Questions tagged [differential-games]
Differential Game Theory studies conflict in dynamical systems described by differential equations.
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Maximizing a function with binary indicators
I am an econ undergrad trying to understand how to maximize this payoff function, which includes binary components. I want to solve this equation using backwards induction, so I want to maximize the ...
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How will the wolf catch the sheep in minimum time?
In $\mathbb{R}^2$, a wolf is trying to catch two sheep. At time $0$ the wolf's at $(0,0)$ and the sheep are at $(1,0)$. The animals are moving continuously and react instantaneously according to each ...
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How can I solve following cooperative differential game?
Consider a game-theoretic model of pollution control. There are 2 players join in the game, N = {1, 2}. Each player has an industrial production site. It is assumed
that the production is proportional ...
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System of quadratic autonomous ODEs - convexity of the solution curve
Crossposted on MathOverflow
Problem:
For a given parameter $a>0$, consider the following autonomous system of ODEs for $(x,y,z): \mathbb R_+\to [0,1)^3$:
\begin{align*}
\dot{x}_t &= (1-x_t) (...
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Playing tag with infinitely many friends
All the countably infinitely many guests of Hilbert's Hotel decide to spend the day playing tag in the park. One player is the runner, and all the others are it. The taggers can agree on a strategy ...
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ODE equivalent to a system of Difference Equations (Discrete to Continuous time)
Consider the following gradient-descent ascent system of equations:
$$\begin{cases}
x_{k+1} = x_{k} - \eta \nabla_{x} g(x_{k}, y_{k}) \\
y_{k+1} = y_{k} + \eta \nabla_{y} g(x_{k+1}, y_{k})
\end{...
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Upper and Lower Games in Zero-Sum Games
I am working on some theory related to controls in the context of stochastic games, and I am a bit confused on some terminologies for zero-sum games.
Suppose we have a zero-sum game with two players.
...
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Do hybrid games exist?
I'm new to game theory. So far, I know that we have games with finite strategy sets and games with continuous strategy sets. I was wondering if there are any games in which some players have finite ...
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Lion and Rabbit in a Cage — references
I am looking for references to a problem with the following approximate statement:
There is a lion and a rabbit in cage $C$. At the initial time they are at locations $I_l$ and $I_r$, they have ...
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Can one solve a mean field game numerically with a finite number of players?
I am analyzing the following problem: given a set of players $x^i_t$
for $i=1,\dots,N$ satisfying the SDE
$$
dx^i_t = \alpha^i_t dt + \sigma dW^i_t
$$
where $W^i_t$ are independent Brownian Motions, ...
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Pursuit-evasion game with n pursuers and one evader
Assume $n$ pursuers ($P_i$) at the vertices of an $n$ sided regular polygon with
the evader ($E$) at the centre. For what all $n$ can be the evader be caught?
Pursuers and evader have same speed
...
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answer
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Meaningful connections between game theory and differential geometry
I'm a 3rd year undergrad in mathematics who has recently developed a burgeoning interest in differential geometry. I'm also quite interested in dynamical systems and game theory, both of which are ...
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Clarification of notation used in differential games
I'm working through Rufus Isaacs's work on differential games and I need clarification on the notation used. Some context: The Value of the game is to be the minmax of the payoff which symbolically is ...
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Pursuit Curve Modification
I've been stuck on this problem of Modified pursuit curve, in which the dog chases the cat with a constant acceleration $a$, starting from rest. The cat moves horizontally with a uniform speed of $v_0$...
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Unit of time and normalization of time preference rates
Consider an infinite horizon cake eating differential game described by
\begin{align}
&\max_{u_1(t)} \int_0^\infty{e^{-r_1 t}\ln(u_1(t))dt}\\
&\max_{u_2(t)} \int_0^\infty{e^{-r_2 t}\ln(u_2(t))...
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Bayesian Stackelberg game
Can anybody provide me a little example of bayesian stackelberg game with the solution.
I know how to solve Stackelberg game using backward induction but have no idea about bayesian.
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Game Theory Reccomendation, Mean Field Theory
I'm about to do a sort of reading course with a mathematics professor wherein I read and teach him about Game Theory. He claims not to know Game Theory. After that, we aim to read about Mean Field ...
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Zero-sum differential game
We consider a zero sum differnetial game.
Let $x \in (0, M] \subset \mathbb R_{++}$ denote the state and $(u,v) \in [0,x]$ the control of player 1 and 2 respectively with $u + v \leq x$.
Denote the ...
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Caracterization of optimal strategy in zero-sum 2 players differential stochastic game
Let a $(\Omega,\Sigma,(\mathcal{F_t})_{0\leq t\leq 1},P)$ a probability space where $\Omega$ is the space of continuous functions $f:[0, 1]\rightarrow \mathbb{R}^n$, $(\mathcal{F_t})_{0\leq t\leq 1}$ ...
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Nash Equilibrium in Normal game as a result of learning in Stochastic game
Lets have a Cournot oligopoly model. With two competitors, there is one Nash equilibrium. The thought experiment that can lead us to this result is for example the Tatonnement process, where there are ...
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Question about viscosity solutions to Hamilton-Jacobi-Isaacs equation。
Recently，I am studying differential game, especially two person zero sum differential game. And I am quite confused that why only the value functions defined in terms of non-anticipating strategies(...
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Introductory level text for differential games
I am interested in studying differential games by myself. An introductory textbook will be great. For introductory, I mean that the book shall have the definitions to concept and theorem (with proof) ...
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Fields of Interest in Game Theory for a Mathematics Dissertation
So In my final year of my Undergraduate Degree (Studying Mathematics and Economics) I have decided to focus my Undergraduate project on Game Theory.
I have done quite a bit of research, and I get ...
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Escaping from a circle of fat lions.
You are surrounded, by X fat lions equally spaced around a circle of radius 200 meters in an open field. While making your escape plan you note several things: they are slow, they can only travel at ...
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Four-Dogs Pursuit [closed]
Four dogs start at the corners of square $ABCD$ (labelled anti-clockwise). Running anti-clockwise, the dog starting at $A$ pursues the dog starting at $B$, which pursues the dog starting at $C$, which ...
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Which way should you run from the lions?
This is a fun problem that I saw somewhere on the internet a long time ago:
Suppose you are at the center of an equilateral triangle with side length $s$. At each of its vertices, there is a lion ...
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Does Tom catch Jerry?
Tom has Jerry backed against a wall. Tom is distance 1 away (perpendicularly). At time t=0, Jerry runs along the wall. Tom runs directly towards Jerry. Tom always runs directly towards Jerry. Tom and ...
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Run away from lions in a cage
I came across an interesting problem:
There is a round cage and you are in it. Also two lions are in this cage
too. The start position is that the distance between you and both lions is
the ...
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A lady and a monster
A famous problem:
A lady is in the center of the circular lake and a monster is on the boundary of the lake. The speed of the monster is $v_m$, and the speed of the swimming lady is $v_l$. The goal ...
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Chased by a lion and other pursuit-evasion problems
I am looking for a reference (book or article) that poses a problem that seems to be a classic, in that I've heard it posed many times, but that I've never seen written anywhere: that of the ...
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