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Questions tagged [game-theory]

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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cooperative games, shapley and additivity

I am studying Shapley values and am interested in understanding cases where additivity does not hold in cooperative games. Specifically, I am looking for a practical example of two cooperative games ...
volperossa's user avatar
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Question 4.31 Heard on the Street - Probability/Game Theory Question

The Question: Two players A and B play a marble game. Each player has both a red and a blue marble. They present one marble to each other. If both present red, A wins \$3. If both present blue, A wins ...
Connor Brown's user avatar
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Voter Count(similar to Bertrands ballot theorem)

This question is from Quantguide: Voter Mayhem2: Two candidates, say A and B, are running for office. Candidate A received n votes, while Candidate B received m votes, with n>m. The n+m votes are ...
Md Kaif Faiyaz's user avatar
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2 answers
86 views

Optimal Strategy : Dice Game

I was asked this question in an interview: You are given a fair dice. You can roll the dice any number of times. Your reward will be the sum of the face value of ...
Md Kaif Faiyaz's user avatar
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0 answers
31 views

Utility function for imperfect information game

I am trying to model a card game (not poker, I swear) using game theory, because I want to build a solver for it, however I don't find the right way to evaluate the payoffs for each player. The ...
adriavc00's user avatar
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1 answer
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How to analyze a variant of the prisoner's dilemma?

In most versions of the prisoner's dilemma I have seen, confessing is the dominant strategy for both prisoners, like this one. However, I am interested in the following case with a change on numbers. ...
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Solve problem path of resistance stochastically stable equilibria

I'm working on Stochastically stable equilibria bout Evolutionary Game theory. In the famous paper by Peyton Young there is an example of a matrix 3x3 with the solution of the paths less resistance. I ...
Filippo Scarparo's user avatar
3 votes
1 answer
95 views

How to formalize game theory in first order logic

In the answer to this question about Zermelo's theorem, Professor Hamkins states that the assertion "B has a winning strategy" can be expressed using the formula $$\forall x_1\exists x_2\...
Petersu's user avatar
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2 answers
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Does Nash Equilibrium say anything about odds?

Edited after good comment from @lulu I'm struggling with the concept of a Nash Equilibrium. I'm getting confused by comparing the prisoners dilemma with a game like rock-paper-scissors. The prisoners ...
Cdl's user avatar
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19 votes
0 answers
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Placing triangles around a central triangle: Optimal Strategy?

Now cross-posted to MathOverflow (link). Question: There is an equilateral triangle. Two players alternate turns placing non-overlapping equilateral triangles of the same size that touch the original ...
Benjamin Dickman's user avatar
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Ali Baba and the 2 urns with black and white balls [duplicate]

Ali Baba is caught by the sultan while stealing his daughter. The sultan is being gentle with him and he offers Ali Baba a chance to regain his liberty. There are 2 urns and m white balls and n black ...
math.enthusiast9's user avatar
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Is it Possible to Define the Price of Stability (PoS) for a Game with Only One Nash Equilibrium?

I have a question regarding the concept of the price of stability (PoS) in game theory. The PoS is typically defined as the ratio of the cost (or utility) of the best Nash equilibrium to the cost (or ...
Lely's user avatar
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1 answer
101 views

Generous Banker

You are at the bank and it is your lucky day. The banker is going pick random positive integer and you are too. You are both allowed to determine the probability distribution on the positive integers ...
Harsh's user avatar
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1 answer
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If A and C are nearer, is B nearer too?

Let $A, B, C$ be three colinear points in $\mathbb{R}^2$, and $P, Q$ any two points in $\mathbb{R}^2$ (on the same line or not). I am trying to prove (or disprove) the following lemma, where $d$ is ...
Erel Segal-Halevi's user avatar
1 vote
1 answer
125 views

What function minimizes the distance to its argument?

Let $d$ be a metric on $\mathbb{R}^n$, and $f: \mathbb{R}^n \to \mathbb{R}^n$ be a function. I am interested in functions that satisfy the following property. For all $x, y\in \mathbb{R}^n$, $$ d(f(x)...
Erel Segal-Halevi's user avatar
1 vote
1 answer
70 views

Game - two players take turn moving a marker to an adjacent square in a 9x9 grid

A marker is placed in the centre of a $9$x$9$ grid. Ann and Beth take turns moving the marker to one of the adjacent squares (one sharing a side) provided that this square has never been occupied by a ...
Abhinav Sood's user avatar
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Even-Nim and Odd-Nim are like Nim in that they are played with piles of stones.

