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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About infinite strategy sets and $\epsilon$-equilibrium from Game Theory: Analysis of Confilct by Roger Myerson
I am studying infinite strategy sets using Myerson's Game Theory: Analysis of Conflict. On Page 143, he defines an $\epsilon$-equilibrium as follows:
Definition For any nonnegative number $\epsilon$, ...
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Auction with independent private values - An example from Game Theory: Analysis of Conflict by Roger Myerson
I have difficulties understanding the equilibrium analysis of the following auction game:
Suppose that there are $n$ bidders in an auction for a single indivisible object. Each player knows privately ...
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Using Variance for Nash Equilibrium
For the mixed strategies, the expected utility (or payoff) is used to find the mixed strategy Nash Equilibrium. The main assumption is players try to maximize their expected payoffs. However, I think ...
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Mixed Strategy Nash Equilibrium
Consider the game of Battle of Sexes, where
$$\begin{matrix}
&&Women\\
&& Football & Resaurant\\
Men&Football&2,1 & 0,0 \\
&Restaurant&0,0 & 1,2
\end{...
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Equilibrium of Perturbed Dollar Auction Game - An Example from Game Theory: Analysis of Conflict by Roger Myerson
I am studying game theory using Myerson's textbook (Chapter 3 - Equilibria of Strategic-Form Games, Section 3.6 - The Decision-Analytic Approach to Games). I have difficulties understanding and ...
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Nash Equillibrium - Depend On The Opponent's Strategy?
Say I have the following pay-off matrix:
For a one-shot game, it is easy to see, that (low, low) is the only Nash Equillibrium in the payoff-matrix.
However, say we're playing an infinitely repeated ...
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Proving the existence of mixed-strategy NE for 2-player zero-sum symmetric game
I am trying to prove the existence of mixed-strategy NE for 2-player zero-sum symmetric game, under the condition that given they have $I$ pure strategies and for the pay-off matrix $A$, $\exists x\in ...
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How do the assumptions $p'+q_ip''<0$ and $p'-c''<0$ ensure the stability of the Nash equilibrium among private firms in basic mixed oligopoly model?
I have two quick question regarding basic oligopoly models:
What is meant by we impose the assumptions to $p'+q_ip''<0$ and $p'-c''<0$ to ensure the stability of the Nash equilibrium among ...
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Signaling Game: Beer and Quiche Payoffs and Extensive Form Representation
I am working on a signaling game scenario named "Beer and Quiche" and need some assistance with understanding the payoffs and representing the game in an extensive form.
In this game, there ...
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Moral Hazard in Teams. Deriving Nash Equilibrium with bonus paying sheme
I have a few mathematical problems with the paper Moral hazard in teams, by Bengt Hölmstrom 1982. Theorem 4
Denote the conditional distribution of $x$, given the action vector $a$, by $F(x, a)$ and ...
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Nash Equilibria in a Game with Three Firms and a Shared Resource
I’m trying to understand the Nash equilibria in a game involving three firms that use water from a shared lake. Each firm can choose to purify the water before returning it to the lake or not purify ...
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Understanding the notations in Bayesian game definition
I am having trouble understanding the definition of a Bayesian game based on the following definition from class. I would appreciate it if you could explain the notations and overall meaning for point ...
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Optimality of the free market and game-theoretic arguments
I recently heard an informal argument that went something like this:
Through individual self-interest and freedom of production and consumption, the best interest of society, as a whole, are ...
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Nash equilibrium in p-beauty contest game where p=1
Setup: players must chose a number between 0 and 100. The winner of the game is the player whose chosen number is closest to the average of all chosen numbers multiplied by "p". Assume that ...
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How to find the equilibrium amount of $p_{2}$ in terms of $p_{1}$?
There are $100$ tons of crops remaining to supply for the two months. The crop holders consider whether to sell crops now or one month later. Holders face the demand curve of each period as below:
$...
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Different payouts of pure strategies in mixed strategies
I have a question with mixed strategies.
The question is as follows, if we're in a strategy profile that is a Nash equilibrium and a player is playing a mixed strategy, can the pure strategies that ...
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Understanding the Nash equilibrium for quadratic utilities
I need help to understand some steps of the article "Who's Who in Networks. Wanted: The Key Player" and I would greatly appreciate if someone can provide me with references or if they can ...
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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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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 ...
