Questions tagged [nash-equilibrium]

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

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Why does Scarf's algorithm only need to examine a small fraction of points in the simplex?

Scarf's algorithm for finding the Brouwer fixed-point searches for the fixed-point in an non-repeating fashion examining a finite number of points. It finds the fixed point however by examining a very ...
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Favorite 3-player zero-sum-game?

I am looking for a $3$-player game that meets the following requirements: It should be a zero-sum-game It should be simple in the sense that the payoff table is not too large (ideally memorisable by ...
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Finding a Nash Equilibrium of a Two-Players Game based on Best Response Functions - Taking a partial derivative from a piecewise, weird, function!

I need to find a nash equilibrium of a two-player game based on their best response function. The problem I have is that the functions I'm dealing with are somehow weird and hard to work with! So, let ...
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Mixed Nash Equilibria

Does it hold for all bi-matrix games that all mixed Nash equilibria have the same expected pay-off ? Or is it necessary that the game is zero-sum ? Finally how can I see and how is it used that the ...
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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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In game theory, are there easy ways to rule out the existence of either (i) all pure Nash, (ii) all completely mixed Nash?

For example, in a rock paper scissors game, there are no pure Nash equilibrium and only a completely mixed Nash equilibrium. in prisoner's dilemma, there is no mixed Nash equilibriumand only a pure ...
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How to find the Side Payment of a TU Cooperative Strategy?

Let's say that we have the game matrix $$ \begin{pmatrix} 3,2 & 5,3 \\ 6,1 & 0,1 \end{pmatrix} $$ We can easily deduce that the value of the cooperative strategy, deduced by $$\sigma = \max\{(...
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Unstable Nash equilibrium (at a boundary point)

Let $x=(x_i)_{1\leq i\leq n}$ be the "actions" of $n$ players, where $x_i\in[0,1]$ is determined by player $i$ seeking to maximize its objective function $\pi_i(x)=\pi_i(x_i,x_{-i})\geq 0$. ...
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Finding Nash balances in Pure-strategy Game theory

I am very new to the game theory, however from videos on youtube I have been learning a bit. However, I am not sure how I can apply the knowledge to this question: ...
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Is it possible for both A1B1 and A2B2 to be in Nash Equilibrium?

The question is as follows: Consider a game with two players A and B. Player A has strategies A1 and A2, while player B has strategies B1 and B2. The payoffs are shown in the table below. ...
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How to decouple/ decompose Couple Riccati Differential Equation (CRDE) in LQ differential game?

I have read some papers about differential games. We need to solve coupled Riccati differential equation in continuous-time (Jacob Engwerda, 2005) or discrete-time (Jank and Abou Kandil). In the ...
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What is the transition matrix for a strategy that uses tit for tat?

Question: Let $S$ be the strategy that it will start with $C$ and continue to do so until the opponent plays $D$ in the previous game. In this case, this strategy will play $C$ with probability 1/3 ...
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Non-Traditional 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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Nash Equilibrium in two zero-sum games with the same strategy for the row players

Consider two zero-sum games where the row players are $R_1$ and $R_2$ and the column players are $C_1$ and $C_2$. Below is the payoff tables: enter image description here The Nash Equilibrium strategy ...
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Why is it optimal to always defect in the iterated prisoner's dilemma with a known finite number of iterations?

From the Prisoner's dilemma Wikipedia page: "If the game is played exactly N times and both players know this, then it is optimal to defect in all rounds." and "For cooperation to ...
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Transform this game into a Strategic Form Game and draw the game table

Find the Subgame Perfect equilibrium strategies. Solution: Stage 1 : player 2 chooses F after C. And player 2 chooses E after A. Stage 2: player 1 chooses C. Then SPNE={ (C), (E after A, F after C)}. ...
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Find pure strategy Nash equilibrium in 3-player game

The answer of this question says that The pure strategy Nash equilibria of this game are: (U ; L; A), (D; R; A) and (D; R; B). But this answer does not make sense for me. I found that (U,L,A), (U,R,A),...
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How was the normal form of this extensive game calculated?

An explanation for just one set of payoffs will be fine (e.g. CH,CH). Because of the overlapping information sets, I am confused as to how the payoffs are calculated. For the CH,CH pair, I don't know ...
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The construction of the reduced strategy form matrix

Question Given the payoffs in the matrix below, two players play a variant of the “Battle of Sexes”(BoS) game, in which each player chooses in between F and C. At first, player 1 makes a choice in ...
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Signaling game and intuitive criterion

I need some help in understanding the intuitive Kreps criterion. Consider the following game, where Nature acts with probability $\frac{1}{2}$ Here's what I found (prove me wrong if I made a mistake) ...
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Show that the game has a unique Nash equilibrium (in pure or mixed strategies)

Each of 15 students simultaneously announces a number in the set {1, 2, . . . , 100}. A prize of 1 TL is split equally between all students whose number is closest to 1/3 of the class average. Show ...
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Find best responses

Question: There are N voters who have positions that can be indicated by the numbers 1 through 7. The number of voter with each position is indicated in the table below: Assume that voters always ...
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Question on n-times game SPNE

Recently I thought on one statement and I'm trying to prove it wrong or true, with the example in the latter Suppose we have a normal-form game $G$ with two Nash equilibria in pure strategies. The ...
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Which pure strategies in each game are dominated?

