Questions tagged [nash-equilibrium]
A basic solution concept in game theory that requires each player to select their best response to the strategies chosen by others.
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finding BNE: when the player with no private information has strictly dominating strategies
I'm solving this question, where I'm supposed to find all the BNE for the following game. There are two players, 1 and 2, where 1 has type a and b, and 2 has just one type (so no private information). ...
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Nash Equilibrium as a Saddle Point
Consider the zero-sum game $A = \begin{bmatrix}5 & 3 \\ 4 & -3\end{bmatrix}$. The interpretation is that Row Player chooses a probability distribution over Top and Bottom $\vec x = \begin{...
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Subgame perfect equilibrium starting with a non-NE
Consider a $2$-player normal formal game that is being repeated twice. If there's only one pure strategy Nash-Equilibrium (call it $(X,x)$), can a subgame perfect equilibrium exist where an action-...
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How to prove that a mixed strategy nash equlibrium does not exist, when a game is dominance solvable?
Analyse and find all the Nash Equilibria (including pure and mixed strategy NE) for the following game table. Explain why if there is none. (Note: You need to present in a clear and easy-to-understand ...
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Is there a concept of gaming efficiency?
Why don't people in the play want to change their strategies when the game is equilibrium? Because the payoff of the current strategy is not less than the payoff (benefit) of changing the strategy, ...
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Given Q=80-4P where MC=0, Q=Q_1+Q_2+Q_3+...+Q_n, find the reaction curve for firm A,B, and C
Does the condtion Q=Q_1+Q_2+Q_3+...+Q_n affect the process of finding the reaction curves and cournot equilibrium or do we treat it as if were given the condition of Q=Q_1+Q_2+Q_3??
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Existence of Nash equilibrium in auctions with common values and complete information
Suppose that a seller wants to allocate a single object among $n \in \{2, 3, \dots\}$ agents.
Agents know the object is worth $v \geq 0$, so that values are common and agents have no private ...
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How to show this game of complete information always has a pure strategy nash equilibrium with positive probability?
Suppose I have the following 3-player binary game with $Y_i\in\{0,1\}$ denote the action of player $i$ and $u_i+\delta_i\sum_{j\neq i}Y_j+\epsilon_i$ is the payoff function of choosing $Y_i=1$ for ...
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Bidding game with asymmetric information
Players $A,B$ are bidding for the outcome of 200 coinflips $X\sim Bin(200,1/2)$.
The winning player gains $X$ minus their bid.
Player $B$ has advantage of knowing the outcome of 20 randomly selected ...
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Game Theory Extensive Game
1st question: What is the SPNE?
My answer: A(BW), B(W): (2,1)
2st question: Show the game at its strategic form
My answer:
B
W
BB
2,1
1,2
BW
1,1
2,1
WW
3,0
0,3
WB
3,0
0,3
3rd question: Find all ...
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Existence and uniqueness of equilibrium points for concave n-person games
I hope you are doing well.
I am reading Rosen's paper.
I am having a difficulty in understanding the following statements on page 530
The set $U(x) \subset E^k$ is determined as follows:
$$
U(x)=\...
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Topkis lattice theorems for strong Nash equilibria?
Apologies in advance if my question is unclear. A well known result of Topkis is that, in short, the set of Nash equilibria of a supermodular game is non-decreasing. Is there an equivalent result for ...
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Finding the mixed-strategy Nash equilibrium in this game
Find the pure and mixed-strategy Nash equilibrium in this game:
\begin{array}{c|cccc}
P_1 \text{/} P_2 & \text{Ll} & \text{Lr} & \text{Rl} & \text{Rr} \\
\hline
\text{T} & (3,2) &...
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Game theory one shot deviation principle
Consider a variant of the alternative offers bargaining game studied
in class. Two players are bargaining over the split of a pie of total
size/value 1; each player likes as big a share as he can get. ...
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About Two Methods of Computing Bayesian Equilibria
Question
I want to compute the Bayesian equilibria for the following Bayesian game:
With probability $p$, player 1 would be of type 1.1. With probability $1-p$, player 1 would be of type 1.2. Player ...
