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 auction when number of items and bidders are the same

For an advertising slot bidding problem, let's say there are two slots and two bidders. If bidder A bids \$10 and bidder B bids \$8, bidder A will win the first slot and pays \$8. How about bidder B? ...
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0answers
9 views

A tough coordination game

Consider the following game in normal form: $$\begin{bmatrix} & x_B=0& x_B=1 \\ x_A=0& 1-\theta_A,1-\theta_B & 0,0 \\ x_A=1 & 0,0 & 1+ \theta_A,1+\theta_B \end{bmatrix}$$ ...
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1answer
57 views

Game theory probability

I have a variant of two-finger morra game, where the winner is determined by the parity of the sum of the two numbers thrown, but the amount won or lost is the product of the two numbers. There are ...
30
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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?
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1answer
1k views

The Notion Of Degenerate Two Player Game

I try to get the intuitive understanding of the notion "degenerate two player game". Definition. A two-player game is called non degenerate if no mixed strategy of support size $k$ has more than $k$ ...
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1answer
54 views

Does it pay to know what you know?

Let's play a game. I ask you question a yes/no question, and you answer. You don't answer with a yes or no though, you answer with a probability of it being yes ($P \in (0,1)$). For example, I might ...
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1answer
19 views

How to compute a mixed Nash Equilibrium where only one payoff is given.

Let's say I have this: $$ \begin{matrix} & A & B \\ X & 1 & 2 \\ Y & 2 & 1 \\ \end{matrix} $$ That is the payoff for me if I make ...
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3answers
40 views

Game is winnable if and only if $n \neq k$

Integers $n$ and $k$ are given, with $n \ge k \ge 2$. You play the following game against an evil wizard. The wizard has $2n$ cards; for each $i = 1, \ldots, n$, there are two cards labelled $i$. ...
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1answer
19 views

Is the Shapely value of this voting game in the core?

Given a voting game where $v(1),v(2),v(3) = 0, v(1,2)= \frac{1} {3}, v(2,3) = \frac{5} {6}$, $v(1,3)= \frac{1} {6}$ and $v(1,2,3) = 1$ I know the Shapely coefficients for a 3 player game, for ...
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0answers
9 views

Confidence interval / incertainty for Shapley Value

I want to know if there is a way to compute a confidence interval (or some measure of incertainty) for Shapley values in a cooperative game. Since the calculation of the Shapley value requires ...
114
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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 ...
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1answer
19 views

Prove that the core of this game is empty

A game (N, v) is simple if for every coalition S is a proper subset of N, either v(S) = 0 or v(S) = 1, In a simple game, a player, i, is said to be a veto player, if v(N \ {i}) = 0. Suppose (N,v) is ...
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1answer
29 views

Find Bayesian Nash Equilibria

A rich, honest, but mischievous father told his two sons that he had placed $10^n$ and $10^{n-1}$ in two envelopes respectively, where n ∈ $\{1,2,3,\ldots,10\}$. The father then randomly handed each ...
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2answers
36 views

IRV failing monotonicity criterion

I am looking for the simplest possible example of instant runoff voting failing the monotonicity criterion. By “simplest possible” I mean the scenario with the fewest number of candidates $(3)$ and ...
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0answers
9 views

When is a multiplayer game factorizable/separable? [closed]

I have typeset the problem in LaTeX here: https://drive.google.com/file/d/0B-6zM2p4CcxINEdKTi1nSklCeFU/view?usp=sharing Basically my question is, when can a multiplayer game be re-written as a ...
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0answers
7 views

Predicting Nash equilibrium after one player enters or leaves

Suppose I have a game with $N$ players, and that the Nash equilibrium can be calculated. If one player enters or leaves the game, is it possible to predict or quickly calculate the resulting Nash ...
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1answer
110 views

Solving the Gobblet game

In 1995 the Connect-4 Game was solved with a brute force approach. Using the standard 6 high / 7 wide grid, first player can force a win in 41 moves. Complexity of the Connect-4 game could be ...
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1answer
727 views

I can't find the Nash equilibrium of this 3x2 game.

Sorry for my English, I am French but i couldn't find help on the French website (so I am here). I have a question about this two-player game: $$ \begin{array}{c|cc} & y_1 & y_2 \\ \hline ...
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1answer
23 views

Problem with proof of the upper hemicontinuity of correspondence

I have a problem with a proof I found here of the upper hemicontinuity of the best-reply correspondence in the Nash Theorem. Below there is the proof, and here my problems: Problems: Is here ...
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1answer
46 views

calculate price based on demands and maximize revenue

I believe I have a simple question which I am struggling to answer. It is as follows: We have 400 items, each item costs £100. Retailer bought these items before the season started. The forecasted ...
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1answer
35 views

2 finger morra game probability

Consider the 2-finger. Bob pays Alice $\$(a + b)$ if $a + b$ is even Alice pays Bob $\$(a + b)$ if $a + b$ is odd Suppose Alice plays $one\; finger$ with probability $\frac 12$ and $two\; fingers$ ...
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1answer
594 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
15 views

Software or code to solve a congestion game for n players

First of all, I am pretty new to game theory so if I say something wrong, please correct me. I have a congestion game (similar to the bookcase of machine job scheduling problem). The jobs are the ...
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0answers
21 views

Mixed strategy, find an equilibrium pair

In this example I need to find equilibrium pair X = ($x_1$, $x_2$, $x_3$), Y = ($y_1$, $y_2$, $y_3$) The matrix looks like this $[0, -1, 2]$ $[3, 1, 0]$ $[-2, 2, 1]$ P(X, $B_1$) = ...
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0answers
19 views

Solve a hypothetical mass-level extinction scenario [closed]

Summary: A single, immortal alien life form that kills by touch seeks to eradicate all life on Earth as efficiently as possible. After killing a victim, he is able to sense the next human nearest to ...
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2answers
1k views

Splitting the dollar Nash equilibrium

I'm working on a game theory problem I can't seem to figure out. Players 1 and 2 are bargaining over how to split $\$10$. Each player names an amount $s_i$, between 0 and 10 for herself. These ...
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1answer
24 views

Working out the value of this two-player game

Now in the solution, I understand how the game tree has been constructed but that's about it. Once the game tree has been constructed I don't understand at all how we work backwards from the ...
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0answers
14 views

What are some techniques of constructing a good utility matrix?

