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

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Prove existence of (Nash) equilibrium

My question is about proving the existence of Nash equilibrium for a game involving two players. $x$ is player 1's strategy and $y$ is player 2's strategy; both strategies are continuous. For each ...
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Mixed strategies in 3x3 game - can strategies be negative?

Heres the payoff for player one. I'm searching for mixed strategies of player two. However I do the algebra, i get:p=4/5, q=3/10 and z=1-p-q=-0,1. Could anybody please explain, how negative ...
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1answer
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Calculating the value of a bi-matrix game.

So I know this question is very simple, however in my text and from what I can find online, the solution tends to simply be given (such as in this example) Example: Let the following bimatrix game ...
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29 views

Pure and mixed strategy in Nash Equilibria with n player

I got confused when I see the following problem: There are n staffs and they want to raise their salary, if any one or more than one of these staffs suggest their boss they want raise salary, all ...
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What is the pure strategy Nash Equilibria of asking your professor to cancel class?

Each student in a class has the option to remain silent or ask the professor to cancel class. If any students asks to cancel class, all students get a payoff of $r$. However, the student that asks ...
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Can a symmetric equilibrium yield superior social welfare in a symmetric game?

Consider a 2-player symmetric game given by a payoff matrix $A\in [0,1]^{n,n}$ for the row player (i.e. the column player matrix is $A^t$). Define the social welfare as the sum of payoffs for both ...
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27 views

Two methods for the Nash equilibrium give different answers; which is correct?

Suppose we have a game, played in which Alice and Bob play mixed strategies: (Sorry about the spacing, but I don't know how to put a table or tab spacing in this text box.) Alice plays Dove with ...
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169 views

Is convergence to a Nash Equilibrium dependent on turn order?

Is convergence to a Nash Equilibrium dependent on turn order? Namely, if you change the turn order or switch between synchronous (all players move at once) and asynchronous turns can the outcome ...
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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 ...
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21 views

Static game with complete and incomplete information

I am currently trying to learn game theory on my own. I have a question regarding the solution methods for static games with complete information vs that of incomplete information. The textbook ...
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Calculating mixed strategy Nash equilibria: using the derivative?

From roaming around and looking for ways to calculate the mixed strategy Nash equilibrium, I learned that a general way to do it is by determining the probability of choosing a strategy in such a ...
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Finding Preto Optimal allocation if utlities are of the form $u_1=x_{11}x_{12}$ and $u_2=2x_{21}+x_{22}$

There are two persons and two goods in an exchange economy. Initial endowment is $$ \omega = (\omega_1,\omega_2) =\left((1,0),(0,1)\right)$$ The utilities of two agents are given by: ...
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Could there be multiple symmetric equilibriums in a symmetric games?

Given a finite symmetric 2 player game with a strategy space $S$, a (mixed-strategy) symmetric equilibrium is a distribution $d\in \Delta(S)$ such that $(d,d)$ is a Nash equilibrium. A known result ...
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Is there a name for the ratio between the optimal social-welfare equilibrium and the worst social-welfare equilibrium of a strategic game?

Suppose you have a $n$ players strategic game, and assume that the "social-welfare"(SW) of the game is defined as the sum of payoffs to the players. Two well known measures about the "efficiency" of ...
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1answer
39 views

convex for nash equilibrium

I have trouble understanding this question, the first question to my understanding is asking me that for a fixed p , (p,q) is nash equilibrium, prove that all (p,q) are convex. and for the ...
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Equilibrium existence proof

Problem: Let $J$ be an integer and let $I$ be an integer multiple of $J$. Let ${\cal I}= \lbrace 1,2,\ldots, I\rbrace$ and ${\cal J}= \lbrace 1,2,\ldots, J\rbrace$. The set $X_{I,J}$ of all ...
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how to find mixed Nash equilibria for 3x3?

A (3,2)(3,0)(2,2) B (1,0)(3,3)(0,3) C (0,2)(0,0)(3,2) p q 1-p-q So what I have done is : 3p+3q+2(1-p-q)=p+3q q=1 this is when A=B p+3q=3(1-p-q) p=-3/4 this is when B=c I don't know ...
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What is the pareto optimal payoff vectors for war of attrition game?

The game works as follows: two player are involved in a dispute over an item. the value of the object to player i is vi>0. time is modeled as a continuous variable that starts at 0 and runs ...
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Nash's Axiomatic Bargaining: Source of problems sets and practice questions.

From where can I practice questions related to the following topic: Nash's Axiomatic Bargaining. Any form of book reference or a link to some online problem set would be highly appreciated.
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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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A conjecture about Nash Equilibria in multiplayer games involving card drafting

A deck of $N$ cards is used to play a 4-player game. The game begins with each player being randomly dealt 7 cards from the deck. They then take turns according to a set of rules, after which a single ...
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Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
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Inverse matrix methods to find Nash Equilibrium

S1 = 100,000 + (1/2)S2 S2 = 70,000 + (2/3)S1 Where S is salary of a company's CEO. Use the inverse matrix method to find the Nash Equilibrium of this problem.
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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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What is the mixed strategy Nash equilibrium in this game?

$$ \begin{array}{cccc} & Q & R & S \\ K & [2,0] &[4,5]&[1,1] \\ L &[3,2]&[1,0]&[0,0] \\ M &[1,1]&[1,0]&[0,0] \\ \end{array} $$ What is the mixed ...
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Mixed Nash equilibrium for non-square matrix game

I'm stuck with understanding the way of finding mixed strategy Nash equilibrium for non-square matrices and want to explain my difficulties with the help of the following example. Let the following ...
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1answer
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Static game of incomplete information.

There is a buyer and a seller. The seller wants to sell a used scooter. The scooter can either be of good or bad quality. The quality of the scooter is only observed by the seller. To the ...
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2answers
135 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
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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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1answer
68 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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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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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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2answers
231 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 ...
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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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--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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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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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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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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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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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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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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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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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
35 views

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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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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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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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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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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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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83 views

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 ...