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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Nash in demand functions!

I am searching for some types of games that are played in linear demand functions. Altough I hear that there is a vast literatrure for games that are played in the intercept or the slope of the demand ...
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Mixed Nash Equilibrium with missing info [closed]

So I've come across this problem in Watson's book: Suppose you know the following about a particular two-player game: S1 = {A, B, C}, S2 = {X, Y, Z}, u1(A, X) = 6, u1(A, Y) = 0, and u1(A, Z) = 0. In ...
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In a game with alternating moves and complete information, the Nash equilibrium cannot be a non-trivial mixed equilibrium?

Where I can find a simple proof for this fact? For example, a trivial bimatrix game with alternating move has the following payoff matrix: \begin{array}{|c|c|c|} \hline & 1 & 2 \\ \hline U &...
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Mixed Strategies in Bayes Nash Equilibrium (Bayesian Battle of the Sexes). Shouldn't it depend on $p$?

I have a question about calculating mixed strategies in a Bayes Nash Equilibrium in a simple 2-player bimatrix game. To demonstrate the issue, consider ``Bayesian Battle of the Sexes.'' Suppose P1 ...
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Cournot oligopoly - first-order condition

I am reading an article that has this description of the first-order condition for a Cournot n-firm game: Take $P(Q) = Q^{-1}$, $\pi_i(q_i, Q) = (Q^{-1} - c_i)q_i$. Then the first-order condition for ...
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How can I build a fixed point theorem argument in pure strategies?

To begin with, I am recalling the Banach Fixed Point Theorem. Let $(X,d)$ be a non-empty complete metric space with a contraction mapping $T:X\to X$. Then $T$ admits a unique fixed-point $x^*$ in $X$ ...
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Pure-Strategy Bayesian Nash equilibrium with general common prior

I'm doing a problem set on the subject of Bayesian Nash equilibrium. I'm asked to find the pure-strategy BNE of the following. I've calculated to matrix shown below. My first concern is if I've ...
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Doubt about Mixed Strategy Nash Equilibrium

Here's the video I will be referring too. Now I am a complete beginner in game theory, so sorry if this sounds like a stupid question, but why would a player want to balance out the payoffs of another ...
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Nash Equilibrium for n-shops Location Game

So if two ice cream shops were to be placed in the location $[0,1]$, inorder to maximize their own pay offs, they both would finally come to the location $[\frac{1}{2}, \frac{1}{2}]$. This is also the ...
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Stag Hare inefficient Nash Equilibrium and level-K thinking

Two players $i,j$; both have two strategies $\{h,s\}$. The payoffs vector of $i,j$: $u(h,h)=(5,5)$ (if both players choose $\{h\}$ then $i$ receives 5 and $j$ receives 5) $u(h,s)=(10,0)$ $u(s,h)=(0,10)...
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Interpretation of Nash equilibrium as a potential stable point of a dynamic process

I'm reading an article called "The Nash equilibrium: A perspective" by Holt and Roth, and the below paragraph caught my attention. When the goal is prediction rather than prescription, a ...
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Can mixed strategies actually predict behaviour of rational actors in non-constant sum games?

I understand how the concept of the mixed NE (mathematically) works. But I don’t understand how we can expect players to behave in a way that would arrive at such an equilibrium. Consider the ...
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Does the value of a pure strategy Nash equilibrium(if exists) equal the value of the mix strategy Nash equilibrium in two-person zero-sum game?

Given a two-person zero-sum game, a mixed strategy Nash equilibrium always exists and all such equilibria have the same value. A pure strategy Nash equilibrium, however, may not exist. My question is:...
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Existence of Symmetric Pure Strategy Equilibrium

I have 2 symmetric players $A$ and $B$. Each of them has 2 decision variables $x_i\in[0, \beta]$ and $y_i\in[0,1]$, where $i\in\{A,B\}$. Their payoff functions are symmetric, i.e., if you swap the ...
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NE equilibrium with lobbying of cournot producers

I have had an exam (exam is now past and submitted, but I want to now understand the solution without waiting) with the following questions: GAME Consider two firms playing the following two-stage ...
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1answer
51 views

Find value of $\beta$ for which there is a strictly dominant strategy

The question is as such: $N$ firms are lobbying for subsidies. Let $h_i$ be the number of hours spent by form $i$ for lobbying, with cost $wh_i^2$ where $w$ is a fixed constant. The subsidies granted ...
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(Why) was von Neumann not a fan of the Nash equillibrium concept?

