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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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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127 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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85 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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52 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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38 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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39 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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154 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
206 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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113 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
53 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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35 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
66 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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40 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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0answers
38 views

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
208 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
134 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, ...
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274 views

Optimal strategy for Jackpot Rock Paper Scissors

Jackpot Rock Paper Scissors is a gambling variant of Rock Paper Scissors, wherein ties result in the wager being carried forward into a jackpot. If a player plays the same hand (rock, paper or ...
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1answer
86 views

Game Theory in relation to economics and sociology [closed]

I know some algebra and calculus, and have been reading about Linear Programming/Game Theory. How are the models in this field, even the infinite calculus models, usable in macro economics. Even ...
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425 views

Game of cards and GCD

Alice and Bob play the game. The rules are as follows: Initially, there are n cards on the table, each card has a positive integer written on it. At the beginning Alice writes down the number 0 on ...
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1answer
40 views

Finding Nash Equilibriums

Nash Equilibrium for this Normal Game 1,1 2,4 1,4 0,8 1,1 1,1 3,0 0,0 7,0 I know ...
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48 views

How to calculate the mixed Nash Equilibrium in the matching pennies game?

The matching pennies game is the following: \begin{array}{|c|c|c|c|} \hline Player1\backslash Player2 & H & T \\ \hline H & (\color{red}{+1}, -1) & (-1, \color{blue}{+1}) \\ \hline T ...
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1answer
63 views

Spinner game outcomes: What has the best chance of winning with your last two coins?

We are discussing probability and odds in my elementary math class. The students came up with two scenarios. They are as follows... In a spinner wheel game based on the days of the week, students bet ...
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57 views

Is the expected utility function linear?

Given the definition of the mixed extension of a finite game as in the link below (only first 7 lines): How to find perfect equilibria in a finite game? We define the expected utility function in the ...
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40 views

How to find perfect equilibria in a finite game?

If we define a game with $n$ persons as below: (i) for each player $i$, he has his strategy set $S_i$, $|S_i|=m_i<\infty$, and denote $S=\Pi_iS_i$; (ii) $u_i:S\rightarrow\mathbb{R}$ is a payoff ...
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44 views

Show a simple strategy.

Imagine that we have 49 cards with the values written on their faces, (they are all visible ) as follows; $$25, 24, 23, 22, ........3, 2, 1, 2, 3, .........23, 24, 25$$ suppose Paola and Victor are ...
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39 views

Help me writing Payoff matrix.

I guess, in order to answer this question, I need to write Payoff matrix. But I cannot write it. And then, I Will able to answer this question by myself. Thank you for helping. (These are just ...
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1answer
62 views

Check my answers: Dominant strategy.

I saw another question on Game theory. My answer for part a the nash equlibria (T, L) and (B,R). for part-b, Player-1's action T is strictly diominated. So Player1 never choose T. For part ...
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1answer
85 views

A question on Game theory

I'm studying Game theory, I saw the question: Consider two players; player A and player B playing the following estimation game. Each player chooses a number from {1, 2, 3}. If the difference ...
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334 views

The Last Man Standing

This is my second question following this post. Three players are playing a game. They all have small amounts of money, let say: player 1 has $\$a$, player 2 has $\$b$, and player 3 has $\$c$, ...
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30 views

state graph and MEx position

I am in a bit trouble. My midterm exam is coming and I do not understand state graphs and MEx positions. I am confused about this concept. Here is an example I give you. Please can you explain to me ...
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35 views

Game theory:Baye's rule for tournament

I'm having challenge with the following computations from the book I'm using. How are the steps obtain from the preceding step? In the expressions below, $E_i$ and $E_j$ are independent, random ...
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3answers
1k views

Optimal strategy for the Rope Climbing Game

Define a two-player, turn-based climbing game as follows. Each turn, players have the option to climb or tie a knot at his current position. If the player chooses to climb, there is a 50% chance ...
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74 views

Game Theory and Uniform Distribution question?

In an Auction , two players are bidding. Their bids will be a unknown fraction of their valuations. The valuations come from a uniform distribution $$[0,1] $$ If Player 2 bids $$ v/2 $$ and Player ...
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30 views

Joint Game Theory?

