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Questions tagged [game-theory]

The study of competitive and non-competitive games, equilibrium concepts such as Nash equilibrium, and related subjects. Combinatorial games such as Nim are under [tag:combinatorial-game-theory], and algorithmic aspects (e.g. auctions) are under [tag:algorithmic-game-theory].

Game theory is the study of mathematical models of strategic interaction between rational decision-makers. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.

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How to arrange sequence of numbers into subcategories to equal 21

I have a bit of a brainteaser for a game my brother and I are playing (link for reference). The objective of the game is to arrange cards into categories where the sum equals 21. The game has four columns and you must arrange the cards into piles in…
scrubs
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How can I build a fixed point argument for Nash Equilibrium in $\mathbb{L}^2$ Banach space?

Consider the sets $Y_1,Y_2\subset\mathbb{L}^2(\Omega,\mathbb{F}_{s},\mathbb{P})$. I am looking for some $y_1$ measurable with respect to $Y_1$ and $y_2$ measurable with respect to $Y_2$ s.t. the pair $(y^*_1,y^*_2)$ constitutes a Nash Equilibrium.…
Nav89
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Doubts on a game that two players take turns to take an element and xor it to their sums.

This is originally a question from https://codeforces.com/contest/1383/problem/B, but this is more of a math problem than an algorithmic one, so I decided to post here. There is a game where two players take turns to take an element from an array…
Learning Mathematics
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Game Theory: Penalty Shot Game

Given a game matrix for the penalty shot game: (1/2,-1/2) (-1,1) (-1,1) (1/3,-1/3) What is the minimax strategy and expected value for this game? I calculated the minimax strategy to be -4 for both the shooter and goalie.…
NuNu
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Is "each player always defects" a Nash equilibrium in Iterated Prisoners Dilemma

For the iterated prisoners dilemma with random ending time, is it the case that "both players defects each round" is a Nash equilibrium? I understood a Nash equilibrium as a set of strategies for which it holds that no player benefits from switching…
user32149
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Bayesian Game with Asymmetric Information and 2 Players

I need some help with the following Bayesian Game. I tried it a few times and watched some tutorials but I just cant figure out how to solve it. Consider a two player game with this payoff matrix where $\theta \in \{0,3\}$ is a parameter known by…
Moritz123
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A probabilistic partisan game

This is a game between two players, A and B. Both players start on $n$ ($n\in\mathbb{Z^+}$) points, and the first player to reach $0$ points wins. A begins and the players alternate taking turns. A turn consists of choosing a number $p$…
Suoria
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Good examples of zero-sum perfect information game?

I am trying to come up with some good examples of zero-sum perfect information games. What I have on the list are: Tic-Tac-Toe Connect $4$ Chess Any thing more? One more question, for Connect4, this game can be solved perfect (1st player is…
kuku
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Finding a Nash equilibrium

I'm doing the exercises at the end of the paper A Brief Introduction to the Basics of Game Theory by Matthew O. Jackson. I would be grateful if somebody could provide me with solutions to it. I'm not sure about question 2: Two hotels are…
Emilia Borko
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No truthful, IR and deterministic online auction can obtain $(2-\epsilon)$-approximation for efficiency theorem proof is unclear to me

The theorem is from Algorithmic Game Theory Textbook by Nisan et al. available here on page 423 https://www.cs.cmu.edu/~sandholm/cs15-892F13/algorithmic-game-theory.pdf To elaborate on the notation, $\theta_i = (a_i, d_i, r_i)$ is the type of agent…
raka
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Stability of Nash equilibria in best response dynamics

If we define a Nash equilibrium as a fixed point of the best-response mapping, i.e. a strategy $x$ s.t. $$x\in BR(x),$$ where BR denotes the (set-valued) best-response mapping, then $$\dot x \in BR(x)-x$$ is generally referred to as best-response…
Mark
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Find maximum gold each miner can mine if both mine optimally.

Problem: Chef and Chefu work as gold miners. There are $N$ gold mines numbered from $1$ to $N$, $i$-th mine having $G_i$ gold. If Chef alone works in mine numbered $i$, it’d take him $A_i$ days to completely mine it. Similarly, if Chefu alone works…
Het
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Convex optimization lagrange multiplier nash bargaining problem

I have an optimization problem that looks like this: let $\mathbf{X} = \begin{bmatrix} x_{0,0} & x_{0,1} \ ... & x_{0,L}\\ x_{1,0} & x_{1,1} \ ... & x_{1,L}\\ \vdots \\ x_{K,0} & x_{K,1} \ ... & x_{K,L}\\ \end{bmatrix}$ $\max_{\mathbf{X}}…
TheMathGuy 314
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changing extensive tree game to normal representation

I was wondering how you can get (RfM) as -0.5, -0.5 I got RrP by [(1, -1) + (1, -1)] / 2 Here is the picture of the problem: Picture on the left: card game in extensive / tree form picture on the right: card game in strategic form / normal…
chrissy kwon
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Game theory:- value of a game?

I haven't found any suitable explanation or even definition for this concept. What is the value of game in game theory? Can anybody explain me with an example.
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