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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109 views

Game Theory: players' gender convention?

What is the Game Theory convention of using gender terms (male/female) for the players? I found only one reference suggesting that odd-numbered players are male and even-numbered players are female. ...
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1answer
65 views

Fair three-way sandwich division

This question discusses fair three-way sandwich division. Mentoined solutions include the Selfridge–Conway discrete procedure and the moving-knife procedure. I posed the question to the guys at the ...
3
votes
2answers
299 views

Existence of Saddle Point of a Matrix (Shapley's Theorem)

A $m\times n$ matrix $M=(a_{ij})_{m\times n}$ with real entries is said to have a pure saddle point at $(i_0,j_0)$ if $\min_j \max_i (a_{ij}) = \max_j \min_i (a_{ij}) = a_{i_0j_0}$. Here the notation ...
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1answer
43 views

Prove $f(x,y) > g(x,y)$ for all $x,y \in [0,1]$

I'm trying to prove the following: $$ 4xy + 4(1-x)(1-y) < \max\{8xy,8(1-x)(1-y),3\} \qquad \forall x,y \in [0,1] $$ In the language from the class, I'm trying to show that: $m_2 < ...
0
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1answer
121 views

Special Counterexample to Kakutani's Fixed-Point Theorem

For reference, here is the statement of Kakutani's fixed point theorem. Let $X$ be a compact, convex subset of $\mathbb{R}^n$ and let $f:X\to \mathcal{P}(X)$ be a set-valued function such that $f(x)$ ...
1
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1answer
86 views

Card game: How much will you pay to gamble?

You turn over the cards 2 at a time, if they are both red, you keep the cards, if they are both black I keep the cards. If one is red and the other is black then neither you nor I get a card. If you ...
2
votes
1answer
126 views

Brouwer's fixed point theorem

Theorem: If $f:D^n\rightarrow D^n$ is continuous then there is $x \in D^n$ such that $f(x)=x$. To prove the theorem we assume that $f$ is cts but has no fixed point, that is $f(x)\neq x$ for all ...
1
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1answer
59 views

Coin based subtraction game

I'm having a problem in Game Theory where I am trying to understand how a subtraction game can be interpreted by a coin based game. From my book: The problem I'm having is if I have 9 coins and the ...
1
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1answer
91 views

Explanation of basic definitions in game theory.

In the article entitled Non-Cooperative Game written by Nash in 1951, he discussed about the symmetries of games. Due to my lack of basic knowledge in permutations and symmetries, I looked up some ...
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0answers
45 views

On a certain type of card game

Suppose two players are playing a card game, which is described as follow. Each player is allowed to construct their own decks of exactly $n$ cards with an additional repeatable card, where each ...
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2answers
94 views

Why should a GE fail to exist in non-convex sets?

In an exchange economy with $2$ goods and $m$ identical Households where each household has utility function $u(x_1, x_2)$, together with positive endowments. If preferences are not convex, then why a ...
4
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1answer
213 views

A game of Chess - Ideal Solution

I am a student of physics. I have learnt some basic group theory, and I am wondering if there is any ideal solution for a given Chess game (like solving Rubik's cube). I know the no. of permutations ...
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0answers
31 views

Prove that the partial derivatives of $(y-g_i+a\sum^n_{j=1} g_j)$ are positive

I have a function: $$\pi_i^1=y-g_i+a\sum^n_{j=1}g_j,$$ where 0 < a<1< na, and I need to prove this: $$\frac{\partial(\sum^n_{i=1}\pi^1_i)}{\partial g_i}=-1+na>0.$$ I am not very ...
6
votes
1answer
203 views

Monkey typing ABRACADABRA and gamblers

Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
0
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1answer
45 views

A little question about the existence theorem of Nash equilibrium in game theory

Recently when I started reading Nash's paper, I found a little question about the linearity of payoff functions. Is it an assumption? Or did I miss some idea about the payoff function and its ...
1
vote
1answer
52 views

