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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How do you seed a “non-binary” tournament?

In a typical tournament, seeding is arranged to provide proportionate advantage to competitors based on their perceived relative ability. (Good teams get to play bad teams, bad teams have to play good ...
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14 views

$N=\{1,2,3,4\}$ abd $v(124)=v(134)=v(234),v(1234)=1$ and $v(3)=0$

Let $N=\{1,2,3,4\}$ abd $v(124)=v(134)=v(234),v(1234)=1$ and $v(3)=0$ otherwise $(a)$ Is $C(v)\neq \phi$ ? $(b)$ Write down a stable set for the game Can anyone please tell how to solve above two ...
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1answer
42 views

How to calculate the payoff in Battleship Game Theory

Consider a 3 by 3 board and suppose that Player I hides a destroyer(length 2 squares) vertically or horizontally on this board. Then Player II shoots by calling out squares of the board, one at a ...
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1answer
34 views

Finding all mixed Nash equilibria in a $3\times 3$ game

I was looking at the exercise $2$ in this file http://isites.harvard.edu/fs/docs/icb.topic1531493.files/Practice%20Problem%20Solutions%20on%20Nash%20Equilibrium.pdf pages 4 to 7. I do not understand ...
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63 views

Prisoner's Dilemma and Centipede Game - what's wrong with this analysis?

This is an analysis which to me seems trivial, but which I very rarely see brought up in any discussion of games like The Prisoner's Dilemma or The Centipede Game which are well known for having ...
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18 views

Sequential equilibrium

Show that one of the SPNEs is not part of any weak sequential equilibrium & Find a weak sequential equilibrium (Perfect Bayesian equilibrium) of this game. enter image description here
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1answer
36 views

Jumping on the Coordinate lattice grid

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show ...
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1answer
77 views

What is the optimal strategy when playing `head or tail` per team

Introduction Once a week, we are playing head or tail in my favorite bar. There are $N$ people in the room and each person is guessing whether ...
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1answer
205 views

Winning strategy of a game

The following is a game on monomials. Let $M(X,Y)$ denote the set of all monomials in $X$ and $Y$, i.e., $$ M(X,Y)=\{X^aY^b\mid(a,b)\in\mathbb{N}^2\}, $$ where $\mathbb{N}=\{0,1,\dots\}$. ...
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34 views

Pursuers (a game)

From pure entertainment I am interested in the following questions: Let $N$ (particularly $N=1$ case is special) pursuers pursue running at max speed $u$ a runner running at max speed $v$. ...
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27 views

Bayesian formulation of the Shapley Value

I wanted to know if there is a Bayesian formulation of the Shapley value in cooperative Games. I'm not sure if my problem really fits the definition of Bayesian games so here is the problem : For a ...
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1answer
40 views

Does it pay to know what you know?

Let's play a game. I ask you question a yes/no question, and you answer. You don't answer with a yes or no though, you answer with a probability of it being yes ($P \in (0,1)$). For example, I might ...
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1answer
51 views

Choosing optimal strategy for real number assignment game

My opponent is dealt a real number $r\in [0, 1]$ uniformly at random and I know that with probability $1-r$ she chooses to discard that $r$ and be dealt a new number in $[0, 1]$ uniformly at random ...
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1answer
33 views

How to simplify decision tree for sequential game

I am working on the exercise 2.8 of Game Theory: an Introduction by Steven Tadelis. I thought to solve this exercise by undertaking the steps sequentially, in order to minimize the loss in case of ...
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2answers
114 views

Finding p positions with 2 subtraction sets in the take-away game [closed]

Find the set of P-positions for the takeaway game with the subtraction sets: $S = {1,3,5,7}$ $S = {1,2,4,8,16,32}$ Who wins each game when there are 100 tokens on the table to start, the first or the ...
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63 views

Game, stealing edges in a graph.

I was inventing a problem for a math contest, I was really pleased with it, but then I found a mistake in my solution and have not been able to solve it. It is as follows: Alice and Bob play a game. ...
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1answer
41 views

What's the difference between a Nash, Correlated, and Extreme equilibrium?

