The study of competitive and non-competitive games, equilibrium concepts such as Nash equilibrium, and related subjects. Combinatorial games such as Nim are under (combinatorial-game-theory), and algorithmic aspects (e.g. auctions) are under (algorithmic-game-theory).

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21 views

Optimal strategy in a bidding game

Note: I do not expect a clean closed-form solution to this, and would be very surprised if one existed, but I figured I'd ask to see what ideas other people had. There is a \$100 bill up for auction. ...
0
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1answer
30 views

Dynamic game of incomplete information

Consider a 2-player game: You and a robber. The robber tells You to give him all your money, otherwise he will kill You. However, the robber could be a 'Good' person (i.e. he would not kill You ...
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1answer
29 views

Dynamic programming approach for multidimensional problem

I use a dynamic programming approach to optimize the behaviour of individuals playing a game.I have one strategy matrix that describes the behaviour of individuals in situation 1, which depends on ...
4
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1answer
67 views

Knight movement on chess field

I had this task in programming competition: There are two knights, which are $(p_1,q_1)$ and $(p_2, q_2)$. $(p,q)$ knight is figure, with p(q)-length first step, and q(p)-length second step in ...
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0answers
23 views

Support lemma - Game theory

Let α be $a$ mixed strategy profile, $a_i ∈ supp(\alpha _i), a_i \notin B_i(\alpha _{−i}), a_i' ∈ B_i(\alpha _{−i})$ and $a_i'$ defined by $\alpha_i'(a_i)=0$, $\alpha_i'(a_i')=\alpha _i ...
2
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1answer
39 views

Is my answer correct? (Devious auction game)

(Taken from here) The question was A man is auctioning a real $20\$$ bill. There are a vast number of bidders. A person may make as many bids as he wants. The starting bid is $5\$$. No $2$ ...
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2answers
31 views

Game Theory: Can someone explain the notation used in the definition of “best response”

I am reading a paper which states that that the best response correspondence of a player is mapping: $B_i(s_{-i}): S_{-i} \Rightarrow S_i$ such that $B_i(s_{-i}) \in arg\ max_{s_i \in S_i} ...
1
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1answer
29 views

Nash equlibrium game theory

Given the following game: Find nash equilibrium (NE) Find subgame perfect nash equilibrium. Main problem i have is with converting this to normal form of the game (because this is i think ...
0
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0answers
21 views

A min-max problem and convex optimization problem.

Let $x^*$ a solution of the convex programming problem $$ \begin{array}{rl} \max & f_0(x)&\\ \mbox{s.t.} & g(x)\leq 0 \end{array} $$ where $f_0:\mathbb{R}^n\to \mathbb{R}$ and the ...
-1
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1answer
35 views

Mixed strategy nash equilbrium

In a mixed strategy Nash equilibrium it is always the case that: a) for each player, each pure strategy that is played with negative probability yields the same expected payoff as the equilibrium ...
0
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0answers
13 views

How to calculate effect of different variables/parameters on a quantity?

I am developing a game for iOS. In the game I have around eight different parameters that directly affect the score of the player. We can say that these eight variables decide the difficulty of the ...
7
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0answers
52 views

Diagonal-free Sudoku grid

I have a Sudoku grid with the property that diagonally adjacent elements are distinct (it is also a torus under the same property). The grid offers new and exciting logical possibilites. My question ...
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0answers
8 views

Prove that a directed tree does not have a path from a descendant to its parent

Prove: Let T = (V, E) be a directed tree. If v is a vertex of V and u is a descendant of v, then there is no path from u to v. My idea is that if u is a descendant of v, then there exist a path from ...
1
vote
1answer
42 views

What is the name of this kind of games?

In game theory, suppose we have a set of players $\mathcal{N}=\{1, 2, \ldots, n\}$, a set of actions $\mathcal{A}_i$ of player $i\in\mathcal{N}$, and a payoff function $u_i$ of player ...
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0answers
67 views

The best strategy to increase StackExchange Reputation [closed]

I do not have a lot of background in game theory, but I am curious how would one formally pose the title problem and mathematically describe possible strategies. Are the problems of this type best ...
0
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1answer
44 views

binomial distributions and their transforming (6.37-6.39)

I'm lost and frustrated. I don't know how the author (Karl Sigmund; The Calculus of Selfishness) transforms 6.37 in the book pages imaged below: $$ P_y = \sigma w^{N-1} + ...
0
votes
1answer
40 views

Game with matches. Very interesting mathematical problem.

