For questions regarding the the Nash equilibrium solution concept in strategic games.

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Can a game with a pure strategy Nash equilibrium also have mixed strategy equilibria? [on hold]

I have questions: If zero-sum game has pure strategy Nash equilibrium (saddle point), can it have also mixed strategy equilibria? What if game is not zero-sum?
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2answers
40 views

Mixed Strategy Nash Equilibrium in this game?

L (q) R (1-q) l (p) [(2, 1), (0, 1)] r (1-p) [(-1, 0), (1,7)] I'm having a lot of trouble understanding what the mixed strategy nash equilibrium is ...
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1answer
47 views

Nash equilibrium in mixed strategies with p = 0

I am currently writing a program to calculate nash equilibria in mixed strategies. My algorithm simply tries a lot of different probabilites and then decides which one is the best. However I came ...
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1answer
1k views

Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question: Is ...
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1answer
3k views

Mixed-strategy Nash equilibria

I didn't find in books, so I'm asking - Mixed-strategy Nash equilibria is always only one or doesn't exist for the one certain game? And I know that there can be several(and can not be at all) pure ...
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1answer
24 views

Is 2nd-price with a discount auction truth-telling?

I know that 2nd-price auction is truth-telling, but 3rd-price auction is not. What If I run the regular 2nd-price auction, in the end, the winner is charged at the 2nd bidding price with a discount, ...
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236 views

Third and average price auction

Third price auction: the winner is the highest bidder but this time instead of paying the second highest bid, he would pay the third highest bid. -assume there are at least 3 bidders. - Average price ...
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1answer
42 views

Can mixed strategies outperform pure strategies?

Let $G$ denote a game with a finite number $n$ of players in which each player $i$ can choose a mixed strategy $\sigma_i$ over a finite set of pure strategies $\Sigma$. Pure strategies can be seen as ...
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5 views

Why Nash product is strictly quasiconcave?

The book "Bargaining theory with applications" mentioned that Nash product is continuous and strictly quasi-concave. But why Nash product is strictly quasi-concave? I haven't found any illustration ...
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77 views

Undestranding Basic Game Theory

Lately I'm studying game theory for an exam. I'm having troubles in understanding some theorems since notes I'm studying on are very brief and concise about sense of definition. In this question I'll ...
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Can the joint decision of car purchasing in a household be formulated as Nash bargaining problem?

Suppose that in a household, husband and wife jointly decide whether to buy a car or not. If they decide to buy, husband and wife contribute their share for the cost of purchasing according to their ...
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1answer
890 views

Find the Nash equilibria for a joint project modeled as a strategic game

I am currently working out of the Introduction to Game theory book by Martin J Osborne and I am looking to solve this given question: Two people are engaged in a joint project. If each person ...
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24 views

Signaling game : response to zero-probability message

We have this signaling game : sender type $t$ is uniformly distributed among $[0, 1]$. She takes action A if $t < \phi$, B if $t > \phi$ if receiver takes action A' when seeing A and B' when ...
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1answer
39 views

Dynamic game of incomplete information

Consider a 2-player game: You and a robber. The robber tells You to give him all your money, otherwise he will kill You. However, the robber could be a 'Good' person (i.e. he would not kill You ...
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1answer
36 views

Nash equlibrium game theory

Given the following game: Find nash equilibrium (NE) Find subgame perfect nash equilibrium. Main problem i have is with converting this to normal form of the game (because this is i think ...
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1answer
46 views

Mixed strategy nash equilbrium

In a mixed strategy Nash equilibrium it is always the case that: a) for each player, each pure strategy that is played with negative probability yields the same expected payoff as the equilibrium ...
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0answers
12 views

Lemke Howson Algorithm Tableau

I am working on an implementation of Lemke Howson Algorithm and I am reading this paper below. http://cnl.gmu.edu/TAVRI/research/LemkeHowson.pdf Can someone please explain why on page 7 they say ...
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1answer
65 views

Subgame perfect Nash equilibrium & perfect Bayesian Nash equilibrium - Game theory

