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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10
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1answer
252 views

Toss a fair die until the cumulative sum is a perfect square-Expected Value

Suppose we keep tossing a fair dice until we want to stop, at which point the game ends and our score is the cumulative sum, or until the cumulative sum is a perfect square, in which case we lose and ...
1
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1answer
50 views

What is the mixed strategy Nash equilibrium in this game?

$$ \begin{array}{cccc} & Q & R & S \\ K & [2,0] &[4,5]&[1,1] \\ L &[3,2]&[1,0]&[0,0] \\ M &[1,1]&[1,0]&[0,0] \\ \end{array} $$ What is the mixed ...
1
vote
1answer
66 views

Aumann-Shapley Uniformly Better Principle

Let $n_1,..,n_r$ be $r$ positive integers, and let $1 \leq k \leq n$, where $n=n_1+...+n_r$. Consider an urn containing $r$ different types of balls, $n_1$ balls of type 1, $n_2$ balls of type ...
2
votes
1answer
153 views

Commutative Algebra and Game Theory

Is there any relationship between commutative algebra and game theory? For example, have any tools in commutative algebra been applied to game theory? A text or reference would be ideal, but I'd be ...
0
votes
1answer
71 views

2 x 2 matrix game and expected value

I have found the optimal strategy for the row player and column player. How do I find the expected value of the game for the row player and determine whether the game is favourable to the row player ...
6
votes
2answers
162 views

Eating chocolate game on grid

Given is a chocolate of size $m\times n$. Anne and Birgitte plays a game, with Anne starting. In each turn, the player has to divide the chocolate into two rectangular parts along the lines, and eat ...
26
votes
6answers
2k views

A beautiful game of gold and silver coins

A stack of silver coins is on the table. For each step we can either remove a silver coin and write the number of gold coins on a piece of paper, or we can add a gold coin and write the number of ...
1
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5answers
324 views

Game Theory - First move vs second move advantage?

This question came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math, and we weren't sure how to get the answer. I apologize in advance if my ...
1
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1answer
58 views

quasi rationality, interesting axiom of revealed preferences

So imagine there is a notion of rationality that captures the idea of "thresholds in preference." For example, let $\mathbb{Z}$ be the integers: $\mathbb{Z} = \{\dots, -10, -9, \dots, 0, 1, 2, ...
4
votes
4answers
751 views

Gambling problem

Question Robert will win $\$1$ with probability $\frac{1}{4}$, win $\$2$ with probability $\frac{1}{4}$, and lose $\$1$ with probability $\frac{1}{2}$ in a bet. Each bet is independent. Determine ...
1
vote
1answer
32 views

Static game of incomplete information.

There is a buyer and a seller. The seller wants to sell a used scooter. The scooter can either be of good or bad quality. The quality of the scooter is only observed by the seller. To the ...
3
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2answers
185 views

Mixed Nash equilibrium for non-square matrix game

I'm stuck with understanding the way of finding mixed strategy Nash equilibrium for non-square matrices and want to explain my difficulties with the help of the following example. Let the following ...
2
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0answers
21 views

dimension of Weber set and selectope (as a operator)

Let $\Omega$ be a finite set of players. For a selector $\alpha:(2^{\Omega}-\{\emptyset\})\rightarrow\Omega$, we define a marginal value operator as a linear operator $m^{\alpha}$ ...
0
votes
0answers
29 views

If there is dummy voter, then SSI(A) and BI(A)=0??

for Shapley–Shubik power index and banzhaf power index, If there is dummy voter A, then SSI(A) and BI(A) have got to be 0? is there any counter example to it? I think there is no counter example to ...
2
votes
1answer
47 views

Game theory books on learning e.g. Fictitious play

I'm looking for some text books on learning in game theory. So far I only found The Theory of Learning in Games by Fudenberg and Levine. Are there others you can recommend?
0
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0answers
52 views

what is the definition of pure strategy of zero sum games?

what is the definition of pure strategy of zero sum games? I tried to google for results, but no clear definition came up . EDIT: I found a definition in a game theory book . told me exactly what I ...
1
vote
2answers
50 views

