Tagged Questions

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).

learn more… | top users | synonyms

0
votes
1answer
13 views

Determining the Likelihoods of Different Game States

Suppose a game is played in which Player 1 must gain two points to win and Player 2 must gain five points to win. Both players start with zero points. In any round, Player 1 has a $1/3$ chance of ...
-3
votes
0answers
25 views

How to write recurrence relations from a verbal description(Question from Oxford math admission test)? [on hold]

Questions are interesting because they only require primary math skill. They have general patterns that developing the problem from specific to general. The 5th and 7th questions in the paper(linked) ...
0
votes
0answers
15 views

Abother question on game theory [duplicate]

There is a polygon with $N$ vertices drawn in the plane. The polygon is strictly convex, i.e., each internal angle is strictly smaller than 180 degrees. The vertices of the polygon are numbered 1 ...
0
votes
2answers
53 views

A tough dice probability conundrum

There are 2 fair die. You randomly roll them at the same time. To win is to have the sum of the die or some combination of the sum or the value of each die cover the numbers 1-9 inclusive. You can ...
0
votes
1answer
29 views

Game Theory Nash equilibrium

How do we do in order to find Nash Equilibrium in a $3$ players game with $3$ possible strategies for each player?
0
votes
1answer
19 views

Proof sprague-grundy value is 0 if and only if it is losing position

So, i take this game theory module this summer, and i encountered this exercise problem, i tried to do this by induction by have terminal position (grundy-value = 0) as base case, but can't figure out ...
75
votes
10answers
8k views

Mathematician vs. Computer: A Game

A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same ...
2
votes
0answers
10 views

Bingo based on number of hits on card?

I was wondering how many possible combinations there were to win based on the number of hits on a bingo card (25 spots, 1-75 etc.)? I know that if you get 1,2,3 hits on the card, there are no chances ...
1
vote
0answers
23 views

Legendre transform and Minimax Theorems.

Denote the class of lower-semi-continuous convex functions $f:\mathbb{R}^n\to \mathbb{R}\cup\{\pm\infty\}$ by $Lscx(\mathbb{R}^n)$ ( so that only function attaining the value $-\infty$ is the constant ...
1
vote
1answer
37 views

Formulating recurrence relation

Alice and Bob worked in a restaurant and received n currency notes in total as tips. Every note has a value (either 1 dollar, 5 dollar or 10 dollar) written on it. The currency notes are arranged from ...
3
votes
1answer
77 views

Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
1
vote
1answer
21 views

Bayesian update from uniform prior to uniform posterior ?!?

I was working through a signaling game problem recently and the proof suggested the following: Actor A has a type: $\ \mathscr{t} \sim Uniform[-1,1]$ Actor A gives signal $\pi^*$ that perfectly ...
0
votes
0answers
27 views

Discrete Structrue

I was stuck with the following problem. Two players A and B play a game where they take turns adding numbers from 1 through 10, and the first person who gets to the target of 100 wins. Assume A ...
4
votes
2answers
82 views

Best strategy for this game. [Nintendo Wii game]

Which is the best strategy for this game? Actually is a Nintendo Wii game. It's a 4 people game. There is a ladder of 10 steps. Each player says a number (they can chose between those numbers ...
0
votes
1answer
23 views

The strategy of row player in $2\times 2$ ordinal game

I am proving the simplest part of theory of moves. Assume that both players are rational, and consider the following $2\times 2$ ordinal game: $$\begin{bmatrix} (3,3) && (2,4) \\ (4,2) ...
1
vote
1answer
50 views

Why is the “rational” solution to the Traveler's Dilemma 2?

Traveler's Dilemma An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical items. An airline manager tasked to settle the ...
1
vote
0answers
20 views

Approximate Solution to Backwards Recurrence of Dynamic Game

Suppose we keep tossing a fair dice until we reach some cumulative sum greater than or equal to $N$. Then, let $S_k$ be the expected value of the final sum, given that the current sum is $k$. We have ...
0
votes
2answers
27 views

Expected value for number

Suppose you have a game with $n$ stages. For every stage $i$, you have $p(i)$ probability to advance to the next stage, and $1-p(i)$ probability to return to stage $1$. You win the game by advancing ...
1
vote
1answer
66 views

Marriage Market Proof (Alternative Proof of Rural Hospitals Theorem)

How do I get (a) + (b) + (c) $\implies$ (d) $\implies$ (e)? (a) Show that for each $m \in M$, if $\mu(m) = \emptyset$ for some stable matching $\mu$, then for the woman-optimal matching, $\mu_W$, ...
1
vote
0answers
11 views

Repeated games with observable actions

I'm going to hold a lecture in a study circle about Repeated games and observable actions and also a little about Repeated games with imperfect public information. We are following the book of ...
5
votes
1answer
65 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
vote
1answer
20 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
0answers
46 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
68 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
26 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
83 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 ...
23
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
vote
5answers
77 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
vote
0answers
30 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
710 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
19 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
votes
2answers
46 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
votes
0answers
15 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
19 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 ...
1
vote
1answer
27 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
votes
0answers
31 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
32 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
43 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
57 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 ...
1
vote
0answers
58 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
votes
0answers
102 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 ...
2
votes
0answers
55 views
+50

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
51 views

Putnam game theory question

There are n>=1 boxes in a line were n is an odd integer. Two players, Connor and Andrew, are playing a game. On 'your' turn, place a stone in a box OR take a stone out of a box and place a stone in ...
4
votes
1answer
45 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 ...
0
votes
1answer
42 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
17 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
votes
0answers
35 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
votes
0answers
31 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
61 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
97 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 ...