A utility matrix is considered to be subjective and arbitrarily defined. Therefore, we run the risk of over-emphasizing or under-emphasizing the possible alternatives. Are there ways to design an ...
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1answer
25 views

Reference Book- Stochastic Games

Any suggestions for an introductory book on Stochastic Games.
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1answer
16 views

Finding the Shapley value solution for a set of players of size n?

I'm revising Game Theory and have come across this question: "The Miners’ Game is defined as follows. There are $n (>2)$ miners who discover a large $(>n/2)$ quantity of gold bars. It takes two ...
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1answer
23 views

Strictly inferior strategies for 2-player game

I wanted to check if this is a mistake in the solution. From what I understand, strategy 2 is strictly inferior to strategy 1 for player 2 if ai2>ai1 for all i where aij represents the entry in ...
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0answers
11 views

Computing payoffs for pure strategies of a game

For the payoffs, I'm not understanding how they have obtained any of the values. E.g if we consider the payoff f(x,delta2) (the top left one). Then does this mean that for the x value (1/4,0,3/4) ...
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0answers
10 views

Finding extreme solutions to zero-sum games

For the matrix B above, I'm not able to understand how they have extended to solution (1,0) to (0,1,0). I understand why this extension is necessary ( because A is a 2 by 3 matrix and so y must have ...
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1answer
66 views

Find all pure and mixed strategies of Nash Equilibrium and Sub-game perfect equilibrium in a simple sequential game

First subgame is a 2-person simultaneous game. The Nash Equilibria in pure strategies are: (No, L) and (No, NL). Player 1 has a dominant strategy of No (so PL1 never mixes strategies in a ...
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0answers
76 views

Finding a Mathematical definition of a Discrete Time Game

Preface: Suppose we have a game world as depicted in the following figure: Where each of the white blocks is passable, And each of the black blocks is a wall and so impassable. Each of the Green ...
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1answer
44 views

Is it possible for a human to learn to play Connect 4 perfectly using a tree search method?

I've seen perfect solvers of the game Connect 4 using various methods. The one that I saw uses alpha beta pruning. Is it possible for a human to learn to play Connect 4 perfectly like these solvers ...
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0answers
15 views

what broad topics in game theory are likely discussed when you say 'game-theoretic analysis' of something?

I actually do not know anything about game theory, and in my current research I think I need to start knowing what it is all about. In the meantime, I always read papers that say they did a ...
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1answer
601 views

Relationship between regular Nim and Lasker's Nim

So I'm trying to do qn $6$ (on pg I-13) about staircase Nim in Game Theory by Ferguson Game Theory, Ferguson and it's asking to prove that $(x_1, x_2, \ldots, x_k) \in P $ only if $(x_1, x_3, x_5, ...
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1answer
29 views

Mixed Strategies for 3x3 matrix

What are mixed strategies in this game? Usually, I can find mixed strategies of 3x3 when there exist a dominant strategy that dominates another one and we eliminate dominant strategy. But in such ...
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1answer
123 views

Winning strategy for solitaire?

I'm talking about the Klondike solitaire, turning three cards at once to the waste and placing no limit on passes through the deck. I know there isn't always a winning strategy, a counterexample can ...
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0answers
14 views

Trembling Hand Perfect Equilibrium

I am looking to find all pure strategies Trembling Hand Perfect Equilibrium. Can anybody help me with this?
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1answer
638 views

Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
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2answers
161 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 ...
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0answers
18 views

Ultimatum game - Bargaining

Consider a variant of the ultimatum game. Let $\omega \in \mathbb{N}_{+}$, (where $\mathbb{N}_{+}$ is the set of non-negative integers), and $A_1=\left \{ (x,y) \in \mathbb{N}_{+} : x+y=\omega \right ...
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Topology of the set of Nash equilibria

Consider a normal form game with $n$ players (and finitely many options per player) defined by finite option sets $A_1,\ldots,A_n$ and payoff matrices $u_1,\ldots,u_n: \prod_{j=1}^n A_j \to ...
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2answers
732 views

Nash equilibrium in first price auction

I'm trying to understand Exercise 18.2 from Martin J. Osborne and Ariel Rubinstein A Course in Game Theory about finding pure Nash equilibria in a first-price auction. There are $n$ players, named ...
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1answer
2k views

Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question: Is ...
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1answer
126 views

Is this Game Theory

A coffee shop wants to bring back a group of customers that are socially connected to each other. It wants to offer them a great discount as an incentive to return. The coffee shop wants to empower an ...
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0answers
66 views

A simple betting game

Consider the following betting game: Two players each have 100 cents to bet. If one player bets more than the other then that player gains a point and the other player loses a point. The goal of the ...
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0answers
13 views

Concavity of the equilibrium

Suppose we have have $n$ players taking action $a_i \in [0,1]$ to generate some value $v(a_1,...,a_n)$ together. The utility for player $i$ given by $\lambda_iv(a_1,...,a_n) - u_i(a_i)$ where $u_i$ ...