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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Second-Price Sealed-Bid Auction

Consider 2 individuals who are interested in one indivisible object. Each player $i$ has a valuation $v_i > 0$ for the object. Assume $v_1 \geq v_2$. In this scenario, each player submits a bid ...
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0answers
235 views

Make the maximum sum in a interval.

Two players $A$ and $B$ are playing a game with array $A$ of $N$ elements in which each player chooses an array interval with maximum sum. The other simultaneously chooses another interval in the same ...
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1answer
363 views

K piles stone game

Their are k piles with total n stones in some order where each pile can have stones greater or equal to zero.Two player A and B plays a game in which player A cant see the configuration of piles but ...
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1answer
61 views

Finding social cost in game theory paper

From theorem 3 in http://cgi.di.uoa.gr/~elias/publications/paper-kp09.pdf Let $w_1$,$w_2$ = 1. I have interpreted the above paper as saying that the social cost is the same thing as the expected cost ...
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1answer
69 views

Flipping the Grid

Suppose a grid of size N * M is having only 1 and 0. The rows are numbered from 1 to N, and the columns are numbered from 1 to M. Following steps can be called as a single move. Select two integers ...
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1answer
43 views

Categorization of PBE refinements into forward/backward looking?

I have recently come across the term forward / backward looking refinement of a Perfect Bayesian Equilibrium. I am, however, unsure about the meaning of this term, and unable to find any information ...
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0answers
205 views

Subgame Perfect Nash Equilibrium Problem

Suppose that there are two incumbent icecream vendors, but that there is a possibility of entry of a third vendor. Specifically, at any location on the beach a third vendor can, after observing the ...
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1answer
51 views

Centipede game of 2 students

PROBLEM We have two students who both begin with 1 point, their utility is equal to the number of points they have. Student $1$ has the first turn, he can decide to end the game and both students ...
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1answer
2k views

Ice cream vendor problem

PROBLEM This is a question considering game theory. Assume there is a beach with $n$ ice cream vendors on it who position themselves along the beach. For an arbitrary $n$, find a candidate Nash ...
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1answer
105 views

Coin game strategy [closed]

PROBLEM Suppose two players play a coin game. They both have a coin and can choose themselves what the probability is that they will play head $H$. So for player $i$, the chance he gets head is ...
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1answer
63 views

Static game question

GAME PROBLEM At a tv game show, two players are handed six cards each. Each of the six cards shows a number, and both players have the same cards. The six numbers are $0,0,3,0,15$. The game is as ...
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1answer
157 views

How to split the difference in game theory?

There are two parties in talks to settle a law suit. The expected value of going to court for the plaintiff is $\$3,155$ The expected value of going to court for the defendant is $-\$10,244$ The ...
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1answer
92 views

Finitely repeated games

Consider the following matrix game: \begin{matrix} & L &M&R \\ T&8,8 & 0,9 & 0,0 \\ C&9,0 & 0,0 & 3,1 \\ B&0,0 & 1,3 & 3,3 \end{matrix} For ...
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0answers
86 views

Proving lower bounds from algorithmic game theory paper (specifically, price of anarchy is lower bounded by 3/2 for $m$ links)

This question is similar to Understanding proofs from paper on Game Theory (Price of Anarchy) This question is about the same proof: proving the lower bound that the price of anarchy (sometimes ...
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1answer
411 views

Question related to the General equilibrium in exchange economies

I need some help in solving this question: Consider the following two-person, two-good economy. Persons $A$ and $B$ each consume two goods xylophones $(x)$ and yams $(y)$. Person $A$ enters the world ...
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1answer
135 views

Generalized Mechanism Design, Stanley Reiter diagram and Vickrey Auction

I am trying to learn this new topic, Mechanism Design and stumbled upon the "Stanley Reiter" diagram (see the top-right side on the page). I have also learned that the Second Price Sealed Bid ...
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1answer
155 views

Two people are looking for each other. Is it faster for both to actively search, or for one to search while the other stays still?