As the title states, what's the difference? As I understand it: The Nash Equilbirum (NE) is a solution concept in non-cooperative games where no player has incentive to unilaterally deviate from a ...
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1answer
28 views

Find the Nash equilibria

A law is passed requiring a monopolistic soft-drink manufacturer to separate the production department and the marketing department. The marketing department chooses the price $P > 0$ to charge for ...
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1answer
11 views

Find all Nash equilibria in pure strategies

We consider the following public good provision game. There are 2 players, each choosing the amount of money $x_i$ ($i$ denotes 1 or 2) they will give to build a public good. We assume that each ...
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1answer
16 views

Find subgame Nash equilibrium

Two players, A and B play the following game. First A must choose IN or OUT. If A chooses OUT, then the game ends, and the payoffs are: A gets 2 and B gets 0. If A chooses IN, then B observes this and ...
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1answer
81 views

value of a matrix game

Suppose I have a matrix $$ \begin{bmatrix} 1&4&2\\ 3&2&1 \end{bmatrix} $$ How do I find the minimax value of the matrix? ( It will be considered as a matrix of a matrix game where ...
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2answers
75 views

Placing circles inside of a regular polygon.

Alice and Bob play the following game: on a table there is a regular $n$-gon. On each person's turn, they are required to place a circle of radius $r$ fully in the interior of the $n$-gon such that it ...
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1answer
32 views

Nash Equilibria of P-Beauty

I'm a little confused with the work I am currently doing in Game Theory. Here is the questions I'm working on: ...
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20 views

Mixed Strategy Probability Distribution

Problem Two firms with equal capacity constraints k but different marginal costs $c_i$ compete in a pay-as-bid auction for a fixed demand of (balancing) energy. The information on capacity ...
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1answer
23 views

Proving a nash equilibria does not exist

At a certain warehouse, the price of tobacco per pound in dollars, $p$, is related to the supply of tobacco in pounds, $q$, by the formula $p=10−(q/100000)$ Thus the more tobacco farmers bring to ...
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2answers
96 views

Game-winning strategy

Player A and Player B are playing a turn-based game. At the beginning of the game there are $N(N \ge 3)$ points in a plane. In each turn one of the players chooses exactly $3$ different points and he ...
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200 views

Factorization game, can we find winning strategy?

I'm thinking about a game theory problem related to factorization. Here it is, Q: two players A and B are playing this factorization game. At very first, we have a natural number $270000=2^4\times ...
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1answer
81 views

Polynomial game problem: do we have winning strategy for this game?

I'm thinking about some game theory problem. Here it is, Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and ...
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27 views

Can't solve matrix for Nash Equilibrium?

So, I have the following 9 by 9 probability matrix. I want to solve it for a nash equilibrium. https://docs.google.com/spreadsheets/d/16Y1FqxRIAHsHpgEz1ckxDt2sEOInOG3zz_wU8kBHvB4/edit?usp=sharing For ...
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2answers
33 views

Definitional question: difference between a correspondence and a function

Is there a difference between a correspondence and a function? For example, in game theory I am told that for a given strategy set, $\Sigma_i$, the best response given by ...
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1answer
26 views

Check that a Nash equilibrium point is given by $\left(0,\frac {1} {2}, \frac {1} {2}\right)$ $\left(0,\frac {1} {2}, \frac {1} {2}\right)$

Given the game matrix \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 2 \end{bmatrix} I already see a Nash equilibrium in pure strategies, which is $a_{11}$, ...
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1answer
30 views

Iterated Best Response to find Pure Nash Equilibria

The context of this question is Game Theory. I've been trying to apply a simplified (?) version of the Iterated Best Response (IBR) technique to find Pure Nash Equilibria (PNE) in games generated by ...
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1answer
33 views

Game Theory - $3\times3 $Matrix - Mixed Strategy

I am trying to solve the following $3\times4$ game: \begin{array}{c|rrrr} & A & B & C & D \\\hline X & -3 & 5 & 2 & -1 \\ Y & 4 & -1 & 1 & -3 \\ Z & ...
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1answer
28 views

