Questions tagged [nash-equilibrium]

For questions regarding the the Nash equilibrium solution concept in strategic games.

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Find all Nash equilibrium given a game

Consider the following: How can we find all Nash equilibrium of the game given that the attacker cannot perform surveillance. I think for a mixed strategy there is but for a pure there isn't, but ...
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Using stric / weak dominant strategy to get 2x2 matrix in Game Theory

I've encountered a problem with a 2x3 game where I need to find Pure NE and Mix NE so I thought it's better first to get a 2x2 matrix. For some reason I find it very difficult to solve this with ...
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Nash Equilibrium: Mixed Strategies calculation

How do I find mixed strategies for this game after eliminating all these dominated strategies? Nash equilibrium game I know that B strictly dominates both C and D for player 2. Deleting C and D for ...
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Solving Mixed Strategies Equilibria

I know that A strictly dominates B for player 2. So after deleting the B for player 2, B strictly dominates A and C for player 1. How do I find the mixed strategies Nash equilibria for this game after ...
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Nash equilibrium - dollar contribution

There are 2 players. Each has an endowment of 10 dollars and chooses how many dollars to contribute to a public pool. Call x1 and x2 the contributions of player 1 and player 2, respectively. If x1 + ...
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Can Nash Equilibrium strategy be exploited, if another player deviates?

Consider a 3 player game (e.g. poker for 3 people). Nash equilibrium is a set of strategies $s_1, s_2, s_3$, such that the expected profit value for each player $s_i$ $E_i(s_1, ..., s_i', ..., s_N)$ ...
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Can braess routes change dynamically with respect to changing demand in a network

Braess Paradox is a counter intuitive phenomenon where removing a link from a network increases the network efficiency. Usually these links are detected over long period of time in an average ...
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UNO Card Game and Game Theory

I've recently been trying to create a computer program which plays UNO against human opponents (and usually wins); however, because I have very little experience in game theory, I have been unable to ...
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Adjusting payoffs has unintuitive effect on optimal strategies in bimatrix games

Consider the game Rock-Paper-Scissors. If we award a win with $1$, a loss with $-1$ and a draw with $0$, we get the following bimatrix game (with rewards ordered as row player, then column player): R ...
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Who wins the Even/Odd game, and by how much?

The Even/Odd game is defined as follows: Both players pick a number in $(0,1]$ for positive $y$. Even's number is called $E$. Odd's number is called $O$. The scorer number is $\Big \lceil \dfrac{E+y+1}...
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Strategy game: draw a random number uniformly between 0 and 1 and if you like redraw. You win if your number is bigger than mine. [duplicate]

You draw a random number uniformly between 0 and 1. If you like it, you can keep it. If you don't, you can have a do-over and re-draw, but then you have to keep that final result. I do the same. You ...
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Why Nash or correlated equilibrium require complete information?

In games of complete information, there are common solution concepts such as Nash equilibrium and correlated equilibrium. The idea is that each player is playing a best response. My question is - Why ...
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How to find the Nash equilibria of a game with continuous strategy spaces

Let $u, w \geq 0$. Let the payoff function $f_1: [0, u] \times [0, w] \to \mathbb{R}$ for player $1$ be defined as $$f_1(a, b) = \frac{u - a}{1+e^{a-b}} $$ for $a \in [0, u]$ and $b\in [0, w]$. ...
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Mixed Strategy Nash Equilibria of 2x3 Game

The setup of this game is very similar to the setup of the game in another question I found (Find all mixed-strategy Nash Equilibria of 2x3 game.): $$\begin{array}{c|c|c|c} & \text{L} & \...
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Computation of Nash equilibrium in a concave game

so we have the game $(N,(S_{i})_{i\in N},(\varphi_{i})_{i\in N})$ where : $\bullet\:N$ is the set of players $\bullet\:S_{i}=[a_{i},b_{i}]$ the set of strategies for each player $i\in N$ $\bullet\:\...
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Does every perfect information game in extensive form have a Nash equilibrium?

I have been struggling to reason whether this is true. Given a perfect information in extensive form (say chess or many such board games), is it guaranteed that there exists a pure strategy Nash ...
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How do you find this pure-strategy Nash Equilibria in this public good game?

