Questions tagged [game-theory]

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Are gaps and loopy games interchangeable in the Surreal Numbers?

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 ...
8 votes
2 answers
396 views

Optimally betting a beta-biased coin

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 $p$...
2 votes
2 answers
50 views

Convergence of naive iteration for a stateful, iterated tabular game

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 ...
10 votes
0 answers
339 views

Examples of games developed purposely to analyze players' strategies for mathematics research

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 ...
0 votes
0 answers
121 views

Optimal strategy of modified Mastermind game

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 ...
0 votes
2 answers
93 views

Points based partial ranking

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 ...
1 vote
1 answer
218 views

How to show that maximizing "chip EV" is the equilibrium strategy in winner-take-all poker

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 ...
19 votes
5 answers
1k views

When is a game tree the game tree of a board game?

This question arises from what I find interesting in the recently asked question What is a chess piece mathematically? My answer to that question was that mathematically, game pieces are in general ...
1 vote
0 answers
55 views

How is the notion of $G$-function related to that of the replicator equation?

Background: While studying "Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics", I am pretty confused about the notion of G-function (fitness-generating function), that is ...
1 vote
2 answers
483 views

Optimizing a game tree

(I asked this on StackOverflow, which garnered no response, but maybe this site is a better choice.) I have a question about game tree planning (I believe this is the correct domain). I am playing a ...
9 votes
0 answers
356 views

For which set $A$, Alice has a winning strategy?

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 ...
2 votes
0 answers
123 views

Go variant: cyclic or not?

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 ...
3 votes
2 answers
535 views

Negative of combinatorial game

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 ...
0 votes
1 answer
70 views

Optimality of a "shopping" heuristic

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^...
14 votes
2 answers
873 views

Who wins infinite Hex?

In this game, you start with a square. Alice tries to connect the top side to the bottom side, and Bob tries to connect the left side to the right side, like in Hex. Unlike in Hex, Alice and Bob use ...
21 votes
4 answers
2k views

Fairest way to choose gifts

Suppose that a parent brings home from a trip $2n$ gifts of roughly equal value for his/her two children. The children get to choose one at a time which gifts they want. What is the fairest way to ...
2 votes
2 answers
218 views

Continuity of Nash equilibrium for a family of games

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 ...
1 vote
0 answers
66 views

Hodge-Helmholtz decomposition for 1-form of strategic game

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. (...
7 votes
1 answer
208 views

How complicated are 3-player clopen determinacy facts?

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 ...
10 votes
2 answers
2k views

Explicit examples of undetermined games

Suppose we have a game between two players in which they take alternating turns. The game can have finite length, length $\omega$ or any transfinite number of steps (however, I'm not concerning games ...
44 votes
9 answers
20k views

How to start game theory?

I recently got interested in game theory but I don't know where should I start. Can anyone recommend any references and textbooks? And what are the prerequisites of game theory?
3 votes
0 answers
125 views

Poker with infinite stack size

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 ...
6 votes
2 answers
1k views

Is perfect play possible in continuous rock-paper-scissors? game "step size" vs. "acceleration"

The first part of my question is simple: Is every game continuous in time and strategy-space also a game of perfect information with a good equilibrium? For example, consider rock-paper-scissors. The ...
9 votes
1 answer
2k views

Are there any interesting connections between game theory and engineering?

I am doing a senior project and it must be based off game theory, but I am having trouble finding any connections to engineering, possibly structural, or architectural, maybe even civil or mechanical. ...
3 votes
2 answers
140 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 ...
1 vote
0 answers
37 views

Nash Equilibria change linearly in (some) game parameters. Already known / follows from a more general result?

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 ...
2 votes
1 answer
190 views

Nash equilibrium at another level

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 ...
1 vote
0 answers
65 views

Representation of an N player game with 2 strategies per player as a matrix and its properties

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 ...
5 votes
2 answers
477 views

A variant of Conway's Game of Life: any cell with more than 3 live neighbours becomes a live cell and no live cell dies. How to make more cells live?

