# Questions tagged [nash-equilibrium]

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

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### Mixed strategy non zero sum game for n players

The game mentioned everywhere is for two players. In the case of more than two players, how the payoff matrix will change ? For 3 players, each having 2 options. The total scenario will be 2^3= 8. So ...
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### Existence of approximate Nash equilibrium (continuous game)

My question concerns continuous games $((S_i)_{1 \leq i \leq n}, (u_i)_{1 \leq i \leq n})$, where $S_i$ are (continuous) compact strategy sets and $u_i$ denotes the utility function of agent $i$. From ...
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### Do all Wardrop Equilibria have the same social cost?

I'm currently studying non-atomic congestion games, and i've come accross the following definition of the price of anarchy: Let $f$ be a Wardrop equilibrium and let $f^*$ be a system optimal flow. ...
Consider a two-player game where Player 1 chooses a strategy $x_1=[1,3]$ and Player 2 chooses $x_2=[0,2]$ Let the payoff functions for P1 and P2 be $u_1(x_1,x_2)=\min\{x_1,x_2\}$ and $u_2(x_2,x_1)=\... 0 votes 0 answers 11 views ### Examples of non-convex Nash equilibrium problems with only inequality constraints? I am trying to find real life application examples for the Nash equilibrium problem of finding$(\overline{x},\overline{y})$with $$\overline{x} \in \begin{array}{cc}argmin_x & f_1(x,\overline{y}) ... • 989 0 votes 0 answers 20 views ### Example of a continuous game without a Nash equilibrium It is well-known that pure Nash equilibria need not exist in continuous games: for e.g. consider two players both playing over [0,1] with payoff functions u_1(x,y) = -u_2(x,y) = (x-y)^2. However, ... 0 votes 1 answer 48 views ### Nash equilibrium of a game I've a doubt about the following game and I'm new with Game-Theory. We have two road managers that aim to maximize their profit. Let be i=1,2 the players and respectively \omega_1,\omega_2\in [0,1]... • 109 1 vote 0 answers 27 views ### Existence of stationary Nash equilibrium of discounted stochastic game N-player discounted stochastic games with finite state and action spaces possess a Nash equilibrium in stationary strategies. This has been proved by Fink (1964) and a closely related result by ... • 23 0 votes 0 answers 22 views ### Could you provide me 10x10 Game with payoffs I have implemented a program which Outputs the Nash Equilibria in pure and mixed strategies and I want some inputs so I can check if my algorithm works fine. I also check the results from here https://... • 29 0 votes 0 answers 35 views ### 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 ... • 6,366 1 vote 1 answer 27 views ### 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 ... • 23 1 vote 0 answers 35 views ### Swapping Objective Function and Constraint When I studied Multivariable Calculus, I discovered that each simple (one objective function and one constraint) constrained optimization problem is associated with another simple constrained ... • 1,554 2 votes 2 answers 84 views ### Prisoner's Dilemma for Bob and Hilal Bob and Hilal fall in a prison. If both/none of them confess that they stole the money, they will both stay 11 months in prison. If one of them confesses but the other does not, the one who confesses ... • 5,166 3 votes 1 answer 142 views ### Blockchain: why choose the longest chain? I am currently learning how the blockchain works, at its mathematical core. I would like to understand precisely why everyone in the blockchain game should trust everyone else to agree on the same ... • 1,962 0 votes 2 answers 91 views ### Game theory - three voters for two candidates nash equilibrium There are 3 voters (x, y, and z) and two candidates (Alice and Bob). For either Alice or Bob to win they need 2 votes. If Alice wins x gets 1, y and z get 0. If Bob wins, x gets 0 and y and z get 1 ... • 11 4 votes 1 answer 103 views ### Algorithms for computing pure strategy Nash equilibria I'm looking through many sources, such as this one that mention algorithms, and time complexity of finding mixed strategy Nash equilibria. But is there any algorithm for finding pure strategy Nash ... • 177 2 votes 1 answer 83 views ### 2 players pick number from 1-100, 10 is subtracted from higher number. What's the best strategy? I am new to game theory and recently came across this question: 2 players pick a number from 1 to 100. From the player with the higher number, we subtract 10 and whoever has the higher number then, ... • 21 1 vote 0 answers 41 views ### Function to describe a game that maps reals to reals I am doing a bit of research into games, and want to know if something like this is possible: Imagine I have a game with N players, each with either the moves \{a\}, \{b\}, \{a, b\}, This can be ... 