Which (pure strategies) in each game are dominated? For each dominated strategy specify the (mixed) strategy that dominates it. The solution manual says that in Game 1, R is dominated by $\sigma_2 = (...
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find the pure strategy Nash equilibrium with 3 players

There are three players and each stakes 10 dollar. The players simultaneously choose a number from the set {1, 2, 3, 4}. If three numbers are different, the player choosing the middle number wins the ...
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Find the Nash equilibrium for the given question

Consider there are two players in a simultaneous game. Each of them chooses and pays integer multiples of one dollar to play the game. The player with the highest bid wins 100 dollar and the loser ...
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How can I find all Nash equlibrium for game?

I am trying to understand how to calculate all Nash equilibria in a 2 player game, but I fail when there are 3 possible options . Also I have I checked answers on this sites and still I dont get it. \...
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Sequential Voting Game

Consider the following model of sequential vote buying. A political leader P needs to convince 2 out of a set of 3 committee members or legislators (called L1,L2,L3) to support her policy project. ...
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Prove that a bimatrix game with diagonal matrices is a Correlated Equilibrium

The question is as shows: Let $[A,B]$ be a bimatrix game such that both A and B are diagonal matrices with nonnegative diagonal entries. Show that any diagonal matrix $(p_{ij})$ such that $(p_{ij}) \...
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Solve Normal form Nash eq. (pure and mixed), especially the probabilities

can someone please help me to solve this question: Consider the normal form game: i) For Player 1 the strategy B is strictly dominated by C. ii) For Player 2 the strategy E is strictly dominated by F....
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A good source to learn refinement of Nash Equilibrium and Bayesian Games

I am studying the computation of various refinements of Nash Equilibrium in pure and mixed strategies, which includes Weakly Perfect Bayesian Equilibrium and Sequential Equilibrium. Also, I want to ...
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Reducing finding a Nash Equilibrium for a 3-player Zero-Sum game to finding a Nash Equilibrium for a two-player zero sum games.

I am currently studying for an exam and got stuck on the following question: We have seen that finding a Nash equilibrium in a two-player zerosum game is significantly easier than general two-player ...
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Optimal Mixed Strategies- Game Theory

I am reading the book Game Theory by E.N.Barron. (p.13) One of the properties for optimal strategies says: If $Y$ (the mixed strategy for the second player) is optimal for II and $y_{j}>0,$ then $E(...
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Difference between Strategic Form and Extensive Form game with perfect Information?

While doing my game theory homework, I cam across this question Is the following game the strategic form of an extensive form game with perfect information? Draw the game tree if your answer is ...
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Game theoretically optimal play can it depend on randomness?

I am studying GTO of poker but my question should apply to general game situations. When I solve some situation using any of the avaialble GTO programs out there, I get a result which say for instance,...
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Nash Equilibrium with a skew

It occurred to me that there is a hidden assumption behind Nash equilibrium. To keep things simple, I am interested in two-player zero-sum perfect-information finite normal-form games given by a ...
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1answer
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Calculating mixed strategy of $3 \times 3$ game

The Question We consider the following zero-sum strategic game in matrix form \begin{array}{c|lcr} & \text{A} & \text{B} & \text{C} \\ \hline A & 0 & +\epsilon & -\delta \\ B ...
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Solving Nash equilibrium

Suppose you are given a payoff matrix dimensions m*n in which player A has m strategies and player B has n strategies and in each cases it results in a different outcome, so how could you find the ...
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Does Nash Theorem imply Zermelo's Theorem?

In 2-Player zero-sum game with every information open and no probabilistic strategy required, Nash Theorem states that one of the players has a strategy, in which the player can maintain a situation ...
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Pure and mixed strategy equlibrium

I am studying Game Theory and have problems with solving the questions regarding the game down below: Consider $v_1 > v_2 > v_3 > 0$ and the following pay off matrix $$ \begin{pmatrix} A/B &...
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Can we find Nash Equilibrium payoffs in degenerate games

In a 2 person, constant sum game, all NE strategies have identical payoff, the 'value' of the game. Typically this is calculated by first calculating one NE strategy. In many algorithms for ...
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Nash Equilibrium - El Farol Bar Problem

Statement of the problem, from Wikipedia: Every Thursday night, a fixed population want to go have fun at the El Farol Bar, unless it's too crowded. If less than 60% of the population go to the bar, ...
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When is Counterfactual Regret Minimization appropriate?

I must be missing something fundamental here. In a 'stacked normal form 2 person zero-sum game' we can solve for the NE strategies and payoffs recursively. If we have access to the full tree we can ...
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Question about proper subgames in game theory.

Consider the following game presented as a tree. As we can see there are $3$ players and the question is to find all N.E and all S.P.N.E. But I have a doubt about one thing. As we can see, if the ...
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Difference between simultaneous and ordered N.E. in the problem.

Consider the following game. There are two players and one of them could be in two states: $A$ with probability $p$ and $B$ with probability $1 - p$. Both players actions are $1, 2,$ and to not ...
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Mix of first price and second price auction problem.

Suppose we have two participants. They play the game with utilities: $u_1 = v_1 - b_2$ if $b_2 \le b_1$ and zero otherwise. For the second participant $u_2 = v_2 - b_2$ if $b_2 \ge b_1$ and zero ...
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What techniques are there for solving extensive form games where we discover the tree on-line

To my knowledge all counterfactual regret algorithms require that we have the tree completed, even if it's only an abstraction of the tree. In other computational strategy settings, there are ...
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Mixed Statregy Nash Equilibrium - probability higher than one?

I have found the Nash equilibrium of the following game but I have doubts that the result makes sense The game is: ...
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Equilibrium from regret matching

I'm reading an article that states that this regret matching algorithm: • For each player, initialize all cumulative regrets to 0. • For some number of iterations: – Compute a regret-matching ...

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