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Questions tagged [nash-equilibrium]

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

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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{...
鈴木悠真's user avatar
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Correlated equlibrium and Ellipsoid against hope algorithm

I am trying to implement the Ellipsoid against hope algorithm for computing correlated equilibria described in the paper Computing Correlated Equilibria in Multi-Player Games by Papadimitriou and ...
Mel's user avatar
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9 votes
1 answer
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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{...
Provoke's user avatar
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10 votes
2 answers
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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 & ...
PowerPoint Trenton's user avatar
-1 votes
1 answer
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Given a 3-player game and 3 utilility matrices which determine the profit of each player. Provide an algorithm to calculate nash equilibrium.

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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8 votes
1 answer
62 views
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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
1 vote
0 answers
51 views
+50

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
2 votes
1 answer
158 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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1 vote
2 answers
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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
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Difference between a pure and a mixed nash equilibrium

There is a game with 3 agents where each agent invest their effort into 2 tasks. Their efforts add up to 1 and they get rewarded a task based on the effort they put in divided by the total effort. The ...
mathsymaths's user avatar
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Finding whether a mixed nash equilibrium exists in a 3 player game

In this game, there are three agents and two tasks. $c_i$ denotes the cost of agent $i$'s effort. (In this case $c_1 = 1, c_2 = 2, c_3 = 3$) $V_j$ denotes the reward for the tasks. (In this case $...
mathsymaths's user avatar
2 votes
1 answer
33 views

Infinite number of nash equilibria in 3x3 game

In the simple game below, there are two strategies, one for each player, that are weakly dominated (another strategy has greater than or equal payoff). If we remove these strategies then I can find a ...
Student123's user avatar
1 vote
0 answers
26 views

Braess paradox: a network counterintuitive NE graph example

I cannot understand here why switching to $CD$ is a dominant strategy?? If $x>4500$ then it may be beneficial for a Nash-Equilibrium to go from $C$ to $B$ and not from $C$ to $D$. And hence $CD$ is ...
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1 vote
1 answer
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Proving Nash's Theorem via Brouwer's Fixed Point Theorem in 2-D

Consider the following theorem and proof: Proposition: For any two player game, there exists a Nash stable point. 2-D Proof. First we embedd the square in $\mathbb{R}^4$ by $$ \Gamma=\left\{\left(...
Maths Wizzard's user avatar
2 votes
0 answers
88 views

Existence of mixed strategy involving a "best" pure strategy

$\textbf{Set up}$: Let $G$ be a two player zero-sum game, and suppose $A$ is a $n \times m$ payoff matrix for this game, so that player $1$ has (pure) strategy set $\{s_1^1,...,s_n^1 \}$ and player $2$...
porridgemathematics's user avatar
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How related are the magnitude of surreal number combinatorial game values and the propensity to win in spite of potential mistakes?

Surreal numbers are used to represent values of positions in partizan games with perfect information and discrete outcomes (win/lose or win/draw/lose). The primary example is Blue-Red Hackenbush, but ...
user10478's user avatar
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2 votes
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What are the arguments for/against interpreting the magnitude of the value of Blue-Red Hackenbush as an objective function?

There appears to be a common assumption in Blue-Red Hackenbush that the Blue player should make whichever move will maximize the position value of the game, and the Red player should make whichever ...
user10478's user avatar
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how do I find a Mixed strategy nash equilibrium for a 5x3 matrix?

(Player A is on the left, Player B is on the top.) I am trying to find a mixed strategy Nash equilibrium for a $5\times 3$ matrix (table below). I've only gone as far as proving that the one strategy ...
MGMatthew F's user avatar
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1 answer
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Nash Equilibrium Notation

Is this representation an accurate one for the Nash Equilibrium? $$\{\nexists s_1^{'} \in S_1 : u_1(s_1^{'},s_2) \geq u_1(s_1,s_2)\} \ \wedge\ \{\nexists s_2^{'} \in S_2 : u_2(s_1,s_2^{'}) \geq u_2(...
Tunay Sabri Yüksel's user avatar
1 vote
1 answer
88 views

Principle of Indifference Contradicting with Nash Equilibrium?

I have come across a game related to a bimatrix, after IESDS it is like this $$\begin{bmatrix} (10,4)&(5,3)\\ (11,1)&(4,6) \end{bmatrix}$$ First off, by using the principle of indifference, we ...
youngeAn's user avatar
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Nash Equilibrium of a Multi-player Game

Original Question: Each of $n$ players secretly choose a natural number. Among the players whose number does not coincide with anyone else, the one with the smallest number wins. State the Nash ...
Ma Ye's user avatar
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Finding the Pareto frontier when the optimization problem is simple?