1 vote 0 answers 41 views ### Does the maximum entropy Nash equilibrium with integer payoffs have rational probabilities? I have a symmetric two-player zero-sum game, represented as an n \times n skew-symmetric payoff matrix M. The components of M are all integers. Are the probabilities in the maximum entropy Nash ... • 293 1 vote 1 answer 34 views ### Counter-intuitive mixed strategy Nash eq question I'm currently in intro game theory, and am having trouble with understanding (conceptually) these two games: \begin{array} {c|c|c} \ & L & R \\ \hline T & 2, 6 & 2, 2 \\ \hline B &... 0 votes 1 answer 62 views ### Nash equilibria problem [closed] I have a problem to solve the next exercise. Two players call a number from 1 to 100. The winnings are distributed as follows: players always receive no more than 100 rubles in total, the most greedy ... • 19 0 votes 0 answers 7 views ### How does this essay find (nrml(x), nrml(y)) Nash-Equilibria on Lemke-Howson Algorithm? Essay: https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=AE0EBA82161430FD61B8915771E19649?doi=10.1.1.110.770&rep=rep1&type=pdf Page 6: I understand all the steps on the Output section ... • 29 0 votes 1 answer 68 views ### How to maximize q and pi? I am desperate about some functions I would like to understand. I hope someone can help me to understand. Lets suppose we have a demand-function like:$$ D_i=\ \ 1/3+\ \ (2q_i-\ \sum_{j\neq i}\ q_j)/(... • 1 0 votes 0 answers 26 views ### A continuous time model where Nash equilibrium is build in a dynamic programming setting or as a system of backward looking SDEs? I am looking for a continuous time model, that builds a game among a continuum of agents who interact strategically and they have mean-variance utility function. In particular mean-variance utility ... 2 votes 0 answers 58 views ### Computational Game Theory, Why aren't Nash Equilibrium calculated with rational numbers instead of floating point numbers? In algorithms like Counterfactual Regret Minimization/Regret Matching, most implementations are coded using floating point numbers to represent (average) strategies, and the value of the game is ... 1 vote 1 answer 58 views ### Nash equilibrium in insurance pricing The insurance market is considered to be a competitive market, so in order to study competition to determine a competitive premium, game theory seems to be a useful tool for studying those situations. ... • 11 0 votes 1 answer 54 views ### Mixed strategy equilibrium in a number guessing game You and me are guessing numbers between 0 and 100. If I guess higher than you (including ties), you pay me £1. Else I pay you £1. If the players played according to uniform distribution, then they ... • 348 0 votes 0 answers 49 views ### optimal bid strategy for uniform distribution Ten players take part in the following auction for a$\$$100 project from some company. If some player i wins, the company will pay him his bid b_i. The ten players submit bids simultaneously, and ... • 441 0 votes 0 answers 25 views ### From Tirole Book: If a single strategy profile survives iterated deletion of strictly dominated strategies, then it is a Nash equilibrium of the game. While I am reading Tirole Game Theory Book , I read the following statement If a single strategy profile survives iterated deletion of strictly dominated strategies, then it is a Nash equilibrium of ... 0 votes 0 answers 17 views ### Max Pure strategies Given Text Consider Player X and Player Y. x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡), 𝑥3(𝑡)) and 𝑦(𝑡) = (𝑦1(𝑡), 𝑦2(𝑡), 𝑦3(𝑡)) be players’ strategies, that is, the portfolio quantities desired (buy or sell) ... • 29 2 votes 1 answer 74 views ### Positive affine transformation doesn’t affect the pure strategy Nash equilibrium While studying a pure strategy Nash equilibrium, I read such a statement “ the positive affine transformations of the payoffs functions do not change the set of pure strategy Nash equilibria” How I ... • 6,366 4 votes 1 answer 109 views ### A lemma of Kakutani’s theorem: prove that Any correspondence having a convex graph is also convex valued. How can prove the following sentence? a correspondence having a convex graph is convex valued as well. In order to explain the functions which used in this proof context, This comes from Kakutani’s ... • 6,366 1 vote 0 answers 29 views ### There exists no Nash equilibrium which is Pareto efficient? Prove or Disprove There exists no Nash equilibrium which is Pareto efficient. My idea is that this sentence is wrong. But I cannot show this. Please share your ideas with me thank you. • 6,366 3 votes 1 answer 59 views ### Find all Nash equilibria This question is from the Dutta’s book Each of the two roommmates simultaneously volunteer an time between 0 and 3 hours to clean their house. (Their choice doesn’t need to be integer) if roommate 1 ... • 6,366 2 votes 0 answers 1k views ### Alternate formulas with NonEquivalent Averages to judge an ending quarter of one season FA Premier League 2019/20. The season was affected by the COVID-19 Pandemic while each team had a so-called quarter of their schedule left. ("quarter" ? Since each team has 4/9 or 5/9 number ... 