique number ...
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1answer
524 views

Finding Pareto optimal solution set in $O(n \log n)$ time

http://cs-people.bu.edu/kvodski/teaching/spring10/lab7.html says: For two points in 2-dimensional space, point ($x_i$, $y_i$) dominates ($x_j$, $y_j$) if $x_i > x_j$ and $y_i > y_j$. Given a ...
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1answer
38 views

How does prior round knowledge affect 2-round voting?

Consider a simple 2-round voting scheme where the 1st round is a decision between two choices $A$ and $B$, and the second round is a decision between the winner of the 1st round and an existing choice ...
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1answer
101 views

A game theory question about tiles, part 2

A recent question asked about the following game: There are six tiles, face down. Three are type A and three are type B. Each player turns over three tiles, and wins if they match. Otherwise, they ...
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1answer
75 views

A game theory question about tiles

So if there are two players playing a tile game where there are two sets of matching tiles $a_1, a_2, a_3$ and $b_1, b_2, b_3$, what would the optimal strategy be to maximize winning probability? Go ...
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1answer
341 views

About game theory for high school students

I am a mathematician with a background in analysis who is teaching at a local high school in his spare time. There is some room for extra curricular math subjects and I want to use it for game ...
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2answers
53 views

Game with two players and 120 points in total

Assume the following game: The game has two players $P_{1}$ and $P_{2}$ and 15 rounds in which they play against each other. Each round gives an amount of points equal to its number, i.e. the ...
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0answers
94 views

How to calculate a Nash equilibrium strategy for toy games quickly?

Given the game of Kuhn poker or Rock-Paper-Scissors, why is it hard to calculate the solution to it? From my very limited understanding, it seems that to solve it you need to employ the counterfactual ...
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0answers
85 views

Nash equilibrium: soccer.

There are five boys playing soccer with a ball and only one port. Initially, the port is occupied by a randomly selected player (the goalkeeper). Each player starts with 10 points. Each player may ...
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1answer
111 views

Two traders don't trust each other; what transactional equation optimises reward and minimises risk?

Years ago while on a Wikipedia browsing binge, I read a maths article about how two (or more) mistrusting parties can reach an transactional equilibrium, but I've wracked my brain and I can't remember ...
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0answers
59 views

Optimizing a population to maximize probability of achieving certain samples.

Preface: I'm reasonably comfortable with mathematics on the whole, but I don't know too terribly much about probability theory, or optimization. I play Magic: The Gathering, and am trying to apply a ...
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1answer
57 views

Games with known outcome but unknown strategy

Is there any two-player game for which it is known that a particular player (not just one of the two players) has a winning strategy but no such strategy is known explicitly? I see that it ...
3
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1answer
84 views

Effecient way to find optimal solution in a 2 player game

I have a function: \begin{equation*} f(a_1,\ldots,a_7,b_1,\ldots,b_4)=-14-7 a_1+30 a_1 a_2-7 a_4-2 a_4 a_5+21 a_6+21 a_7+16 a_1 b_1-24 a_1 a_2 b_1+6 a_4 b_1-6 a_4 a_5 b_1+6 a_1 b_2-6 a_1 a_2 b_2+8 a_4 ...
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1answer
168 views

Solving a linear programming problem: Are my formulations correct?

QUESTION J (PTY) LTD is a fertilizer manufacturing enterprise that produces two types of fertilizers, namely white and gray. The white fertilizer is for crops like maize, sorghum, etc while the gray ...
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2answers
84 views

A Recurrence Equation From a Game

$a_n=a_{n-1}(a_{n}-a_{n-2}+1)$ The above equation is defined in $[0,m]$ st. $a_{0}=0$ and $a_m=1$. It turned up as I was trying to analyze a simple richman game. I have managed to solve the equation ...
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3answers
180 views

Recommended math background for game theory

I recently got interested in some game theory applications to poker. I want to try some of them out programmatically, but a lot of the math is a bit confusing. I learn math on my own fairly quick and ...
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1answer
77 views

Game Probability Problem

Consider a game played by two people $X_1$ and $X_2$. Against the general population, $X_1$ wins with probability $0.51$ and $X_2$ wins with probability $0.49$. We have no knowledge of the ...
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3answers
96 views

solutions poker texas hold'em

Is there any equation that characterizes the poker game in terms of variables such as the strength of the hand, the amount of betting money in the pot, etc? Is there any solution that says what the ...
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1answer
62 views

Simulating Mixed Nash Equilibria

I have a $N$ person game where each person has a set of $M$ discrete strategies. I know from the theory that at least one mixed strategy Nash Equilibrium exists. Can someone please tell me how do I ...
53
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0answers
1k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
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2answers
310 views

Determining the number of valid TicTacToe board states in terms of board dimension

I am attempting to find a closed form equation in terms of $n$, for the number of valid Tic-Tac-Toe board states (ignoring symmetry), where the board has dimension $n \times n ,\; 0 \lt n,\;n \in \Bbb ...
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1answer
62 views

What is the relative strength of each of the players in this game?

