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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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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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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308 views

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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2answers
58 views

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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1answer
93 views

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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53 views

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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1answer
181 views

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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234 views

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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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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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
80 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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557 views

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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151 views

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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208 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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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
149 views

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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148 views

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
83 views

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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525 views

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
238 views

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
406 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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684 views

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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135 views

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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3answers
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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
181 views

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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318 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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3answers
618 views

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
455 views

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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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 ...