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

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Do all equilibria in 2 player zero sum games have the same distribution over outcomes

I know that in a 2 player zero sum game all equilibria give each player the same expected value, but is it the case that they also induce the exact same distribution over payoffs? Or might there be ...
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1answer
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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 ...
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2answers
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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
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1answer
55 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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2answers
32 views

Why you randomize your opponents payoff in a mixed nash equilibrium?

I wanted to understand the justification more intuitively -- if that is possible. For example, I'm in a abstract game with another opponent and there is no pure strategy equilibrium: why do I ...
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1answer
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The potential function of Prisoner's Dilemma

As in the famous example of "Prisoner's Dilemma" like this If the potential function is defined as: (V(q,q), V(q,c), V(c,q), V(c,c)) q = quiet, c = confess, V is the potential. So should the order ...
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0answers
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--Should I bother to vote?? — so thats toughest one Ive ever faced… :/ any help to solve that?

So thats the question Im facing, Im still stack at Bayesian equilibriums, any help would be very much appreciated Thanks very much for your time guys :)
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Help to solve this problem, the result doesn't seem right :/ silly mistake somewhere probably

Suppose that Coke and Pepsi are the only firms producing cola. Their products are not identical, but are very close substitutes. Let $P_c$ denote the price of Coke and $P_p$ the price of Pepsi. Demand ...
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1answer
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So I have the following question, dont have much info on class notes and not sure how to tackle it, any suggestions, any help?

A seller has a single item for sale (which she values at zero). There are two potential buyers. The seller decides to use the following auction format to sell the object: each bidder submits a sealed ...
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1answer
29 views

Finding Nash Equilibria for this Bimatrix Game

Consider the following Bimatrix Game a b c d -----|------------------------------------------- T (1,4) ( 4,3) (0,2) (1,0) B ...
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1answer
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Decide the Nash Equilibrium

If two people collaborate on a work: the (3,3) means if neither of them do any work, then, they have to be put on detention for 1 hour and then both of them still have to finish the same 2-hour work ...
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4answers
231 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 ...
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3answers
679 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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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 ...
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0answers
24 views

Finding mixed nash equilibria

I understand how to find equilibria in two player games. I don't understand how to find mixed Nash equilibria in Load balancing games? For example, games which involve a specific number of machines ...
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2answers
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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 ...
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1answer
238 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 ...
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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, ...
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1answer
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Finding Nash Equilibriums

Nash Equilibrium for this Normal Game 1,1 2,4 1,4 0,8 1,1 1,1 3,0 0,0 7,0 I know ...
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1answer
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Is there any dominant strategy? [closed]

Is there any dominant strategy for the below matrix? I think no... But I am not sure. please check my answer:) $$\left (\begin{array}{ccc} (0,0) & (0,1) \\ (1,0) & (-1,-1) \end{array}\right ...
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0answers
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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 ...
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1answer
43 views

Show a simple strategy.

Imagine that we have 49 cards with the values written on their faces, (they are all visible ) as follows; $$25, 24, 23, 22, ........3, 2, 1, 2, 3, .........23, 24, 25$$ suppose Paola and Victor are ...
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1answer
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Help me writing Payoff matrix.

I guess, in order to answer this question, I need to write Payoff matrix. But I cannot write it. And then, I Will able to answer this question by myself. Thank you for helping. (These are just ...
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1answer
48 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 ...
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1answer
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A question on Game theory

I'm studying Game theory, I saw the question: Consider two players; player A and player B playing the following estimation game. Each player chooses a number from {1, 2, 3}. If the difference ...
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2answers
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A little question about payoff functions being continuous.

In the mixed extension of a finite game $G$, why are the payoff functions of players continuous? Does it has something to do with being von Neumann and Morgenstern utility functions? Is there other ...
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126 views

Pure vs mixed strategy Nash Equilibria

Just learning about Nash Equilibria. The pure strategy one is explained as an outcome where both/all players feel like they couldn't have done better given what the others were doing. Mixed strategy ...
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1answer
64 views

Finding Nash Equilibria of a finte game of 2 players.

In a finite game, suppose player 1 has strategies $\{\alpha_1,\alpha_2\}$ and player 2 $\{\beta_1,\beta_2\}$ with payoffs as below. \begin{array}{c|c|c} &\beta_1&\beta_2\\ \hline\\ ...
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1answer
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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) = ...
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1answer
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Explanation of basic definitions in game theory.

In the article entitled Non-Cooperative Game written by Nash in 1951, he discussed about the symmetries of games. Due to my lack of basic knowledge in permutations and symmetries, I looked up some ...
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1answer
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A little question about the existence theorem of Nash equilibrium in game theory

Recently when I started reading Nash's paper, I found a little question about the linearity of payoff functions. Is it an assumption? Or did I miss some idea about the payoff function and its ...
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0answers
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Author of Best response

I used the concept of a best response, so I want to refer to the first author of the concept of the best response function. What I find is (Fudenberg & Tirole 1991, p. 29), however, nash already ...
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2answers
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Which Nash equilibrium is this?

There are two stores, A and B with the following demand functions, The open circles indicate an open interval, the filled circle is a closed interval. Both firms wants to maximize profit, and ...
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1answer
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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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1answer
102 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 ...
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1answer
224 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 ...
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1answer
391 views

Ice cream vendor problem

PROBLEM This is a question considering game theory. Assume there is a beach with $n$ ice cream vendors on it who position themselves along the beach. For an arbitrary $n$, find a candidate Nash ...
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1answer
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Nash Equilibrium in Cournot Duopoly

If you have a demand function Q1= 20 + 3/4 p2 - p1 and Q2= 20 + 3/4 p1 - p2, how does one go about solving for the Nash equilibrium if you know there are no costs? I've tried solving for Q and then ...
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0answers
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Proving lower bounds from algorithmic game theory paper (specifically, price of anarchy is lower bounded by 3/2 for $m$ links)

This question is similar to Understanding proofs from paper on Game Theory (Price of Anarchy) This question is about the same proof: proving the lower bound that the price of anarchy (sometimes ...
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2answers
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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 ...
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0
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Find the Nash equilibria in pure strategies in the following games.

Payoffs on the left correspond to Player 1 and payoffs on the right correspond to Player 2—for instance, in (U, L), player 1 gets a payoff of 3 and player 2 gets a payoff of 4. Explain in detail why ...
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Consider 2 identical players (i.e. i = 1, 2) with utility function:

Consider 2 identical players (i.e. i = 1, 2) with utility function: πi = b(qi + q-i) - cqi. Where qi is equal to one if player i contributes to the public good and is zero if she does not, q-i is the ...
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2answers
470 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 ...
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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 ...
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1answer
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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 ...
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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... ...
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1answer
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Finding Mixed Strategy Nash Equilibria

Okay, so I was working through this problem: Now, I understand the computations. What I don't understand is why the solution says that each player will play H with probability p=2/3. I would have ...
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1answer
62 views

Simulating Mixed Nash Equilibria

I have a $N$ person game where each person has a set of $M$ discrete strategies. I know from the theory that at least one mixed strategy Nash Equilibrium exists. Can someone please tell me how do I ...
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1answer
91 views

Efficient way to compute min/max

I have a function: \begin{equation*} f(a_1,\ldots,a_7,b_1,\ldots,b_4)=-\tfrac12 a_1 + a_1 b_1+\tfrac12 a_1 b_2-\tfrac12 a_1 a_2 b_2 + 2 a_1 a_2 - \tfrac32 a_1 a_2 b_1 - \tfrac12 a_4 + a_4 b_1 + ...