For a week or so I have been struggling with the topics around the concept of subgame perfect Nash equilibrium (SPNE) and the perfect Bayesian Nash equilibrium (BNE). Namely: Is it possible to apply ...
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3answers
22k views

How to compute ALL Nash equilibria in an example of a 3x3 matrix

I am trying to understand how to compute all Nash equilibria in a 2 player game, but I fail when there are more than 2 possible options to play. Could somebody explain to me how to calculate a matrix ...
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1answer
16 views

Algorithms for computing Nash equilibria

Excuse me, since I am modeling a situation into a nonzero-sum n-player non-cooperative game. I wonder if there is any algorithm for computing its Nash equilibria?
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2answers
157 views

Splitting the dollar Nash equilibrium

I'm working on a game theory problem I can't seem to figure out. Players 1 and 2 are bargaining over how to split $\$10$. Each player names an amount $s_i$, between 0 and 10 for herself. These ...
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32 views

Finding the expected revenue of a first price auction

I'm currently trying to solve for the expected revenue of a first price auction involving n players who draw their values v independently from F with support $[\underline{v}, \bar{v} ]$ and positive ...
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1answer
95 views

I can't find the Nash equilibrium of this 3x2 game.

Sorry for my English, I am French but i couldn't find help on the French website (so I am here). I have a question about this two-player game: $$ \begin{array}{c|cc} & y_1 & y_2 \\ \hline ...
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0answers
59 views

How can I find the Nash equilibrium for this game?

Sorry for my English, I am French but i couldn't find help on the French website (so I am here). I have a question about this two-player game: $$ \begin{array}{c|cc} & y_1 & y_2 \\ \hline ...
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2answers
47 views

Nash Equilibria in simultaneous game with four players

Four parliamentary parties are working on a necessary but highly unpopular law. Each party decides whether to put forward the law on its own behalf. If $n$ parties will put forward the law on its own ...
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1answer
48 views

How do I show that this game played on a Markov chain has a unique Nash equilibrium?

There are $k$ stages in this game, and each stage is worth one unit of utility to a player (of which there are $n$). Each player $i$ finishes stages at a rate $\lambda_i$ (in a continuous time Markov ...
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2answers
48 views

Static game with complete and incomplete information

I am currently trying to learn game theory on my own. I have a question regarding the solution methods for static games with complete information vs that of incomplete information. The textbook ...
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1answer
92 views

Nash equilibrium in first price auction

I'm trying to understand Exercise 18.2 from Martin J. Osborne and Ariel Rubinstein A Course in Game Theory about finding pure Nash equilibria in a first-price auction. There are n players, named ...
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1answer
45 views

Mixed strategy Nash equlibria (defending territory)

Hi I am trying to figure out the MSNE of this game. I am confused wether there are 2 MSNE or 3. Player 2 would put (1/2) on xx and yy and then player 1 would either put (1/4)on all or (1/2) on xxx ...
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1answer
66 views

Best response dynamics in Bertrand duopoly game

Question: Consider Bertrand's duopoly game in which the set of possible prices is discrete. Does the sequences of prices under best response dynamics converge to a Nash equilibrium when both prices ...
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0answers
20 views

Not every correlated equilibrium is equivalent to a Nash equilibrium?

This is a statement made under Theorem 3.4.13 on page 84 in the book by Yoav Shoham and Kevin Leyton-Brown, Multiagent Systems. Could someone explain this to a lay-man and also elaborate on why they ...
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1answer
362 views

Equilibrium existence proof

Problem: Let $J$ be an integer and let $I$ be an integer multiple of $J$. Let ${\cal I}= \lbrace 1,2,\ldots, I\rbrace$ and ${\cal J}= \lbrace 1,2,\ldots, J\rbrace$. The set $X_{I,J}$ of all ...
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1answer
228 views

Lemke-Howson pivoting in degenerate bimatrix games

I'm working on an implementation of the Lemke-Howson algorithm for finding Mixed Nash Equilibria of bimatrix games, and I'm running into a roadblock when the algorithm is fed a degenerate game. For ...
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1answer
20 views

