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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questions
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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
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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 ...
13
votes
1answer
212 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
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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 ...
10
votes
1answer
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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. ...
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 ...
7
votes
1answer
262 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
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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):
...
6
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1answer
461 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
70 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
votes
2answers
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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
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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:
$$...
6
votes
1answer
123 views
Are symmetric equilibria continuous with respect to the payoff matrix?
Assume a two player symmetric game where the payoff for the row player is given by:
$$
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
684 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
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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
votes
2answers
73 views
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
votes
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. ...
5
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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
1answer
158 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
336 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
1answer
155 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
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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 $r_{B,L}$...
5
votes
1answer
192 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
159 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
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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 &...
4
votes
2answers
733 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
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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
2answers
47 views
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
votes
1answer
49 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
votes
1answer
43 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
308 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
1answer
332 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
2answers
87 views
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
98 views
Perfect Bayesian Equilibrium in a two stage game with incomplete information
I would like to solve a game where firms have private information about their own type, but only know the distribution of the other firm's type. They interact in two stages, where the strategies ...
4
votes
1answer
174 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
0answers
58 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).
...
4
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0answers
59 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
1answer
160 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
70 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
231 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
94 views
Stag Hare inefficient Nash Equilibrium and level-K thinking
Two players $i,j$; both have two strategies $\{h,s\}$.
The payoffs vector of $i,j$:
$u(h,h)=(5,5)$ (if both players choose $\{h\}$ then $i$ receives 5 and $j$ receives 5)
$u(h,s)=(10,0)$
$u(s,h)=(0,10)...
3
votes
2answers
188 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
172 views
Nash Equilibrium for n-shops Location Game
So if two ice cream shops were to be placed in the location $[0,1]$, inorder to maximize their own pay offs, they both would finally come to the location $[\frac{1}{2}, \frac{1}{2}]$. This is also the ...
3
votes
1answer
108 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
108 views
Is the Nash product really maximised ex post?
In my game theory class this term, we studied Nash bargaining. It is only now when starting to prepare for the exam that I have come to realise there is something I fundamentally don't understand, and ...
3
votes
1answer
120 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
70 views
Non-Bayesian Mechanism Design
Suppose we have a mechanism where a finite number of agents possess private information that is not drawn from a probability distribution.
The agents' types are given and fixed but agents only know ...
3
votes
1answer
44 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
112 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
50 views
Who is the first person/paper to introduce “mixed strategy”?
Who is the first person/paper to introduce "mixed strategy"? The PNAS by Nash used this notion without citing anyone.
Does the earlier book: Von Neumann, J., and Morgenstern, O., The Theory ...