Even-Nim and Odd-Nim are like Nim in that they are played with piles of stones. However, in Even-Nim, a move consists of removing a positive even number of stones from a pile, while in Odd-Nim, a move ...
user avatar
2 votes
1 answer
39 views

Game Theory problem: 2x3 matrix where one of the cells tends to infinity

I'm currently revising for a Linear Programming exam and found this problem on a past paper: Parts A and b are clear to me. My issue comes with parts c and d. My rough solution looks like this: At ...
c1maths 's user avatar
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Conway's Angel Problem: Strategy for Devil to catch $1-$Angel

I am learning about Conway's Angel Problem, which is in the image below. How can the Devil devise a strategy that will successfully capture the $1$-Angel, or an angel of power $1$, which is also a ...
GSmith's user avatar
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2 votes
1 answer
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Standard definition of a game in game theory

Sorry for my naive question, but at the moment I can't quite figure it out. I'm consulting various documents on game theory in order to get the standard definition of what a game (and an associated ...
u31672873's user avatar
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2 answers
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A very basic Game theory doubt

I was going through the Blockchain Documentation and on section 1.8 they talk about Nash Equilibria My very modest understanding of the term is that it is a set of strategies for each player where ...
Anuj Jha's user avatar
1 vote
1 answer
37 views

Stackelberg duopoly problem

I have a task to use the Stackelberg model, and I'm trying to find the best response function to the 2nd player; it says it should look like a derivative from a function like $q_2\cdot P(Q)-cq$. For ...
Almer's user avatar
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2 answers
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The existence of pure strategy nash equilibrium and best responses

I am studying game theory with Fudenberg and Tirole, and not so familiar with this area of economics. Suppose there are only two players in a strategic game with complete information. Then, is the ...
Wooyong Park's user avatar
2 votes
2 answers
49 views

Closed formula for Shapley value of elementwise multiplication

Let's assume we have a set of players $ N = \{1, 2, \ldots, n\} $. Each player $ i $ contributes a value $ v_i $. The value of a coalition $ S \subseteq N $ is given by the product of the ...
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1 vote
0 answers
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Strategy for correctly sorting proceduraly sampled set of numbers

Consider the following single player game. Let $N$ be some natural number. The game then proceeds as follows. On step $1\leq k\leq N$, we sample $X_k\sim\mathcal U([0,1])$ and give it a ranking $1\leq ...
Gabriel Golfetti's user avatar
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1 answer
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How to solve for Nash Equilibrium?

Image: I am currently studying for a college exam next week in Games Theorie. Unfortunately the example questions are very different to the course material and im stuck on this one. I would solve the ...
woodenbook's user avatar
2 votes
1 answer
74 views

Understanding Nash Equilibria in a Bimatrix Game

I am currently studying game theory and I came across a problem involving a bimatrix game. The bimatrix is given by: $$ (A, B) = \begin{pmatrix} (4, 2) & (0, 0) \\ (0, 0) & (1, 3) \end{...
user avatar
3 votes
2 answers
259 views

Necessary and sufficient condition for the strategy to be unique [closed]

Suppose $A$ is the $m \times n$ game matrix for a two-person zero sum game. Suppose row player uses the strategy $x\in P^m$, what is the condition for the strategy for the column player to be unique? ...
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3 votes
1 answer
185 views

prisoner's dilemma bimatrix

I have a question about the following derivation Consider the prisoner's dilemma with the following bimatrix: $$ (A, B) = \begin{pmatrix} (-5, -5) & (-1, -10) \\ (-10, -1) & (-2, -2) \end{...
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3 votes
2 answers
133 views

Bimatrix Game: Nash Equilibrium and Safety Levels

I am studying the following example but don't understand how the solution works: Consider the following bimatrix game: $$ (A, B) = \begin{pmatrix} 4 & 2 & 0 & 0 \\ 0 & 0 & 1 & ...
user avatar
-1 votes
1 answer
29 views

Given a 3-player game and 3 utilility matrices which determine the profit of each player. Provide an algorithm to calculate nash equilibrium. [closed]

There is a 3-player game that player-1 has n actions, player-2 has m actions, and player-3 has p actions. Furthermore, 3 utilility matrices n×m, n×p, and m×p which determine the profit of each player ...
Reza's user avatar
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4 votes
1 answer
141 views

Why don't we apply the principle of indifference here?

The Problem You and a friend play a game where you both select an integer $1-100$: The winner receives $\$ 1$ from the loser. The winner is the one who chooses a number that is either exactly two ...
Abhay Agarwal's user avatar
1 vote
1 answer
97 views

Finding the Values of x for Different Numbers of Nash Equilibria in a Bimatrix Game

I am currently studying game theory and I've come across a problem involving a bimatrix game that I'm having trouble with. I would appreciate any help or guidance. The Problem: Consider the following ...
prob1 yuma's user avatar
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0 answers
28 views

Counterexample of projection game

Consider projection name $H$ with projection function $\sigma$ with $val(H)\in [0.1,0.9]$, how to construct a $H$ s.t. $val(H^{\otimes 2})\geq val(H)-\epsilon$ for arbitrary $\epsilon>0$? There is ...
Lagranngekmno4's user avatar
1 vote
1 answer
25 views

Is the Even–Paz cake-cutting protocol a finite algorithm that guarantees proportional and connected pieces?

