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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Strategic game with complete informaation

Consider the following strategic game with complete information played by three players. Each player $i ∈ {1, 2, 3}$ chooses her action from $A = \{1, 2, . . . , 10\}$. Utility functions, mapping each ...
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Why Sequenial Equilibrium (SE) imposes no restrictions on the off-equilibrium beliefs in the Spence's model?

I read some lectures on the Spence's model. Some (see e.g. P31 of lecture PPT from MIT game thoery course) mention that SE imposes no restrictions on the off-equilibrium beliefs but without proof. I ...
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Perfect Bayesian Equilibium - Application to game with inconsistent beliefs / no common prior

Does the concept of a Perfect Bayesian Equilibrium apply only to incomplete games with a common prior / consistent belief? In both Bonanno's "Game Theory" and Osborne's "A Course in ...
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Are these two definitions of Bayesian Nash Equilibrium equivalent?

Consider a standard game $\Gamma$ with incomplete information. There are $n$ players indexed by $i=1,...,n$. $S_i\equiv \{s_{i1},...,s_{iJ}\}$ is the set of actions of player $i$. $S\equiv \times_{i=1}...
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Extension of Harsanyi Transform for Two-sided Incomplete Information Games to Beliefs with Zero Probability

In the textbook I'm reading "Game Theory - Giacomo Bonanno", one requirement to applying the Harsanyi transform to convert a two-sided incomplete information game to an imperfect information ...
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Why does the belief over information sets with probability zero matter in Perfect Bayesian Equilibrium?

I'm struggling to understand why the notion of "belief revision" is an important concept. In particular, why does the belief over information sets with probability zero matter? When ...
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Equivalence from correlated/communication equilibrium to Nash Equilibrium?

Taking into account the seminal papers of Forges and Imre Bárány, they proove a very strong result that gives an exact connection among the communication and the correlation equilibrium solution ...
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Sequential and Perfect Bayesian Equilibrium: an example?

My question is quite simple. Could someone given an example of how to determine a Sequential Equilibrium given a set of Perfect Bayesian Equilibria? The definition of sequential equilibrium where ...
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Does Subgame Perfect Nash Equilibria (SPNE) allow for credible threats?

Consider the following extensive-form game: In one alternative, Player 2 chooses G and E and Player 1 chooses D. However, Player 2 can increase her gain by making a credible threat and switch from G ...
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Correlation device that induces a specific transition probability

Taking a look at this paper of Forges and Vida the authors define a correlation device in page $102$, that is a standard probability space $\left(\Omega,\mathcal{B},\mu\right)$, They assume that the ...
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What if Bergemann and Morris setting used mixed (or bbehavioral) actions instead of pure actions as reccomendations?

Once again, I will refer to the setting of Bergemann and Morris (2016) and write here the payoff formula of player $i$ from the perspective of the information designer. The payoff formula is the ...
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Bergemann and Morris information designer and decision rule concept

Taking a look in the paper of Bergemman and Morris in 2016, they refer to the desicion rule as mapping $$\sigma:\Theta\times T\to\Delta(A)$$ The explanation to understand the notion of it is given as ...
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Price competition; finding the equilibrium expression for price and profit

This question deals regarding the price competition between two firms developing products that directly compete on the market. Fundamentally basing on the game theory and the Nash equilibrium, the aim ...
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Bertrand competition with homogenous good and Hotelling's spatial model

Q. There are a 1000 costumers uniformly distributed on [0,3]. Each wants to buy 1 ice-cream. There are two firms which produce ice-cream costlessly and firm i charges p_i. Consumer's effective price ...
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Bayes correlated equilibrium of Bergemann and Morris

The paper of Bergemann and Morris proves a theorem based on some foundations about the information sets and their expansions. I am trying to understand theorem one intuition, more precisely I cite the ...
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Correlated equilibrium definition

Where can I find the definition of correlated equilibrium apart from here and the seminal paper of Aumman ?
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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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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 ...
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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 $\...
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Prove 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 ...
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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 ...
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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 ...
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How to find pure strategy NE if you have a n X n matrix (n players) [closed]

Consider the down below which I have trouble with solving. I am not used to find NE for n players, but rather for a simple $2 x 2$ matrix or $3 x 3$, but how does one find NE when you have N players? ...
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Cournot nash equilibrium

The market demand for a good is described by the inverse demand function $P(Q) = 120 - Q $ where $Q$ is total quantity demanded and $P(Q)$ the market price. Two firms $i =1,2$ have identical cost ...
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Pure Nash equilibrium in bidding game?