Choose among two actors randomly and place the chosen actor at the origin. Place the other actor in the unit circle uniformly at random. Both actors move at the same speed. Both actors are said to ...
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1answer
521 views

Reduce the payoff matrix using (weakly) dominated strategies

Below is the payoff matrix of a game. Use the principle of elimination of (weakly) dominated strategies to simplify the payoff matrix. What is the optimal solution of the game for the row player? ...
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0answers
71 views

Optimally Efficient Tournament Strategies

Consider a fair symmetric game between two players that always results in exactly one of the players winning, i.e. there are no ties. When two players $P$ and $Q$ play each other, $P$ wins with a ...
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2answers
73 views

Calculate the Ranks of Candidates based on Votes and Total Candidates

What is the formula to calculate the rank of each candidate when I have the total candidates and votes secured by each? I've managed the percentage part, but calculating the rank has me stuck. Id be ...
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1answer
1k views

questions on information set definition

The definition of "information set" is An information set is a set of decision nodes, all belonging to the same player, over which that player cannot distinguish. ...
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2answers
296 views

Problem regarding filling squares inside a $n\times n$ grid.

Assuming a $n\times n$ square grid, what is the most number of squares that can be filled in such that there are no completed rows, columns, or diagonals? Is there a formula to calculate this? ...
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0answers
23 views

Game with no Right option is an integer

Lessons in Play, Chapter 5, number 15. Prove that if G either has no right options (Or no left options), then G is an integer.
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1answer
81 views

Optimal social cost = 1 … Game Theory paper

Say you have $n = m$ identical links, where each agent $i$ has a unit of traffic to push from source to sink, that is, $w_i = 1$. I've seen it stated, but not shown, that the optimal solution has ...
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3answers
34k 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 ...
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0answers
31 views

Playing Connect 6?

Say I'm playing Connect 6, a variation of Connect 4, and I get to go first. For a $n \times n$ board, which position should I place my chip on to maximize my odds of winning? Note than unlike Connect ...
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1answer
189 views

Partisan/Partial Game Theory

There are enough resources available on the internet regarding "impartial" game theory. But I cannot seem to find much information regarding "partial" game theory. Can someone name some such resources ...
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1answer
317 views

Placing K knights in an nxn board such that no two attack each other

This is a problem from spoj A and B are playing a very interesting variant of the ancient Indian game 'shatranj(also known as chess)' on a 'maidaan'(chessboard) n×n in size. They take turns ...
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0answers
56 views

Is there a textbook treatment of Ky Fan's minimax theorem and its generalizations?

Theorem 2 in Ky Fan(1952) is a powerful tool in zero-sum games, which states: Let $X$ be a compact Hausdorff space and $Y$ an arbitary set (not topologized). Let $f$ be a real-valued function on ...
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3answers
595 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 ...
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2answers
546 views

What is worth of a stalk in red-blue Hackenbush??

I was studying about Red-Blue hackenbush from this link http://www.link.cs.cmu.edu/15859-s11/notes/Hackenbush.pdf http://math.ucsd.edu/~wgarner/math168a/blueredhackenbush.htm this url shows a ...
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1answer
15k views

pure strategy vs mixed strategy

Apparently, I'm not understanding this simple concept. What are the differences between the two? Can a person have multiple pure strategies that change throughout the game?
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1answer
104 views

Understanding proofs from paper on Game Theory (Price of Anarchy)

I'm trying to distill the arguments in the paper "Worst-Case Equilibria" (http://cgi.di.uoa.gr/~elias/publications/paper-kp09.pdf). But there are some things I do not understand and would appreciate ...
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1answer
242 views

Question on the equivalence of behaviour strategy and mixed strategy for a player with a single information set

Prove that if a player in an extensive-form game has only one information set, then his set of mixed strategies equals his set of behavior strategies. This is the exercise $6.4$ on page $246$ in ...
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1answer
57 views