Questions about Auctions

I am having a hard time figuring out a problem. In a first price auction with a reserve price R and values of the bidders are U[0,1], how do we find expected revenue given the strategy of both of them ...
2
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1answer
74 views

Deriving statistical distributions from games

The normal distribution can be derived from basic principles and calculus The Normal Distribution: A derivation from basic principles. Are there other distributions that can be derived like this from ...
2
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2answers
87 views

A Special Type Of Guess The Number Game

There is Guess Number game like this: In this game, the player must find a hidden positive number by at most $T$ guesses (or turns). The parameter $T$ together with a health parameter $H$ is ...
5
votes
2answers
107 views

Can you incentivise competitors to handicap accurately, and also try to win?

A problem I ran into for real. A group of friends of widely differing abilities wants to hold a handicap cycling race, so that if everyone does about as well as expected, there would be a perfect dead ...
4
votes
1answer
100 views

What is a good strategy for this dice game? [duplicate]

I learned the following dice game from another forum. It was not solved there. The dice game is as follows. You start tossing six dice. After each toss you must put aside at least one of the dice ...
0
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1answer
43 views

How to recognize a proper sub game

The extensive form game in both diagrams appears the same, why the difference in the number of subgames?
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0answers
44 views

Waiting Time For Computer Cluster

There are $n$ computers. Computer users stay on their computers for a certain amount of time, $t$, throughout the day. Computer users come and go. How long will I have to wait, min/max, for a computer ...
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5answers
5k views

Help: rules of a game whose details I don't remember!

In a probability course, a game was introduced which a logical approach won't yield a strategy for winning, but a probabilistic one will. My problem is that I don't remember the details (the rules of ...
0
votes
2answers
56 views

Help me come up with a function

I have some numbers and corresponding numbers: 0 = 0 1 = 0 2 = 1 3 = 0 4 = 2 5 = 1 6 = 3 7 = 0 8 = 4 9 = 2 10 = 5 11 = 1 12 = 6 13 = 3 14 = 7 15 = 0 16 = 8 17 = 4 ...
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2answers
93 views

Expected value and optimal strategy for red/blue game

Firstly please excuse my ignorance if I'm posting this to the wrong exchange site. If this doesn't belong here let me know and I'll move it. Now as for my question, today during a short course that I ...
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0answers
48 views

Why is the feasible set of utility values (in bargaining problem) convex?

Let $S := \{x \in \Bbb{R}^n \mid x \ge 0, \sum_{i=1}^n x_i = 1\}$ be the set of mixed strategies. For a bimatrix game with pay-off matrices $A$, $B$ we denote $C := \{ (u, v) \mid \exists (x,y)\in ...
3
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0answers
52 views

News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
0
votes
1answer
55 views

About mixed strategy Nash Equilibrium

Does a payoff matrix like $$\begin{array}{c|cc} &B& B&\\ \hline A& (0,1)& (1,1)\\ A& (1,1)& (0,1) \end{array}$$ has infinite number of mixed-strategy Nash ...
0
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1answer
71 views

Game theory reference for a beginner

I need to use game theory to model interaction in a network. What are some books or lectures that a beginner in game theory could use to understand the theory well?
0
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1answer
32 views

calculate win chance and win or not

I've got a logical problem with my mathematics skills. so far, I have to calculate a prizing system to say "you won" or "you lose". here my way until now: u = max. users p = prizes w = winchance in ...
2
votes
1answer
82 views

Game theory: proof V is convex and compact

Consider a non-cooperative game $(N,S_i,u_i)$. $\bullet \ N = \{1,...,n\}$ is the set of players. $\bullet$ For every player $i$, the set $S_i$ is the finite set of pure strategies. $\bullet$ For ...
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1answer
71 views

Implications of axioms of expected utility theory

Axioms for Expected Utility: Let $\succ $ be a binary relation on $X$. A1. $\succ $ is asymmetric and negatively transitive. A2. Independence of Irrelevant Alternatives: If $p,q,r \in X$ and if ...
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2answers
57 views

