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

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Splitting the dollar Nash equilibrium

I'm working on a game theory problem I can't seem to figure out. Players 1 and 2 are bargaining over how to split $\$10$. Each player names an amount $s_i$, between 0 and 10 for herself. These ...
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0answers
16 views

Compute the set of subgame perfect equilibria for this game (mixed strategies help)

In the above game there are 2 proper subgames (not including the whole game itself). I know there's 9 sub-game perfect equilibrium i have to figure out. I've managed to work out 4 of those, the ...
4
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1answer
2k views

Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question: Is $(\frac{1}{3},\frac{1}{3}...
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2answers
794 views

Nash equilibrium in first price auction

I'm trying to understand Exercise 18.2 from Martin J. Osborne and Ariel Rubinstein A Course in Game Theory about finding pure Nash equilibria in a first-price auction. There are $n$ players, named ...
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1answer
35 views

nash equilibrium and best response dynamics

I have a very basic question : How is nash equilibrium found in real life problem. I mean, in theory it exists and we can prove it but how this knowledge will be applied in real life especially with ...
2
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1answer
37 views

A bin-assignment infinite 2-player zero-sum game

What is known about the following infinite 2-player zero-sum game? There are $k$ bins. Each player has 1 unit of mass and, simultaneously, divides it arbitrarily among the $k$ bins. The player wins ...
2
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2answers
38 views

While finding an optimal strategy for a mixed nash equilibrium, why do we not consider strategies which are never a best response?

"A strategy cannot be plausibly chosen by a rational player if and only if it is never a best response." I understand the logic behind neglecting the strategies that are strictly dominated. But why ...
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1answer
797 views

I can't find the Nash equilibrium of this 3x2 game.

Sorry for my English, I am French but i couldn't find help on the French website (so I am here). I have a question about this two-player game: $$ \begin{array}{c|cc} & y_1 & y_2 \\ \hline ...
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3answers
859 views

Optimal Strategy for Rock Paper Scissors with different rewards

Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward. If you win with Rock, you get \$9. Your opponent loses the \$9. If you win with Paper, you get \$3. ...
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0answers
31 views

Visualizing Nash Equilibria of a 4 dimensional matrix

Are there any good ways to visualize Nash equilibria of a 4-d matrix? I have created an game theory model which consists of of four players (P1; P2; P3; P4) who can all choose between a set of 27 ...
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2answers
3k views

Does chess have more Nash equilibria than you can find through backwards induction?

All equilibria found with backwards induction on a tree of a perfect information game are Nash equilibria, but in general the reverse is not true: ...
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1answer
27 views

How to compute a mixed Nash Equilibrium where only one payoff is given.

Let's say I have this: $$ \begin{matrix} & A & B \\ X & 1 & 2 \\ Y & 2 & 1 \\ \end{matrix} $$ That is the payoff for me if I make ...
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0answers
28 views

Software or code to solve a congestion game for n players

First of all, I am pretty new to game theory so if I say something wrong, please correct me. I have a congestion game (similar to the bookcase of machine job scheduling problem). The jobs are the ...
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0answers
23 views

Mixed strategy, find an equilibrium pair

In this example I need to find equilibrium pair X = ($x_1$, $x_2$, $x_3$), Y = ($y_1$, $y_2$, $y_3$) The matrix looks like this $[0, -1, 2]$ $[3, 1, 0]$ $[-2, 2, 1]$ P(X, $B_1$) = 3$...
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0answers
15 views

Concavity of the equilibrium

Suppose we have have $n$ players taking action $a_i \in [0,1]$ to generate some value $v(a_1,...,a_n)$ together. The utility for player $i$ given by $\lambda_iv(a_1,...,a_n) - u_i(a_i)$ where $u_i$ ...
2
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2answers
37 views

The Nash Demand Game

I am having troubles to figure out how to find all pure strategies and two mixed strategies (one in which the shares received by players are always $0$, and one in which $\omega$ is often - but may be ...
2
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1answer
44 views

Maximum of minimums

Suppose $v_1,\ldots, v_k \in \mathbb{R}^n$ are vector with all coordinates non-negative. How to explicitly calculate: $$ \max_{x\geqslant 0, ||x||_1=1} \min_{1\leqslant i \leqslant k} <x,v_i>$$ ...
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1answer
37 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 ...
2
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1answer
70 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
37 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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0answers
22 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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0answers
33 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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1answer
35 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 ...
0
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0answers
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 ...
0
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0answers
19 views

Finding All Nash Equilibrium

$N \geq 2$ firms simultaneously, and independently, decide to enter a market. The pay-off for each firm entering the market is, $$\pi(n) = \begin{cases} \pi_{M} - I & \text{if } n = 1 \\ -I &...
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1answer
21 views

How to solve mixed strategy Nash equilibrium.

Lets say I have following problem: Zero sum game. Payoff matrix for player one: -1 4 4 2 6 -2 I start by writing equations for each strategy. ...
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1answer
136 views

How to interpret negative probability for a strategy in mixed nash equilibrium?