Charles A. Holt and Alvin E. Roth's The Nash equilibrium: A perspective notes: "In a personal communication with one of the authors, Nash notes that von Neumann was a “European gentleman” but was not ...
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1answer
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Why do we need a restriction of a game to prove the given statement?

Consider a game $G$. We have to prove that is $s$ is a Nash Equilibrium of $G$, then it is also a Nash Equilibrium of the game formed by removing strictly dominated strategies of $G$. I looked at the ...
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Is the Nash product really maximised ex post?

In my game theory class this term, we studied Nash bargaining. It is only now when starting to prepare for the exam that I have come to realise there is something I fundamentally don't understand, and ...
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1answer
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Pure Nash Equilibria 3 players game

I'm trying to solve this pure-strategy Nash equilibria of this game below: I highlighted the best pay off for player 1 and 2. But I don't get it when it comes to player 3. The correct answer is (A) ...
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1answer
43 views

Pure and Mixed Nash Equilibrium algorithm gives different results

I have a game represented by following table: It is clear that there is a pure Nash equilibrium at 4,2 (both players do not cooperate, player 1 awarded 4 points and player 2 awarded 2 points). Now ...
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What is the subgame perfect pricing policy for seller?

Suppose there is a seller S who is selling to 3 different potential customers H, M, L The good is a durable good and it's utility in terms of dollar equivalent is as in the picture below: So if H ...
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Why is infinite recursion on the common knowledge assumption necessary?

If something is common knowledge in a game, that means that every player knows it, and every player knows that every player knows it, and so on. Are there cases where only one such level of knowing ...
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REE with asymmetric information and Bayesian Nash equilibrium?

What is the difference between a Rational Expectations Equilibrium (REE) with asymmetric information and Bayesian Nash Equilibrium (BNE)? Since agents in both cases play some game or have a strategic ...
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How to find mixed optimal strategies in this zero-sum game?

I'm trying to solve this problem from last year final exam in game theory: Consider the zero-sum game $G=(X, Y, g)$ where $X=Y=[0,1]$, and $$\forall (x,y) \in X \times Y: g(x, y)=\max \{x(1-2 y), y(1-...
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Who is the first person/paper to introduce “mixed strategy”?

Who is the first person/paper to introduce "mixed strategy"? The PNAS by Nash used this notion without citing anyone. Does the earlier book: Von Neumann, J., and Morgenstern, O., The Theory ...
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Finding Bayesian Nash Equilibrium

I'm recently new to Game Theory and I've recently started teaching myself about Bayesian Nash Equilibirum. I've stumbled across a problem set that I can't seem to wrap my head around concerning ...
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How would a 2D model of Hotelling's law outweigh the benefits of a 1D model?

I'm not sure if this goes here, or on a math exchange. I could move it if you guys want... Let's examine Hotelling's law on a 1D plain with two shops. Both shops would do society a favour if placed ...
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Confusion about the convexity of the best response correspondence

I am recently reading the proof of the existence of the Nash Equilibrium. As a math student, I do understand the use of Berge's maximum theorem and Kakutani's fixed point theorem, but I am not sure ...
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Is it possible to find a nash equilibrium that is not an equilibrium in weakly dominant strategy?

I know that it is possible to have a Nash equilibrium which is not an equilibrium in dominant strategy, but is it also applicable for equilibrium in weakly dominant strategy (i.e. a Nash equilibrium ...
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Difference between equilibrium and k-rationalizability

I'm reading a Structural Models of Nonequilibrium Strategic Thinking: Theory, Evidence, and Applications by Crawford, Costa-Gomes and Iriberri. They write the following: In two-person games, a ...
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1answer
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How to read nash equilibrium from a normal form?

