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).

learn more… | top users | synonyms

14
votes
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
1k 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 ...
8
votes
3answers
3k 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 ...
2
votes
2answers
257 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
2answers
281 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 ...
430
votes
25answers
54k 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 ...
45
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 ...
79
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 ...
25
votes
7answers
8k 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?
8
votes
5answers
5k 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 ...
15
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 ...
9
votes
1answer
210 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 ...
7
votes
5answers
459 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 ...
4
votes
1answer
422 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 ...
5
votes
8answers
9k 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
2answers
3k 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 ...
3
votes
1answer
101 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 ...
2
votes
0answers
194 views

What is the (expected) outcome of this hybrid auction?

A certain hybrid auction can be accurately modelled as follows. There are $n$ risk-neutral, rational participants $i=1,2,\ldots,n$, and a guy called Zerro: $i=0$. Each, except Zerro, has a private ...
1
vote
0answers
130 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
146 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 ...
3
votes
1answer
78 views

Why the set of outcomes generated by a fixed strategy of one player in Gale-Stewart game is a perfect set?

In the proof that there is a payoff set $X$ such that the Gale-Stewart game is not determined(see here, Proposition 3.1.). I don't know why $X$, the set of all outcomes generated by a fixed strategy ...
3
votes
1answer
387 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 ...
3
votes
1answer
737 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 $\mathbf{S}_i$ $=$ ...
2
votes
1answer
610 views

Saddle points in zero sum game

We only had one lecture about the subject and already have quite difficult questions, could someone please help me? The matrix looks something like this: \begin{matrix} 3 & 2 & 1 & 4 ...
1
vote
1answer
86 views

Finding mixed nash equlibrium

In the following game I found one pure nash equilibrium: $(R, r)$: $\begin{array}{r|ccc} A\backslash B & l & m & r\\ \hline L & (-10, 4) & (10, 0) & (-1, -1)\\ M & (0, 10) ...
0
votes
1answer
98 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
83 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 ...
56
votes
3answers
6k views

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 ...
47
votes
8answers
12k views

Why do people lose in chess?

Zermelo's Theorem, when applied to chess, states: "either white can force a win, or black can force a win, or both sides can force at least a draw [1]" I do not get this. How can it be proven? ...
18
votes
1answer
1k views

Number of moves to solve a flood-it/sock-dye game

[ Question based on the sock dye game ] [ Update: It appears that this game is better known as "Flood it" and is NP-hard. Also, "the number of moves required to flood the whole board is $\Omega(n)$ ...
12
votes
3answers
585 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 ...
5
votes
2answers
375 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 ...
5
votes
2answers
966 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
2answers
344 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 $\$c$, ...
4
votes
1answer
116 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 ...
12
votes
3answers
630 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
289 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 ...
7
votes
1answer
382 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
1answer
2k views

Number Game: 31 - Winning Strategy?

My Maths teacher taught us how to play a game called 31 on Friday. Not once did my Maths teacher lose. I want to know why. I'll explain the game... 31 is a game between two people. Let's say you've ...
7
votes
1answer
325 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 ...
5
votes
2answers
19k 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
665 views

Game theory Computing pure Nash equilibrium probability

We have a $2$-player game and each player has $n$ strategies. The payoffs for each player are in range $\left[0,1\right]$ and are selected at random. Show that the probability that this random game ...
5
votes
2answers
2k 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. ...
4
votes
3answers
782 views

prerequisites for understanding game theory

I am from programming background but with very limited knowledge of maths. I am very much eager to learn and apply game theory to understand dynamics of International Politics and economics. But I am ...
3
votes
2answers
116 views

probable squares in a square cake

There is a probability density function defined on the square [0,1]x[0,1]. The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area. Now Alice and Bob play a ...
3
votes
2answers
2k views

Does chess have more Nash equilibria than you can find through backwards induction?

All equilibria found with backwards induction on a tree of a perfect information game are Nash equilibria, but in general the reverse is not true: ...
3
votes
1answer
566 views

what resources would help someone understand Game Theory proofs?

Please note that my knowledge of math proofs is little to none. I was wondering what resources, free or paid, would allow me to understand the math proofs specifically related to Game Theory? I ...
1
vote
1answer
75 views

What is the sprague-grundy value of these games?

This is a follow-up question of my previous question : Optimal strategy for this Nim generalisation? Consider the following game: There are a number of piles of stones. On each turn a player can ...
1
vote
1answer
142 views

You are Johnny Depp 3!

An extension of this question. As @Jared stated in his answer the solution is: we assume that the head pirate chooses between multiple possible proposals that maximize his profit by rewarding ...
1
vote
1answer
660 views

Understanding common knowledge in logic and game theory

For $k = 2$, it is merely "first-order" knowledge. Each blue-eyed person knows that there is someone with blue eyes, but each blue eyed person does ''not'' know that the other blue-eyed person ...