Game theory expected value

We play a game involving two players. Each player calls a number 1 or 2. If the sum of these numbers are odd (i.e. equal to 3), then player 1 gets 3 points and player 2 loses 3 points. If the sum of ...
1
vote
1answer
49 views

How to note down an infinite game where strategy gets changed at a certain point

I have to write a proof concerning a grim trigger in an infinite Prisoners' Dilemma game. I can write the utility for both players cooperating infinitely. But suppose at some turn $n$ one player ...
0
votes
1answer
118 views

Nim Sum Game Variant

Suppose there are black and white balls in a box. The initial number of white balls is m and the initial number of black balls is n. This is a two player game. Each player can do the following taking ...
2
votes
0answers
131 views

Game theory: connect four?

Through Allis' solution etc... P1 can force a win if he places the first stone in the highlighted region. Assuming both players have perfect information, it will take 41 turns maximum (if I recall ...
2
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0answers
113 views

Game theoretical approach to other branches of mathematics

Are there some methods and ideas derived from game theory that are successfully applied to better (or more intuitively) understand theorems and proofs or tackling problems from other areas of ...
4
votes
4answers
197 views

A conjecture about Nash Equilibria in multiplayer games involving card drafting

A deck of $N$ cards is used to play a 4-player game. The game begins with each player being randomly dealt 7 cards from the deck. They then take turns according to a set of rules, after which a single ...
1
vote
0answers
80 views

Putnam game theory question

There are $n\ge1$ boxes in a line where $n$ is an odd integer. Two players, Connor and Andrew, are playing a game. On a turn, you can place a stone in a box OR take a stone out of a box and place a ...
4
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1answer
53 views

Optimal decision in game of Memory

Suppose two players are playing a game of Memory with $2n$ tiles consisting of $n$ distinct pairs. (To play, you publically reveal two tiles. If they match, you keep them and take another turn; if ...
2
votes
1answer
163 views

Counting all possible board positions in Quoridor

I'm trying to figure out how many possible board positions there are for the game Quoridor. I think sorting out the legal positions from the illegal positions will be difficult, so to start I'm trying ...
0
votes
1answer
69 views

How to write induction proof of Sprague-Grundy function for subtration game?

So lets say that S={1,2,3} I find the sequence of Sprague-Grundy function. How do I justify my answer using induction?
3
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0answers
50 views

Simple undetermined games

We know that, under AC, there exists a game in which two players play finite numbers and neither one has winning strategy. There are also such undetermined games when we consider players playing ...
0
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0answers
48 views

Monty Hall problem extended with expectations i.e. prior probabilities

I am fascinated by the Monty Hall problem and its variants such as N-doors version here. Now suppose expectations. How does the Monty Hall problem changes with expectations? Simple example ...
2
votes
0answers
96 views

What is the largest value one can get in game 2048 without new tiles appear

This is a simplified version of the famous game 2048. Given a 4x4 grids with some values chosen from {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048}. A value of 0 indicates that the position in ...
3
votes
0answers
108 views

Maximizing dot-product score by asking queries

Let $a>b>0$, and let $T=\{a,b\}^n$ be the set of all $n$-tuples each entry of which is $a$ or $b$. Let $X\subseteq\{0,1\}^n$ with $|X|>1$, and let $f:T\rightarrow X$ be a function. For each ...
0
votes
0answers
37 views

Mixed strategy approximations in a large game

I'm new to game theory and was thinking about how to find mixed strategies for a particular game, and furthermore how to find good mixed strategy approximations of a game that may be too large to ...
1
vote
1answer
32 views

dominated strategy can never be used in mixed strategy nash equilibrium

dominated strategy can never be used in mixed strategy nash equilibrium. how to prove it? Sounds like obvious, but how to write the proof mathematically correct?
1
vote
1answer
141 views

Second-price sealed-bid auction uniformly independent with unknown value

a disclaimer before the question: this is a homework problem. I just want some help/push in the right direction, I'm kind of stuck! The problem is as follows: In a second-price sealed-bid auction for ...
0
votes
1answer
50 views

Find the nim-sum of all numbers from 1 to ((2^n)-1) where n >1 is a natural number

I started learning nim sum , the examples given in class were all two number kind of problem . What should I do with this kind of problem ?
1
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1answer
41 views

$3\times 3$ matrix game where PI has a totally mixed optimal strategy, but also has an optimal strategy which is not totally mixed.

