Questions tagged [game-theory]

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27 votes
7 answers
6k views

Why is game theory formulated in terms of equilibrium instead of winning strategies?

Game theory, on the outset, seems to invite the questions, "what can I do to win" or "how do I beat my opponent?" So many people who are not familiar with game theory look to game ...
3 votes
1 answer
270 views

Game theory approach to Trans Europa

The other day I was playing a game called Trans Europa (or Trans America) which is quite graph theoretic in flavour. The game takes place on a triangular lattice graph with certain distinguished ...
-1 votes
1 answer
140 views

Why do two potentials of a game only differ by a constant? [closed]

Can someone explain to me the proof on page 7/20 of the original paper about potential games (https://www.cs.tau.ac.il/~mansour/sem-game-02-03/monderer-potential-96.pdf)? It is about why two ...
8 votes
1 answer
222 views

Name of a game : Remove two chips from a vertex or one chip from both ends of an edge

Consider a finite graph $\Gamma$ with a positive number $n_v\geq 0$ of chips stacked at each vertex $v$ of $\Gamma$. Two players play in turn with moves consisting either of removing two chips from a ...
4 votes
1 answer
1k views

Who wins this two player game of making squares?

Two players take turns coloring edges on an $n$-by-$n$ grid. Both players use the same color. Every time a player surrounds a square of the grid, they mark that square with their name and go again. ...
24 votes
9 answers
25k views

Looking for a mathematically rigorous introduction to game theory

I am looking for the best book that contains a mathematically rigorous introduction to game theory. I am a group theorist who has taken a recent interest in game theory, but I'm not sure of the ...
5 votes
1 answer
501 views

Can we determine the game-theoretically best first move by White in chess without solving chess?

In turn-based board games with high branching factor (such as chess) are there any arguments that could ascertain the ideal first move but not solve the entire game? I am asking because solving chess ...
6 votes
1 answer
231 views

Ordering preference for two zero mean Gaussian outcomes

Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...
-1 votes
1 answer
319 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) ...
1 vote
1 answer
273 views

Is there an equilibrium for this non-zero-sum game?

The game $G(N,M)$ is played: $N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$....
5 votes
2 answers
261 views

Is there a dominant strategy for this game?

Alice and Bob have $N_A$ and $N_B$ warriors under their command, numbered $1$~$N_A$ and $1$~$N_B$ respectively. Alice has $1$ fighting power at her disposal, and Bob has $b$ ($b\gt 0$). Before the ...
2 votes
0 answers
120 views

Existence and uniqueness of solution of a nonlinear system

I need a proof of the following result to calculate a Nash equilibrium in the Showcase Showdown game. For all $n>1$, the system of equations $$\left\{ \begin{aligned} (1+e^{x}(-1+x))^{n-2}&=\...
45 votes
2 answers
3k views

Can the fugitive escape?

A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively. In a fugitive move, the fugitive can travel no more than ...
1 vote
1 answer
277 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 ...
2 votes
1 answer
128 views

special case of the minimax theorem

The minimax theorem of von Neumann says that for any payoff matrix $A$, we have \begin{equation} \max_x \min_y x^T A y = \min_y \max_x x^T A y. \end{equation} In the above, $x$ and $y$ are probability ...
1 vote
2 answers
240 views

Should mixed strategies in normal form games be interpreted as measurable functions or probability vectors?

I have recently been stuck trying to understand how game theorists extend a normal form game (matrix game) into a game with mixed strategies (so called mixed extension). I feel like I am missing ...
1 vote
2 answers
122 views

Variant of Parthasarathy's minimax theorem

Does there exist a variant of Parthasarathy's minimax theorem [1] that relaxes the assumption that the spaces $X$ and $Y$ are $[0,1]$? [1] https://en.wikipedia.org/wiki/Parthasarathy%27s_theorem
86 votes
27 answers
19k views

Which popular games have been studied mathematically?

