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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Hat 'trick': Can one of them guess right?

There are $n$ boys and $n$ girls. Each of them is given a hat of only 4 possible (known) colors and doesn't know its color. Now each can only see all the colors of hats of those of the other gender and no contact is allowed, then each is asked to…
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A beautiful game of gold and silver coins

A stack of silver coins is on the table. For each step we can either remove a silver coin and write the number of gold coins on a piece of paper, or we can add a gold coin and write the number of silver coins on another piece of paper. We stop…
Jackie Poehler
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A Category Theoretical view of the Kalman Filter

Some basic background The Kalman filter is a (linear) state estimation algorithm that presumes that there is some sort of uncertainty (optimally Gaussian) in the state observations of the dynamical system. The Kalman filter has been extended to…
Emily
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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 like this (without computer) \begin{matrix} …
user106371
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Secretary problem - why is the optimal solution optimal?

I have read about this problem: http://en.wikipedia.org/wiki/Secretary_problem But I want to see how it is proven that the "optimal" solution is indeed optimal. I understand how to prove that if the optimal solution is of the form "wait for $t$…
Gadi A
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Infinite 'hex': is a win always achievable?

In the game hex, at least one player always wins because they can form a chain of hexagons across the board. This led me to wonder, what happens if we generalise to infinitely many points? Specifically, if every point in a unit square (including…
Wen
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Can the lion protect the sheep from three wolves?

Generally, in pursuit-evasion games, there's one prey and one or many pursuers. I'd like to know how extending the food chain would change the dynamics of such games. Specifically, let's consider a closed, circular shape arena in $\mathbb{R}^2$.…
Eric
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Is Rock Paper Scissors with two players but three contestants a fair game?

In a game of Rock Paper Scissors with two players, there are $3$ outcomes every round each with multiplicity $3$. Player 1 can win. Player 2 can win. Both players can draw. If we were to assign a winning result to a third "Player" when Player 1…
Axoren
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Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff the program halts? The rules are the same as…
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Game theory textbooks/lectures/etc

I looking for good books/lecture notes/etc to learn game theory. I do not fear the math, so I'm not looking for a "non-mathematical intro" or something like that. Any suggestions are welcome. Just put here any references you've seen and some brief…
becko
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Optimal strategy for the Rope Climbing Game

Define a two-player, turn-based climbing game as follows. Each turn, players have the option to climb or tie a knot at his current position. If the player chooses to climb, there is a 50% chance that he advances his position by $1$, and a 50% that…
Alexander Gruber
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best strategies for 'Squid Game' episode 6 game 4 "marble game" players

Two players each get $n=10$ marbles. Every alternating turn, one player (say, the player whose turn it is) hides an amount of own marbles in fist, and, the other player must guess if hidden amount is odd or even, and, that other player (i.e., the…
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"Infinito", a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to solve this game? How can I find the best move each…
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Game involving points on $[0,1]$

You're given a list of $22$ points in $[0,1]$ (not necessarily distinct), and you're asked to select, at every iteration, $2$ points to be substituted by their midpoint. After $20$ iteration, you should end up with $2$ points. Is there a selection…
Katy
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Determining the number of valid TicTacToe board states in terms of board dimension

I am attempting to find a closed form equation in terms of $n$, for the number of valid Tic-Tac-Toe board states (ignoring symmetry), where the board has dimension $n \times n ,\; 0 \lt n,\;n \in \Bbb Z $. Tic-Tac-Toe Rules: The $X$ token moves…