I am asked to give an example of a $3\times 3$ matrix game where PI has a totally mixed optimal strategy, but also has an optimal strategy which is not totally mixed. So I know that ...
1
vote
1answer
52 views

Updating in game with normal distribution

In a game from the following paper, it is stated that Player $i$ observes a private signal $x_i = \theta + \epsilon_i$. Each $\epsilon_i$ is independently normally distributed with mean $0$ and ...
1
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2answers
51 views

Strategy In auction bidding

I was at an auction recently where three autograph signatures of Marilyn Monroe were up for auction, not as a single lot, but in three separate lots. The three lots were virtually identical, and all ...
1
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1answer
87 views

Battle Ship Winning Algorithm - Optimal Strategy

I have an $8 \times 8$ grid. I have three ships that are $4$ long, $3$ long, and $2$ long. Is there an algorithm that can ensure a win every time? Oh! Most importantly, you must know the number of ...
0
votes
1answer
170 views

Relationship between regular Nim and Lasker's Nim

So I'm trying to do qn $6$ (on pg I-13) about staircase Nim in Game Theory by Ferguson Game Theory, Ferguson and it's asking to prove that $(x_1, x_2, \ldots, x_k) \in P $ only if $(x_1, x_3, x_5, ...
15
votes
8answers
2k views

Understanding the solution of a riddle about lions and sheep.

I heard a riddle once, which goes like this: There are N lions and 1 sheep in a field. All the lions really want to eat the sheep, but the problem is that if a lion eats a sheep, it becomes a sheep. ...
2
votes
1answer
61 views

Does the first player never lose this numbers game?

There is an even number of numbers in a row. Two players cross out numbers one by one from left or right. It is not allowed to cross out a number in the middle. Only left or right. After all numbers ...
0
votes
3answers
94 views

Game Theory Voting Utilities

! So far, I've managed to come up with this solution: ! But as far as here...I can convert this into payoffs, however I'm unsure of how to figure out the Nash equilibria as when we convert from ...
1
vote
0answers
37 views

Prove that a mixed strategy in two player, zero sum, matrix game must exist (alternative proof)

So I am having a trouble with this game theory proof. I feel pretty good with my answer for part 1, but I am not really sure how to get started on the rest of it. Any help would be appreciated. Let ...
1
vote
2answers
37 views

Can a game with negative expectation still have a positive utility?

Intuitively, I think not. But I can't clearly prove why. Specifically, I've been thinking about lottery games, where the expectation is obviously negative. But can the utility of hitting the jackpot ...
3
votes
1answer
92 views

Resources for understanding game trees?

I am trying to make an AI to solve the popular game $2048$, and I think that the theory of game trees would help me quite a bit in this endeavor. The only issue is that most of the results I've found ...
2
votes
1answer
66 views

Coin-tossing games

Suppose that you start off with $100$ dollars. You toss a coin $10$ times and guess it right $5$ times and lose $5$ times (the order of the outcomes is not known). It is known that every time you ...
1
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2answers
146 views

Probability that 20 sided die beats 12 sided die with reroll

Alice rolls a 12 sided die (the faces labeled 1 through 12) and Bob rolls a 20 sided die (the faces labeled 1 through 20). After seeing their roll (but not the other person's roll), each person can ...
-1
votes
1answer
277 views

Finding winner of flipping game

Alice and Bob play a game with N non-negative integers. Players take successive turns, and in each turn, they are allowed to flip active bits from any of the integers in the list. That is, they ...
0
votes
0answers
52 views

Game theory (2 player, random number) question

So we have a game in which 2 players, P1 and P2, are randomly given a number $x_1,x_2\in [0,1]$. Each player first antes \$1, are given their numbers, and then P1 can choose to bet any value $B$, or ...
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0answers
39 views

Simple Dice Game - Optimal Mixed Strategy

Just started an introductory course in game theory, and here is a problem we have been talking about. So here is the description of the game. Two players, each player starts by placing \$1 each into ...