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

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13
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1answer
234 views

Optimal strategy for Jackpot Rock Paper Scissors

Jackpot Rock Paper Scissors is a gambling variant of Rock Paper Scissors, wherein ties result in the wager being carried forward into a jackpot. If a player plays the same hand (rock, paper or ...
9
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3answers
673 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. ...
9
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3answers
356 views

Is there experimental evidence that people ever play mixed Nash equilibrium in real games?

Have any studies been done that demonstrate people (not game theorists) actually using mixed Nash equilibrium as their strategy in a game?
7
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2answers
282 views

Finding the payoff matrix of a game

A two player zero-sum game can be represented by a $m\times n$ payoff matrix $M$ having $m$ rows and $n$ columns with values in $[0,1]$. The value $M(x,y)$ represent the payoff given to player $1$ ...
6
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4answers
228 views

Can Nash Equilibrium be more than two?

In the Prisoner's Dilemma example, we know that there is only one Nash Equilibrium. That is both of them confess. Is it possible that there are two Nash equilibrium in one example? Can you roughly ...
6
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2answers
335 views

What is the Nash Equilibrium of the Monty Hall Problem?

The Monty Hall problem or paradox is famous and well-studied. But what confused me about the description was an unstated assumption. Suppose you're on a game show, and you're given the choice of ...
5
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1answer
265 views

Algebraically finding a Nash equilibrium

Here's the problem that relates to a whole class of problems to which I am trying to figure out a general solution. Given two players 1 and 2 who can select a number from the interval $[0, 1]$, ...
5
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1answer
415 views

Finding Nash equilibrium aka finding where lines intersect

I am tagging this as multivariable calculus because it potentially involves taking partial derivatives. I am working on some mathematical treatment for Cournot duopoly models (not homework, just ...
5
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1answer
179 views

When do $\epsilon$-Nash equilibrium strategies converge to Nash equilibrium strategies?

Suppose I have a game on $n$ players and a sequence of strategy profiles $(s_1^{(1)},\dots,s_n^{(1)}), (s_1^{(2)},\dots,s_n^{(2)}), (s_1^{(3)},\dots,s_n^{(3)}), \dots$. Each ...
4
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2answers
249 views

Newspaper competition

A newspaper launches a competition. It said that readers should submit one number between 1 and 1000. A £2000 prize would be awarded to the person that got the closest to 2/3 of the mean of all the ...
4
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2answers
2k views

Cournot Nash Equilibrium Between Two Firms

Suppose we have two firms with specialized, but similar products. Suppose market demand for the two products is: $$p_1(q_1,q_2)=a-bq_1-dq_2$$ $$p_2(q_1,q_2)=a-bq_2-dq_1$$ where $d \in (-b,b)$. Suppose ...
4
votes
2answers
469 views

3x3 Nash Equilibrium?

I'm trying to figure out a Nash Equilibrium for a 3x3 zero-sum game, and it's not following normal patterns (or I'm making a huge oversight, in which case I'll feel stupid!). Can anyone help me? The ...
4
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1answer
230 views

Meaning of a partial derivative here?

I am given a 'tariff' function for two countries, $i=1, 2$. Both players can select a tariff between 0 and 100. If player $i$ selects $x_i$ and player $j$ selects $x_j$, country $i$ gets a payoff of ...
4
votes
1answer
91 views

Nash Equilibrium of cheating a test($N$-player game)

Consider a classroom with $N$ students. All the students are taking a test. Each student has 2 strategies. They can either "cheat" or be "honest"(meaning they don't cheat). The payoffs are as follows ...
4
votes
1answer
717 views

Finding mixed Nash equilibria in continuous games

I'm taking my first (graduate-level) game theory class. I understand how to find Nash equilibria in simple games, such as those given in finite tables, and can see (usually) how to find the mixed ...
4
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1answer
728 views

Analytically solving (calculating Nash equilibrium for) 3-player extensive form games

