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].

learn more… | top users | synonyms

0
votes
0answers
59 views

How many distinct strict ordinal 2x2x2 games exist?

Consider the same type of strict ordinal games as described in How to simply show that there are "78 'strict ordinal' 2x2 game matrices" and add a third player with two strategies (...
0
votes
0answers
23 views

Sustaining cooperation

The question poses: Explain why better monitoring of treaty compliance can be more effective at sustaining cooperation I'm assuming this question is talking about one's reputation for example? If my ...
1
vote
1answer
28 views

Optimal point selection to maximize length of closest points interval

Consider the following game: there are $n$ players, who pick $x_i\in I = [0,1]$ in turns, $1\leq i\leq n$. For each selection $x = (x_1,\dots,x_n)\in I^n$, the player's $i$ reward is the length of the ...
1
vote
2answers
39 views

IRV failing monotonicity criterion

I am looking for the simplest possible example of instant runoff voting failing the monotonicity criterion. By “simplest possible” I mean the scenario with the fewest number of candidates $(3)$ and ...
0
votes
0answers
45 views

Combinatorial Allocation Problem

The problem I am trying to solve is that there are $m$ distinct items to sell through a combinatorial auction and bids have been received. But for any pairs of bids $b_i(X)$ and $b_i(Y)$, the subsets $...
1
vote
0answers
214 views

Game theory book by Tirole,Fudenberg, zero up to first order of $\epsilon$,equilibrium

In this book on Game Theory, on page 186,I do not understand the very end of the page: [T]he incentive to deviate$-$the left hand side of equation 5.18$-$ is $0$ in first order of $\epsilon$, so that ...
6
votes
1answer
53 views

Prove that the sum $a_1+a_2+…+a_n+b_1+b_2+…+b_n$ cannot equal to $0$

We are given an $n \times n$ board, where $n$ is an odd number. In each cell of the board either $+1$ or $-1$ is written. Let $a_k$ and $b_k$ denote the products of the numbers in the $k$-th row ...
5
votes
1answer
52 views

Show that all the cards contain the same number.

Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible ...
2
votes
0answers
81 views

Game Theory Duel Problem

We have the following duel problem: http://mathoverflow.net/questions/75318/the-duel-problem (You can read about it here). We have $P:\frac12, \frac23, \frac34, 1$, Q: $\frac14, \frac13, \frac12, 1$. ...
0
votes
0answers
30 views

Probabilistic Games

I've been working on a problem in my probability class. Suppose there are two players, $i$ and $j$, and they are playing a game where each player has $k$ strategies (i.e. player $i$ has strategies $A$ ...
2
votes
1answer
22 views

What probability p corresponds to an expected number of 10 turns

The problem below is from a problem set for a Game Theory course. We never really touched on much probability, probability distributions, etc so I was surprised when I saw this question... "In ...
0
votes
0answers
88 views

Game theory question inflation and macro

Suppose the Federal Reserve can fix the inflation level ˙p by an appropriate choice of monetary policy. The rate of nominal wage increase W˙, however is set not by the government but by an employer-...
0
votes
2answers
84 views

Optimally played game

Consider a game played on the tree above, where the "cost" at a leaf is paid by P1 to P2. Thus P1 wants the number to be as small as possible where P2 wants it to be large. What is the cost paid by ...
0
votes
1answer
90 views

Schools for Game Theory/Algebraic Geometry [closed]

I am looking into both Game Theory and Algebraic Geometry for graduate study and potentially doing a thesis in. I wanted to know if there are areas of mathematics that rely on both algebraic geometry ...
2
votes
1answer
44 views

Proof: Board Game Strategy

We have a sequence of squares, extending infinitely up and infinitely to the right, and a coin is in one of the squares. Player A and then Player B take turns moving the coin. The players always have ...
0
votes
1answer
39 views

Game Theory Mixed Strategy Nash Equilibrium

I have been trying to solve this particular game in terms of mixed strategies, but I am unable to find the strategy using expected payoffs. Is there a way to solve this particular problem? There are ...
4
votes
1answer
166 views

Can't EF game theory be applied to finite languages WITH function symbols?

