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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Has the Nash Equilibrium led 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 its creation, the Nash Equilibrium has been applied to "...
15
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1answer
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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 ...
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2answers
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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. ...
13
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1answer
223 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
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2answers
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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 ...
8
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2answers
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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 ...
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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):
...
7
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1answer
281 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
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1answer
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Relaxing the notion of Nash Equilibrium
Consider a game with $N$ players, each indexed by $i=1,...,N$. Every player $i$ has to choose a $J\times 1$ vector of actions $a_i\equiv (a_{i,1},...,a_{i,J})$ where each $a_{i,j}$ can be zero or one. ...
6
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1answer
53 views
Proove that for every Nash equilibrium $\sigma^*$, the probability distribution $p_{\sigma^*}$ is a correlated equilibrium
This is a classic theorem in game theory, that is left as an excersice in my textbook. Can anybody proove it? I can not thing of anything excpet from the definition of the correlated equilibrium in ...
6
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1answer
132 views
Effect of bounding action space on the set of equilibria
Suppose $N$ players play a game, where each player's action space is $[0,1]$.
Each player has an identical continuous utility function $u:[0,1]\times [0,1]^{N-1}\rightarrow\mathbb{R}$, where the first ...
6
votes
1answer
166 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 ...
6
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1answer
498 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
1answer
353 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 ...
6
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1answer
71 views
(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 ...
6
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1answer
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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:
$$...
6
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1answer
128 views
Are symmetric equilibria continuous with respect to the payoff matrix?
Assume a two player symmetric game where the payoff for the row player is given by:
$$
A = \left( \begin{array}{cc}
a_{1,1} & a_{1,2} &\cdots & a_{1,n}\\
a_{2,1} & a_{2,2} &\cdots &...
6
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1answer
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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
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2answers
349 views
Correlated equilibrium intution
What is the difference between the correlated equilibrium with the mixed strategy Nash equilibrium? Even further, how is this related to the Bayesian correlated equilibrium with complete or incomplete ...
5
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2answers
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In a game with alternating moves and complete information, the Nash equilibrium cannot be a non-trivial mixed equilibrium?
Where I can find a simple proof for this fact?
For example, a trivial bimatrix game with alternating move has the following payoff matrix:
\begin{array}{|c|c|c|}
\hline
& 1 & 2 \\ \hline
U &...
5
votes
3answers
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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 ...
5
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2answers
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Cournot oligopoly - first-order condition
I am reading an article that has this description of the first-order condition for a Cournot n-firm game:
Take $P(Q) = Q^{-1}$, $\pi_i(q_i, Q) = (Q^{-1} - c_i)q_i$.
Then the first-order condition for ...
5
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1answer
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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. ...
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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 ...
5
votes
2answers
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Does Nash Equilibrium predict the existence of vaccine reluctance?
I was listening to a lecture on Nash Equilibrium, which stated that a Nash Equilibrium by definition occurs at a point that no players in the game have an incentive to change their strategy - everyone ...
5
votes
1answer
77 views
Bayesian Nash Equilibrium in a Duopoly Cournot Competition
I am having a hard time to solve a Bayesian Nash equilibrium game in a duopoly cournot competition setting. So, I have two firms with given production quantities, let's say $q_1$ and $q_2$ (i.e., not ...
5
votes
2answers
85 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 ...
5
votes
1answer
463 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\...
5
votes
1answer
170 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 ...
5
votes
2answers
522 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
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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 ...
5
votes
1answer
162 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,...
5
votes
1answer
187 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 $...
5
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0answers
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Providing an example in cooperative - games and coalitions
Here is the paper from chich I previously posted another definition here Definition of a $k-$strong Nash Equilibrium
I am trying to construct an example to understand the idea of the following ...
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0answers
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How to find an optimal strategy in an auction?
I have asked this question in mathematics forum as well but since I have not recieved an appropriate answer yet, I ask it here as well.
Consider an auction of sculptures by four artists: A, B, C and D....
4
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2answers
799 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
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1answer
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symmetry of equilibria with heterogeneous players
I have a question about game theory terminology. I am working on a model in which players are heterogeneous in two dimensions, and there are four types of players. For example one type of players ...
4
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2answers
298 views
Showing existence of a Nash equilibrium in pure strategy
Consider a game with $N$ players, each indexed by $i=1,...,N$. Every player $i$ has to choose a $J\times 1$ vector of actions $a_i\equiv (a_{i,1},...,a_{i,J})$ where each $a_{i,j}$ can be zero or one. ...
4
votes
1answer
116 views
Finitely repeated Prisoner’s Dilemma with switching cost
I'm doing this finitely repeated Prisoner's dilemma with switching costs but I have trouble showing the fact that $\varepsilon$ had to be $1 < \varepsilon < 2$. I do see why and that it is a ...
4
votes
1answer
60 views
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 ...
4
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1answer
49 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
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1answer
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Defining the set of strategies, mixed strategies and the simplex set
Suppose that we have a two players game, where $(S^i)_{i=1}^2$ denotes the set of pure strategies for each one. The set of mixed strategies of player $i$ is denoted by $\Sigma^i=\Delta(S^i)$ while $\...
4
votes
2answers
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Does the value of a pure strategy Nash equilibrium(if exists) equal the value of the mix strategy Nash equilibrium in two-person zero-sum game?
Given a two-person zero-sum game, a mixed strategy Nash equilibrium always exists and all such equilibria have the same value. A pure strategy Nash equilibrium, however, may not exist.
My question is:...
4
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1answer
371 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 $\epsilon_i$.
...
4
votes
2answers
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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 ...
4
votes
1answer
37 views
Definition of a $k-$strong Nash Equilibrium
Consider a game $G=(N, (A^i)_{i\in N}, (g^i)_{i\in N})$, $N=\{1,2,\dots,n\}$, $A=\Pi_{i\in N}A_i$ is the set of actions and $g^i:A\to \mathbb{R}$ is the payoff function. The latter can be extended ...
4
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1answer
216 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
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0answers
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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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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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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 ...