Card Game Probability, the 15th card

Playing a card game in which 2 52 card decks are combined to create the pile and from this pile each player is dealt 14 cards. One rule of the game is if any player is dealt 3 doubles (a double being ...
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18 views

How to find convergence with a learning rule that depends on the outcome of a game?

my first post here and really excited about the community. In a game theory set in which agents choose from a finite set of actions with a probability distribution, how can I look for convergence ...
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44 views

Find a Mixed-strategy Nash equilibrium in an all-pay auction

This is an all-pay auction (Highest bidder wins the object, all players pay what they bid, player 1 wins all ties): Player 1 has $300$ dollars, Player 2 has $500$ dollars, the object being auctioned ...
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1answer
59 views

Factor Game Calculation

Game: There are two players. The game starts off with the numbers 1 - 30 and player1 chooses a number and gets that number added to their score; however, the sum of all the available factors of ...
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21 views

Find a Nash Equilibrium point in mixed strategies using the simplex algorithm

Here is the game matrix A. First I note that $C_4<C_1$ and $C_4 <C_3$ So I can remove $C_1 $ and $C_3$ since they are dominated strategies. Then considering only $C_2,C_4$, I note that $R_3$ ...
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1answer
55 views

Math for game theory

I have read a bunch of book on Game Theory, and I find that one of the best is the the book of Osborne and Rubinstein. Nevertheless, all books which I found are not made for mathematician, meaning ...
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1answer
244 views

Game theory, olympiad question

I've seen the following question in the brazilian olympiad for university students, and I couldn't solve it. Thor and Loki play the game: Thor chooses an integer $n_1 \ge 1$ , Loki chooses $n_2 \gt ...
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46 views

Are there different forms of the Mexican standoff?

The Mexican standoff is a stalemate situation where three mutually-hostile gunmen have each trained their gun on the next gunman. If any gunman shoots first, the remaining gunman is now free to shoot ...
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1answer
94 views

Construct a game with only pure strategy nash equilibrium.

I'm trying to construct a normal-form game with $2$ players such that the game has exactly $4$ Nash Equilibria From the above properties, I know the game has to be a $4 \times 4$ matrix game, and it ...
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7 views

Continuity of utility function in normal form games

I want to characterize the utility functions of normal form games. Let $G$ be a game with a finite number of players $k$ given by the action sets $S_1,\ldots,S_k$ and utility function $u:S_1\times ...
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Preference relation

Let $A=\{a,b,c\}$ and $\preceq$ is quasi-linear order on $\mathcal{L}(A)$. We also know that $a\prec b\prec c$ and for every lotery $L \notin \{[a],[c]\} $ we have $L\approx [b]$. Is $\preceq$ ...
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1answer
20 views

The core -symmetric players

We have $n$-persons ($n\ge 3$) cooperative game. And we know that player $1$ and $2$ are symmetric. So for each element $(x_1,x_2,...,x_n)$ from the core we have $x_1=x_2$ ? Is that true ? Never seen ...
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53 views

Game Theory Projects for Undergrads

I am looking for some project ideas for beginning math students in the topic of game theory. I am not very knowledgeable in the topic so it would be great if I could get an good introductory source to ...
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1answer
34 views

Win/Lose ratios and selection strategies

Imagine the following scenario: You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...
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1answer
25 views

Security level and equilibrium payoffs in $3-$person zero sum game.

Let the security level ($p-$payoff, $M-$set of all strategies) $$B_i:=\sup_{\sigma_i \in M_i} \inf_{\sigma_{-i}\in M_{-i}}p_i(\sigma_{-i},\sigma_i)$$ Now I consider $3-$person zero sum game. The ...
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2answers
42 views

When solving linear equations what does ${0x_n = 0}$ mean? What if the system is used to find Nash equilibrium?

When solving systems of linear equations one sometimes gets result like ${0x_n = 0}$ what does it mean for solving the system? Is it error on part of the solver or just feature of the assignment? ...
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25 views

How are lottery winnings calculated?

I'm pretty familiar how most chance games payouts are calculated - the ratio shoul be inversely proportional to the probability of winning, minus house edge. If we bet the same amount on the same ...