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, ...
20 votes
3 answers
677 views

Escaping from infinitely many pursuers

The fugitive is at the origin. They move at a speed of $1$. There's a guard at $(i,j)$ for all $i,j\in \mathbb{Z}$ except the origin. A guard's speed is $\frac{1}{100}$. The fugitive and the guards ...
7 votes
2 answers
411 views

Chasing game on the Go board

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 ...
228 votes
9 answers
24k views

John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends. Maybe this is an appropriate time to ask a ...
3 votes
1 answer
864 views

Optimal auction for risk-averse seller

Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim ...
3 votes
0 answers
74 views

Projective plane finite game

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 ...
7 votes
1 answer
565 views

JUSTICE & INJUSTICE — two 2-player finite games

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 ...
2 votes
0 answers
70 views

Equilibrium for a game with mixed strategies on a compact ultrametric space

Let $(X,d)$ be a compact ultrametric space. Hartig and de Vink considered the following ultrametric on the set $P(X)$ of probability on $X$: $$\hat d(\mu,\nu)=\inf\{r>0:\forall x\in X\;\;\mu(B_r(x))...
2 votes
1 answer
2k views

Matrix tic tac toe

So we have a 3x3 matrix and two players, a player that only puts in ones and a player that only puts in zeros. A coin flip is used to decide which player goes first. The first move is always to fill ...
3 votes
1 answer
385 views

Equilibrium strategy for half-street [0,1] poker game, no-limit (from "The mathematics of poker")

I have been trying to understand Example 14.3 from p154 of Bill Chen's and Jerrod Ankenman's book The Mathematics of Poker without much success. In this section they are analyzing what they refer to ...
113 votes
54 answers
53k views

Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in the game's structure, optimal strategies, practical strategies, analysis of the game ...
4 votes
0 answers
150 views

Two-player item picking game

Two players $A$ and $B$ play this game: There are $n$ items, where the $i$th item is of value $a_i$ to player $A$ and is of value $b_i$ to player $B$. Two players take turns picking items, and each ...
-3 votes
1 answer
311 views

What is a good formalization of this classic math puzzle? [closed]

Here is a classic math olympiad problem (but this is NOT my question!): Each of the girls A and B tells the teacher a positive integer but neither of them knows the other's number. The teacher writes ...
3 votes
0 answers
135 views

Can you escape from two lions in a closed arena?

You're at the center of a circular arena. A pair of lions are at the border, planning to catch you. One of them moves as fast as you, but the other moves slower than you. The three of you are confined ...
3 votes
1 answer
272 views

Difficulty of 3-color forest Hackenbush

"Forest Hackenbush" (for lack of a better name) is the particular case of the game of Hackenbush where the initial position (and therefore all subsequent positions) is a (finite) forest (:= disjoint ...
9 votes
6 answers
2k views

Surprising applications of the theory of games?

I am currently studying the applications of games in quantum information theory and related fields and I am aware of its uses in places like model theory and set theory. So I was curious, what are ...
1 vote
0 answers
97 views

Game with Turing machines

Introduction The following game is quite nice: Alice has, in secret, constructed a polynomial $P \in \mathbb{Z}[x]$. On day $n=1,2,3,...$, she secretly writes down $P(n)$ on a piece of paper. Each day,...
3 votes
0 answers
368 views

How to promote a blog?

Math behind might be interesting. Quite recent bloggingg activity might have interesting math model. The point is that bloggers compete for subscribers and at the same time cooperate gaining ...
0 votes
0 answers
110 views

Game on a square grid (part II)

Related to this question, where there the solution was unexpected for us. Let $n,m$ be positive integers, $n \le m \le n^2/2$. The board is $n \times n$ square grid. Phase 1: Two players, $A,B$ make $...
6 votes
1 answer
324 views

Do random asymmetric games have more complicated strategies than random symmetric games?

Let $\Delta \subset \mathbb R^n$ be the locus of vectors whose entries are nonnegative and sum to $1$. For $M$ an $n\times n$ matrix over $\mathbb R$, let $x_M \in \Delta$ be the vector $x$ that ...
1 vote
2 answers
253 views

Do restricted Nim-like games have winning strategies?

Considering a Nim-like game to be: There are three piles $A,B,C$, and the amount of their elements are $|A|=2, |B|=5, |C|=6$; There are 2 players. Each time a player can either take $x (1\leq x \leq ...
3 votes
1 answer
292 views

Games and the right mathematical framework for GANs

Generative Adversarial Networks were introduced in http://papers.nips.cc/paper/5423-generative-adversarial-nets and has more than 20000 citations. It is an important topic within deep learning. Are ...

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