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.

Filter by
Sorted by
Tagged with
4
votes
0answers
73 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
278 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
3answers
321 views

SPNE of a normal form game

If we have a sequential game which is equivalent to a simultaneous move game because of its information structure, then the NE we find are also SPNE?
3
votes
2answers
120 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
194 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
302 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
111 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
129 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
151 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
81 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
49 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
224 views

Disagreement in Strategic Bargaining

Construct a pair of startegies for the ultimatum game ($T=1$ bargaining game), that constitutes a Nash Equilibrium and together support the outcome that there is no agreement reached by the two ...
3
votes
1answer
114 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
52 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 ...
3
votes
1answer
113 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 ...
3
votes
1answer
255 views

Necessary indifference conditions in mixed equilibrium

Suppose we are playing a game where the Action set for Player 1 is $(a,b)$, for Player 2 is $(c,d)$, and for Player 3 is $(L,M,R)$. Assume that for Player 3, the action $M$ is weakly dominated by some ...
3
votes
2answers
5k views

Identifying Nash equilibria in extensive form game

Is there a systematic way of identifying all (pure strategy) Nash equilibria (not just the subgame perfect ones) in an extensive form game? In the following Entrant v Resident example, there are three ...
3
votes
1answer
220 views

Trembling hand perfection and weakly dominated strategies

It is well known that players cannot use weakly dominated strategies in a trembling hand perfect equilibrium. My question, however, is a little different: does iterated deletion of weakly dominated ...
3
votes
2answers
658 views

Pareto optimality and Externalities

Let's consider 5 farmers, each of them has 2 cows to put into the field. So every farmers can put 0,1 or 2 cows. I denote the three stategies by $q_i$, i=0,1,2. Now, the payoffs ( i.e. the amout of ...
3
votes
1answer
40 views

Interpretation of Nash equilibrium as a potential stable point of a dynamic process

I'm reading an article called "The Nash equilibrium: A perspective" by Holt and Roth, and the below paragraph caught my attention. When the goal is prediction rather than prescription, a ...
3
votes
1answer
172 views

Rosen's unique equilibrium conditions: Multi dimensional strategies?

I was wondering if the uniqueness of equilibrium conditions in n-person games as published in Rosen's 1965 paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games....
3
votes
1answer
93 views

Mixed Strategies in Bayes Nash Equilibrium (Bayesian Battle of the Sexes). Shouldn't it depend on $p$?

I have a question about calculating mixed strategies in a Bayes Nash Equilibrium in a simple 2-player bimatrix game. To demonstrate the issue, consider ``Bayesian Battle of the Sexes.'' Suppose P1 ...
3
votes
0answers
43 views

Asymmetric Nash Bargaining

The Nash bargaining solution selects the unique solution to the maximization problem $\max_{s_1, s_2 } (s_1 - d_1) (s_2 - d_2)$ such that the solution satisfy the following axioms : Invariance ...
2
votes
2answers
487 views

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. • ...
2
votes
3answers
675 views

Zero sum game, constant sum game

Given any bilateral zero-sum game G, show that strategy profile σ is a Nash equilibrium for G if, and only if, it is a Nash equilibrium for the constant-sum game G' obtained from G by adding any fixed ...
2
votes
2answers
78 views

Can mixed strategies actually predict behaviour of rational actors in non-constant sum games?

I understand how the concept of the mixed NE (mathematically) works. But I don’t understand how we can expect players to behave in a way that would arrive at such an equilibrium. Consider the ...
2
votes
1answer
1k views

Finding Bayesian Nash Equilibrium

I'm recently new to Game Theory and I've recently started teaching myself about Bayesian Nash Equilibirum. I've stumbled across a problem set that I can't seem to wrap my head around concerning ...
2
votes
2answers
128 views

Pure-Strategy Bayesian Nash equilibrium with general common prior

I'm doing a problem set on the subject of Bayesian Nash equilibrium. I'm asked to find the pure-strategy BNE of the following. I've calculated to matrix shown below. My first concern is if I've ...
2
votes
1answer
100 views

Incentive compatibility: Weakly dominant strategy versus Nash equilibrium?

