# Questions tagged [game-theory]

The game-theory tag has no usage guidance.

241
questions

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### Brinksmanship: how to achieve the best outcome by a single statement [closed]

This game is taken from Schelling's Game Theory: How to Make Decisions by R.V. Dodge, in which contenders practice brinksmanship to their own advantages. It goes as follows:
Anderson, Barnes, and ...

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**2**answers

40 views

### Cyclic inequality for 2 dimensional simplex elements

Let $p=(p_{1},p_{2},p_{3})\in\Delta$, with $\Delta:=\lbrace p\in(0,1)^{3}\ |\ p_{1}+p_{2}+p_{3}=1 \rbrace$. I aim to prove (not knowing whether it is true though) that
\begin{equation}
p_{1}^{p_{3}-p_{...

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49 views

### Conditions for optimal stationary strategies in MDPs

I have a specific markov decision process (MDP) which is generated from a problem in another domain. What I would like to show is that under the limit of means criterion (no discounting) the optimal ...

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170 views

### Can Alice ever fare the worst in this variant of the truel game?

In the well known classic three way duel puzzle, 3 players Alice, Bob and Carol take turns to shoot each other until only one survives. In his/her turn, a player can either choose to shoot or pass$^{1}...

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257 views

### Game on a square grid

Not research level, comments are welcome.
Consider the following game:
The board is the vertices of an $n$ by $n$ square grid.
Two players take moves in turns.
A move is picking two vertices and ...

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**1**answer

56 views

### Proving the existence of a symmetric Bayesian Nash equilibrium

I am currently faced with the following question:
Consider the public goods game. Suppose that there are $I > 2$ players and that
the public goods is supplied (with benefit of 1 for all players) ...

**25**

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**1**answer

725 views

### The lion and the zebras

The lion plays a deadly game against a group of $N$ zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the $N$ zebras may ...

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160 views

### Help with a definition of a two-person game in a referenced paper

In the paper "Finding Mixed Nash Equilibria of Generative Adversarial Networks" the authors write in equation (1) on page 2:
Consider the classical formulation of a two-player game with
finitely ...

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**1**answer

244 views

### When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality:
$$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$
For ...

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**1**answer

74 views

### Dubious matrix monotonicity

Coming from a problem in game theory, I arose at some dubious monotonicity like property for matrices of the following art. Let $H=\lbrace h\in\mathbb{R}^{n}\colon h_{1}+\dots+h_{n}=0\rbrace$. I'm ...

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**1**answer

102 views

### Nash equilibria for “presidential election” game

Suppose, in a country there are $m$ different social issues, positions on which are being indexed with numbers $[-1; 1]$, with radicals on the opposing ends and moderates in the center. In this ...

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30 views

### Suggestions for two-choice game played in ladder graph

I was just working on counting all the possible Nash Equilibrium solutions for a two-choice game played on a ladder graph (I got my results and all that for a generic number of players).
And I was ...

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39 views

### Maschler's bargaining set-an incomplete step in a proof

I have a problem with the concept of the bargaining set which is given below in some detail.
Let $N=\{1,\ldots,n\}$ be a set of players and $v:2^N\to \mathbb{R}$
a superadditive game (meaning $S,T \...

**3**

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**1**answer

204 views

### Is following function a metric on the set of isomorphism classes of graphs with countably many vertices?

Suppose $\Gamma_1(V_1, E_1)$ and $\Gamma_2(V_2, E_2)$ are simple graphs with countably many vertices. And suppose $A_1$ and $A_2$ are initially empty sets. Suppose two players play the following game: ...

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39 views

### Core-nonemptiness

In the snippet below in the theorem 3.2.6.
there is given a characterization of the core like this:
$NE,IR, RGP$. However, I believe that core might be empty so
it doesn't satisfy $NE$. What's wrong ...

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**1**answer

47 views

### Perturbation of the value of a general-sum game at a equilibirium

Consider a general-sum game with $N$ players. Let $u_i(a_1, \ldots, a_N)\colon \prod_{i=1}^N A_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes ...

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**1**answer

121 views

### Prenucleolus vs. nucleolus

I want to find a cooperative, characteristic function TU game $v$ (of at best of 3 or 4 players;2 players seem impossible to me) for which the prenucleolus is different from the nucleolus.
I do not ...

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**1**answer

80 views

### Can backward induction be used to solve any game? [closed]

I'm new to game theory and I would like to know, if you can model any game through a payoff tree, couldn't you find the subgame perfect equillibrium for all games through backward induction?

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167 views

### Evasive maneuver game

First of all, I want to state that I'm not an expert in the game theory and searching the references for the game I just made up. Solving this game by itself seems like a decent project for strong ...

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### Being sparse with information in an impartial game - is this game known?

You might known the thesis of Alfthan where he showed that in the game of Memory, your best move may contain turning a card that is known anyway.
I came up with a game to model the effect (denying ...

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61 views

### Extension of Standard English Peg Solitaire

An entire analysis of standard English Peg Solitaire has been given. See
Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (2001) [1981], Winning Ways for your Mathematical Plays (paperback) (2nd ed.), A ...

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54 views

### Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...

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108 views

### General way to find Nash equilibrium in continuous game

I'm really interesting how to find Nash equilibrium in a continuous game with two players in the general case.
Let's consider a game with continuous utility functions $F_1, F_2 : [0, 1] \times [0, 1]...

