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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14
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5answers
2k views

Probability of dice sum just greater than 100

Can someone please guide me to a way by which I can solve the following problem. There is a die and 2 players. Rolling stops as soon as some exceeds 100(not including 100 itself). Hence you have the ...
13
votes
5answers
2k views

A lady and a monster

A famous problem: a lady is in the center of the circular lake and a monster is on the boundary of the lake. The speed of the monster is $v_m$, and the speed of the swimming lady is $v_l$. The goal of ...
22
votes
6answers
8k views

Help: rules of a game whose details I don't remember!

In a probability course, a game was introduced which a logical approach won't yield a strategy for winning, but a probabilistic one will. My problem is that I don't remember the details (the rules of ...
3
votes
1answer
999 views

Game about placing pennies on table

This problem is from The Art and Craft of problem solving book: Consider the following two player game. Each player takes turns placing a penny on the surface of a rectangular table. No penny can ...
7
votes
1answer
600 views

Bounded sequence with divergent Cesaro means

Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)? More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that $$\lim_{M\rightarrow\...
5
votes
1answer
124 views

Game: two pots with coins

Rules of the game with two players. First player puts any number of coins in the first pot. Then second player, knowing that number, puts any amount of coins in the second pot. Then they in turns (...
2
votes
2answers
417 views

Mixed strategy nash equilibria in from $2\times N$ bimatrix form

I'm looking for a way of finding (manually!) mixed strategy Nash equilibria in a $2\times N$ game. Calling player 1 the player with two strategies and player 2 the one with $N$ strategies, I've ...
46
votes
3answers
3k views

A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe

Consider a 9 by 9 matrix that consists of 9 block matrices of 3 by 3. Let each 3 by 3 block be a game of tic-tac-toe. For each game, label the 9 cells of the game from 1-9 with order from left to ...
32
votes
8answers
14k views

Game theory - self study

I want to self study game theory. Which math-related qualifications should I have? And can you recommend any books? Where do I have to begin?
11
votes
1answer
490 views

Toss a fair die until the cumulative sum is a perfect square-Expected Value

Suppose we keep tossing a fair dice until we want to stop, at which point the game ends and our score is the cumulative sum, or until the cumulative sum is a perfect square, in which case we lose and ...
2
votes
2answers
435 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 ...
506
votes
25answers
58k views

Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...
88
votes
18answers
8k views

Fastest way to meet, without communication, on a sphere?

I was puzzled by a question my colleague asked me, and now seeking your help. Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and ...
13
votes
3answers
5k views

Secretary problem - why is the optimal solution optimal?

I have read about this problem: http://en.wikipedia.org/wiki/Secretary_problem But I want to see how it is proven that the "optimal" solution is indeed optimal. I understand how to prove that if the ...
5
votes
1answer
580 views

Which side has winning strategy in Go?

Go is actually a finite two-person game of perfect information and cannot end in a draw. Then by Zermelo's theorem, it is exactly one of the two has winning strategy, either Black or White. So my ...
3
votes
2answers
551 views

Game involving tiling a 1 by n board with 1 x 2 tiles?

Consider a $1$ by $n$ tiled rectangle. You want to play a game with one opponent in which you place $1$ by $2$ "dominoes" on this rectangle. The player who places the last domino wins. Which player ...
-1
votes
2answers
193 views

How to divide the money based on the below conditions? [closed]

Two cheeky thieves have collectively decided to rob a bank. They have carefully counted a total of exactly $100$ dollars in the bank vault. Now they must decide how to divide the booty. But there is ...
5
votes
8answers
13k views

Deal or no deal: does one switch (to avoid a goat)?/ Should deal or no deal be 10 minutes shorter?

Okay so this question reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon Deal or no deal. For the uninitiated: In playing deal or no deal, the ...
3
votes
1answer
115 views

Name for a certain “product game”

Let $G,H$ be two (combinatorial impartial) games. Consider the following new game $P$: The positions are the pairs of positions of $G$ and $H$. A move in $P$ is a move in $G$, or a move in $H$, or a ...
3
votes
1answer
511 views

Game Theory Matching a Deck of Cards

Moderator Note: This question is from a contest which ended 1 Dec 2012. Suppose we have a deck of cards labeled from $1$ to $52$. Let them be shuffled in a random configuration, then made visible. ...
3
votes
2answers
6k views

Winning strategy for a matchstick game

There are $N$ matchsticks at the table. Two players play the game. Rules: (i) A player in his or her turn can pick $a$ or $b$ match sticks. (ii) The player who picks the last matchstick loses the game....
1
vote
0answers
157 views

Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
6
votes
1answer
163 views

Invariance of strategy-proof social choice function when peaks are made close from solution

A question emerging from reading Schummer, J., & Vohra, R. V. (2002). Strategy-proof Location on a Network. Journal of Economic Theory, 104(2), 405–428. The setting is as follows: A finite set ...
5
votes
2answers
12k views

fair and unfair games

so, as far as i understand there are two types of mathematical games: fair and unfair. fair games are games where both (all) players have exactly the same chance of winning (outcome of the game is ...
4
votes
2answers
271 views

How many legal states of chess exists?

