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For questions on mathematical game theory (winning strategies, Nash equilibria, ...)

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This mathematical game occurred to me. It may be quite basic for experts, but it seems to lead to some interesting questions. Turns are indexed by elements of $\mathbb{N} = \{1,2,\ldots, \}$. In turn $...
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I’m working with the Eisert–Wilkens–Lewenstein (EWL) formalism and I’m trying to understand whether structural results on equilibria exist beyond what is usually presented. What I’ve checked: Eisert–...
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It is not hard to find the Nash-equilibrium of an attack-defense game with finite energy levels by solving a finite number of equations, but how to solve an infinite number of equations? In this game, ...
Y. Yang's user avatar
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Consider a finite strategic form game. Let $U_i(\sigma_i; \sigma_{-i})$ denote the real-valued payoff to player $i$ when using mixed strategy $\sigma_i$ while his opponent also uses mixed strategy $\...
Your neighbor Todorovich's user avatar
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Introduction Elementary linear algebra gives us the motivating example of Hodge theory — that given two matrices $A,B$ acting on finite dimensional vector spaces such that $AB=0$, we may decompose ...
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Note: This question is motivated by some work on tug-of-war games in connection with the infinity Laplacian. If further details on the background of the problem are desired, please let me know and I ...
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One of the evolution equations used in evolutionary game theory is the Replicator type II dynamics $$ x_i(t+1)=x_i(t)\frac{(Ax(t))_i}{x^T(t)Ax(t)} $$ where $A$ is an $M\times M$ payoff matrix with ...
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References: Applications of limited information strategies in Menger’s game by Clontz Almost compatible functions and infinite length games by Clontz and Dow Def. 3.7 of [1] $\mathcal{A}(\kappa)$ ...
Jakobian's user avatar
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Consider the following cooperative game played on the circle $S^1$, which we identify with $[0, 1]$ with its endpoints identified. Let $0 < \alpha < 1$ be an irrational number. A total of $N \...
Nate River's user avatar
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So I’ve been obsessing over this for a while now and could really use some help. I’m trying to figure out the minimum number of moves it would take to win the Google Snake game if you played perfectly....
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As discussed here The typical form of a functional for the Euler-Lagrange equation is $$ J[\tau] = \int_a^b L(e,\tau(e),\tau'(e)) de. $$ In the link above they discuss a situation where the bounds of ...
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An odd number $N \geq 3$ of players are playing a game - they bet on the outcome of a biased coin that comes up heads $p > \frac{1}{2}$ of the time, where $p$ is known to all of the players in ...
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Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which. He has \$1 with which to bet with. On ...
Nate River's user avatar
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$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or ...
Nate River's user avatar
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I am trying to reproduce Figure 1 in the paper "Chaos in learning a simple two-person game" (English) Proc. Natl. Acad. Sci. USA 99, No. 7, 4748-4751 (2002) (MR1895748, Zbl 1015.91014), by ...
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¶1. In classical game theory, a normal form game between $k$ players is given by finite sets of options $A_1,\ldots,A_k$ and payoff functions $u_1,\ldots,u_k : \prod_i A_i \to \mathbb{R}$ (all of ...
Gro-Tsen's user avatar
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Consider the following problem: We have $k$ rooms, each equipped with a light that cycles through three colors – red, green, and blue – in a cyclic order. Initially, all lights are set to red. Each of ...
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Let $f: \{0,1\}^{m + k} \to \{0,1\}^n$ be some function, where $m = n \log n$ and $k < \log \log \log \log n$. Consider the following game between Alice and Bob that determinates by $f$, $c_A$ (...
Alexey Milovanov's user avatar
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Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is ...
Nate River's user avatar
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$N \geq 2$ players play a cooperative game on the integers $\mathbb Z$. All of them start from $0$. At each turn, they are simultanously given the same yes or no question to answer. The questions ...
Nate River's user avatar
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Given a subset $S\subseteq \mathbb{R}^n$, the metric projection associated with $S$ is a function that maps each point $x\in \mathbb{R}^n$ to the set of nearest elements in $S$, that is $p_S(x) = \arg ...
Erel Segal-Halevi's user avatar
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Background In evolutionary game theory, one can what kinds of different strategies yield the most payoff to players that play the same game repeatedly. Consider, for instance, the iterated Prisoner's ...
Max Lonysa Muller's user avatar
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My question: how to prove or disprove the following two conjectures? Conjecture 1: (Gomoku large conjecture) there is no draw on infinite board for Gomoku with any initial opening with finite stones, ...
hzy's user avatar
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This is a crosspost from Math stackexchange as I left the question open a while and bountied it but received no answers. If the game of Hex is played on an asymmetric board (where the hexes are ...
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I have been developing a card game called "Infinity", which involves a unique play mechanic based on card interactions. In this game, each card displays a set of symbols, and players match ...
mathoverflowUser's user avatar
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I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves ...
Gro-Tsen's user avatar
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Consider the following three-player game: Alice chooses an integer congruent to $0$ mod $3$, Bob chooses an integer congruent to $1$ mod $3$, and Chris chooses an integer congruent to $2$ mod $3$. (...