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88 views

### Pursuit-evasion with many slow pursuers

Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...

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### Reference Request: A Set-Valued Minimax Theorem?

Suppose that $\mathcal{C}$ and $\mathcal{D}$ are subsets of $L^2(X,\Sigma,\mu)\cap L^{\infty}(X,\Sigma,\mu)$, where $\mu$ is a finite-measure on $(X,\Sigma)$. Let $F:L^2(X,\Sigma,\mu)\times L^2(X,\...

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### What is the Bruss-Yor concept of no information?

A few years ago, a question related to a paper of Thomas Bruss and Marc Yor on the so-called last arrival problem received some attention on this forum.
What I'd like to know now is:
What are the ...

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**2**answers

2k views

### Guessing each other's coins

I recently thought about the following game (has it been considered before?).
Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an ...

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**1**answer

83 views

### Round robin with 5 player game [closed]

Im trying to write some software for pairing players in card gaming tournaments, each player must play on a table with each other player exactly 1 time, currently there is going to be 25 players with ...

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153 views

### Combining a Nim-variation and Wyrthoff's game. How to find a winning strategy? [closed]

Wythoff's game is a variation of the classical Nim - There are two heaps and the players take turns either taking any amount from one heap, or the same amount of both heaps. The winner is the one ...

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134 views

### final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...

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127 views

### A game theory problem mixed strategy over a continuous set

I have two players $A$ and $B$, the action of $A$ is $x_A\geq 0$ and the action of $B$ is $x_B\geq 0$. Let $c_0\in(0,1)$, $c_3>0$ and $c_2>c_1>0$ be constants. The payoff functions of $A$ and ...

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83 views

### Zero-sum games where getting information helps the opponent more

You may know of the paper on the "Memory" game - sometimes the best strategy is turning known cards (here: https://www.math.kth.se/xComb/x1.pdf). Here is a simpler toy example: You and your opponent ...

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136 views

### Satisfier-Falsifier games

In a Maker-Breaker game, there is a finite set of elements $X$, and a family $F$ of subsets of $X$ called the "winning sets". Two players, Maker and Breaker, take turns picking untaken elements from $...

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890 views

### What is the fairest order for stage-striking (and is it the Thue-Morse sequence)?

Here's a fair-sequencing problem that doesn't quite match the usual fair-division problems. I think that, like those, the answer should also be the Thue-Morse sequence ("balanced alternation"), ...

**3**

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**1**answer

73 views

### Optimal Strategies for a “Blind” Graph Coloring Game

By the "blind" graph coloring game I denote the following problem, which is played by two players:
player A has $k<n$ colors at hand to color the $n$ vertices of a graph $G$, but that player has ...

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**0**answers

96 views

### A community effort: equilibrium in quitting games [closed]

This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...

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vote

**1**answer

633 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 ...

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### Strong Nash Equilibria in repeated games

Suppose we have a simultaneous game, that has a strong Nash equilibrium (SNA), i.e. a weak Pareto efficient Nash equilibrium (no deviation of any subset of player brings a benefit to them).
Now ...

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**0**answers

111 views

### Piece rank probability in this Stratego-like game

There's this game in a 9x8 board where 2 players take turns moving pieces. The players have pieces ranked 1-21. Players can't see the opponent's pieces' ranks, just positions. Pieces landing on the ...

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503 views

### Generalization of Penney's game (Penney's paradox)

The standard Penney's game is played by two players. Player $A$ chooses a sequence of $k>2$ bits and $B$ (seeing $A$'s selection) chooses a different sequence of $k$ bits. A fair coin is flipped ...

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2k views

### Who first chose the names Alice and Bob for players A and B? [closed]

Who first chose the names Alice and Bob for the players (or observers) A and B?

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1k views

### Forcing and Family Contentions: Who wins the disputes?

The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, $V$, a model of $ZFC$. Just like many other couples they sometimes argue over a statement, $\sigma$, expressible ...

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### 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 ...

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1k views

### Cops, Robbers and Cardinals: The Infinite Manhunt

Cops & Robbers is a certain pursuit-evasion game between two players, Alice and Bob. Alice is in charge of the Justice Bureau, which controls one or more law enforcement officers, the cops. Bob ...

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**1**answer

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### On the limit of assessments that are not sequentially rational

I have asked this question in Mathematics StackExchange, but there is no response yet. I've just realized that here is the right forum for asking research level questions... :'(
In game theory, in ...

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246 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 ...

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**1**answer

70 views

### Effective way to find Nash equilibrium

Is there any good algorithm for finding Nash equilibrium point, for one and repeated game theory? Thansk a lot for giving me some guidance.

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3k views

### Alice and Bob playing on a circle

I want to solve this problem:
Let there be $n \ge 2$ points around a circle. Alice and Bob play a game on the circle. They take moves in turn with Alice beginning. At each move:
Alice takes ...

**19**

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**1**answer

624 views

### Who wins the Rubik's cube game?

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube (to make the problem easier).
Solver and Spoiler take turns making 90 degree twists (starting with Solver). ...

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**1**answer

159 views

### Algorithms for Fixing Sudokus

Suppose someone got stuck solving a Sudoku and asks you to figure out, what went wrong. Unfortunately that person only sends you a copy of the instance, where you neither see which of the numbers ...