I have a fairly simple question. How many legal states of chess exists? "Legal" as in allowed by the rules and "state" as an unique configuration of the pieces. I'm not asking for the number of ...
4
votes
1answer
1k views

Set of rationalizable strategies

Consider a guessing game with ten players, numbered 1 through 10. Simultaneously and independently, the players select integers between 0 and 10. Thus player i's strategy space is $\mathbf{S}_i$ $=$ $\...
0
votes
1answer
197 views

Game Theory question about a financial pyramid scheme

Salut, fellow game theorists. I have to solve 6 Game Theory problems and fell almost hopeless. Would appreciate any guidance with this one. A company Zest is actively promoting its services. ...
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 ...
116
votes
1answer
4k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where $...
17
votes
7answers
2k views

Game theory textbooks/lectures/etc

I looking for good books/lecture notes/etc to learn game theory. I do not fear the math, so I'm not looking for a "non-mathematical intro" or something like that. Any suggestions are welcome. Just put ...
12
votes
3answers
604 views

Formula for picking time closest to (but after) target

Let's say you have an arbitrary length of time. You are playing a game in which you want to push a button during this time span after a light comes on. If you do so, you win ($+1$), if not, you lose (...
9
votes
5answers
607 views

Good non-mathematician book on Game Theory

I'm looking for a good book on Game Theory. I run a software company and from the little I've heard about Game Theory, it seems interesting and potentially useful. I've looked on Amazon.com but ...
7
votes
4answers
222 views

100-Sided Dice “Blackjack” Game

I am attempting to determine two variables in this game: The optimum strategy: (What number the bettor should stay at) The expected value given perfect play: (The percent return on a bet when using ...
11
votes
2answers
657 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
8
votes
4answers
115 views

Guess the number despite false answer

This is the Guess-The-Number game with a twist! Variant 1 Take any positive integer $n$. The game-master chooses an $n$-bit integer $x$. The player makes queries one by one, each of the ...
6
votes
3answers
608 views

How do I calculate the odds of a given set of dice results occurring before another given set?

Dice odds seem simple at first glance, but I've never taken a Calculus based statistics course or game theory, and I think I may need to in order to solve some of the things I'm trying to solve. I can ...
13
votes
3answers
41k views

How to compute ALL Nash equilibria in an example of a 3x3 matrix

I am trying to understand how to compute all Nash equilibria in a 2 player game, but I fail when there are more than 2 possible options to play. Could somebody explain to me how to calculate a matrix ...
5
votes
2answers
5k views

Computer software for solving mixed strategy Nash equilibrium

Is there any computer software available for solving for mixed strategy Nash equilibria for two players given each player's payoff matrix?
5
votes
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 $\$c$, ...
5
votes
2answers
1k views

Modified pirate game

The pirate game is a popular problem that is often asked in interviews (which is how I stumbled upon it). The problem asks There are 5 rational pirates, A, B, C, D and E. They find 100 gold coins. ...
4
votes
1answer
173 views

What is a good strategy for this dice game? [duplicate]

I learned the following dice game from another forum. It was not solved there. The dice game is as follows. You start tossing six dice. After each toss you must put aside at least one of the dice ...
3
votes
0answers
195 views

Alice and Bob make all numbers to zero game

Alice and Bob are playing a number game in which they write $N$ positive integers. Then the players take turns, Alice took first turn. In a turn : A player selects one of the integers, divides it ...
1
vote
1answer
62 views

The application of Nimbers to Nim strategy

I've been reading about combinatorial game theory, and some works start with the game of Nim. After that, they introduce Nimbers, which are numbers that represent Nim games. So far so good. I get ...
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 ...
12
votes
3answers
709 views

Best Strategy for a die game

You are allowed to roll a die up to six times. Anytime you stop, you get the dollar amount of the face value of your last roll. Question: What is the best strategy? According to my calculation, for ...
9
votes
5answers
307 views

How can the observed strategies* in this actual auction be explained?

This is a "real world" question. Recently I witnessed the separate auctions of about 30 houses. The place where I went uses the following rules. The following describes the procedure for the ...
8
votes
2answers
3k views

Do Symmetric Games with Nash Equilibria always have a symmetric Equilbrium?

Define a game with S players to be Symmetric if all players have the same set of options and the payoff of a player depends only on the player's choice and the set of choices of all players. ...
7
votes
1answer
350 views

Optimal strategy for slice weighing game

I watched an interesting contest on a Swedish game show the other night. I have tried to find an english name of the contest but haven't found any. Two contestants were each given one large sausage ...
7
votes
1answer
639 views

Monkey typing ABRACADABRA and gamblers

Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
7
votes
2answers
386 views

You are Johnny Depp 2!

An extension of this question repeated below. A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. ...