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Mixed Strategy Nash Equilibirum, Contributing to a Public Good
I'm a economics undergrad student currently studying the basics of Game Theory. I'm trying to solve the following mixed strategy game:
-Two players, Player 1 and Player 2
-Available actions: Each ...
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Are there Sequential Equilibria that are not Trembling Hand Perfect Equilibria, that don't involve Weakly Dominated strategies?
I'm taking an advanced course (graduate level) in Game Theory. My class has covered several equilibrium concepts, up to Sequential Equilibrium.
The professor said that there are already more ...
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On and off equilibrium path game theory
I am super confused about on and off equilibrium path equilibria. Is it safe to say Nash equilibria can be on or off the equilibrium path due to the fact that one player could be irrational and ...
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Sequential Bertrand competition Nash equilibrium when marginal costs are zero
How do I find the Nash equilibrium in sequential Bertrand competition between 2 firms if both their marginal costs are zero?
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Find the Pure Nash Equilibria in this Two-Player Strategic Game
Players 1 and 2 are involved in a joint project. Each player i independently chooses an effort $c_{i}$ that can be any number in the interval from 0 to 1;
that is, $0 \leq c_{1} \leq 1$ and $0 \leq c_{...
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How to find BNE of the exchange game?
Each of two players receives a ticket t on which there is a
number in [0,1].
The number on a players ticket is the size of a prize that he may
receive.
The two prizes are identically and independently ...
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Bertrand Duopoly game
Consider the Bertrand’s Duopoly game with demand D(p) = α − p. Assume that each firm is restricted to choose a price is an integer. Also, assume that each firm has a constant marginal cost c is an ...
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Proof of existence and uniqueness
Please help me out with the conditions for existence of a pure strategy Nash equilibrium. The game is one of two players with symmetric strategies. After that, please help me out with the conditions ...
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Iterative deletion and Nash equilibria in a 3 players game
I was waked with finding all the Pure Nash Equilibrium in this game. After using the "circle the best response" method of finding each players rational choices it seems as though there is no ...
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About infinite strategy sets and $\epsilon$-equilibrium from Game Theory: Analysis of Confilct by Roger Myerson
I am studying infinite strategy sets using Myerson's Game Theory: Analysis of Conflict. On Page 143, he defines an $\epsilon$-equilibrium as follows:
Definition For any nonnegative number $\epsilon$, ...
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Auction with independent private values - An example from Game Theory: Analysis of Conflict by Roger Myerson
I have difficulties understanding the equilibrium analysis of the following auction game:
Suppose that there are $n$ bidders in an auction for a single indivisible object. Each player knows privately ...
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Using Variance for Nash Equilibrium
For the mixed strategies, the expected utility (or payoff) is used to find the mixed strategy Nash Equilibrium. The main assumption is players try to maximize their expected payoffs. However, I think ...
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Mixed Strategy Nash Equilibrium
Consider the game of Battle of Sexes, where
$$\begin{matrix}
&&Women\\
&& Football & Resaurant\\
Men&Football&2,1 & 0,0 \\
&Restaurant&0,0 & 1,2
\end{...
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Equilibrium of Perturbed Dollar Auction Game - An Example from Game Theory: Analysis of Conflict by Roger Myerson
I am studying game theory using Myerson's textbook (Chapter 3 - Equilibria of Strategic-Form Games, Section 3.6 - The Decision-Analytic Approach to Games). I have difficulties understanding and ...
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Nash Equillibrium - Depend On The Opponent's Strategy?
Say I have the following pay-off matrix:
For a one-shot game, it is easy to see, that (low, low) is the only Nash Equillibrium in the payoff-matrix.
However, say we're playing an infinitely repeated ...
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Proving the existence of mixed-strategy NE for 2-player zero-sum symmetric game
I am trying to prove the existence of mixed-strategy NE for 2-player zero-sum symmetric game, under the condition that given they have $I$ pure strategies and for the pay-off matrix $A$, $\exists x\in ...
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How do the assumptions $p'+q_ip''<0$ and $p'-c''<0$ ensure the stability of the Nash equilibrium among private firms in basic mixed oligopoly model?