I am working on the following problem: Find integer $\vec{x}\geq 0$ within the solution space $\mathbf{A}\cdot \vec{x} \leq -1$ such that none of $(x_1, \ldots, x_n)$ can be made any smaller. In ...
user326210's user avatar
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1 vote
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Can a weakly dominated strategy be optimal in a zerosum game?

Can a weakly dominated strategy be optimal in a zerosum game? I saw an earlier question posted which was using the game $$A = \left(\begin{array}{} 1& 1 \\ 1 &0 \end{array}\right)$$ But, in ...
mathwizard2234's user avatar
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0 answers
42 views

Mixed strategy Nash equilibria for an all-pay auction between two players?

I am actually unsure if this game would be considered a minimum effort game or an all-pay auction, so please forgive me if I'm misusing these names for the title. In a game where 2 players each choose ...
rwbycwbe's user avatar
1 vote
0 answers
68 views

Typo in Nash's paper?

Nash : Bargaining Problem On page 156, Assumption 4, is there is typo? Shouldn't it read, "If A, B and C are as in assumption 2, ..... which is just as desirable as B..." ? (There is no non-...
cartman's user avatar
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0 votes
1 answer
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Finding pure NE with two players choosing numbers up to 1000 and winning based on the lower number

Here's the problem: Let's consider a game for two players in which each player selects a non-negative number, with the maximum value not exceeding 1000. Player 1 chooses even numbers, while Player 2 ...
Tymofii256's user avatar
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101 views

Random Selection Game And Mixed Strategy Nash

I came across this question in QuantGuide: You and your friend play a game where you both select an integer 1−100. The winner receives $1 from the loser. The winner is the player who selects the ...
Md Kaif Faiyaz's user avatar
3 votes
1 answer
103 views

How do I find Nash Equilibria if I have Payoff functions instead of Payoff Matrix in MATLAB?

I am trying to implement the noncooperative game theory in my problem, where I framed two objective functions $J_1$ and $J_2$ for maximizing the power. I plotted the $ P_1$ and $P_2$ in MATLAB as ...
aman2909's user avatar
0 votes
1 answer
73 views

How to compute Nash equilibria for this 3x3 matrix

\begin{matrix} &P2 \\ P1&0,0 & 7,6 & 6,7 \\ &6,7 & 0,0 & 7,6 \\ &7,6 & 6,7 & 0,0 \end{matrix} Then: For P1: \begin{matrix} & A &B&C \\ A&0,0 ...
Try_hard's user avatar
2 votes
1 answer
251 views

Weak dominance in second-price auction

Consider a sealed-bid second price auction with two bidders. Standard textbooks claim that bidding one's true valuation $v_i$ weakly dominates bidding lower than one's valuation $b_i<v_i$. But I am ...
PGupta's user avatar
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0 votes
1 answer
93 views

Finding Nash Equilibrium in Mixed Strategies: Confusion with Dominant Strategies

I'm working with a two-player normal-form game represented by the following matrix: U D L 2,0 2,0 R 6,1 3,2 I attempted to find the Nash equilibrium in mixed strategies and concluded with the ...
Itay Etelis's user avatar
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0 answers
46 views

Perfect Bayesian Nash Equilibrium - Pooling / Sep

Given the following question - I'm required to find all the Perfect Bayesian Equilibrium and specifically state whether they are pooling / separating. Struggling to understand what would be the types ...
BOB123's user avatar
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0 answers
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Explanation of the "Nash Equilibrium Existance" Proof

We were shown the following proof for the existence of "Nash equilibrium" in a lecture in our "Game Theory" course. There are two equalities that I don't understand why they are ...
Ariel Yael's user avatar
0 votes
1 answer
78 views

How to prevent Nash-EQ solver getting stuck in an infinite loop?

I would like to write a Nash equilibrium solver for a simplified Poker game. I was reading that the way some Nash equilibrium solvers for Poker work is that they iterate until convergence: For a given ...
mercury0114's user avatar
1 vote
1 answer
172 views

Optimal strategy in modified guessing 2/3 of the average.

