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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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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107 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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57 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
56 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 ...
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1answer
77 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
160 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
81 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
177 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
76 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
95 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 ...
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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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1answer
286 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 ...
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2answers
514 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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42 views

Extended Probabilities in Risk

I'm working on a paper furthering some research into probabilities in the RISK game. The paper my research is based from is Osborne's paper on the use of Markov Chains, ...
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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 ...
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1answer
202 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 ...
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2answers
76 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 ...
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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. ...
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4answers
374 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 ...
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1answer
185 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 ...
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1answer
106 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 ...
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1answer
357 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 ...
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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 ...
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2answers
443 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?
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1answer
158 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
29 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) ...
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1answer
82 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 ...
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2answers
222 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 ...
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1answer
167 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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33 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 ...
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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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43 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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93 views

Convex combination of correlated equilibria

Prove that any convex combination of correlated equilibriua is also a correlated equilibrium.
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0answers
42 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
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118 views

Routing Game : Modified Pigou's example

Suppose that we modify Pigou’s example so that the lower edges have cost function $c_1(x)=1$ and $c_2(x) = (x/n)^d$ for some $d ≥ 1$. What is the price of anarchy of the resulting selfish routing ...
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1answer
70 views

A combinatorial game about stones

There are some piles of stones. Two players move in turn. One can remove a stone from a pile or merge two piles in a move. The player that removes the last stone wins. With the number of stones in ...
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0answers
103 views

Mathematical reason for 2-player turn-based games

I've been reading Games, Puzzles, and Computation which analyzes games through game theory and complexity theory. The authors introduce something called "Constraint Logic" as a way of modeling games ...
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2answers
2k views

A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe

Consider a 9 by 9 matrix that consists of 9 block matrices of 3 by 3. Let each 3 by 3 block be a game of tic-tac-toe. For each game, label the 9 cells of the game from 1-9 with order from left to ...
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82 views

Optimal auction for risk averse seller

Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim ...
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1answer
180 views

Air Strike Game

This is an Air Strike Game with the solution, I have added some questions regarding the solution and I would appreciate if someone could answer them. Army $A$ has a single plane with which it can ...
4
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1answer
192 views

Pirate Game (modified)

http://en.wikipedia.org/wiki/Pirate_game What happens if you remove the order of seniority? Whenever a pirate dies, you randomly pick the next pirate to propose a distribution. Here's my solution ...
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1answer
70 views

Expected revenue in first bid auction.

We find expected revenue in first bid auction by following method. let us say $V_1$ and $V_2$ denotes maximum amount that player 1 and player 2 willing to pay. $V_1,V_2 \in [0,1]$ In case when we ...
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1answer
62 views

Generlized Büchi Games and Closed under superset Muller Games

For a unique infinite play $p$ in a 2-Player game $G=(V_0,V_1,E)$. Let $$ \inf(p) \subseteq V_0 \cup V_1 $$ be the set of vertices which occur infinitly often in $p$. Generlized Büchi (GB) Games ...
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2answers
222 views

Exercise in Mechanism Design

I found an exercise with solution in the field of Mechanism Design. The problem is I don't understand the solution. Exercise. Use the characterization of incentive compatible direct-revelation ...
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2answers
133 views

Fastest way to get \$1 million of two different currencies in a video game

This question actually relates to a video game, I came across the scenario and I realized I had no idea how to go about solving something like this or even what branch of mathematics it falls under. ...
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3answers
125 views

Is there any research field dedicated to estimating a “game” itself in game theory?

Game theory stuffs usually provide how a "game" works and then tries to figure out solutions - but I am wondering if there is any research field dedicated to estimating the full rules of a game. So ...