In theorem 8.2 of "Cake-cutting algorithms" by Robertson and Webb, it is stated that No finite algorithm can guarantee each of $n$ players at least $1/n$ of the cake using only $n - 1$ cuts ...
Abelaer's user avatar
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2 votes
1 answer
121 views

Are pure Nash Equilibria better than Mixed Nash Equilibria

Let's consider this 3x3 game: \begin{matrix} &A&B&C \\ A&1,1 & 10,0 & -10,1 \\ B&0,10 & 1,1 & 10,1 \\ C&1,-10 & 1,10 & 1,1 \end{matrix} Player 1 is ...
FluidMechanics Potential Flows's user avatar
1 vote
1 answer
35 views

Find the optimal solutions to a system of linear equations?

I have a linear optimization problem $\mathbf{A}\cdot \mathbf{x} < \mathbf{0}$, where $\mathbf{A}$ is a particular square matrix for my application, and $\mathbf{x} \geq \mathbf{0}$. I want to ...
user326210's user avatar
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2 votes
2 answers
140 views

Game of Pigeons - Probability Puzzle

The Problem: Alice and Bob take turns drawing a pigeon from a sack which initially contains $W$ white and $B$ black pigeons. The first person to draw a white pigeon wins. After each pigeon drawn by ...
Devansh Agarwal's user avatar
2 votes
3 answers
83 views

Topological game on $(0,1)$

I consider a « game » on a topological space with $2$ players. I will describe the game and tried to prove that one of the player has no winning strategy in the sense that the other player can always «...
G2MWF's user avatar
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2 votes
1 answer
163 views

Visualizing Best Response Functions in a 4-Player Game: Seeking Nash Equilibrium

I have a game with 4 players, each of whom must minimize a cost function. The strategic leverages of the 4 players are indicated by the variables: $s_i$, $s_j$, $c_i$, $c_j$. I would like to find the ...
Mark's user avatar
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10 votes
3 answers
224 views

What is the minimal number of pieces to surround n pieces in a Go game?

Go is a game of black and white pieces on a lattice of $19\times 19$. Pieces have liberty by having empty spaces next to them and are killed if the liberty are occupied by the opponent. Pieces are ...
ZhenRanZR's user avatar
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1 vote
2 answers
44 views

Game Theory Pure strategies in a sequential game with perfect information

Suppose there are two companies, A and B, each produce and sell a product. Each wants to increase their market share. Company A intends to do some of either: – Spend 10% of profits on advertising, – ...
John Smith's user avatar
5 votes
0 answers
103 views

$2$-for-$2$ asymmetric Hex

If the game of Hex is played on an asymmetric board (where the hexes are arranged in a $k\times k+1$ parallelogram), the player who wants to connect the closer pair of sides can force a win, ...
volcanrb's user avatar
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0 answers
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Pivotal's definition on arrow's impossible theorem

I'm trying to understand Arrow's impossible theorem, and I understand the idea of using extremal lemma. The pivotal is defined as the first individual who change his preference of from ranking $b$ to ...
Chang Henry's user avatar
1 vote
2 answers
94 views

Existence of Nash Equilibrium in a Game with Mixed Strategy Spaces

I am considering formulating an applied research problem as a simultaneous zero-sum game with two players. The first player's set of actions is an infinite and compact subset of $\mathbb{R}^n$, while ...
graphtheory123's user avatar
1 vote
0 answers
40 views

Valididity of information based solution to Monty Hall problem

The hypothesis: the probability that you will win by using the best strategy is equivalent how well you would do if you were given the minimum amount of information Monty needs to know. Ex. In the ...
Michael Wang's user avatar
0 votes
3 answers
141 views

The 50 game between two players, selecting numbers between 1 and 10 inclusive + variations

Let's play a game with two players, with player 1 going first. The players take turns selecting a number between 1 and 10 inclusive. The person who says the number that makes the sum reach or exceed ...
user1013124's user avatar
1 vote
0 answers
53 views

Optimal Strategy For Mastermind-Like Game

A few days ago my brother challenged me to a game in which we both pick a 4 digit number and type it into a google doc. We then exchange guesses to find the others number. If we have a number in the ...
Felix Shainker's user avatar
5 votes
1 answer
97 views

Characterize the mixed strategy Bayesian Nash equilibria for this game. [closed]

A two-player game where Player 1 can choose either U or D and Player 2 can choose either L or R. Player 1 is either cooperative with probability $P$ or uncooperative with probability $1−P$. Player 1 ...
Toshani Singh's user avatar
2 votes
0 answers
35 views

Max-min and min-Max relationship in ZeroSum Games

In the attached Zero Sum Game, I have solved for two Mixed Nash Equilibrium, $(l,m)$ and $(m,r.)$ In $(l,m)$, the payoff is $8/5$ to $P_1$, and $-8/5$ to $P_2$. Here, $P_1$ mixes between Top and ...
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