I am trying to get the mixed strategy in Nash equilibrium for the following matrix. $$\begin{pmatrix} 0 & 3 & 4 & 5 & 6 \\ 3 & 0 & 5 & 6 & 7 \\ 4 & 5 & 0 &...
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1answer
71 views

How can I find the Nash-equilibrium of the following zero sum game?

I want to find the Nash-equilibrium of the following zero sum game. $$A=\begin{bmatrix}0&2&-1\\-2&0&3\\1&-3&0\end{bmatrix}$$ I used the Minimax Theorem. $$min_{x \in X} max_{...
2
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1answer
104 views

Game theory: Finding Nash equilibrium in $3\times 3\times 3$ matrices

I tried to find how to solve $3\times 3\times 3$ matrix to find Nash equilibrium but I could not find anything on the web. Maybe I am searching with wrong keywords... I understand how to solve Nash ...
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0answers
18 views

Difference between Sequential and Weak Sequential Equilbria

This is in reference to the Game theoretic concepts as Nash equilibrium refinements. Sequential equilibrium are often defined as satisfying two conditions: consistency and sequential rationality. ...
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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". ...
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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 ...
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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  ...
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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 ...
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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} &...
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0answers
26 views

Differential equation, Cournot competition, Game theory

I would like to solve differential equation derived from differentiating $u_i(q_1,q_2)=q_ip(q_1+q_2)-c_i(q_i)$ by $q_1$ and $q_2$, resp. and putting them equal to zero,and taking $q_2$ to be $r_2(q_1)$...
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0answers
28 views

Best voting strategy

Let us assume there are two parties in a country with population $p$. There are $c$ constituencies. Each party knows exactly who are their supporters and can choose where each supporter votes. Note ...
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1answer
50 views

Is this Nash-Equilibrium valid?

The game is as follows: $$\begin{array}{c|c|c|} &A&B\\ \hline A&2;3&2;3\\ \hline B&-1;2&1;2\\ \hline C&-1;3&4;2\\ \hline \end{array}$$ I've written a program that ...
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2answers
8k views

cournot competition with N-firms

The question is as follow: Here is how we can think of N-firm Cournot competition. Assume all the firms have the same marginal cost C > 0. Firm 1 chooses Q1, Firm 2 chooses Q2, and so on. The market ...
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0answers
58 views

Unique Nash Equilibrium

If a game has a unique Nash Equilibrium, then does it have a unique Mixed Nash Equilibrium as well, where this MNE is the unique NE? The game I have in mind is the following (but I am more curious ...
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0answers
48 views

Game theory, Duopoly problem

. Suppose two firms compete in micro-chip industry. Each period firm 1 produces q1 chips and firm two produces q2 chips and the firms face a demand curve of P = 1000−20Q, where Q = q1 +q2. Both firms ...
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1answer
105 views

Cournot Duopoly Question [closed]

(Repeated Games). Suppose two firms compete in micro-chip industry. Each period firm 1 produces q1 chips and firm two produces q2 chips and the firms face a demand curve of P = 1000−20Q, where Q = q1 +...
0
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0answers
34 views

Perfect Bayesian Equilibria of the following game

Consider the following game between a monopolist firm and a consumer. Consumer's income is $1$, and he needs to allocate it between period 1 and period 2 consumption to maximize his utility $u(c_1,c_2)...
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1answer
27 views

Gaining intuition as to why maximal lotteries use randomness to break general ties

The maximal lottery is a voting system based on choosing an optimal candidate game-theoretically. If a winner isn't clear (there is no condorcet winner), then it will return probabilities as to which ...
0
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1answer
67 views

Nash equilibrium for n players game

There is a question that I am trying to solve but I am not sure about my approach and is hoping I could get some help. Here is the question: There are $n$ companies sharing a water reservoir, let's ...
13
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3answers
41k views

How to compute ALL Nash equilibria in an example of a 3x3 matrix

I am trying to understand how to compute all Nash equilibria in a 2 player game, but I fail when there are more than 2 possible options to play. Could somebody explain to me how to calculate a matrix ...
0
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0answers
27 views

numerical solution for nash equillibrium

I have the following setup. $\pi_1=f_1(q,r)$ and $\pi_2=f_2(q,r)$ are the real valued payoff/profit functions of the two players. Player 1 gets to pick $q$ and player 2 gets to pick $r$. I also know ...
0
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1answer
38 views

Nash equilibrium for two players game.

Consider a game for two players, say "Player A" and "Player B". The two sets of strategies are denoted by $A$ and $B$, available to the players. Consider a symmetric situation where the players have ...
0
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1answer
88 views

Find a Nash equilibrium solver

The solvers I know so far are designed only to allow payoffs as given numbers. But is there a solver allowing users to type payoffs as variables?
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1answer
27 views

What is the optimum strategy? Game Theory.

If you invest Rs 1000, you would get a paid back of Rs 4000 at the end of the day, provided atleast 90% of the students invested. If the number of students investing falls below 90%, all those ...