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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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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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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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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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How to find nash equilibrium in this model? [duplicate]

I am only looking for an intuitive answer, so I won't provide any specific equations. In this model: firm A and firm B compete on prices. Firm A is always better off choosing a smaller price than ...
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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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1answer
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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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2answers
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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 ...
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1answer
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Non-negative deviations from Nash Equilibrium

I know that in a Nash Equilibrium, no player can profitably deviate from the equilibrium strategy assuming that the strategies of the other players remain the same. My question is, what if a player ...
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Take It Or Leave It Strategy: Social Optimum

Here is what I understood Using Backward Induction, I inferred that buyer offers a price, say, $P$ and the seller will sell only if $P \geq c(I)$. Setting the lowest possible Price that will ensure ...
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Finding correlated equilibria

I am a bit confused in my game theory class when it comes to finding correlated equilibria. I understand how to write the constraints using probability distributions, but I don't know how to find the ...
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Disagreement in Strategic Bargaining

Construct a pair of startegies for the ultimatum game ($T=1$ bargaining game), that constitutes a Nash Equilibrium and together support the outcome that there is no agreement reached by the two ...
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Computing pure strategy Nash equilibria in finite games

I am trying to compute the (pure strategy) Nash equilibria of some discrete auctions. More precisely, let us define the strategy of each player as a function mapping from every valuation that they ...
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Prisoner's dilemma as a Bayesian one-shot game

What happens if we assume that there is incomplete information to the prisoner's dilemma game? For example, suppose we have the following matrix with the utilities $T>R>P>S$ and $2R>S+T$ ...
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1answer
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Solving a two stage game by backward induction: which is the equilibrium notion?

Take a two-stage game with complete information and simultaneous actions in each state: (1) Player 1 and 2 simultaneously choose action $a_1\in A_1$ and $a_2\in A_2$ respectively. (2) Player 1 and ...
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Static game with complete but imperfect information

I am confused on the concept of static game with complete but imperfect information and its consequences on the equilibrium definition. Suppose we have 2 players. Each player $i$ chooses action $Y_i\...
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If a mixed strategy is strictly dominated, then there is a strictly dominated pure strategy in its support?

I am looking at the proof of NE survives the iterated removal of strictly dominated strategies (MWG, ex 8.D.2) and in the solution manual, authors say something like if a mixed strategy is strictly ...
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1answer
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Nash Equilibria in Target Destroying-Guarding Game

Army A has a single plane which can strike one of three possible targets, A, B and C. Army B has one anti-aircraft gun that can be assigned to one of the three targets to guard it. The value of each ...
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1answer
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Nash equilibrium two players in a joint effortful project

I have a problem where there are two agents in a joint project, Each agent $i$ puts in effort $x_i$ $(0 \leq x_i \leq 1)$ which cost each $c(x_i)= x_i^2$. The outcome of the project is $$ f(x_1, x_2)...
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1answer
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All-Pay Auction Mixed Strategy Equilibrium

I am currently struggling with this exercise. Professor Nash announces that he will auction off a 20 dollars bill in a competition between two students chosen at random. Each student is to privately ...
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1answer
62 views

Proving the existence of Nash Equilibrium using alternate approaches

Most of the standard books/papers/reading materials prove/state the existence of a Nash Equilibrium by appealing to Sperner's Lemma, or to Brouwer's/Kakutani's FPT. However, I've recently come to know ...
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2answers
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Find all of the Pure and Mixed Strategy Nash Equilibria

When I do the basic calculations for mixed probability, I get that the Column player always plays B. However, I am getting a negative probability for the row. Any help is appreciated.
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Location game between 3 firms

I could solve Question 25 (simultaneous location game between 2 firms), but I'm confused between options b) and d) for Question 26 (sequential game involving a third firm). I have attached both ...
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1answer
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Intuition behind a cournot duopoly nash equilibrium producing a higher output than a monopoly?

