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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Matrix Saddle Points and Dominance

I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it can be used in games. I wanted to know how to prove a problem like ...
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1answer
35 views

Computing a revenue for VCG auction

I would like your help with the following question regarding computing a revenue for a seller of an VCG (vickrey clarke groves) auction, I'm really new to this auctions\game theory so I'd really ...
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3answers
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Definitions of noncooperative and cooperative games.

These days I have read many descriptions of a noncooperative game like the one below. A noncooperative game is a game in which players are unable to make enforceable contracts outside of the ...
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4answers
86 views

How does one explain basic probability theory to a layman?

I have recently been involved in a number of discussions with people with little or no background in mathematics when we considered a problem of the following shape. A random event is going to ...
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1answer
53 views

Reverse Hex board game winning strategy

I just wanted to know the winning strategy to this question: In a reverse Hex board game I know it means where the player who first forms a path between his/her edges loses. Find a winning ...
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35 views

Game theory problem… I think…

I need some help with the following: Let's say I'm running a store of electronic devices (call it Store $A$). and let's say that right next to me, there's another electronic devices store (store $B$) ...
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1answer
37 views

Game theory - Pure ESS test

Let $A \in \mathbb{R}^n$ describe a symmetric game with $n$ strategies. For the sake of clarity, I call symmetric game a two-player game where payoff matrices are the same for both players. Suppose ...
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4answers
168 views

How do you create a nonlinear game that the player can always win?

I thought a lot about this question — and initially, I intended to ask this on gamedev.stackexchange.com — but due to its rather theoretical aspects, I think it might be more appropriate to address a ...
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3answers
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Mathematical research of Pokémon

In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some ...
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1answer
78 views

Guessing a number among K

Consider two players $a$ and $b$. Player $a$ moves first and picks a number $n\in\{0,1,2,...,K\}$. Then moves player $b$ who guesses at the number picked by $a$. If the guess is correct, $b$ wins a ...
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1answer
52 views

Winning or Non-losing strategy for A or B

Find a winning or a non-losing strategy for the following game: Consider $25$ sticks arranged in a $5$ x $5$ square. Players alternately take any number of sticks from a single row or column. At ...
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2answers
87 views

Hex game winning strategy

I was teaching myself how to play a hex board game by reading some books a couple days ago. I learned how to do $2$ x $2$ and $3$ x $3$ hex games by starting at the principal diagonal. I wanted to ...
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2answers
32 views

Why you randomize your opponents payoff in a mixed nash equilibrium?

I wanted to understand the justification more intuitively -- if that is possible. For example, I'm in a abstract game with another opponent and there is no pure strategy equilibrium: why do I ...
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2answers
36 views

Finding optimal mixed strategy

I have to find an optimal mixed strategy for the 'column' player, who mixes with the probabilites $q_1,q_2,q_3$. What is known is the optimal mixing of the 'row" player. The game is a zero-sum game, ...
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0answers
37 views

What is the optimal strategy?

There are $m+n+1$ cards numbered $1,2,\ldots m+n+1$. Participants A and B respectively get $m$ and $n$ cards. Meanwhile, they only know what they get. The remaining card is face down on the desk. ...
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1answer
33 views

The potential function of Prisoner's Dilemma

As in the famous example of "Prisoner's Dilemma" like this If the potential function is defined as: (V(q,q), V(q,c), V(c,q), V(c,c)) q = quiet, c = confess, V is the potential. So should the order ...
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1answer
36 views

Chase game with doubling cube

Consider the following 2-player game: Each player has some score. Taking turns, each player gets added to his score a uniform random on (0,1). If after this addition that player is ahead by at least ...
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0answers
70 views

Application of Markov Chain to Game of Life Board Game

I need to calculate the expected outcomes for the Game of Life. I believe that if I multiply the probability of landing on a particular square with the payoff of said square and add up all these ...
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1answer
28 views

Finding Nash Equilibria for this Bimatrix Game

Consider the following Bimatrix Game a b c d -----|------------------------------------------- T (1,4) ( 4,3) (0,2) (1,0) B ...
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1answer
24 views

Decide the Nash Equilibrium

If two people collaborate on a work: the (3,3) means if neither of them do any work, then, they have to be put on detention for 1 hour and then both of them still have to finish the same 2-hour work ...
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Can Nash Equilibrium be more than two?

In the Prisoner's Dilemma example, we know that there is only one Nash Equilibrium. That is both of them confess. Is it possible that there are two Nash equilibrium in one example? Can you roughly ...
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1answer
33 views

An algorithm to rate players in team?

I would like to design an algorithm to rate players in a team sport. One team of N players plays a match against another team of N players. The individual players will possibly change, from match to ...
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0answers
74 views

What is the lowest possible score in $2048$, $4\times4$ and $5\times5$ board if the computer plays to try to make you win the game?

What is the lowest possible score in $2048$, $4\times4$ and $5\times5$ board if the computer plays such that it wants you to win the game? I got to the $32$ tile in a $5\times5$ board now with a low ...
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36 views

Two person zero sum problem, help/guidance needed..

I'm a computer science student and I have this problem I need to solve for my games theory course. I don't have an example to follow, or use as guidance, and my colleagues are not very helpful( as in, ...
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5answers
128 views

game theoretic die rolls

Suppose player X has a 6 sided die and player Y has a 10 sided die. They each get two rolls and they can each choose to stop rolling on either one of the rolls, taking the number on that roll. Whoever ...
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1answer
54 views

Finding pure strategy and pay off matrix in game theory

"A two person games begins with the random selection of an integer $x$ from the set {$1,2,3$}, each choice is equally likely. Then the two players, not knowing the value of $x$, simultaneously select ...
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de Bono's L-game modification

I am trying to find out if a simple modification od de Bono's L-game is still infinite if two players are perfect. Modified rule is that there no neutral pieces but, there is one piece for each player ...
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2answers
430 views

simple games with cute winning strategies?