When it comes to proving that a mechanism e.g. auction is incentive compatible this is the approach I'm using: I break down all the cases that might happen if the agent reports an untruthful value to ...
2
votes
1answer
565 views

Bertrand-equilibrium with discrete price set

Consider a market for a homogenous product with three producers, firms A, B and C. The firms have constant marginal costs which are equal to $c = 20$ for each firm. Consumers always buy from the firrm ...
2
votes
1answer
39 views

Computing optimum efforts

Consider the following cost function: $$c(e_1, e_2) = (\beta_1e_1 + \beta_2e_2)^2$$ The value function is: $$v = v_0 - [l_1(1-e_1) + l_2(1-e_2)]$$ How do I compute the optimum efforts $e_1$ and $...
2
votes
1answer
913 views

Show that an equilibrium in strictly dominant strategies is a unique Nash equilibrium

I am new to game theory and I came across this line, " A strategy profile (s1, . . . , sn) in which every si is dominant for agent i (strictly, weakly, or very weakly) is a Nash equilibrium." But why ...
2
votes
1answer
29 views

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 ...
2
votes
1answer
56 views

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, ...
2
votes
1answer
721 views

Difference between Nash equilibrium and Pareto Efficiency

Nash Equilibrium is defined as a solution concept referring to a best outcome which players won't want to unilaterally deviate given the response of other players doesn't change. To me it seems like ...
2
votes
1answer
127 views

Solving a two stage game by backward induction: which is the equilibrium notion?

Take a two-stage game with complete information and simultaneous actions in each state: (1) Player 1 and 2 simultaneously choose action $a_1\in A_1$ and $a_2\in A_2$ respectively. (2) Player 1 and ...
2
votes
1answer
84 views

Proving the existence of Nash Equilibrium using alternate approaches

Most of the standard books/papers/reading materials prove/state the existence of a Nash Equilibrium by appealing to Sperner's Lemma, or to Brouwer's/Kakutani's FPT. However, I've recently come to know ...
2
votes
1answer
277 views

Nash Equilibrium in 2 bidder auction

I am trying to find Nash Equilibrium of an auction with two bidders in which the highest bidder wins the object but both bidders pay the losing bid. Here every bidder follows the same bidding strategy ...
2
votes
2answers
946 views

Nash Equilibrium in a bargaining game

I've got a question to ask about the Nash Demand game from my assignment. Sarah and Ruth find \$100 on the ground and decide to split it between them in the following manner. Each individual ...
2
votes
1answer
421 views

Rosen's uniqueness theorem: Why is the Jacobian Square?

Rosen's paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, 1965) presents a condition for the uniqueness of the Nash ...
2
votes
1answer
71 views

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. ...
2
votes
1answer
66 views

What is the subgame perfect pricing policy for seller?

Suppose there is a seller S who is selling to 3 different potential customers H, M, L The good is a durable good and it's utility in terms of dollar equivalent is as in the picture below: So if H ...
2
votes
2answers
81 views

What trick can be used to calculate mixed-equilibria?

In continuous games, the probability distributions over the players' strategy spaces are infinite. How then is it even possible to then derive a mixed-strategy nash equilibrium? One would have to ...
2
votes
1answer
142 views

Stone-Geary preferences and competitive equilibrium

Does anybody know if a competitive equilibrium obtains under Stone-Geary preferences; are there multiple equilibria problems; do such preferences admit an analysis with more than one type of ...
2
votes
1answer
961 views

Symmetric mixed-strategy equilibrium: Entering markets [Solved]

Question: Suppose three identical, risk-neutral firms must decide simultaneously and irreversibly whether to enter a new market which can accommodate only two of them. If all three firms enter, all ...
2
votes
1answer
273 views

Choosing the nondominant strategy in a duopoly

Would a company ever choose a nondominant strategy in a duopoly? Let's take this specific example (2007 AP MicroEcon B #2). Two airlines, Airtouch and Windward, are scheduling flights for either ...
2
votes
1answer
80 views

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 ...
2
votes
1answer
54 views

Find value of $\beta$ for which there is a strictly dominant strategy

The question is as such: $N$ firms are lobbying for subsidies. Let $h_i$ be the number of hours spent by form $i$ for lobbying, with cost $wh_i^2$ where $w$ is a fixed constant. The subsidies granted ...
2
votes
2answers
392 views

Bertrand Duopoly Equilibrium for Discrete Prices

There are two identical firms, $1$ and $2$, with zero marginal costs. They produce homogenous product, which is demanded by a unit mass of identical consumers, each of which has inelastic unit demand ...
2
votes
1answer
144 views

Repeated Game SPNE

I approached this question in this way: $(P_1,P_2), (R_1,R_2), (S_1,S_2)$ are the Nash Equilibria of the Stage 1 game. For the given strategy to be sustained as SPNE, there should be no way unilateral ...