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

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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 ...
2
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2answers
174 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 ...
1
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1answer
83 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, ...
0
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5answers
198 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 ...
1
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1answer
277 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 ...
4
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2answers
549 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 ...
4
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0answers
573 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 ...
3
votes
2answers
245 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. ...
9
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3answers
864 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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1answer
59 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 ...
2
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1answer
2k 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 & ...
1
vote
1answer
37 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 $4\alpha^...
5
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2answers
280 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 $\mathbb{R}$....
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0answers
146 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 example,...
0
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2answers
107 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 ...
1
vote
1answer
303 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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2answers
611 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 ...
1
vote
2answers
845 views

Third and average price auction

Third price auction: the winner is the highest bidder but this time instead of paying the second highest bid, he would pay the third highest bid. -assume there are at least 3 bidders. - Average price ...
0
votes
1answer
56 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 of player1 ...
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0answers
52 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 & ...
1
vote
1answer
77 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, ...
0
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1answer
57 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
65 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 ...
2
votes
2answers
434 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 ...
2
votes
0answers
252 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, ...
13
votes
1answer
378 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 ...
0
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2answers
498 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 ...
0
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1answer
62 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 ...
2
votes
2answers
262 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 &...
2
votes
1answer
207 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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1answer
87 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 ...
2
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0answers
55 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 ...
0
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1answer
48 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 ...
0
votes
1answer
51 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 ...
2
votes
1answer
96 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 c, ...
1
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1answer
137 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 ...
5
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2answers
384 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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0answers
47 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 ...
21
votes
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 ...
0
votes
1answer
172 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 ...
1
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2answers
42 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 ...
1
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2answers
145 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 ...
2
votes
1answer
112 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 ...
2
votes
1answer
131 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 ...
1
vote
1answer
54 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 ...
4
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3answers
1k 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 non-...
1
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1answer
43 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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0answers
77 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 ...
0
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0answers
59 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 ...
1
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0answers
328 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 ...