I am confused. A Game is, generally, defined by: $\mathcal{G}=(\mathcal{P}, \mathcal{A}, \mathcal{U})$ where $\mathcal{P}$ is the set of players, $\mathcal{A}$ is the set of actions $\mathcal{U}$ is ...
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71 views

Probability Theory $\Rightarrow$ Game Theory?

It is a very simple question. I would like to learn Game Theory but I am not that good at Probability Theory. I would like to know it is necessary to be good at probability theory in order to learn ...
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1answer
62 views

puzzle on [13,10,3] perfect Hamming code over $\mathbb F_{3}$

The soccer betting form contains a list of 13 games. There are three possible outcomes for each game: “the first team won”, “the second team won” and “draw”. Each betting form allows to chose one ...
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1answer
108 views

Unbalanced game: probability of winning over an infinite number of possible match sequences

We have 2 players, A and B, competing. The probability that A wins a match is p, making the probability that B wins a match (1-p) = q. The game is won by player A as soon as he gets one more win than ...
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1answer
42 views

How to solve this problem? Distributed Game theory?

I have this problem: We dispose of some resources, say $\{f_1, f_2, \dotsc, f_m\}$; We have some agents or players, say $\{\mathrm{p}_1, \mathrm{p}_2, \dotsc, \mathrm{p}_n\}$; Every player has some ...
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285 views

How practically relevant is game theory?

I usually don't care too much about the practical relevance of nice mathematics :-) But this time, as I am looking to find some areas where I can apply maths and possibly collaborate with ...
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1answer
37 views

Why all games are not Potential?

A definition given in wikipedia of an exact potential game as follow: A game $G=(N,A=A_{1}\times\ldots\times A_{N}, u: A \rightarrow \mathbb{R}^N)$ is: an exact potential game if there is a ...
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49 views

Game Theory - Voting

In this setup there are 4 candidates running. For a candidate to be eliminated, the candidate needs to receive less than 1/3 of the votes when paired up with another candidate. This process ...
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49 views

Is it possible to represent any arbitrary game as a 2 player game?

[I'm sorry that I wasn't more specific. Please bare with me I'm a curious novice and a new comer here to stack exchange.] original question: "Is it possible to represent any arbitrary game as a 2 ...
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120 views

Proof that 12 in a row tic-tac-toe is a tie game?

How can be it proved that tic-tac-toe on an infinite grid (winning with 12 in a row, a column or a diagonal) can always end in a tie (with optimal strategies of both players)? There is a hint: to use ...
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3answers
255 views

Precise definition of a “game of incomplete information” (Game Theory)

Question: In game theory, what is the precise definition of a "game of incomplete information"? What I've found so far: In the standard first year graduate economics textbook on microeconomics ...
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70 views

The Fundamental Theorem of Matrix Games, and the “indifference” method of solving games

In the following we will consider two-person zero-sum games. Let $A = (a_{ij})$ be the payoff-matrix of such a game. In this book the fundamental theorem of such games is states as: Theorem: Given ...
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59 views

A little question about payoff functions being continuous.

In the mixed extension of a finite game $G$, why are the payoff functions of players continuous? Does it has something to do with being von Neumann and Morgenstern utility functions? Is there other ...
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89 views

Game theory: inheritance distribution

An old man is dying and wishes to split his $2^n$-dollar fortune between his two sons. It shall be distributed this way: (1) The older brother should propose a way to split the money. If the younger ...
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24 views

Minimum number of steps to guess an item in a database

Let's say I have a database of N*N size (N rows N columns) My friend wants me to guess the location of an item. We start by binary guess, means I ask him if it is in upper half, he says yes or no, if ...
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1answer
189 views

Best strategy for rolling 20-sided and 10-sided dices

There are a 20-sided (face value of 1-20) dice and a 10-sided (face value of 1-10) dice. A and B respectively roll the 20 and 10-sided dices. Both of them can roll the dice twice. They may choose ...
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826 views

Pure vs mixed strategy Nash Equilibria

Just learning about Nash Equilibria. The pure strategy one is explained as an outcome where both/all players feel like they couldn't have done better given what the others were doing. Mixed strategy ...