Im thinking of games of two players ($A$ goes first and $B$ second) like the following: There are 35 chips in a table, during each turn a player can remove 1,2,3 or 4 chips. Prove player $B$ can ...
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13 views

Dual proof in Zero sum games

Say Player 2 is $P: max \sum_1^n x_j $ subject to $Ax \leq b$. I believe the Dual problem for this would be $D: min \sum_1^n y_j$ subject to $A^T y \geq 1$. Player 1's problem would be $max \{min ...
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284 views

What is the highest possible score in 2048 hard?

There is a variant of the popular game 2048, called 2048 hard or 2048 impossible, which automatically places each new tile in the hardest possible location. Is this variation possible to solve, and if ...
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2answers
66 views

Alice and Bob card game

I came across this puzzle online in an online Princeton thingy (course?): Alice writes down two integers between 0 and 100 on two cards. Bob gets to select one of the two cards and see its value. ...
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3answers
679 views

Optimal Strategy for Rock Paper Scissors with different rewards

Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward. If you win with Rock, you get \$9. Your opponent loses the \$9. If you win with Paper, you get \$3. ...
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41 views

need help with zero sum game

Tom chooses an integer in {1,2,3} and Bob chooses an integer in {2,3,4}. If the chosen numbers are the same, no money changes hands If the numbers are different the person who picks the bigger number ...
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1answer
68 views

Solving a 3x3 payoff matrix

I need some help solving the value of this payoff matrix and finding the optimal strategy: $$ \begin{matrix} 1 & 2 & 4 \\ -1 & 5 & 3 \\ 3 & 3 & ...
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1answer
36 views

Show why answer no longer holds when inequality changed

This below is a Nash equilibrium problem, I'm stuck in the math part. I solved the first part but I'm confused on the second one. I believe there is a mistake on denominator and it should be ...
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2answers
89 views

Banach-Mazur Game: Proof about winning strategies

I have to hold a presentation about the Banach-Mazur-Game to undergraduates this week. It should all stay very simple, so I will mainly only talk about the "original" Banach-Mazur Game on ...
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61 views

Guess the number of a liar

Sue picks a number from 0 to 3. Tom asks questions about the number, with yes/no answers. For example, "Is it odd" or "Is it 3?" If Sue picked X, she is allowed to lie at most X times. For ...
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2answers
32 views

Game Theory - Setting Up Column Player's Optimal Stategy

Above is my question. Could someone please help me with the first part? I should be ok once I have set up the linear programming problem, but I don't even know what $x_1, x_2 \ \text{and} \ x_3$ are ...
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24 views

Finding mixed nash equilibria

I understand how to find equilibria in two player games. I don't understand how to find mixed Nash equilibria in Load balancing games? For example, games which involve a specific number of machines ...
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24 views

Value of Zero-Sum Game

My question is the following: Let $A \in Mat_{n}(\Bbb R)$ be the payoff matrix for player 1 in a 2-player zero-sum game, and suppose that $A$ is invertible, symmetric and such that $A^{-1} e \ge ...
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99 views

If a game ends after finite number of moves, does it mean that at least one of the players has winning strategy?

Let us consider a game played by two players and if the game reaches some of the ending positions, one of the players is declared a winner. Let us assume that the game has to end after finitely many ...
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1answer
109 views

Optimal strategy for dominoes game

Here is the basic principle of the game I'm trying to find an optimal strategy for: Two players (say P and Q) are given a 2x3 grid and a domino. Then P chooses one way of positioning the domino on ...
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2answers
70 views

Third and average price auction

Third price auction: the winner is the highst bidder but this time instead of paying the second highst bid, he would pay the third highst bid. -assume there are at least 3 bidders. - Average price ...
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1answer
49 views

Prove the assertion on the game theory.

if a dominant strategy for player1 is added to finite normal form game then the payoff to player1 at any equlibrium of the new game must be at least as great as any nash equlibrium payoff ...
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0answers
30 views

How do we determine the saddle point in game theory?

I am a little confused of why this works. If a$_i$$_j$ is positive, row player pays the column, and vice versa. \begin{bmatrix} 3 & -5 & 6\\ -2 & 1 & 8 \\ 3 & -6 & ...
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1answer
57 views

Is it possible to prove that this game is always winnable?

I was at my lunch table today and was trying to come up with a card game, and here is what I came up with: Let there be a standard deck of $52$ cards called $\mathbb{D}$, with four suits: spades, ...
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34 views

A question on game theory strategies. Private/public info

I'm currently writing my thesis in econ and have encountered a bit of game theory which im not too well acquainted with. The problem is as follows: Suppose there are two players, In the first round ...
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Matrix multiplication in game theory doesn't add up? Min y^T*Ax

I'm studying game theory and something seems weird to me. My book says y is the probability of the row player and x is the probability of column player, both x and y are vectors. A = [a$_i$$_j$] is ...
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2answers
126 views

Puzzle with pirates

That one I'm pretty low on ideas of how to approach it. Five pirates of different ages have a treasure of 50 gold coins. On their ship, they decide to split the coins using this scheme: The oldest ...
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0answers
125 views

What is the optimal strategy for this 2 player game?

Let some finite array of integers is given initially. There is a number k which is initially '0'. In a move, a player will select a number from the array arr[i] and change k to gcd(k,arr[i]). Also, ...