Let $\mathcal{M}$ and $\mathcal{N}$ be two structures in a language $\mathcal{L}$. We define the finite determined game $G_n(\mathcal{M},\mathcal{N})$ as a game with $n$ rounds where in each round ...
2
votes
1answer
31 views

Two player game about maximizing earnings subject to an interesting condition

Me and my friend had a bet. We each pick an integer between $1$ and $100$ inclusive and reveal it at the same time. Whoever picks the higher number has his number halved. Then the person whose number ...
1
vote
0answers
4 views

Support of $x$ also belonging to the set of best responses when $x$ is a best reponse

If $ \mathbf x $ belongs to the set of best responses to $\mathbf y$ i.e. to $BR(\mathbf y) = arg max \mathbf x \cdot A \mathbf y$, why do all of the pure strategies in the support of $\mathbf x $ ...
0
votes
0answers
27 views

What are the general algorithm and precise mathematical language that can optimise the nodes in a graph?

Recently I came across this via social media Out of curiosity (and because I am a visual learner) using the paragraph in the article, I end up drawing some kind of mixed graph, as shown I then ...
1
vote
1answer
16 views

Fair division of bills

Suppose at a restaurant my friend ordered \$30 worth of pizza, and I ordered \$20. The restaurant is having a promotion so that we could get the second order at half price (the second order can't be ...
0
votes
0answers
18 views

Having more than one Nash-Equilibirum

For my term paper, I'm trying to explain a symmetric game in the music industry. The two players two different singers rehearsing for a duet. $$ \begin{matrix} & {\bf Song 1} & {\bf Song2} &...
2
votes
1answer
41 views

Shooting with Probability, Game Theory

Two people are standing in front of each other in a rail with distance of $2$ meters. Player $A$ stands on point $-1$ and Player $B$ stands on point $1$. They each have only one gun with one bullet. ...
1
vote
1answer
170 views

Nash equilibrium indifference principle

In the Hebrew wiki page on Nash equilibrium there is a reference to an indifference principle which means that once we know the other player uses the equilibrium strategy then the first player can use ...
3
votes
1answer
57 views

What are the optimal mixed strategies for this game?

Fix $k < n$ positive integers, and two players play the following game: each player picks a positive integer between 1 and $n$. If the two numbers picked are within $k$ of each other, the larger ...
0
votes
1answer
38 views

Playing roulette using martingale

A player with unlimited money decides to play roulette. He bets $1$ on red, if he loses, he bets $2$, if he loses again he bets $4$ and so on till he wins. Prove that he is guaranteed to make a profit....
1
vote
0answers
27 views

Differential equation, Cournot competition, Game theory

I would like to solve differential equation derived from differentiating $u_i(q_1,q_2)=q_ip(q_1+q_2)-c_i(q_i)$ by $q_1$ and $q_2$, resp. and putting them equal to zero,and taking $q_2$ to be $r_2(q_1)$...
0
votes
1answer
56 views

Calculate the expected value of this game

Every night, different meteorologist gives the probability of rain for the next day. To judge their predictions, we use the following scoring system: if a meteorologist predicts rain with the ...
2
votes
0answers
51 views

Queen moves — The Squared Chain Puzzle

Karl Scherer made the interesting Squared Chain Puzzle. Start with a $7\times7$ board, with a queen somewhere. Make a legal move with the queen, placing coins over all squares visited. For subsequent ...
0
votes
0answers
52 views

Probability of winning a certain amount of money in this roulette martingale

Given this classic roulette system: A gambler starts by betting $1$ unit on red. Every time he loses his bet he will double his wager until he wins. Afterwards he will start this scheme all over ...
0
votes
0answers
43 views

Game theory problem: the Man and the Lion

I have been reading this solution to the somewhat famous problem about a man and a lion closed in a circular arena: they both move at the same speed, and the lion is trying to eat the man. The ...
0
votes
1answer
44 views

Matchstick game problem

I'm going through past exam papers and this question is proving to be tricky. I can't seem to solve it and have no clue how to, I tried checking numerous ways like odds,evens,primes etc but i'm sure ...
0
votes
0answers
34 views

