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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Question regarding a Bayesian game and the number of proper subgames
I have to find all Nash and subgame perfect Nash equilibria of the following Bayesian game:
By writing the whole game in normal form I've been able to find the Nash equilibria which are $(L,L')$ and $...
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Can game theoretic concepts be applied to any groups of strategies collectively partitioning the strategy space?
It is clear that players of a game can almost always create trivial variations on strategies without breaking game theoretic conclusions. For example, a player playing Rock Paper Scissors can play ...
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Does decreasing a player's external regret always (monotonically?) decrease that player's cost function?
In a game in which, at each time step, a player declares a mixed strategy, then an adversary assigns a cost to each of that player's pure strategies, and then the mixed strategy is applied and that ...
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Geometric Interpretation of the Potential Function of a Game
One geometric interpretation of (at least one term of) the potential function I've come across is as the Riemann-approximated area under an individual player's cost as a function of the number of ...
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Why are Mixed Strategy Nash Equilibria special cases of Correlated Equilibria and Coarse Correlated Equilibria?
In a Mixed Strategy Nash Equilibrium, each player constructs their own probability distribution over the set of their respective possible strategies.
In a Correlated Equilibrium or a Coarse Correlated ...
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Asymmetries in Equilibrium Utility
In this lecture, the professor says that all Nash Equilibria have the same utility in non-atomic selfish routing, whereas this is not guaranteed in atomic selfish routing. It is unclear how general ...
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Rationalizable strategies/ Nash equilibrium
For the question below, how can we solve it generally for every value of θ? As the θ is not discrete, I am not sure how to apply iterated elimination of dominated strategies in this question. And is ...
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Slight Uncertainty of Continuation in Repeated Prisoner's Dilemma
In a repeated prisoner's dilemma with some probability δ of continuing after each round, a Subgame Perfect Nash Equilibrium may be found which induces cooperation instead of defection in each round. ...
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Does introducing mixed strategy Nash Equilibria override the pure strategy Nash Equilibria?
I am a bit confused as to why pure strategy Nash Equilibria are often used in analyzing a game alongside mixed strategy Nash Equilibria. While I understand that pure strategy Nash Equilibria can be ...
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Existence of Nash Equilibrium
I know there are two proofs of the existence of Nash equilibrium: One by using Kakutani's fixed point theorem in Nash (1950), and the other by using Brouwer's fixed point theorem in Nash (1951).
I saw ...
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Using Equilibria Payoffs in Battle of the Sexes Subgame
In this lecture on the Matchmaker Game, the professor says that the payoffs from the pure strategy Nash Equilibria of the Battle of the Sexes subgame can be rolled back via backward induction to the ...
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Mulitiple equilibria in zero sum games
How could we prove that for all two-player zero-sum games: if 𝑅𝑖, 𝐶𝑗 and 𝑅𝑛, 𝐶𝑚 are Nash equilibria, then so are 𝑅𝑖, 𝐶𝑚 and 𝑅𝑛, 𝐶𝑗;
Can we also proof all Nash equilibria have the same ...
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Every Nash equilibrium is Subgame perfect Nash equilibrium
Every Nash equilibrium is Subgame perfect Nash equilibrium.
This statement is wrong. I show this by an counter example.
Is this example enough and good to disprove this statements?
Please share your ...
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Sequential version of all-pay auction
Sequential version of the all-pay auction. Two bidders alternate in bidding. A prize of \$5 is auctioned. At each move of the game, the bidding player decides whether to raise the current bid by \$1 ...
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How to find all mixed strategy Nash equilibrium
So I have been taught how to find a single mixed strategy Nash equilibrium in a 2 player game by ensuring both players are indifferent to which strategy is played.
So for example:
...
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Perfect Bayesian Nash Equilibrium
Suppose a seller has a product which can be of quality H (high) or quality (L). There is a buyer who values H at (vH) and L at (vL), but can not distinguish between H and L without consuming it. ...
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How realistic is the conclusion that players do not change their mixing proportions in response to changes in their own payoffs?
A major lesson from game theory seems to be that in simultaneous move games, a player does not change mixing proportions in response to changes in their own payoffs. Rather, their opponents change ...