Let's say we extend the popular half-street Kuhn poker variant to 3 players. The rules would be as follows: ...
4
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1answer
422 views

Iterated prisoners dilemma with discount rate and infinite game averages

Suppose we have two players who are perfectly rational (with their perfect rationality common knowledge) playing a game. On round one both players play in a prisoners dilemma type game. With payoffs ...
3
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2answers
1k 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: ...
3
votes
1answer
223 views

Game Theory Voting

I am having some difficulty in solving the following problem. I was wondering whether someone would be kind enough to sketch a solution or even better to solve the whole game. Thanks Suppose that ...
3
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0answers
98 views

Unexpected hanging paradox maxmin strategies

I have a question about strategies of the players of Unexpected hanging paradox (I am very sorry for a long topic, topic exist already for a while, during this time I try to develop idea how to solve ...
3
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1answer
666 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 ...
2
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2answers
2k views

Cournot-Nash Equilibrium in Duopoly

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way of solving the following question: There are 2 firms in an industry, ...
2
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2answers
83 views

Is equilibrium selection in zero sum game trivial?

Does a zero sum game always has a unique payoff, whatever the nash equilibrium selected is ? even with mixed strategies ? If so, what is the proof ?
2
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1answer
47 views

Check my answers: Dominant strategy.

I saw another question on Game theory. My answer for part a the nash equlibria (T, L) and (B,R). for part-b, Player-1's action T is strictly diominated. So Player1 never choose T. For part ...
2
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1answer
56 views

What are the optimal strategies for the “prime-game”?

A and B are playing the following game : A and B choose a number from 1 to 100, not knowing the number chosen by the opponent. A wins if the sum of the chosen numbers is prime, otherwise B wins. ...
2
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2answers
79 views

In the next matrix, why is (55,0) not a Nash Equilibrium?

My book says that the next matrix has no Nash Equilibriums. Still, Im a little confused about row 3, column 2. Reasoning from player 2's perspectivo, he could say "if player 1 chooses row 3, I Will ...
2
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2answers
99 views

nash equilibirum help! seems tricky

Any advice for finding all nash equilibrium for this symmetric game? (B,B) looks like one but I feel like there are more. I tried looking for strictly dominant strategies, but only A weakly dominates ...
2
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1answer
157 views

Is there an example of zero-sum game that has a Nash equilibrium which is not subgame perfect?

As a refinement of Nash equilibrium, it is known that not all Nash equilibria are subgame perfect. But it seems to me in zero-sum games of perfect information, Nash equilibrium coincides with subgame ...
2
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2answers
542 views

Are all Nash equilibrium pure strategies also Nash equilibrium mixed strategies.

while going over wiki page on Battle of the Sexes game I found something funny. This game has two pure strategy Nash equilibria, one where both go to the opera and another where both go to the ...
2
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1answer
144 views

vickery auction question(second-price auction)

The question is as follow, Alice and Bob would both like to own the same manuscript. The manuscript is worth 5 million to Alice and worth 3 million to Bob. The present owner of the manuscript ...
2
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2answers
208 views

Mixed strategy nash equilibria in from $2\times N$ bimatrix form

I'm looking for a way of finding (manually!) mixed strategy Nash equilibria in a $2\times N$ game. Calling player 1 the player with two strategies and player 2 the one with $N$ strategies, I've ...
2
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1answer
36 views

Game theory: Mixed Strategies and Nash Equilibrium

So I've recently become interested in game theory, and I've visited this site to help me understand what exactly game theory is and the applications of it. In the lesson, they use an example of ...
2
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1answer
55 views

Is the Nash Equilibrium example in a “Beautiful Mind” accurate?

I was wondering if the Nash Equilibrium example shown in the movie A Beautiful Mind is accurate? and if not, what's wrong with it? Thanks
2
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1answer
41 views

Does Unpredictable Four have an optimal solution?