2 votes 0 answers 72 views ### Find all Nash equilibrium given a game Consider the following: How can we find all Nash equilibrium of the game given that the attacker cannot perform surveillance. I think for a mixed strategy there is but for a pure there isn't, but ... 1 vote 0 answers 28 views ### Using stric / weak dominant strategy to get 2x2 matrix in Game Theory I've encountered a problem with a 2x3 game where I need to find Pure NE and Mix NE so I thought it's better first to get a 2x2 matrix. For some reason I find it very difficult to solve this with ... 0 votes 0 answers 20 views ### Nash Equilibrium: Mixed Strategies calculation How do I find mixed strategies for this game after eliminating all these dominated strategies? Nash equilibrium game I know that B strictly dominates both C and D for player 2. Deleting C and D for ... 0 votes 0 answers 10 views ### Solving Mixed Strategies Equilibria I know that A strictly dominates B for player 2. So after deleting the B for player 2, B strictly dominates A and C for player 1. How do I find the mixed strategies Nash equilibria for this game after ... 0 votes 0 answers 19 views ### Nash equilibrium - dollar contribution There are 2 players. Each has an endowment of 10 dollars and chooses how many dollars to contribute to a public pool. Call x1 and x2 the contributions of player 1 and player 2, respectively. If x1 + ... 1 vote 1 answer 82 views ### Can Nash Equilibrium strategy be exploited, if another player deviates? Consider a 3 player game (e.g. poker for 3 people). Nash equilibrium is a set of strategies s_1, s_2, s_3, such that the expected profit value for each player s_i E_i(s_1, ..., s_i', ..., s_N) ... • 341 0 votes 0 answers 9 views ### Can braess routes change dynamically with respect to changing demand in a network Braess Paradox is a counter intuitive phenomenon where removing a link from a network increases the network efficiency. Usually these links are detected over long period of time in an average ... 0 votes 0 answers 329 views ### UNO Card Game and Game Theory I've recently been trying to create a computer program which plays UNO against human opponents (and usually wins); however, because I have very little experience in game theory, I have been unable to ... 0 votes 0 answers 12 views ### Adjusting payoffs has unintuitive effect on optimal strategies in bimatrix games Consider the game Rock-Paper-Scissors. If we award a win with 1, a loss with -1 and a draw with 0, we get the following bimatrix game (with rewards ordered as row player, then column player): R ... 2 votes 0 answers 102 views ### Who wins the Even/Odd game, and by how much? The Even/Odd game is defined as follows: Both players pick a number in (0,1] for positive y. Even's number is called E. Odd's number is called O. The scorer number is \Big \lceil \dfrac{E+y+1}... • 3,853 2 votes 1 answer 343 views ### Strategy game: draw a random number uniformly between 0 and 1 and if you like redraw. You win if your number is bigger than mine. [duplicate] You draw a random number uniformly between 0 and 1. If you like it, you can keep it. If you don't, you can have a do-over and re-draw, but then you have to keep that final result. I do the same. You ... 0 votes 0 answers 29 views ### Why Nash or correlated equilibrium require complete information? In games of complete information, there are common solution concepts such as Nash equilibrium and correlated equilibrium. The idea is that each player is playing a best response. My question is - Why ... 0 votes 1 answer 120 views ### How to find the Nash equilibria of a game with continuous strategy spaces Let u, w \geq 0. Let the payoff function f_1: [0, u] \times [0, w] \to \mathbb{R} for player 1 be defined as$$f_1(a, b) = \frac{u - a}{1+e^{a-b}} $$for a \in [0, u] and b\in [0, w]. ... • 316 0 votes 0 answers 101 views ### Mixed Strategy Nash Equilibria of 2x3 Game The setup of this game is very similar to the setup of the game in another question I found (Find all mixed-strategy Nash Equilibria of 2x3 game.):$$\begin{array}{c|c|c|c} & \text{L} & \...
so we have the game $(N,(S_{i})_{i\in N},(\varphi_{i})_{i\in N})$ where : $\bullet\:N$ is the set of players $\bullet\:S_{i}=[a_{i},b_{i}]$ the set of strategies for each player $i\in N$ \$\bullet\:\...