This is a real life problem. A group of people meet once a week to play a game between two teams. Each round 2 people are randomly appointed captains. Each captain takes turns picking people to be on ...
9
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2answers
550 views

Determine the winner of a tic tac toe board with a single matrix expression?

Assume a tic-tac-toe board's state is stored in a matrix. $$ S=\begin{bmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ \end{bmatrix} $$ Here, $X$ is mapped to $1$, $O$ is ...
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1answer
78 views

What is a sample space supposed to be?

In this paper, Robert Aumann claim that(page 508): But as shown at the bottom of page 520, all these sample spaces don't admit uncountable independent random variables. What's the implication of ...
2
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1answer
213 views

Axelrod's Prisoner's Dilemma source code

I'm analyzing the prisoner's dilemma and want to reproduce Axelrod's historic computer tournaments (Robert Axelrod, "Effective Choice in the prisoner's dilemma", Journal of conflict resolution). Does ...
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1answer
82 views

Shapley value: an alternative representation

It is my belief that the more common representation of the Shapley value is given by $$ \phi_i(v)=\sum_{S\subseteq N-i} \frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup\{i\})-v(S)) $$ where $v \in ...
0
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2answers
77 views

Decide the most favorable candidate

Consider an election voting process where people need to elect a representative among n number of candidates. Is there an approach to determine the most favorable option? Voting just a single person ...
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1answer
62 views

Correlated Equilibrium

I have a question about the definition of the correlated equilibrium. I see that some authors define it as "expected payoff of playing the recommended strategy is no less than playing another ...
3
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1answer
100 views

How much money should we take?

I'm a new user so if my question is inappropriate, please comment (or edit maybe). We want to define a dice game. We will be the casino and a customer will roll dice. I will assume customer is man. ...
17
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4answers
376 views

Salvaging a damaged cable

Let's say we have a cable of unit length, which is damaged at one unknown point, the location of which is uniformly distributed. You are allowed to cut the cable at any point, and after a cut, you'd ...
2
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1answer
186 views

Von Neumann's minimax theroem and Carathéodory's theorem

In J.F. Mertens(1986)(Paywall), there's a neat proof of a version of Von Neumann's minimax theroem. But I can't understand how Carathéodory's theorem is invoked. Suppose, in a two-person zero sum ...
2
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1answer
112 views

Up to how much would one pay to play this game? 30 red and 30 blue marbles

There are 30 red marbles and 30 blue marbles. Your opponent may arrange these marbles in any way he/she chooses into 2 urns. You then pick one of these 2 urns. You get 10 dollars if you draw red and 0 ...
7
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1answer
372 views

Determinant game - winning strategy

I came across this problem while looking at Putnam problems a while ago: "Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008 x 2008 array. Alan plays ...
2
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2answers
37 views

is there a difference between counting those better than you vs counting those worse than you in relative scoring games with ties allowed

This is my first question so let me know if I am doing something wrong. Imaging a relative scoring game. What I mean by this is a game with a set number of players... lets just say 100 where the ...
1
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3answers
503 views

Chance of Winning In Tic Tac Toe

I'm sure everyone knows how to play the game of tic-tac-toe. I have just been wondering what's the probability of winning if one player started his or her move by putting his mark in the middle?
2
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1answer
174 views

Duality and the Minimax Theorem

I review LP duality by reading Lecture 7: The LP Duality Theorem. I get the idea how to find the dual LP from primal LP, but my basic knowledge is not enough for finding dual LP for the LP in chapter ...
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0answers
31 views

Uniqueness of equilibrium points in bimatrix games

I'm searching for some results about the uniqueness of equilibrium in a bimatrix game. In all articles that I can find the study is about existence of two matrix $A,B$ that have a given couple (x,y) ...
4
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1answer
87 views

Ehrenfeucht–Fraïssé game, how can I understand it?

My course of "Formal Methods" deals with Ehrenfeucht–Fraïssé games, particularly regarding the inexpressibility in FO logic. At the moment I've fully understand what this games are and how they are ...
4
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2answers
236 views

Bounded sequence with divergent Cesaro means

Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)? More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that ...
1
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1answer
179 views

SPNE of infinitely repeated game

Let $G$ be a game with finitely many players and $\underline{v}= (\underline{v}_i)$ be the minmax payoff profile. Denote by $G_{\infty}(\delta)$ the infinitely repeated game whose stage game is $G$ ...
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0answers
34 views

Cyclic Matching Pennies Game

I read about a game, a variant of matching pennies where there are 3 players. What if there are say 6 players located on a circle? Player $1,3,5$ want to match with both of his $2$ neighbours and ...
0
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1answer
82 views

How to mathematically express in a payoff matrix that “not losing” isn't the equivalent of winning

My uncle was watching a documentary on the revolutionary war and one of the historians said, "Washington realized he didn't need to win the war, he only needed to not lose it." Is it possible to ...
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0answers
45 views

Recursive core of coalition game

Can someone please explain the recursive core concept possibly with an example? http://arno.unimaas.nl/show.cgi?fid=5152 I don't understand how the recursion works. Thank you
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0answers
93 views

Convex combination of correlated equilibria

Prove that any convex combination of correlated equilibriua is also a correlated equilibrium.