I am having trouble with question 2 of this exercise. Consider a public good game with two consumers who have identical marginal valuation 0 < θ < 1 for the public good. The public good is ...
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Under which conditions is the subgame perfect nash equilibrium unique?

Suppose that we have a finite complete information game, under which conditions is there a unique subgame perfect nash equilibrium (if it exists)? Does it matter if the game is perfect information?
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"Nash equilibrium" involving altruists, self-sacrificiers, lovers, haters and teams...

In a Nash equilibrium, no player has anything to gain by changing only their own strategy. And we know that Nash has proved that there is a Nash equilibrium for every finite game. Players are usually ...
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Existence of Nash equilibrium in n-person concave game.

In order to show the existence of Nash equilibrium in a continuous game for concave objective functions (Theorem.1. (Rosen 1965) ), i found a result that says that the function $\Gamma $ is upper ...
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Game theory extensive form game

I have a question regarding the following game. How do we find all the SPNE( Subgame perfect Nash equilibrium)? Click here to see the game
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What are some results about non-existence of Nash Equilibrium solutions?

The Nash Equilibrium probem is finding $x=(x_i)_{i=1}^n \in C \subset \mathbb{R}^n$ such that $$ x_i \in argmin_{x \in C_i} f_i(x),$$ where $C_i$ depends only on $x_i$, $f_i:\mathbb{R}^n \mapsto \...
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Circle of prisoners

I do not know if the following game is well-defined: Suppose that there are $n$ prisoners standing in a circle one after other $n\geq 2$ everyone holding a silent gun. When a bulb lights all either ...
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Applications of Kakutani's fixed point theorem for two different functions

Kakutani's fixed point theorem described in Theorem 2.6 in the paper (as shown below) and its application to prove the minimax theorem in Theorem 3.2 in the same paper seem like I can make the ...
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How to calculate optimal strategy for bimatrix game

Consider the bimatrix game given by $$A = \begin{bmatrix} 3 &2 &-2 \\ -1 &3 &0 \\ 1 &-2 &0 \\ \end{bmatrix} $$ $$ B = \begin{bmatrix} -2 &3 &-1 \\ -1 &-2 &2 \\...
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Difference between Pareto optimal redistribution and strict pareto optimal redistribution

Can someone explain the difference between Pareto optimal redistribution and strict pareto optimal redistribution? Because I know the definition but I do not understand it.
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Pareto optimal redistribution in binary exchange economy

Consider binary exchange economy with two goods and two agents, whose preferences are defined as follows: $\textbf{x}\succ \textbf{y}$ iff $x_{1}x_{2}>0 $ and $y_{1}y_{2}=0 $. In Edgeworth's box ...
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Nash equilbrium in a 5x5 game

Consider a zero-sum game with the following matrix for row player: $$ A= \begin{pmatrix} 0 & 1 & -1& -1&1 \\ -1&0&1&-1&-1\\ 1&-1&0&-1&1\\ 1&1&1&...
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Best response to a given game matrix

Consider the following game matrix: $\begin{aligned}[] \begin{array}{|c|c|c|} \hline & A & B \\ \hline C & 5,0 & 1, 2 \\ \hline D & 1,2 & 7,4\\ \hline \end{array} \end{...
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Nash equlibria payoff matrix

Can someone help me to find the Nash Equlibria, I have the following payoff matrix: F C D B F |40 80 70 70 C |90 70 30 90 B |80 90 70 46 I usually have Bimatrix payoffs, but here I am not sure how ...
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Property of action pairs in the support of a Nash equilibrium

I was thinking about what properties action pairs in the support of Nash equilibrium strategies have. Specifically, let $(x, y)$ be a mixed-strategy Nash equilibrium of some general-sum game $(A,B)$ ...
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Prove existence of Correlated Equilibrium

How can I prove that every finite game has at least one Correlated Equilibrium? I know there is the idea that it has already been shown that every finite game has a Nash equilibrium and then from ...
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to find all mixed Nash equilibria in a 3x3, how many "probability zero choice" combinations do you have to check?

Consider a game with player P1 having choices A, B, and C, and player P2 having choices X, Y, and Z, and the associated 3x3 payoff matrix. I understand how to find the pure Nash equilibria, if they ...
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A potential game and Nash equilibria

Consider the $n-$player game with the strategy sets $A_i = [ 0, \infty)$ for all $i$. For a strategy profile $a = (a_1, \dots , a_n)$, define $\pi_a = a_1a_2\dots a_n$ and $\sigma_a = a_1 + \dots +...
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Is a Correlated equilibrium with independent random variables (= product distribution) a Nash equilibrium?