Suppose you have a set of matches. You arrange them in 9 rows such that the first row has one match the second two matches the third three and so on until the ninth row which has nine matches. There ...
2
votes
1answer
46 views

Subgame perfect Nash equilibrium & perfect Bayesian Nash equilibrium - Game theory

For a week or so I have been struggling with the topics around the concept of subgame perfect Nash equilibrium (SPNE) and the perfect Bayesian Nash equilibrium (BNE). Namely: Is it possible to apply ...
2
votes
1answer
72 views

move a cuboid from start to destination

Consider the field with infinitely many boxes, "S" means start, "D" destination, and I already found a way to move a $1\times 2\times 4$-cuboid (as you can see on the right at this picture) from the ...
0
votes
1answer
71 views

Gambling to pay off debt?

Someone told me something interesting today. They said they were going to take their bonus check from work, to the casino because they have "better odds" of paying off more debt then if they would ...
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0answers
20 views

A minimum settlement for a bargaining problem

Question: Alpha and Beta are 2 companies. Now Alpha thinks that Beta has violated Alpha's trademark. Beta denies that. Now, Alpha is threatening to go to the court and claim 5,000,000 EUR from Beta ...
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1answer
66 views

Brouwer's fixed point continuous function

Can anyone point me out the continuous functions without brouwer fixed point's for the following sets $$A = \{x \in \mathbb{R}^2 | x_1,x_2 \geq 0 \text{ and }x_1^2+x_2^2 = 1 \}$$ $$B = \{x \in ...
0
votes
1answer
51 views

Game theory question- no idea how to proceed

A monopolist sells two products, X and Y . There are three consumers with asymmetric preferences. Each consumer buys either one unit of a product or does not buy the product at all. The per-unit ...
0
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0answers
20 views

Mixed Strategy Nash Equilibrium in this game?

L (q) R (1-q) l (p) [(2, 1), (0, 1)] r (1-p) [(-1, 0), (1,7)] I'm having a lot of trouble understanding what the mixed strategy nash equilibrium is ...
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1answer
16 views

Algorithms for computing Nash equilibria

Excuse me, since I am modeling a situation into a nonzero-sum n-player non-cooperative game. I wonder if there is any algorithm for computing its Nash equilibria?
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1answer
42 views

Game theory question- boxes

There are two players 1 and 2, and the game begins with player 1 selecting one of the boxes marked 1 to 16. Following such a selection, the selected box, as well as all boxes in the square of which ...
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1answer
31 views

How much is it worth to participate in a second price auction?

You have a valuation for an object (say $v_a$), which you don't know yet but you know is distributed U[0,1]. You will be competing in a second price auction against a completely identical guy as you, ...
0
votes
1answer
34 views

What is a two person constant sum game?

I read that a two-person constant-sum game is a two-player game in which, for any choice of both players strategies, the row player's reward and the column player's reward add up to a constant value ...
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3answers
39 views

Looking for the function of x for a line that approaches, but never reaches 100

I'm looking for the function of x for a line that intersects at (0,0) and (100,80), and as x goes off into infinity, the line approaches, but never touches 100. See image attached. I am writing a ...
1
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1answer
23 views

A proof related to diameter of a simplex S

Question: Prove that the diameter $\mathcal p(S)$ of a simplex $\mathcal S$ equals the greatest Eucledian distance between two vectors in the simplex. My opinion: We all know what every vector in the ...
0
votes
1answer
29 views

Mixed strategy Nash Equilibrium

How do I solve this problem by using mixed strategy Nash equilibrium? \begin{pmatrix} (2,0)& (1,1)&(4,2)\\ (3,4)&(1,2)&(2,3)\\ (1,3)&(0,2)&(3,0) \end{pmatrix} I tried to ...
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vote
0answers
23 views

Has two player five-card draw poker been solved?

I know that some other types of Poker have recently been solved with computers but has five-card draw poker been solved and if so, is there any place for mathematical analysses in the game? I need to ...
2
votes
0answers
51 views

Cournot competition: profit maximizer vs. market share maximizer

Today during an informal conversation with an established business researcher, I learned such a fact: In the classical Cournot competition model, if one player is a profit-maximizer, the other ...
4
votes
1answer
82 views

Do you need true randomness to beat the two-envelope game?