According to the answer key for a problem set, there is no pure strategy Nash equilibrium in the following problem. Yet I can't see why not. Could it be an error in the answer key? Here's the problem: ...
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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. ...
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Auctions and finding nash equilibrium of a dynamic game

Suppose we have a sequential version of an Auction game: • Player 1 places a bid. • Player 2 observes player 1’s bid, then places a bid. • The player with the highest bid wins the item at auction. • ...
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Characterising a set of outcomes containing the collection of pure strategy Nash equilibria

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. ...
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Nash equilibrium with three players

Consider the game below played by three players. Player 1 chooses one of the rows (T vs B). Player 2 chooses one of the columns (L vs R). Player 3 chooses one of the three tables (A vs B vs C). Each ...
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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 ...
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Game Theory: Continuity in equilibrium profits?

Consider 2 agents $A_i$. $A1$ moves before agent $A2$. Each of their utility functions is continuous in each agents' decision $0<s_i\in \mathbb{R}$ and a parameter $x$. Additionally, each agent's ...
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Strong sequential equilibria and the existence of others

I am working on the following game and I have to find all strong sequential equilibria here. I determined that here any belief derived from a fully mixed strategy gives a distribution (1/2, 1/2) over ...
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If a best-response dynamic converges, does it converge to a Nash equilibrium?

Consider a game with a finite number of players and finite action space. Suppose we consider a sequential iterative game-playing process in which, in each period, players myopically select actions ...
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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....
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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. ...
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1answer
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Differences between best response, dominant strategy and Nash equilibrium

I can't seem to get the differences of these terms. I watched this video that has the differences of best response and Nash equilibrium: But then I heard about dominant strategies from another video ...
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Can't find the SPNE

For a homework assignment, I need to find the subgame perfect equilibrium. The assignment asserts that there is only one subgame perfect equilibrium in this problem, but I am stuck between two ...
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Is there really a Nash equilibrium in this example?

I was watching this video on Coursera and worked out the example before the solution was presented. The example begins at 4:20 The presenter says that the Nash ...
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How to set up the payoffs properly for a division of labor game

I'm admittedly a novice when it comes to game theory (currently a few lectures into Yale's intro course lectures), so hopefully people will indulge me what may be a dumb question. I was trying to ...
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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 ...
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Will the outcome of a game always be a Nash Equilibrium?

Consider this game between two players. This game has two Nash Equilibria: (U, C) and (D, R). Suppose we ask the players to play this game once. What should our prediction of the game's result be? If ...
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Rationalizable strategies and Weak Dominance

Can I find the rationalizable strategies for a game where none of the players has strict dominance but only weak dominance?
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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 ...
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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 ...
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Fehr-Schmidt, Ultimatum game, Subgame-Perfect Nash Equilibrium

I'm studying the different variations of the ultimatum games. I've spent some time on this following game: Assume now that each player does not only care about the amount of money she receives, but ...
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Subgame-perfect Nash equilibrium perfect information

This might be a stupid question but please bear with me. I'm trying to solve this game but I'm in doubt on how to represent the strategy profile of the game. The game looks like this in extensive-form....
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Set of rationalizable strategies for this 4 x 4 matrix

I would like to find the set of rationalizable strategies for this 4x4 game: The first thing I did was try and find all PSNE. I found two, the ones I bolded. Thus, my answer to this question is that ...
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Mixed Strategy Nash Equilibrium for this particular 3x3 matrix

Suppose I am given the following matrix: I would like to find all MSNE I started by doing the double underline method to find any PSNE. I discovered that none exist. I then looked at which strategies ...
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A question about Nash Equilibrium

I have some trouble with Nash Equilibrium. The specific question as follows. Suppose that there are $2N$ people in the village, of which $N$ residents live in the first district, and each person ...
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Finding pure-strategy subgame-perfect Nash equilibria

I'm interested in finding the pure-strategy subgame-perfect Nash equilibria of the game below. What is confusing me is that after player A chooses between reducing and not reducing his end payoffs, ...