In the original "Guessing 2/3 of the average" game, guessing 0 is said to be the optimal strategy assuming that everyone is rational. In this modified version of this game, players can input ...
Hebuball's user avatar
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0 answers
59 views

surviving iterative deletion of strictly dominated strategies and determining all pure and mixed strategy NE

enter image description here I am trying to solve this and used the following: \begin{bmatrix}(0,1)&(0,0)&(10,4/5)\\(3,1)&(1,2)&(0,1)\\(1,5/2)&(2,3/2)&(0,2)\end{bmatrix}, where ...
A. Jahanyar's user avatar
1 vote
1 answer
157 views

Nash equilibrium in $p$-beauty contest game where $p=1$

Setup: players must chose a number between $0$ and $100$. The winner of the game is the player whose chosen number is closest to the average of all chosen numbers multiplied by $p$. Assume that in a ...
Arif's user avatar
  • 11
-1 votes
1 answer
127 views

Is a steady state necessarily Nash? [closed]

A steady state is Nash if it is strict and pure. And there can be multiple steady states, but not all of them satisfy the conditions of a Nash Equilibrium. Is this right?
Dirk's user avatar
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0 votes
1 answer
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Calculating Nash equilibrium in mixed strategies for non-quadratic normal form games

I am wondering how to calculate the Nash equilibrium in mixed strategies if the normal form game is NOT represented by a square matrix but a non-square matrix. I have read several books about how it ...
sawmaths's user avatar
0 votes
1 answer
597 views

Mixed strategy Nash equilibrium of 3 person game

I'm working on an exercise in which I have to find all mixed Nash equilibria. The payoff matrix is the following Player one chooses rows with probabilities (p,q,r,s) , player chooses row L with ...
Vics's user avatar
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0 votes
1 answer
31 views

Equilibrium values in test-taking game

I'm having a lot of trouble computing the equilibrium values for this game. Suppose we have a game where a player with true skill $\theta \sim U[0, 1]$ can choose to take a test or not, which costs $c$...
physman28's user avatar
1 vote
0 answers
31 views

Is this a subgame perfect equilibrium?

I have a multiple leader, multiple follower Cournot game below: Sequence: In stage 1, leaders independently decide their supply quantities to each of the followers, and selling prices are determined ...
wenxiu0000's user avatar
0 votes
0 answers
46 views

Define subgame perfect equilibrium

I am defining an equilibrium for a multiple-leader, multiple-follower Cournot game where the selling prices are functions of the equilibrium quantities. I require that, in the equilibrium, 1) each ...
wenxiu0000's user avatar
1 vote
1 answer
48 views

Mixed Nash Equilibrium of cyclic game of order 4.

I am looking at a symmetric zero-sum game with 4 strategies and the following payoff matrix that has no pure Nash Equilibrium since it forms a circulant graph.$$\begin{pmatrix} 0 & 1 & 0 & ...
Max M.'s user avatar
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0 votes
1 answer
427 views

How to write down a SPNE (Game Theory)

This may be a very simple and silly question, but since I am unfamiliar with Game Theory and am currently working towards a test for which my college lecturer gave very few materials to work on / ...
Anna's user avatar
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1 vote
1 answer
28 views

Can the same support hosts multiple isolated Nash equilibria?

As from the title, in a strategic game in normal form, can a given support (e.g. actions [1,3,5] for the first player, [2,4] for the second, [1,3] for the third one..) hosts multiple isolated Nash ...
Antonello's user avatar
  • 261
1 vote
1 answer
123 views

Existence of approximate Nash equilibrium (continuous game)

My question concerns continuous games $((S_i)_{1 \leq i \leq n}, (u_i)_{1 \leq i \leq n})$, where $S_i$ are (continuous) compact strategy sets and $u_i$ denotes the utility function of agent $i$. From ...
Aclid's user avatar
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1 vote
1 answer
62 views

Do all Wardrop Equilibria have the same social cost?

I'm currently studying non-atomic congestion games, and i've come accross the following definition of the price of anarchy: Let $f$ be a Wardrop equilibrium and let $f^*$ be a system optimal flow. ...
George Moneftsis's user avatar
1 vote
1 answer
426 views

Finding Nash Equilibria and Best Response in a Strange Game

Consider a two-player game where Player 1 chooses a strategy $x_1=[1,3]$ and Player 2 chooses $x_2=[0,2]$ Let the payoff functions for P1 and P2 be $u_1(x_1,x_2)=\min\{x_1,x_2\}$ and $u_2(x_2,x_1)=\...
Kai Breese's user avatar
0 votes
1 answer
80 views

Nash equilibrium of a game

I've a doubt about the following game and I'm new with Game-Theory. We have two road managers that aim to maximize their profit. Let be $i=1,2$ the players and respectively $\omega_1,\omega_2\in [0,1]$...
Lorenzo's user avatar
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