A well-known (non-)paradox in probability involves a two-envelope game played between two players, $A$ and $B$: $A$ selects two distinct (real) numbers, $x$ and $y$, writing each one down on a card ...
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1answer
27 views

Probability of number of drawing cards in a scenario being equal to that in another scenario

I came across the following question in a book:- $Q.$ Cards are drawn one by one at random from a well shuffled pack of $52$ cards. $(a)$Find the probability that exactly $n$ cards are drawn before ...
2
votes
0answers
29 views

Finding the core of a coalition game

I need to find the core of a 3-player coalition game graphically, given that $v(\phi)=0$, $v(1) = 9, v(2)=8, v(3) = 9, v({1,2}) = 14, v({1,3})=15, v({2,3}) = 13, v({1,2,3}) = 21$ So I'm following the ...
4
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0answers
62 views

Wizard against two dwarfs: guess the whole function

An evil wizard plays the following game with two dwarfs $A$ and $B$: he thinks of a function $f:\mathbb{R}\to\mathbb{R}$ (which is not required to have any regularity properties, such as ...
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0answers
22 views

Unfair coin tossing game,target,optimal fixed investment,

Suppose the player has capital 1\$. He chooses a number $f\in[0,1]$.He tosses an unfair coin repeatedly, which wins for him, with probability $p$, a gain $q\times f \times$ current capital \$,where ...
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0answers
26 views

Finding the expected revenue of a first price auction

I'm currently trying to solve for the expected revenue of a first price auction involving n players who draw their values v independently from F with support $[\underline{v}, \bar{v} ]$ and positive ...
0
votes
1answer
17 views

Example of a matrix which is copositive plus but not PSD.

This came up in our game theory course. While doing the Lemke's algorithm for solving LP, it was said that the process terminates when the matrix $M$ is copositive plus. Now copositive plus has a ...
1
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1answer
41 views

game theory question

For a non-cooperative bimatrix game the feasible set is $$\{(u,v)=(\mathbf{p}^TA \mathbf{q},\mathbf{p}^TB \mathbf{q}):p \in X^*, q \in Y^*\}$$ graph the non-cooperative feasible set for the Battle ...
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0answers
36 views

Forming a differential equation from game

I was wondering if someone could help me form a differential equation from the following game: A population consists of two types of diets, fish and veg. People play a with every other person and the ...
1
vote
1answer
27 views

show no equilibrium pairs exist in a non cooperative game using pay-off set?

I am trying to understand the following exersice from the solutions of my professor and I really don't understand what she is doing. The exersice is the following: Suppose the matrix below is a pay ...
1
vote
0answers
24 views

calculating expected gains.

A game costs \$100 to play. Toss a coin repeatedly, and win \$1 if you get heads for the first time, \$2 if you get heads both of the first two times, \$4 all of the first three times, \$8, and so ...
3
votes
1answer
59 views

Why “Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong” is paradoxical?

In a paper(see here) by Adam Brandenburger and H. Jerome Keisler, they give a game-theoretic impossibility theorem akin to Russell’s Paradox: Ann believes that Bob assumes that Ann believes that ...
0
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1answer
71 views

I can't find the Nash equilibrium of this 3x2 game.

Sorry for my English, I am French but i couldn't find help on the French website (so I am here). I have a question about this two-player game: $$ \begin{array}{c|cc} & y_1 & y_2 \\ \hline ...
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0answers
27 views

Quasiconcavity of $g(x)=xf(K-x)$

The function $f(x)$ is strictly increasing, finite, positive and twice continuously differentiable on the compact interval $[0,K]$, and $f(0)=0$. I'm trying to either find a counterexample to, or a ...
5
votes
0answers
80 views

Fastest way to meet, without communication, in a toroidal palace?

I was interested by a similar question asked here, but wanted to pose a slightly different variant that avoids some of the pitfalls and ambiguities in the first question in order to ask something more ...
2
votes
0answers
30 views

Cournot Oligopoly in Bayesian Game Theory

I have this Cournot game in which $n$ firms produce quantities $q_1, \ldots, q_n$ with respective marginal costs $c_1, \ldots, c_n$. They all sell at price $P=1-(q_1 + \cdots + q_n)$. For any $i$ ...
0
votes
0answers
38 views

Nash equilibrium in mixed strategies with p = 0

I am currently writing a program to calculate nash equilibria in mixed strategies. My algorithm simply tries a lot of different probabilites and then decides which one is the best. However I came ...