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All-pay auction question
Two players take part in the following auction for a £1000 prize. The two players submit bids simultaneously, and the higher bid wins the prize (if bids are identical each gets £500). Both the winner ...
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MWG 8.B.7 - Any strictly dominant strategy must be a pure strategy
This question from MWG 8.B.7
Any strictly dominant strategy must be a pure strategy.
How can I show this?
My explanation is as follows:
Suppose we have a strictly dominant strategy, $\sigma_i$ . ...
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Why is pre-specification of punishment order necessary to manipulate compliance?
According to the book Thinking Strategically by Avinash K. Dixit and Barry J. Nalebuff, a government can coerce every citizen to register for the military by threatening to punish only the first ...
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Find SPNE in the two stage game
Consider a firm in a two period world. The firm operates as a monopolist in the first period. In the second period, another firm will enter and the two firms will act as Cournot duopolists (i.e., in ...
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How can we prove that an equilibrium in dominant strategies is a Nash equilibrium?
So far I have that an equilibrium in dominant strategies is one where two strictly dominant strategy profiles meet (e.g. to report the other prisoner in the Prisoner's Dilemma). Going by the ...
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Can a dominance solvable game have a mixed strategy equilibrium?
Prelude: we get this question about specific finite games. Let us answer it generally. (examples 1, 2)
Suppose there is a two-player finite game (a "matrix game") where both players ...
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Is there a mixed strategy in dominance solvable game?
If a game is dominance solvable, is there a mixed strategy NE? If there is, how do I solve that?
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Nash Equilibrium with Constraints on Decision Variables
I am trying to solve a two player game with constraints on decision variables. The general structure looks something like this:
$$\max_{x_1} f(x_1, x_2)$$
$$\max_{x_2} g(x_1, x_2)$$
subject to
$$x_1 + ...
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(Game Theory) Why is voting for your worst alternative a weakly dominated action?
I don't fully understand why voting for your worst alternative is a weakly dominated action.
The question comes from a question I'm working on:
"Assume there are three candidates, A,B and C,...
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Subgame perfect Nash equilibrium when there is a tie in payoffs seems problematic
My question follows from this question: https://math.stackexchange.com/questions/2132846/game-theory-subgame-perfect-nash-equilibrium-in-a-sequential-game-with-identica from Maths stackexchange.
Based ...
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Convex Preference in Nash Equilibrium
Arrow Debreu (AD) uses the convex preference (A4 among their four assumptions, also see the assumption IIIc in AD 1954 ECTA) to make general equilibrium (GE) exist, unique, and well-behave.
What ...
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Best response to convex combination of strategies
Suppose that several pure strategies in a 2-individual game have pure strategy best responses. Can we say that best responses to convex combination of those pure strategies still lie in the convex ...
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Two-dimensional hotelling equilibria
Customers are heterogeneous with regard to their preference for quality $q$.
Specifically, a customer's utility from buying a product of quality $q$ at price $p$ is $V-p + \lambda q$, with $\lambda$ ...
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What does it mean when an economist talks about "equilibrium"
In economics, there are many equilibrium concepts, like equilibrium under perfect competition, Monopolist equilibrium, competitive equilibrium, general equilibrium, nash equilibrium, equilibrium price,...
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Is a Monopoly equilibrium also a Nash equilibrium?
Consider a monopoly with price power in the market and the demand is a function of price. Can the result of such a monopoly problem be called a nash equilibrium?
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Nash equilibrium in strictly mixed strategies
I have the following statement which I have been said it is false, but I don't understand why:
"All finite games have at least one Nash equilibrium in strictly mixed strategies, as long as there ...
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bargaining game information/equlibria/gametree/normalform
I have a game with two players, player one offers player two one of two cars, car 1(M) has value 2 and car 2 (H) has value 1. Player two can accept (A) or reject (R) the offer. Now I have to answer ...
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Bayes Nash equilibrium distribution
I would like to define a decision rule that is induced by a BNE distribution in a game with a continuum of agents.
For that, I have a decision rule $\varphi:\Theta \rightarrow\Delta(\Delta(A))$ that ...
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Game Theory Model needed to model the question whether "not taking the covid vaccine is free-riding"
I am a student and completely new to Game Theory, in fact, it is an additional course for me, I am actually from an entirely different field. I am asked to choose a Game Theory approach to model the ...
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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 ...