Symmetric game with selecting numbers

Two players select numbers from the set 1,2,...,N. Denote x the number selected by Player 1 and y the number selected by Player 2. Here is the payoff: If $x < y$ => Player 1 wins and gets 1 dollar....
3
votes
1answer
52 views

Game theory: power and mod

Given two non negative integers $a, b$. Two players alternate turns. If at any state of the game the two integers are $a\le b$ then the player with the turn can either replace $b$ with $b\bmod a$ or $...
1
vote
1answer
905 views

Osgood Box (Doctor Who) Dilemma

Today's episode of Doctor Who featured an interesting dilemma, centered around an object called the "Osgood Box". I'm not well-versed enough in game theory to recognize it as an existing problem, but ...
0
votes
0answers
28 views

Best voting strategy

Let us assume there are two parties in a country with population $p$. There are $c$ constituencies. Each party knows exactly who are their supporters and can choose where each supporter votes. Note ...
2
votes
2answers
42 views

Object and number trick

You are telling your friends Adam and Steve a trick and here is how it goes: Tell Adam to take out $3$ objects from his pocket which all the objects have a different amount of letters. Lets ...
0
votes
0answers
15 views

Class-participation problem modeled with game theory

I'm taking a class and the teacher has set up a system of class-participation to encourage us, the students, to, well, participate more actively. The system is as follows: each student is given 20 ...
0
votes
1answer
50 views

Is this Nash-Equilibrium valid?

The game is as follows: $$\begin{array}{c|c|c|} &A&B\\ \hline A&2;3&2;3\\ \hline B&-1;2&1;2\\ \hline C&-1;3&4;2\\ \hline \end{array}$$ I've written a program that ...
2
votes
1answer
80 views

Winning strategy - nim variation

i was reading about different variations of nim game and i'm trying to find winning strategy to one of them: There are n empty places on the circle. Two players are placing their "coins: on empty ...
0
votes
0answers
50 views

Expected Value in multi-round game

Say I have a game structured as follows: 1) if you decide not to play, I reward you $16 Otherwise: 2) you can flip a coin OR you can throw a ball into a basket (numbered 1 thru 10 with the ...
1
vote
0answers
45 views

Can anyone explain this more clearly?

I'm new to CGT so i might need help but could anyone simplify this and explain it to me please- "set f ⊕ f = 0 for any f. (A nice correspondence can be made if we think back to the original game of ...
1
vote
1answer
50 views

Formulating a game in an economic setting

I'm trying to teach myself Game Theory, and have come across the following question: Suppose that a company, $L$, produces left shoes only, and a company $R$ produces right-shoes. If $L$ charges $...
1
vote
1answer
64 views

The application of Nimbers to Nim strategy

I've been reading about combinatorial game theory, and some works start with the game of Nim. After that, they introduce Nimbers, which are numbers that represent Nim games. So far so good. I get ...
3
votes
0answers
54 views

Prove that the set of $(m \times n)$-matrix games is dense and open

Show that the set of $(m \times n)$-matrix games with unique optimal strategies is dense and open. Let $\mathbb{R}^{nm}$ be a $nm$-vector space of all matrix games and let $M \subset \mathbb{R}^{...
0
votes
0answers
31 views

How to construct potential function in game theory

Can someone explain me how can I find a potential function for a potential strategic game? I know the theory and definitions, but can't figure out the reasoning process. Potential function that I ...
3
votes
1answer
33 views

Partitioning integers into sets

Try to find out if it is possible to partition 6 consecutive positive integers into two sets, $A_1$ and $A_2$ such that the product of the elements in $A_1$ is equal to the product of the elements in $...
0
votes
1answer
31 views

Finding the numbers on cards

This question is a bit confusing. I decided to start by showing that all cards don't contain the same number. Lets say I pick a subset of $10$ cards then they have $10-19$ $10+11+12+13+..+18+19=145$ ...
1
vote
1answer
109 views

Cournot Duopoly Question [closed]

(Repeated Games). Suppose two firms compete in micro-chip industry. Each period firm 1 produces q1 chips and firm two produces q2 chips and the firms face a demand curve of P = 1000−20Q, where Q = q1 +...
0
votes
0answers
35 views

Game theory participation constraint

What are the implications of a participation constraint that is unsatisfied? I am confused by a problem I am working on.