Gro-Tsen's user avatar
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The class of surreal numbers (commonly called $No$) is not complete: it contains gaps. Some people have studied the "Dedekind completion" of the surreal numbers in order to do limits and ...
Farran Khawaja's user avatar
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Summary: Consider a stateful, two-player zero-sum game: at each state, two players pick moves simultaneously, and the reward and next state depends on those moves. We can attempt to solve such a game ...
Geoffrey Irving's user avatar
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Background This question is about games that were created, developed, deployed and popularized1 by researchers because they wanted to learn more about some mathematical structure, and did so by ...
Max Lonysa Muller's user avatar
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9 votes
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This question is inspired by How to optimally bet on a biased coin? by Nate River but generalized slightly. I decided the generalization might be interesting enough to be its own question. A number $p$...
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The following game is a modified version of the popular game Mastermind described here in which you are only given information about the total correct guesses you have made, and nothing about how many ...
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I want to rank a population $P=\{P_1,\ldots,P_n\}$. I am given a set $R=\{R_1,\ldots,R_k\}$ of partial rankings. The partial rankings may have varying sizes (e.g. the first ranking ranks only 8 ...
Max's user avatar
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1 answer
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This question is based on poker, but you don't need to know anything about poker to analyze it. A while ago I asked over on math.SE how to prove that the probability of winning a head up poker match ...
Davis Yoshida's user avatar
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Cross-posted from MSE: https://math.stackexchange.com/questions/4775193/for-which-set-a-alice-has-a-winning-strategy Alice and Bob are playing a game. They take an integer $n>1$, and partition the ...
Veronica Phan's user avatar
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I would like to know if a cycle of moves is possible in the Go variant, Savage Go. That is, you capture my stones, I capture your stones, you capture my stones... The game never ends. A position is ...
Mark Steere's user avatar
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4 votes
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I am having problem understanding what negative of a combinatorial game $G$ exactly means in combinatorial game theory. Does it mean that if I have normal game, if I create inverse, i.e., $-G = \{-G^R ...
Nick's user avatar
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Suppose the following situation: we have to buy $n$ goods $g_1,\,\dots,\,g_n$ starting at day 1 and we can't buy more than one good per day. On day $d$ the prices are $p_1^d,\,\dots,\,p_n^d;\quad p_i^...
Manfred Weis's user avatar
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The question may be too vague, but ultimately in search of various (counter)examples or theorems to exhibit the following: Do continuous families $t\mapsto G_t$ of "games" (say each $G_t$ is ...
Chris Gerig's user avatar
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This question is motivated by the attempt of decomposing (in a direct sum sense) a strategic game into components in the spirit of Hodge decomposition. Preamble Combinatorial setting Candogan et al. (...
DavideL's user avatar
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Say that a clopen 3-player game is a well-founded tree $T\subseteq\omega^{<\omega}$; intuitively, starting with player $1$ and continuing cyclically, the players $1,2,3$ alternately play natural ...
Noah Schweber's user avatar
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In two-player No Limit Texas Hold 'Em (NLHE), the optimal strategy depends on the "effective stack size," that is maximum amount of money held by one of the players (in the sequel I'll just ...
Davis Yoshida's user avatar
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3 votes
2 answers
342 views

$N$-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 ...
kehagiat's user avatar
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EDIT: The key thing that I am wondering about is the linearity of the P2 strategy, not the constancy of P1. (The latter is straightforward.) Question: Is the following result already known? Or is it a ...
Vojtěch Kovařík's user avatar
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2 votes
1 answer
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This is a variant of the Nash equilibrium. Let's say that there are 3 prizes: A Ferrari, a diamond watch, and a new boat. There are 6 players. 3 players with a motive while 3 players with another ...
Hass Boyouk's user avatar
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Is there any well-studied representation of a N player game with 2 strategies per player as a matrix? Intuitively, I think that each strategy can be represented as a binary digit, and each strategy ...
wavosa's user avatar
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5 votes
2 answers
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In Conway's Game of Life, we got an infinite board (two-dimensional orthogonal grid of squares). Every cell in the board interacts with its eight neighbors, which are the cells that are horizontally, ...
Blanco's user avatar
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7 votes
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In Go (Weiqi), two players take turns placing stones on the vacant points of a board. Once placed, stones can only be removed from the board if a stone or a group of stones are surrounded by their ...
richard jameson's user avatar
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This is a 2-person game. Let $\ P\ $ be any arbitrary projective space (of any dimension $\ \ge2$ and any cardinality, etc., but typically, let it be a finite plane over a field). Let $\ S_0\subseteq ...
Wlod AA's user avatar
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7 votes
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There is a non-empty finite set $\ K,\ $ say, of plates. Initially, there are $\ p_0(k)\ $ stones on the $k$-th plate, where $\ p_0(k)\in\mathbb Z_{_{\ge0}}\ $ for each $\ k\in K.$ So far, it is like ...
Wlod AA's user avatar
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