For example, on the section with the title "sequential games in normal form" of this wikipedia page, there is a table with all its SPNE and nash equilibrium labelled. How are the NEs obtained? It ...
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Is a mixed strategy ever the best response to a pure strategy?

Suppose you are playing a game against an opponent whom you know only uses pure strategies. My question is, is there any such game in which using a mixed strategy in response is better than all the ...
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Existence of symmetric trembling hand perfect equilibria

Consider symmetric and finite game. By Nash (1950), the game must have at least one symmetric equilibrium (proof). Also, it must have at least one trembling hand perfect equilibrium (proof). ...
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Bertrand Duopoly Equilibrium for Discrete Prices

There are two identical firms, $1$ and $2$, with zero marginal costs. They produce homogenous product, which is demanded by a unit mass of identical consumers, each of which has inelastic unit demand ...
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Unique Nash-equilibria in multi-unit auctions with uncertain participation

Setup Consider a one shot sealed bid multi-unit auction where $N$ bidders compete for $K$ identical objects and each bidder $i$ has demand $d_i\in \{1,\dots,K\}$. Bidders receive private i.i.d. ...
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Trembling hand perfection and weakly dominated strategies

It is well known that players cannot use weakly dominated strategies in a trembling hand perfect equilibrium. My question, however, is a little different: does iterated deletion of weakly dominated ...
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1answer
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Incomplete information in multi stage game

I would like to solve a game where firms have private information about their own type, but only know the distribution of the other firm's type. They interact in two stages, where the strategies ...
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Argue that no further mixed Nash Equilibria can exists

I'm looking at the following Normal-Form Game: ...
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Fill out Normal-Form Game to obtain exactly one mixed equilibrium

I'm given the following incomplete Normal-Form Game: | L | R +-------- O|1,?|7,? U|?,2|?,1 First I was asked to fill out the missing pieces to obtain a game ...
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Difference between Nash equilibrium and Pareto Efficiency

Nash Equilibrium is defined as a solution concept referring to a best outcome which players won't want to unilaterally deviate given the response of other players doesn't change. To me it seems like ...
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1answer
42 views

Non-Bayesian Mechanism Design

Suppose we have a mechanism where a finite number of agents possess private information that is not drawn from a probability distribution. The agents' types are given and fixed but agents only know ...
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Maximum-support Nash equilibria in zero-sum games

Context: I would like to know how likely a player is to pick a specific action, provided that he plays optimally and the action is optimal. Phrased like this, the question is ill-defined. But is there ...
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I cannot find for this Simple shared effort level game

Each player can contribute to the project with non-negative effort. Player 1's utility is $u_1=e_1(1+e_2-s\cdot e_1)$ where $s\in [0,1]$. Player 2's utility is $u_2=e_2(1+e_1-e_2)$ For case 1, ...
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The difference between a Nash equilibrium and Bayesian Nash equilibrium [closed]

What is the basic difference between Nash equilibrium and Bayesian equilibrium?
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Fehr & Schmidt, ultimatum game, inequaltiy aversion, perfect subgame Nash equilibrium

I am preparing for an exam. I have found an old exam but I have no solutions for it, so I tried to solve it, but I dont know if I did it correctly and need therefore your help. The problem looks as ...
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NASH equilibrium [closed]

How to approach questions like these: In a two player static game with a discrete strategic space that permits each player to chose one of the four possible strategies what is the maximum number of ...
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Optimal pareto in two-person game

For what values $x$, $y$ the profile $(D,L)$ is Pareto optimal? \begin{array}{c|ccc} & L & R \\ \hline U & x,5 & x+2,y \\ D& 1,-1 & x,0 \\ \end{array} Is correct $x<1$ ? ...
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Condition for a Nash equilibrium

Consider two people have a mutually advantageous relationship. That is, if both dedicate more effort to the relationship both improve. Specifically, each individual chooses a level of effort $x_{i}\...
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Asymmetric Nash Bargaining

The Nash bargaining solution selects the unique solution to the maximization problem $\max_{s_1, s_2 } (s_1 - d_1) (s_2 - d_2)$ such that the solution satisfy the following axioms : Invariance ...