I have two quick question regarding basic oligopoly models:
What is meant by we impose the assumptions to $p'+q_ip''<0$ and $p'-c''<0$ to ensure the stability of the Nash equilibrium among ...
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Nash Equilibria in a Game with Three Firms and a Shared Resource
I’m trying to understand the Nash equilibria in a game involving three firms that use water from a shared lake. Each firm can choose to purify the water before returning it to the lake or not purify ...
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Understanding the notations in Bayesian game definition
I am having trouble understanding the definition of a Bayesian game based on the following definition from class. I would appreciate it if you could explain the notations and overall meaning for point ...
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Optimality of the free market and game-theoretic arguments
I recently heard an informal argument that went something like this:
Through individual self-interest and freedom of production and consumption, the best interest of society, as a whole, are ...
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Nash equilibrium in p-beauty contest game where p=1
Setup: players must chose a number between 0 and 100. The winner of the game is the player whose chosen number is closest to the average of all chosen numbers multiplied by "p". Assume that ...
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How to find the equilibrium amount of $p_{2}$ in terms of $p_{1}$?
There are $100$ tons of crops remaining to supply for the two months. The crop holders consider whether to sell crops now or one month later. Holders face the demand curve of each period as below:
$...
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Different payouts of pure strategies in mixed strategies
I have a question with mixed strategies.
The question is as follows, if we're in a strategy profile that is a Nash equilibrium and a player is playing a mixed strategy, can the pure strategies that ...
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Understanding the Nash equilibrium for quadratic utilities
I need help to understand some steps of the article "Who's Who in Networks. Wanted: The Key Player" and I would greatly appreciate if someone can provide me with references or if they can ...
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Can game theoretic concepts be applied to any groups of strategies collectively partitioning the strategy space?
It is clear that players of a game can almost always create trivial variations on strategies without breaking game theoretic conclusions. For example, a player playing Rock Paper Scissors can play ...
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Geometric Interpretation of the Potential Function of a Game
One geometric interpretation of (at least one term of) the potential function I've come across is as the Riemann-approximated area under an individual player's cost as a function of the number of ...
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Why are Mixed Strategy Nash Equilibria special cases of Correlated Equilibria and Coarse Correlated Equilibria?
In a Mixed Strategy Nash Equilibrium, each player constructs their own probability distribution over the set of their respective possible strategies.
In a Correlated Equilibrium or a Coarse Correlated ...
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Asymmetries in Equilibrium Utility
In this lecture, the professor says that all Nash Equilibria have the same utility in non-atomic selfish routing, whereas this is not guaranteed in atomic selfish routing. It is unclear how general ...
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Rationalizable strategies/ Nash equilibrium
For the question below, how can we solve it generally for every value of θ? As the θ is not discrete, I am not sure how to apply iterated elimination of dominated strategies in this question. And is ...
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Slight Uncertainty of Continuation in Repeated Prisoner's Dilemma
In a repeated prisoner's dilemma with some probability δ of continuing after each round, a Subgame Perfect Nash Equilibrium may be found which induces cooperation instead of defection in each round. ...
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Does introducing mixed strategy Nash Equilibria override the pure strategy Nash Equilibria?
I am a bit confused as to why pure strategy Nash Equilibria are often used in analyzing a game alongside mixed strategy Nash Equilibria. While I understand that pure strategy Nash Equilibria can be ...
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Existence of Nash Equilibrium
I know there are two proofs of the existence of Nash equilibrium: One by using Kakutani's fixed point theorem in Nash (1950), and the other by using Brouwer's fixed point theorem in Nash (1951).
I saw ...
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Every Nash equilibrium is Subgame perfect Nash equilibrium
Every Nash equilibrium is Subgame perfect Nash equilibrium.
This statement is wrong. I show this by an counter example.
Is this example enough and good to disprove this statements?
Please share your ...
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Sequential version of all-pay auction
Sequential version of the all-pay auction. Two bidders alternate in bidding. A prize of \$5 is auctioned. At each move of the game, the bidding player decides whether to raise the current bid by \$1 ...
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How to find all mixed strategy Nash equilibrium
So I have been taught how to find a single mixed strategy Nash equilibrium in a 2 player game by ensuring both players are indifferent to which strategy is played.
So for example:
...
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