I am just wondering if someone could explain the descriptive, not mathematical intuition behind why a cournot equilibrium for a duopoly produces a higher level of output than a monopolist but lower ...
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Mixed Nash equilibrium

I have game table posted below: $$\begin{matrix} &\#2 \\ \#1 & \begin{array}{c|c|c|c} &D &E &F \\ \hline A &4,4 &6,6 &2,6 \\ \hline B &6,4 &2,2 &0,4 \\ \...
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Nash equilibrium for Bertrand Model with Spatial Differentiation

Consider a town with consumers represented by a closed interval $[0,2]$ with the consumers spread continuously and uniformly. There are two stores, $A$ and $B$ who sell the same product at $p_A$ and $...
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Existence of nash equilibria in finite games

I was going through the proof of existence of a Nash Equilibria in finite normal form games (Proof via Brouwer’s theorem) and got a question regarding the requirement of finiteness for the number of ...
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1answer
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Example of a game with no Nash equilibria but at least one correlated equilibrium

In this answer there is the offhand remark Of course, a game with no Nash equilibria may have a correlated equilibrium, but I'm not aware of any simple examples where this is the case. Can ...
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Bayesian-Nash equilibrium in a first-price auction

In a famous textbook example of a Bayesian-Nash equilibrium, there is a first-price auction with two independent players. Each player $i$ values the item as $v_i$, which is distributed uniformly in $[...
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1answer
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Nash equilibrium of sequence of games

My setting is the following. I have a sequence of games $\lbrace G_n \rbrace$ in which the strategy space is $S=[0,1]^2$, there are two players $(I=\lbrace 1,2 \rbrace)$, and payoff functions are ...
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Are there Nash Equilibria that aren't mixed strategies?

We can consider only finite games if it makes a difference, but are there nash equilibria that can't be characterized as mixed equilibria?
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Why is the symmetric grim trigger not a Nash?

Consider the stage game: Let $\delta\in(0,1)$ be the discount factor. Let $G$ be the symmetric grim trigger strategy profile. The payoffs are then $$E_{A}(G) = E_{B}(G) = \sum_{i=0}^{\infty}3\delta^{...
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Tit-For-Stat Strategy Best Replies

Let $\delta\in(0,1)$ be the discount factor. Consider the stage game in the infinitely repeated prisoner's dilemma game: The goal is to derive conditions on $\delta$ such that the symmetric tit-for-...
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Incentive compatibility: Weakly dominant strategy versus Nash equilibrium?

When it comes to proving that a mechanism e.g. auction is incentive compatible this is the approach I'm using: I break down all the cases that might happen if the agent reports an untruthful value to ...
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Rationalizable strategies in a game

Consider a game in which, simultaneously, player $1$ selects any real number $x$ and player $2$ selects any real number $y$. The payoffs are given by: $u_1 (x, y) = 2x − x^2 + 2xy$ $u_2 (x, y) = 10y ...
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1answer
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Nash Equilibrium in 2 bidder auction

I am trying to find Nash Equilibrium of an auction with two bidders in which the highest bidder wins the object but both bidders pay the losing bid. Here every bidder follows the same bidding strategy ...
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1answer
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Rosen's unique equilibrium conditions: Multi dimensional strategies?

I was wondering if the uniqueness of equilibrium conditions in n-person games as published in Rosen's 1965 paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games....
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1answer
232 views

Bertrand-equilibrium with discrete price set

Consider a market for a homogenous product with three producers, firms A, B and C. The firms have constant marginal costs which are equal to $c = 20$ for each firm. Consumers always buy from the firrm ...
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How to derive a Nash equilibrium pure strategy in a linear Cournot Model [duplicate]

Suppose there are $N$ firms each with the same positive marginal cost $c$. How would I go about finding a pure strategy Nash Equilibrium for the firms? Suppose the Inverse Demand curve is defined: $p=...
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Deviating from Cournot-Nash

Suppose player $1$ and $2$ are playing a simultaneous move game where with continuous strategies $x_1$ and $x_2$. The Cournot equilibrium is $x_1^*,x_2^*$. The following diagram purports to show that ...
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Compute Nash Equilibrium in median voter game

Consider a spatial model in which two candidates A and B compete for office. The policy space ranges from -1 to 1 and each candidate can take one of three positions, -1, 0, and 1 (so that they have ...
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Game theory software

I was wondering what software/libraries everyone uses to simulate games? For instance finding the Nash Equilibrium. I see that Gambit is a popular one, but I was wondering if there are any other good ...