I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...
15 votes
0 answers
523 views

Identification of a curious function

The following question was asked on MSE but there were no replies. During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\...
27 votes
1 answer
1k views

Players alternate moving a $\{\swarrow,\uparrow,\rightarrow\}$ piece on a chessboard

Edit $4.$ $-$ Proposing to reopen the question (the related competition should be over by now). Edit $3.$ $-$ I have just found out that the linked competition (see the "Edit $1$.") is still ...
22 votes
5 answers
3k views

Irreversible chess

Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ...
2 votes
0 answers
161 views

Finding an optimal strategy for a combinatorial sequential game

We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...
16 votes
5 answers
4k views

Is a fair lottery possible?

I'm trying to come up with a scheme for a lottery where each individual has roughly the same chance of becoming the winner, regardless of the number of tickets one holds. So no individual should have ...
7 votes
2 answers
1k views

Mathematics of GANs (generative adversarial networks)

Generative Adversarial Networks were introduced in http://papers.nips.cc/paper/5423-generative-adversarial-nets and has more than 20000 citations. The paper introduced key paradigm changes which ...
1 vote
0 answers
133 views

Nim variant with minimum number of objects?

I'm wondering where I can find in the literature (if it exists) a discussion of a Nim variant where we impose the additional condition on Nim that we can remove only up to $c$ objects before the game ...
3 votes
2 answers
493 views

How to prove that $a + b + 4 \sqrt{1 + a^{2} + b^{2}} \leq 4 \sqrt{a^{2} + b^{2}} + \sqrt{1+b^{2}} + \sqrt{1+a^{2}} + 2 $ for all $a, b > 0$?

I'd like to prove the following: $$ a + b + 4 \sqrt{1 + a^{2} + b^{2}} \leq 4 \sqrt{a^{2} + b^{2}} + \sqrt{1+b^{2}} + \sqrt{1+a^{2}} + 2 $$ for all $a, b \in \mathbb{R}_{>0}$. Question: is ...
3 votes
2 answers
200 views

A "Markov game"

I call games similar to the one I describe below to be Markov games. I am selecting just that one or rather a 1-parameter series of games. The open challenge is to find out which of the players $\ 0\ $...
8 votes
2 answers
372 views

A game of singletons

Alice and Bob play the following zero-sum game, parametrized by two integers $m$ and $k$: Alice picks $m$ sets, each of which has $k$ items. Bob colors some items in green. Bob's score is the number ...
12 votes
1 answer
356 views

An averaging game on finite multisets of integers

The following procedure is a variant of one suggested by Patrek Ragnarsson (age 10). Let $M$ be a finite multiset of integers. A move consists of choosing two elements $a\neq b$ of $M$ of the same ...
7 votes
2 answers
965 views

Drawing lines and removing squares - an Alice and Bob game

Thought about the following while in a Complex Analysis lecture: Let there be a $N \times N$ grid of squares and two players $A$ and $B$. First, $A$ needs to draw a line $l$ that needs to intersect ...
4 votes
1 answer
180 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 ...
1 vote
1 answer
140 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 ...
6 votes
1 answer
132 views

Does is it change an auction's incentives when causing the winner to pay more makes losers pay less?

Say there are three roommates moving into an apartment with three rooms. Two of the apartment's rooms are identical, but the third one is valued higher by all three parties (say it's bigger and has a ...
15 votes
4 answers
803 views

Fair cake-cutting between groups

The cake-cutting game is usually played between individuals. What if we try to play it between groups? A certain land has to be divided between two states. ‎There are $n$ citizens in each state. ...
1 vote
1 answer
235 views

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 ...
1 vote
2 answers
48 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_{...
9 votes
0 answers
218 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}...
5 votes
1 answer
404 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 ...
2 votes
1 answer
605 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 ...
27 votes
1 answer
979 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 ...
0 votes
2 answers
172 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 ...
2 votes
1 answer
86 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 ...
1 vote
0 answers
40 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 ...
1 vote
0 answers
49 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 votes
1 answer
230 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: ...
74 votes
11 answers
26k views

Does War have infinite expected length?

My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers. The ...
1 vote
1 answer
153 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 ...
0 votes
1 answer
371 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 ...
1 vote
1 answer
495 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?
3 votes
1 answer
229 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 ...