1
vote
3answers
30 views

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 ...
0
votes
2answers
26 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 ...
0
votes
1answer
38 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\\ ...
1
vote
1answer
64 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
1answer
59 views

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 ...
0
votes
1answer
33 views

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 ...
2
votes
1answer
55 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. ...
1
vote
1answer
74 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 ...
3
votes
1answer
168 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 ...
0
votes
1answer
268 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 ...
1
vote
0answers
59 views

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 ...
1
vote
2answers
4k 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 ...
3
votes
2answers
419 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 ...
2
votes
2answers
74 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 ...
4
votes
1answer
72 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 ...
1
vote
2answers
72 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... ...
0
votes
1answer
60 views

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 ...
0
votes
1answer
60 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 ...
0
votes
1answer
168 views

Air Strike Game

This is an Air Strike Game with the solution, I have added some questions regarding the solution and I would appreciate if someone could answer them. Army $A$ has a single plane with which it can ...
0
votes
1answer
96 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 ...
5
votes
1answer
161 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 ...
2
votes
2answers
90 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
votes
1answer
133 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 ...
1
vote
0answers
90 views

Algorithm to verify that a weak Nash equilibrium is an ESS, or a strict Nash equilibrium

Is there any algorithm that might assist me in checking whether a weak Nash equilibrium in a signalling game is also an Evolutionarily Stable Strategy, or a strict Nash?
1
vote
1answer
46 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 ...
3
votes
0answers
85 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 ...
7
votes
2answers
244 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 $1$ ...
1
vote
2answers
1k 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, ...
0
votes
1answer
47 views

Theorem that stable equilibria in iterated games are equivalent to coalition-based static equilibria

Consider an $n$-player nonzero sum finite game $G$. I have a vague recollection of a wonderful paper proving an equivalence between (1) steady state Nash equilibria of $G$ played countably many times ...
-4
votes
1answer
378 views

Nash equilibria and best response functions

a) Let $G=(A,u)$ be a strategic game such that, for each $i \in N$ $A_i$ is a nonempty, convex, compact subset of $R^{m_i}$ $u_i$ is continuous For each $a_{-i}$, $u_i(a_{-i}, . )$ is quasi-concave ...
0
votes
1answer
177 views

Prove set of Nash equilibria is closed?

Is this even possible with just the formal definition of a Nash equilibrium, that is, without any additional conditions, such as the utility function is continuous? Thanks.
6
votes
2answers
310 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 ...
2
votes
0answers
120 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 ...
4
votes
1answer
605 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 ...
2
votes
2answers
466 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 ...
3
votes
1answer
549 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 ...
4
votes
1answer
687 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: ...
0
votes
0answers
73 views

Can the Nash bargaining solution be applied in repeated game?

I am trying to develop a model involving two agents who interact strategically to set an optimal time for a joint work. These agents will have to meet repeatedly. I want to derive the optimal time for ...
3
votes
2answers
780 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: ...
1
vote
1answer
132 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 ...
1
vote
1answer
64 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 ...
0
votes
1answer
201 views

subgame perfect nash equilibrium for war of attrition

the question is as follow: suppose that two players are playing war of attrition, that means both of them could choose either to fight or quit, if either one of them quit, the game ends, and if ...
1
vote
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 ...
0
votes
1answer
1k views

cournot equilibrium and stackelberg equilibrium question

Question is as follow: there are 2 firms that want to enter the apple juice market in country A. There are no existing firms in the market or potential entrants. They need to decide on yearly ...
2
votes
1answer
169 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 ...
1
vote
1answer
196 views

Question on mixed nash equilibrium!

The question is as follows: Think of the Golden Ball game. Now player 1 is money-minded and jealous, and player 2 is very good-hearted, so the payoff matrix is follows: ...
2
votes
2answers
82 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 ?
0
votes
2answers
577 views

Subgame Perfect Nash Equilibrium

My homework question is summarized below: There are 7 players (say P1,P2,...,P7) trying to split 100 dollars. The game starts with P1 proposing an allocation of the 100 dollars to each ...
4
votes
1answer
353 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 ...
0
votes
1answer
569 views

Finding Nash Equilibria with Calculus

The problem is summarized as: There are two players. Player 1's strategy is h. Player 2's strategy is w. Both of their ...