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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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Finding Nash equilibrium [on hold]

$n\geq 2$ individuals have to decide how much capital($y$) to put into their firm, they use the following method. Simultaneously each partner $i$ submits a real number $s_i \geq 0$, the amount of ...
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1answer
106 views

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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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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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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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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1answer
131 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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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
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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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Public good contribution

An economy has $n$ consumers. Each consumer belongs to one of the two possible types, type $1$ and type $2$. Consumers have preferences over a private good $x$, and a non-excludable public good $G$. ...
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1answer
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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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3 by 3 Pure Strategy Nash Equilibrium

I got this question in the exam to find the PSNE of the following payoff matrix: \begin{bmatrix}(2,2)&(4,1)&(2,1)\\(1,5)&(6,6)&(1,1)\\(1,2)&(1,1)&(4,4)\end{bmatrix} I found ...
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2answers
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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 ...
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1answer
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Stone-Geary preferences and competitive equilibrium

Does anybody know if a competitive equilibrium obtains under Stone-Geary preferences; are there multiple equilibria problems; do such preferences admit an analysis with more than one type of ...
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Subgame Perfect Equilibrium with Pure Strategies in Sequential Games [closed]

If I have a sequential game, i.e. in each node (that I will call $t$) only one player choose an strategy from a finite space of strategies, Is it true there always exist a subgame perfect equilibrium ...
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1answer
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Non-cooperative Nash Equilibrium in political game

I have difficulties deriving the non-cooperative Nash Equilibrium of this problem. The objective function is to maximize the expected total rent over the two periods, that is: \begin{align} \max_{...
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2answers
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Can a game with a unique pure strategy Nash equilibrium also have a mixed strategy equilibria?

Consider an arbitrary 2x2 simultaneous game with complete information. Say that the model has only one pure-strategy Nash equilibrium. For example (first pay-off refers to Player 1): ...
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1answer
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Existence of pure strategy Nash equilibrium

I understand the reason why mixed strategy Nash equilibrium exists. But what are the conditions for the existence of pure strategy Nash equilibrium?
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1answer
177 views

How to demonstrate that a game always have a subgame-perfect equilibrium in pure strategies?

If I have an specific extensive game, with only a finite set of strategies, how can I demonstrate that the game always have a subgame-perfect equilibrium in pure strategies? My first intuition was to ...
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symmetric Nash equilibrium2

How one can prove a symmetric bi-matrix game(having two players with two strategies each) always has at least one symmetric Nash equilibrium. I found it here Theorem 2 by Nash (1951)http://www.lsi....
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Zero sum game, constant sum game

Given any bilateral zero-sum game G, show that strategy profile σ is a Nash equilibrium for G if, and only if, it is a Nash equilibrium for the constant-sum game G' obtained from G by adding any fixed ...
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1answer
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Is there a systematic approach to find equilibria in sequential games?

I know that one can use backward induction to find one particular subgame perfect NE. And I know that wherever possible one can represent the game in normalform and then find all NE. But is there a ...
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1answer
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Mixed strategies: Nash equilibrium

I'm working on a game theory problem.I'm having trouble understanding what the mixed strategy nash equilibrium is exactly in this game. The game is :Two players have to choose how distribute a piece ...
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1answer
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Computing optimum efforts

Consider the following cost function: $$c(e_1, e_2) = (\beta_1e_1 + \beta_2e_2)^2$$ The value function is: $$v = v_0 - [l_1(1-e_1) + l_2(1-e_2)]$$ How do I compute the optimum efforts $e_1$ and $...
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Nash Equilibrium and Dominant Strategy

If I have a game that goes as follow: Player 1 is the row player and player 2 is the column player. I think that the Nash Equilibria should be (10, 5) and (5, 10), since neither of the player has ...
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Reference for truthful Nash on cartesian domain implies strategy-proofness

Consider a mechanism $M: \mathcal{R} \rightarrow X$, where $\mathcal{R}$ is a domain of preference profiles $R = (R_1,\dots, R_n)$, and $X$ is a set of outcomes. I believe that the following is a ...
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1answer
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Symmetric mixed-strategy equilibrium: Entering markets [Solved]

Question: Suppose three identical, risk-neutral firms must decide simultaneously and irreversibly whether to enter a new market which can accommodate only two of them. If all three firms enter, all ...
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Nash Equilibrium of modified Keynes' beauty contest

Recently I conducted a small game among students of our institute. The game was based on Keynes' beauty contest game. The participants had to guess a number between 0 to 100 and the participant whose ...
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0answers
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Why must the wage barganing be derived at steady-state?

In wage bargaining theory, in the context of matching theory, firms and workers can negotiate a Nash equilibrium by maximizing a function of firms' and workers' surplus - with the purpose of allowing ...
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1answer
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Nash Equilibrium in a bargaining game

I've got a question to ask about the Nash Demand game from my assignment. Sarah and Ruth find \$100 on the ground and decide to split it between them in the following manner. Each individual ...
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1answer
238 views

Rosen's uniqueness theorem: Why is the Jacobian Square?

Rosen's paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, 1965) presents a condition for the uniqueness of the Nash ...
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Menu-pricing with three consumer groups

I want to analyze the following setting: An entrepreneur (with monopoly power) sells a product in two periods. In period 1 there are two consumer groups (denoted by 1 and 2) and in period 2 there is ...
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1answer
1k views

Calculating Nash Equilibrium prices for Bertrand duopolists

I am attempting to solve the following problem. Suppose that firms' marginal and average costs are constant and equal to c and that inverse market demand is given by $P = a - bQ$ where $a,b > 0$. ...
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1answer
333 views

how to find Cournot equilibrium for 2 firms having different MC?

When MC function is different for both the firms, how will MR = MC work ?
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Does the concept of Nash-equilibrium conflict with the concept of market equilibrium in the lemon market

Consider a version of Akerlof's Lemon market with two types of sellers. One type sells Quality cars the other type sells Lemons. Buyers' reservation prices are $r_{B,Q}$ for a Quality car and $r_{B,L}$...
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Rosen's Diagonal Strict Concavity condition

Consider a game with $n$ players, with strategy space $S \subset \mathbb{R}$, where $S$ is bounded set, and player's $i$ payoff function $\pi_i:S^n \rightarrow \mathbb{R}$. Rosen's condition (J. B. ...
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Can't solve this matrix for Nash Equilibrium?

So, I have the following 9 by 9 probability matrix. I want to solve it for a nash equilibrium. https://docs.google.com/spreadsheets/d/16Y1FqxRIAHsHpgEz1ckxDt2sEOInOG3zz_wU8kBHvB4/edit?usp=sharing For ...