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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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
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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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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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42 views

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
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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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1answer
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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(...
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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 of Games ...
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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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1answer
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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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1answer
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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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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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Evolutionary stable strategies

I am new to evolutionary game theory so I can't figure out whether I'm looking at things correctly. I have the following payoff matrix: $$ A = \begin{matrix} 3 & 0 \\ 5 & 1 \end{matrix} $$ ...
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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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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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1answer
238 views

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
39 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 ...
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1answer
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Rationalizability or Nash equilibrium

In the 'chicken' game, (stop,go) and (go, stop) are the two pure strategy Nash equilibrium profiles. If the game is played once, does it make more sense to use the rationalizability solution concept? ...
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Interpretation of Solution Concepts

I was wondering whether there is a neat overview over different interpretations of game theoretic solution concepts such as Nash equilibrium, Sequential Equilibrium and the like. Textbooks I found ...
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Game Theory: finding nash equilibria of an extensive form game (game tree) [duplicate]

How do you find the pure Nash equilibria from this commitment game? The SPNE is (U, ps) Do you find the NE by finding the best response (BR) of a player to a specific strategy of the other player? ...
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1answer
225 views

Definition of Bayesian Nash equilibrium

I have a basic doubt on the definition of Bayesian Nash equilibrium. Consider the following game: 1) $N$ players. 2) Each player $i$ has a type, assigned by nature and denoted by $\epsilon_i$. ...
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1answer
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Repeated Game SPNE

I approached this question in this way: $(P_1,P_2), (R_1,R_2), (S_1,S_2)$ are the Nash Equilibria of the Stage 1 game. For the given strategy to be sustained as SPNE, there should be no way unilateral ...
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Nash equilibrium - mistake in proof of paper?

I have a question regarding the proof of Proposition 1 in Besley and Ghatak (2007) in Appendix A of their paper. It is a quite highly cited paper but I believe there is a mistake in the proof of their ...
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Cournot competition subgame perfect Nash equilibrium with two products

QUESTION: Assume there are two types of products, labelled $l$ and $n$. Firms compete in the market by choosing which product to sell and then choosing the quantities. Let $Q_n$ and $Q_l$ denote the ...
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2answers
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What trick can be used to calculate mixed-equilibria?

In continuous games, the probability distributions over the players' strategy spaces are infinite. How then is it even possible to then derive a mixed-strategy nash equilibrium? One would have to ...
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Sequential Bertrand game with differentiated goods, how to write the strategies of firm 2 [closed]

In a Bertrand competition with differentiated goods where firms set the prices sequentially, we have the following demand functions: q1 is quantity of goods demanded for firm 1 q2 is quantity of ...
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Rationalizable action profiles in nice symmetric games

Suppose we have a nice symmetric game with $n$ players, i.e. each player's action space is the same compact interval of the real line. I am tasked with identifying all of the rationalizable action ...
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What does these “strategy notations” mean?

In a sequential game, where there are 8 pizzas. Player 1 decides number of pizzas he wants. Let's call it S1 (strategy of player 1), and S1 = 5 means player one decided to get 5 pizzas. Then player ...
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Necessary indifference conditions in mixed equilibrium

Suppose we are playing a game where the Action set for Player 1 is $(a,b)$, for Player 2 is $(c,d)$, and for Player 3 is $(L,M,R)$. Assume that for Player 3, the action $M$ is weakly dominated by some ...