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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3
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65 views

Compute shooting targets for the gunmen

This is an extension of the well known "3 gunmen puzzle": N gunmen with hitting probabilities in (0,1] take turns to shoot at each other. Firing order is fixed (gunman 1 shoots first, then gunman ...
3
votes
1answer
33 views

Representations Verse Solution Concepts

In Game Theory, we generally refer to "normal form" and "extensive form" as representations. And, we generally describe "Nash Equilibrium," "strictly dominated strategies," "maxmin strategies," "...
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2answers
43 views

Can a modified $19\times19$ checkerboard be tiled with decominoes ($10\times1$ rectangles)

Consider a $19\times19$ checkerboard with the center square removed, the four corner squares removed, and with four extra squares-one above the center square of the top row, one below the bottom ...
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0answers
53 views

Nim game real life applications

I've learned how to prove and apply the Nim game strategy in discrete mathematics, but I was wondering if there is any real life examples and application for this theory. I searched online and didn't ...
2
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0answers
109 views

Game theory, Book by Tirole and Fudenberg, Never a weak best response,unclear example

In this book, I have the following problem: on page 446, there is a sentence: Note that $(0.9,0.9)$ is not removed by NWBR, as D is not dominated after C is deleted. I do not understand this "as". ...
2
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1answer
77 views

Pick a number that is better than your friends

Consider the following game. There are $n$ players, each one has to pick a (real) number $x$ between $0$ and $100$. There is one round to the game. The winner is the person whose number is closest ...
1
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1answer
74 views

A game with coin toss

So I have a game between two players, p1 and p2. Someone(nature?) tosses a biased coin with 80% chance on head. p1 observes the outcome of it and write on a piece of paper head/tail(not neccessarily ...
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0answers
19 views

does individually strategy proof implies coalitionally strategy proof?

Suppose $F$ is a social choice function \begin{equation*}F:N\rightarrow A\end{equation*} where $N=\{1,...,n\}$ is the set of agents and $A$ is a finite set of outcomes. suppose that $F$ is ...
0
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0answers
47 views

Finding an equilibrium to this game

So this question was given in an exam: One of Player 1 and Player 2 need to wake up in the morning to receive a package. Neither wants to wake up early and that is the cost to each player. Both the ...
0
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0answers
32 views

Game theory - coalition game question

The following is a question from a past exam that I am studying: For a 3-person game of perfect information. Let S denote the set {1,2,3}. First player A chooses i ϵ S. Then player B, knowing i ...
2
votes
1answer
79 views

Probability in the board game “Istanbul”

I tried this game yesterday with a couple of friends (really interesting, although I did not win, I would definitely recommend it) here is a small piece of the game: A player first calls out a ...
1
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1answer
53 views

Optimal strategy in chicken game

Consider a one-shot, simultaneous chicken game, as described here: https://en.wikipedia.org/wiki/Chicken_(game) Assume that I'm playing this game against a player that I consider to be of similar ...
2
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0answers
51 views

Replicator equation for mixed strategies?

The the replicator equation is usually defined for pure strategies. More specifically, the replicator eqn for $n$ strategies is given by: \begin{equation} \dot x_{i} = x_{i} \left( \sum_{j=1}^{n} a_{...
3
votes
1answer
71 views

Algorithm for assigning users to “buckets” according to users' preferences and ranking

Suppose there is a set of $n$ users which must each be assigned to one, and only one, of $k$ mutually exclusive "buckets". However, the number of users allocated to the $i$-th bucket must be no lower ...
1
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0answers
63 views

Bayesian Equilibrium

We have to answer the following question, and I can make some progress on the first couple of parts but get stuck in finishing them off. The third part I'm not sure where to start! *Consider the ...
0
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0answers
44 views

How do I prove using strong form induction a statement regarding winning strategies in this coin game?

Consider a game in which, initially, there is a pile of n coins placed on a table. There are two players who alternate turns. Each player, on her or his turn, removes either one, two, or three coins ...
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0answers
10 views

obtaining cases from the integral equation

I'm struggling with the recalculation of the formal model from the published article by Epstein and O'Halloran (1994), and I am failing miserably when it comes to understanding of their calculation, ...
2
votes
1answer
106 views

Bayesian game theory

We have to answer this question and I think I have done part (a) right but get stuck at part (b). Since $-0.5 \le \varepsilon_i \le 0.5 \ \forall i$, I seem to get a solution of the NE being TR, which ...
5
votes
1answer
97 views

Winning Strategy with Addition to X=0

Problem: Two players play the following game. Initially, X=0. The players take turns adding any number between 1 and 10 (inclusive) to X. The game ends when X reaches 100. The player who reaches 100 ...
0
votes
1answer
38 views

Expected value, discrete random variable,discount factor

This example is taken from HERE ,page 424: What is the expected value $X(v)$ of this series: $X(v)=E(\Sigma_{t=0}^{\infty}\delta^t p_t(v))$ where $p_t(v)\in[0,1]$ with $\delta\in(0,1)$? Are all data ...
1
vote
1answer
27 views

Disprove that the given strategy pair is a solution to the game.

Problem: For the following matrix game, prove or disprove that the given strategy pair is a solution to the game. \begin{align} A &= \begin{bmatrix} -1 & 2 & -3 \\ 3 & -4 & 2 \\...
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0answers
21 views

Payoff Matrix/ How to fill positions in the matrix — Game Theory.

The Problem state: P1 and P2 each have three cards: a king, a queen, and a jack. They play their cards, one at a time, with the high card winning the trick (K>Q>J) and the playing of equal cards ...
2
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0answers
37 views

Why doesn't the frequency of a strategy reach zero under the replicator dynamics?

Background The replicator equation with $n$ strategies is given by the differential equation: \begin{equation} \dot x_{i} = x_{i} \left( \sum_{j=1}^{n} a_{ij}x_{j} - \phi \right) \qquad i = 1, \...
0
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0answers
14 views

Information sets in Extensive Form Game with imperfect information

I have constructed an extensive form game with imperfect information given in the attached image. I am however a little uncertain as to whether my information sets are actually admissible if I, for ...
2
votes
1answer
19 views

When does a matrix game and the sign flipped matrix game have the same nash equilibria?

Given a game $G$, we can construct another $G'$, by a positive scaling i.e. $\lambda \in \mathbb{R}_{++}$, s.t. each entry of $A$ is scaled by $\lambda$ Obviously, $G$ and $G'$ have the same nash ...
0
votes
1answer
62 views

Mixed Nash equilibria in $n$-player games

I'm reading up on Game Theory. So far, I feel like I have a pretty good understanding on two-player games and their properties. Consider a two person game where the payoff matrices are $A_{m\times ...
0
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0answers
17 views

Subgame perfect nash equilibrium in a non-Prisoner Dilemma Game

working on this for hours but I can't find any solution yet ... I got a 3x3 matrix of a non-Prisoner dilemma game It looks like that:      C2      D2  ...
0
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0answers
58 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 (...
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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
27 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
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0answers
43 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 $...
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0answers
213 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
78 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
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0answers
28 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
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0answers
87 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
74 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
85 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
27 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
38 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
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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
26 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
15 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
40 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. ...