Designing a game for peer evaluation

Say you have a class of $100$ students and you would like the students to grade each other's work. Is it possible to design a game theoretic scheme in which a rational student would mark fairly? The ...
2
votes
1answer
208 views

Largest white rectangle on board

Given a string rectangular board which is divided into unit cells. Each cell is initially painted black or white. The character board[i][j] represents the cell with coordinates (i, j). Each of those ...
2
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1answer
122 views

Play with pairs of numbers

Two players are playing a game. The game is played on a sequence of positive integer pairs. The players make their moves alternatively. During his move the player chooses a pair and decreases the ...
1
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1answer
112 views

Card game question with conditional probability

If I have a standard deck of 52 cards with ace being 1 and jack being 11, queen 12, and king 13. I know the expected value of any card I draw to be 7. However, how does this change with each ...
0
votes
1answer
134 views

Dice roll game fair price

You and I each roll 1 die each at the same time. I win if I roll a six on one roll, and then a five on the next. You win if you roll two sixes in a row. Who would you bet your money on? Note that You: ...
1
vote
1answer
67 views

Game of chocolates

Two players A and B play a game alternatively and A starts the game. Their are 2 boxes of chocolates and we are Given the number of chocolates in both the boxes, let them be c1 and c2, the player ...
0
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0answers
21 views

Bad coditioned games

Given a bimatricial game $(A,B)$, what can I say about distances between its nash equilibria and the nash equilibria of another perturbed game $(A + \delta A, B + \delta B)$. In other words, do ...
2
votes
1answer
98 views

A game of kettle

I just got inspired by a situation in my office: Let us assume there are $n$ players. Each player must in some time go to the kitchen and prepare some tea. Each player loses $t_1$ to fill the kettle ...
17
votes
4answers
521 views

A Category Theoretical view of the Kalman Filter

Some basic background The Kalman filter is a (linear) state estimation algorithm that presumes that there is some sort of uncertainty (optimally Gaussian) in the state observations of the dynamical ...
406
votes
25answers
52k views

Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...
3
votes
1answer
75 views

Mathematics of a Simple Counting Game

I wonder how can one think mathematically about the following game: People sit in a circle. One of them says "One!". Then somebody (no matter who - he/she can even be the former person) says ...
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votes
1answer
129 views

A Nim game variant: Odd number [closed]

Consider the variant of Nim where the allowed moves are the removals of an odd number of stones from a heap. Who's the winner and what is the winning strategy in normal play (player unable to move ...
0
votes
1answer
52 views

How to maximize your score in this game?

Consider the following game: On a table, there are two glass bowls with $x$ and $y$ beads each. In each move you can do the following: Remove a bead from the first bowl and put it in the second ...
2
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1answer
59 views

What are the optimal strategies for the “prime-game”?

A and B are playing the following game : A and B choose a number from 1 to 100, not knowing the number chosen by the opponent. A wins if the sum of the chosen numbers is prime, otherwise B wins. ...
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1answer
23 views

Understanding a component of an equation of finite game repetition

I have the following equation: T = total times the game is repeated t = current period delta = discount rate u_i(a_i^t...) = the state payoff to player i I'm attempting to understand the ...
2
votes
1answer
158 views

Plotting the best response

Observe the following matrix; The pure strategy and mixed strategy nash equilibria are The best response plot is given below Can someone explain how this graph was plotted. I would much ...
5
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1answer
136 views

How to simply show that there are “78 'strict ordinal' 2x2 game matrices”

In "Theory of Moves", Steven J. Brams analyses two-player games with two strategies per player, where each player can totally rank his payoffs, although payoffs need not be comparable among players. ...
4
votes
1answer
79 views

How to design a cost sharing auction format for collective bidding?

Problem goes like this: There's one resource which can only be utilized by a single set of agents $A_i$ (at any one time) out of $n$ predefined (disjoint) sets of agents. Each agent wants to use the ...