It is well known that each Nash equilibrium is also a Correlated equilibrium but not the other way around (in general). The difference between Nash equilibria and Correlated equilibria is that in NEs ...
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Coarse correlated equilibria conditions for this two player game

In the lecture we talked about the Correlated Equilibria (CE) conditions for this Traffic light game: $$\begin{bmatrix}1,1 & 1,0\\0,1 & 100,100\end{bmatrix}$$ for a correlated strategy profile ...
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Fictitious play for Matching Pennies (zero-sum game)

I am trying for an exercise to use Fictitious Play to find a mixed Nash equilibrium for the Matching Pennies zero-sum game with matrix: $$A = \begin{bmatrix}1 & ...
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Why are optimal strategies in a two-player zero-sum game convex?

The professor told us today, that the set of optimal strategies of the row and column player in a two-player zero-sum game is convex, but he doesn't mention why that is. What does it even mean that a ...
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2 answers
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The LP formulation of a 2 player 0-sum game

I am studying Nash equilibria for 2 player 0-sum games from the book "Multiagent Systems" by Shoham and Leyton-Brown. In chapter 4 there is the following Linear Program (LP) formulation for ...
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Mixed nash equilibrium equilibrium $2\times4$ players

I have to solve this exercise, but I don't know how to find mixed strategies nash equilibria with $2\times4$ matrix, I've only done it with $2\times2$ ones. Also I don't know how I could prove that it ...
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2 votes
1 answer
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Is there a Nash equilibrium in Bertrand model with cost advantages?

The Nash equilibrium of the classical Bertrand model is that the price of both firms equals to their marginal cost. Now, if one of the firms has a cost advantage, i.e., the marginal cost of the firm $...
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Do Subgame Perfect Nash Equilibria (SPNE) allow for credible threats?

Consider the following extensive-form game: In one alternative, Player 2 chooses G and E and Player 1 chooses D. However, Player 2 can increase her gain by making a credible threat and switch from G ...
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Need help finding specific undefined equilibrium problem

So I have this method for equilibrium problems which I want to test on undefined problems to see how the behavior is. For simplicity consider a 2-player equilibrium problem of finding $(\overline{x},\...
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Game Theory - Correlated Equilibrium Example

I struggle to understand the meaning of a correlated equilibrium. I found this example of a game on the internet: $$ \begin{array}{c|c|c|c} & L & C & R \\ \hline T & 2, 2 & 0, 3 &...
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Find the equilibrium point for a single neural network layer

Given a matrix $\mathbf{W}\in\mathbb{R}^{n\times n}$ and a vector $\mathbf{b}\in\mathbb{R}^{n}$, find the equilibirum for the following single-layer neural network: $$ f(\mathbf{x})=\text{ReLU}(\...
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Online solving for nash equilibrium with imperfect knowledge of actions

For simplicity sake, let's say $M$ is a two player, constant sum game. This gives us uniqueness of NE payoffs and 'interchangeability' of strategies. We can solve for Nash equilibrium using linear ...
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How to understand the Nash equilibrium [closed]

I was reading some course notes, and I don't quite understand the mathematical definition of Nash equilibrium How should I interpret the mathematical definition? Also, why is (D,D) the unique Nash ...
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Elimination of weakly dominated strategies - example

I am supposed to solve a game by iterated elimination of weakly dominated strategies: $$ \begin{array}{c|c|c|c} & L & C & R \\ \hline T & 2, 1 & 1, 1 & 0, 0 \\ \hline M & 1,...
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How many Nash equilibrium in the furthest from the average game

Game: 3 persons pick a non-negative number up to one hundred (inclusive). The one who has the furthest from the mean of these three numbers wins 1000$. My idea: I think that we need to select cases ...
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Does the regularized optimization problem has solutions sufficiently close to those of the original problem?

Consider the following bilinear saddle point problem: \begin{align*} \min_x \max_y &~ f(x,y)\\ \text{where} &~ f(x,y) = x^\top A y + b^\top x + c^\top y, ~\text{and}~ x,y,b,c \in \...
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