A message from our CEO about the future of Stack Overflow and Stack Exchange. Read now.

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.

Filter by
Sorted by
Tagged with
20
votes
4answers
2k views

Has the Nash Equilibrium lead to any significant economic discoveries?

The Nash Equilibrium provided a new look at certain economic problems and won the Nobel Memorial Prize in Economic Sciences in 1994. Since it's creation, the Nash Equilibrium has been applied to "...
15
votes
1answer
269 views

Submodularity property in congestion games?

Let $G$ be a $n$-players and $m$-elements congestion game. For an equilibrium $e$, denote by $$SUP(e)\triangleq<sup_1(e),sup_2(e),\ldots, sup_n(e)>$$ Where $sup_i(e)$ contains the support of ...
14
votes
1answer
189 views

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 ...
12
votes
2answers
3k views

What is the definition of a “Stackelberg leader-leader equilibrium”?

I have encountered the equilibrium concept of "Stackelberg leader-leader equilibrium" while reading Product Line Rivalry (AER, Brander and Eaton (1984). They say "we define a Stackelberg strategy as ...
9
votes
1answer
986 views

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. ...
8
votes
2answers
2k views

How infinite Nash equilibria are possible in a game?

I was studying games when one of the players seems to be indifferent between two or more pure strategies because he gets the same payoff with each strategy. We say that there are infinite Nash ...
7
votes
1answer
239 views

Is there always a pure Nash equilibrium in a resource selection game?

Denote $[r]\triangleq\{1,2,\ldots,r\}$. Consider a game with $n$ players, $[n]$, each has $m$ strategies, $[m]$. Each player $i$ has an associated payoff function, which considers only his selected ...
6
votes
3answers
973 views

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 ...
6
votes
3answers
5k views

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): ...
6
votes
1answer
399 views

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 $[...
6
votes
2answers
63 views

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 ...
6
votes
1answer
216 views

Arguing uniqueness of Bayes-Nash equilibrium in an auction setting

In an auction setting with interdependent values, let $\theta_i$ denote the type of player $i$ and $m_i$ that player's message (a bid, essentially). I have calculated the best response function as: $$...
6
votes
1answer
118 views

Are symmetric equilibria continuous with respect to the payoff matrix?

Assume a two player symmetric game where the payoff for the row player is given by: $$ A = \left( \begin{array}{cc} a_{1,1} & a_{1,2} &\cdots & a_{1,n}\\ a_{2,1} & a_{2,2} &\cdots &...
6
votes
1answer
644 views

How does the core relate to strong equilibrium?

An allocation is in the core if there's no coalition that blocks it. A strong equilibrium (Aumann, 1959) is a Nash equilibrium in which no coalition, taking the actions of its complements as given, ...
5
votes
1answer
8k views

Difference between Sequential and Weak Sequential (Weak Perfect Bayesian) Equilibria?

This is in reference to the Game theoretic concepts as Nash equilibrium refinements. Sequential equilibrium are often defined as satisfying two conditions: consistency and sequential rationality. ...
5
votes
1answer
942 views

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 ...
5
votes
1answer
139 views

Monopolistic and Bertrand (Nash) Competition

Can we view the monopolistic competition equilibrium (a la Dixit-Stigliz) as the limit case of a Bertrand competition with an infinite number of firms providing differentiated products, where the ...
5
votes
1answer
316 views

Is a Nash equilibrium anything more than what it is?

(Sorry for the fuzzy title, could not think of something more informative. Feel free to suggest improvements) This question is somewhat of a generalization of "Osborne, Nash equilibria and the ...
5
votes
2answers
453 views

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 $r_{B,L}$...
5
votes
1answer
174 views

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 ...
5
votes
1answer
145 views

Are symmetric equilibria monotone?

Assume a two player symmetric game is given by $n\times n$ payoff matrix $A$ for the row player (and $A^t$ for the column player). Let $B$ be a matrix such that $\forall i,j\in [n]:B_{i, j}\geq A_{i,...
4
votes
2answers
592 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 ...
4
votes
1answer
35 views

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 ...
4
votes
1answer
119 views

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 ...
4
votes
1answer
135 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\...
4
votes
1answer
139 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 ...
4
votes
0answers
55 views

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. ...
4
votes
0answers
118 views

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 $...
4
votes
0answers
62 views

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 ...
4
votes
0answers
181 views

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^{...
3
votes
2answers
161 views

Comparing Nash equilibria

Suppose two players play the following game: \begin{array}{cc} & L & R \\ U & 1,1 & 0,0 \\ D & 0,0 & 4,4 \end{array} Is there any way to compare the top-left Nash ...
3
votes
1answer
174 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 $\epsilon_i$. ...
3
votes
1answer
40 views

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 ...
3
votes
1answer
97 views

If a game admits a unique Nash equilibirum, does common knowledge of rationality implies Nash equilibirum?

In a highly controversial paper by Robert Aumann(see here), it is stated as a theorem: In PI games, common knowledge of rationality implies backward induction. If we stick to the strong and ...
3
votes
1answer
89 views

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 ...
3
votes
1answer
106 views

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_{...
3
votes
1answer
66 views

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 ...
3
votes
1answer
56 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 ...
3
votes
1answer
118 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 ...
3
votes
2answers
2k views

Identifying Nash equilibria in extensive form game

Is there a systematic way of identifying all (pure strategy) Nash equilibria (not just the subgame perfect ones) in an extensive form game? In the following Entrant v Resident example, there are three ...
3
votes
2answers
60 views

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 ...
3
votes
2answers
594 views

Pareto optimality and Externalities

Let's consider 5 farmers, each of them has 2 cows to put into the field. So every farmers can put 0,1 or 2 cows. I denote the three stategies by $q_i$, i=0,1,2. Now, the payoffs ( i.e. the amout of ...
3
votes
1answer
139 views

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....
3
votes
0answers
53 views

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). ...
3
votes
0answers
37 views

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 ...
2
votes
3answers
570 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 ...
2
votes
1answer
79 views

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 ...
2
votes
1answer
336 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 ...
2
votes
1answer
39 views

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 $...
2
votes
1answer
812 views

Show that an equilibrium in strictly dominant strategies is a unique Nash equilibrium

I am new to game theory and I came across this line, " A strategy profile (s1, . . . , sn) in which every si is dominant for agent i (strictly, weakly, or very weakly) is a Nash equilibrium." But why ...