Number of rounds to find out the winner in nondeterministic game

Imagine, I'm organizing competition for AI Yahtzee players (or some other game, poker, backgammon etc.). I want to find out who plays better: player A or player B. If they play just one game, winner ...
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1answer
565 views

Hackenbush game strategy for stalk

There are some piles of numbers. Numbers are divided in 2 groups, A and B. Player x plays with group A and player y plays with group B. x makes the move first. On each step a player chooses a pile and ...
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1answer
1k views

Game theory : A and B are playing divide the dollar

A and B are playing divide the dollar. Denote A’s strategy by a (the amount he claims) and B’s strategy by b. If they can agree on a division of the dollar (a + b ≤ 1) they walk away with the share ...
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1answer
104 views

In the game shown below what strategies can player 2 adopt in a subgame-perfect equilibrium?

What difference would it make to possible predictions of how rational states play this game if player 2 does not know player 1’s move when it chooses (i.e. node2 and node3 belong to the same ...
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2answers
1k views

3x3 Nash Equilibrium?

I'm trying to figure out a Nash Equilibrium for a 3x3 zero-sum game, and it's not following normal patterns (or I'm making a huge oversight, in which case I'll feel stupid!). Can anyone help me? The ...
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1answer
405 views

Prove the dominant strategy of Game Theory

A row $r$ of the payoff matrix is said to dominate a row $s$ if $a_{rj}\geq a_{sj}$ for all $j$ = 1,2,......,$n$. Similarly, a column $r$ of the payoff matrix is said to dominate a column $s$ if ...
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1answer
68 views

simultaneous probability elicitation from multiple agents without an exogenous banker

A proper scoring rule is a function $f:[0,1]\times\{0,1\}\to \Bbb R$ such that, if a subject will receive a reward of $f(x,0)$ for reporting his estimate of the likelihood of an event as $x$ if the ...
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2answers
366 views

Game Theory Problem with Dice

I need solution to this game theory problem. It seems impossible to me. Two players (1 and 2) play the following game. Player 1 must write the numbers from 1 to 18 on the sides of 3 dice without ...
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1answer
97 views

Is this a novel game theory measurement?

A measurement of skill versus chance in games. Overview I present here what I believe to be a novel approach to measuring the amount of skill versus luck in various types of games. The method gives ...
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2answers
104 views

In the next matrix, why is (55,0) not a Nash Equilibrium?

My book says that the next matrix has no Nash Equilibriums. Still, Im a little confused about row 3, column 2. Reasoning from player 2's perspectivo, he could say "if player 1 chooses row 3, I Will ...
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1answer
75 views

Choosing strategies to maximize ones payoff and minimize others payoff

In game theory, the aim of every player is to pick strategies that maximizes their payoff. This is done irrespective of the payoff of others. But in real life competitions, there arises a secondary ...
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1answer
409 views

Man, Woman, Dog, seeking stable relationship.

There is a classic problem in combinatorics dealing with a stable pairing between a set of men and a set of women as spouses. (Gale-Shapely algorithm) ...
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1answer
45 views

probabilistic behaviour

I am trying to understand what 'probabilistic behaviour' in a 'deterministic model' means. I am reading this paper http://www.ulb.ac.be/sciences/use/publications/JLD/16.pdf but i find myself unable ...
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1answer
238 views

Proximal functions

I am a little bit new to proximal functions and I am currently stuck with the following problems How would I derive the prox function for the regularizer (h(x) function) : $\alpha\sum_{k+} $ and for ...
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2answers
69 views

Duals of Linear Programs

We are trying to find the dual of the following linear program. $$ \max_x \ ax_1 \ + x_2 $$ such that: $$ v_1x_1 - v_2x_2 \geq b_1 \\ v_1x_1 - v_2x_2 \geq b_2 \\ x_1 \geq 0 \\ x_2 \geq 0$$ ...
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3answers
136 views

How to discourage contestants from entering a lottery twice?

Suppose you have a lottery. And you want to prevent participants from buying multiple tickets. What would be the best way to discourage this? For example, increasing the win-chance for all previously ...