The rules of Unpredictable Four are quite simple. One player (the crazy) tries to be unpredictable, while still achieving a goal -- and the other player (the psychic) tries to predict them. However, ...
2
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1answer
98 views

Plotting the best response

Observe the following matrix; The pure strategy and mixed strategy nash equilibria are The best response plot is given below Can someone explain how this graph was plotted. I would much ...
2
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1answer
180 views

mixed strategy nash equilibrium question!

Suppose the game consists of only $2$ players, player $1$ and player $2$, and each of them has only $2$ strategies to choose between. This gives a $2$ by $2$ payoff matrix. Player $2$ has no ...
2
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0answers
32 views

How to find perfect equilibria in a finite game?

If we define a game with $n$ persons as below: (i) for each player $i$, he has his strategy set $S_i$, $|S_i|=m_i<\infty$, and denote $S=\Pi_iS_i$; (ii) $u_i:S\rightarrow\mathbb{R}$ is a payoff ...
2
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0answers
131 views

Comparing Nash equilibrium and Pareto optimal actions

Suppose that $(x_{i}, x_{j})$ identify actions for two players $(i,j)$. If we define Pareto optimal actions by $$h(x_i) +h(x_j)- \eta[p(x_i)+p(x_j)]=2\gamma$$ and Nash equilibrium actions by ...
1
vote
2answers
7k 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 ...
1
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2answers
3k 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 ...
1
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2answers
69 views

Third and average price auction

Third price auction: the winner is the highst bidder but this time instead of paying the second highst bid, he would pay the third highst bid. -assume there are at least 3 bidders. - Average price ...
1
vote
1answer
757 views

Finding Nash equilibria using Support Enumeration

Chapter 3 of the Book "Algorithmic Game Theory" introduces an algorithm (page 8 of that PDF) to find mixed Nash equilibria for a bimatrix game $(A, B)$, which I struggle to understand. ($M$ and $N$ ...
1
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1answer
67 views

Correlated Equilibrium - Transforming a non-linear objective function into a linear one

I am trying to transform a non-linear objective function into a linear one, in order to create a LP. How might I go about to do this (I have never taken a course in linear programming). I have that I ...
1
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1answer
168 views

Nash equilibria of mixed strategies

I am given the following game to find nash equilibria in pure and mixed strategies: $\begin{pmatrix}& & Litte John &\\ & & c & w \\Big John & c & (5,3) & (4,4) \\ ...
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1answer
1k views

Symmetric nash equilibrium

I was reading this paper on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how ...
1
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1answer
54 views

Finding mixed nash equlibrium

In the following game I found one pure nash equilibrium: $(R, r)$: $\begin{array}{r|ccc} A\backslash B & l & m & r\\ \hline L & (-10, 4) & (10, 0) & (-1, -1)\\ M & (0, 10) ...
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1answer
96 views

Game Theory - Nash equilibrium question

Consider a game in which 2 players transmit packets in a network with a selected power $x ∈ [1, A]$ and $y∈ [1, A]$, respectively. The utility of the players can be expressed as: $$u_{i} (x,y) = ...
1
vote
2answers
74 views

Where's the Nash Equilibrium here? $ \begin{pmatrix} (2,-2) & (2,-2)\\ (1,-1) & (3, -3) \\ \end{pmatrix} $

I just opened a book on Game Theory, so I'm totally new to this. My book says that the only Nash Equilibrium in the example below is (2, -2) -first row, first column-, and I really don't see why... ...
1
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1answer
47 views

Question on the construction of mapping from space of strategy profile into itself in Nash(1951)

To appeal to Brouwer fixed point theorem, Nash(1951) constructed a continuous mapping $\operatorname{T}$ from strategy profile space into inself: For player $i$, the probability of a pure strategy ...
1
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1answer
36 views

Show why answer no longer holds when inequality changed

This below is a Nash equilibrium problem, I'm stuck in the math part. I solved the first part but I'm confused on the second one. I believe there is a mistake on denominator and it should be ...