Sub-game perfection when an agent is indifferent

If looking at one of the sub-games the player is indifferent between two actions. How does the backward induction work to recognize sub-game perfection? I.e., suppose player 3 has two action $A_3 = ...
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22 views

House allocation with existing tenants

In a house allocation with existing tenants model using the TTC mechanism, consider the incentive of an agent to misreport his/her preferences. Can it ever be that misreporting the true preferences by ...
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1answer
38 views

Mixed Nash equilibrium in two players game three stategies

I have this problem about finding the mixed Nash equilibrium. The payoff matrix is the following A(p) B(q) C(1-p-q) A 4 0 0 B 0 4 0 C ...
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32 views

Nash equilibrium in marriage market stable matching

Consider a marriage market where matchings are formed using the DA women-proposing algorithm. However, there is an additional constraint that only matches among agents that have previously gone out on ...
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1answer
108 views

Find the Nash equilibria in pure strategies in the following games.

Payoffs on the left correspond to Player 1 and payoffs on the right correspond to Player 2—for instance, in (U, L), player 1 gets a payoff of 3 and player 2 gets a payoff of 4. Explain in detail why ...
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1answer
264 views

Dominant-Strategy Equilibrium vs Nash Equilibrium

What's the difference between dominant-strategy solution and Nash Equilibrium? I could not tell the difference judging from the definitions. It would be appreciated if these concepts can be ...
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1answer
79 views

Pareto optimality - Game theory

Good morning, I have this game theory problem. 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 ...
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1answer
70 views

Cournot Duopoly Game - Nash equilibrium

I have this problem about Cournot Duopoly game. Actually I don't know if I have understood the "real sense" of the problem. I consider CD game described by the following payoff fucntions: $$ ...
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1answer
84 views

Calculating Nash equilibrium in mixed strategy in a game where a Nash equilibrium in pure strategy exists

Let's say I want to calculate Nash equilibrium with mixed strategies for a two-players game, in which there is no Nash equilibrium with pure strategies (no dominant strategy for any of the two ...
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2answers
79 views

if $G$ has no Nash equilibrium in pure strategies then $G$ has single Nash equilibrium in mixed strategies.

Let $G=(S,T,\pi _1 ,\pi_2)$ be a 2 player game with strategies $T$ for player 1 and $S$ for player 2 such that $|T|=|S|=2$, and payoff functions $\pi _1 ,\pi_2$. prove that if $G$ has no Nash ...
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2answers
39 views

Nash Bargaining Equilibrium with exponential utilities

I'm trying to derive the answer to the following question: Two players play the classic divide-the-dollar game, which is an imperfect information version of the ultimatum class of games. Utility ...
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2answers
282 views

Mixed strategies in 3x3 game - can strategies be negative?

Heres the payoff for player one. I'm searching for mixed strategies of player two. However I do the algebra, i get:p=4/5, q=3/10 and z=1-p-q=-0,1. Could anybody please explain, how negative ...
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0answers
26 views

Prove existence of (Nash) equilibrium

My question is about proving the existence of Nash equilibrium for a game involving two players. $x$ is player 1's strategy and $y$ is player 2's strategy; both strategies are continuous. For each ...
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1answer
116 views

Calculating the value of a bi-matrix game.

So I know this question is very simple, however in my text and from what I can find online, the solution tends to simply be given (such as in this example) Example: Let the following bimatrix game ...
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1answer
92 views

Pure and mixed strategy in Nash Equilibria with n player

I got confused when I see the following problem: There are n staffs and they want to raise their salary, if any one or more than one of these staffs suggest their boss they want raise salary, all ...
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1answer
114 views

What is the pure strategy Nash Equilibria of asking your professor to cancel class?

Each student in a class has the option to remain silent or ask the professor to cancel class. If any students asks to cancel class, all students get a payoff of $r$. However, the student that asks ...
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1answer
60 views

Can a symmetric equilibrium yield superior social welfare in a symmetric game?

Consider a 2-player symmetric game given by a payoff matrix $A\in [0,1]^{n,n}$ for the row player (i.e. the column player matrix is $A^t$). Define the social welfare as the sum of payoffs for both ...