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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Two person cooperative non-zero sum game

This a two person cooperative non-zero sum game. The shaded area is the negotiation set. $s_a$ and $s_b$ are the security levels for $A$ and $B$ respectively. I do not understand the part I have ...
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38 views

Big Two's classification according to game theory

Is the game Big Two, as described in https://www.pagat.com/climbing/bigtwo.html, classified as: a game of perfect or imperfect information? deterministic or stochastic? EDIT: I am fairly certain ...
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83 views

Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
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1answer
28 views

Representation of a game with simultaneous movements

In Game Theory, can you use a tree structure to represent a game with simultaneous movements or you have to use a matrix form? In a sequential game it is logical to use a tree, as every node ...
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1answer
28 views

Randomised strategies

I don't understand what is meant my assigning probabilities to randomised strategies. Randomised strategies are themselves probability distributions over the pure strategies.
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1answer
27 views

Value of Zero sum game

In part iii) I am unsure as to why we subtract 1 from the value of the game (underlined in green)
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32 views

Visualizing Nash Equilibria of a 4 dimensional matrix

Are there any good ways to visualize Nash equilibria of a 4-d matrix? I have created an game theory model which consists of of four players (P1; P2; P3; P4) who can all choose between a set of 27 ...
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31 views

Set of equilibrium points generically finite and odd

Let $\Sigma$ denote a finite product of unit simplices and $E:\mathbb{R}^k \rightrightarrows \Sigma$ an upper-hemicontinuous and compact valued correspondence with graph $\Gamma$. By $\pi: \Gamma \to ...
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18 views

How you can model the other players replies in a game theoretic model?

In a game theory field, the payoff function of a player n is basically a function of the other players responses which are considered as constants. I'm trying to solve the maximization of the payoff ...
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50 views

An Interesting Variation to the “Pebbling a Checkerboard” Puzzle

Pebbling a Checkerboard (or chess board) was a puzzle proposed by Maxim Kontsevich in 1985, which was very interesting and fun to try, and you can find a great video on it at: https://www.youtube.com/...
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1answer
17 views

Average discounted payoffs after T periods formula.

I am being told that; The average discounted payoffs after T periods is given by; $$\pi_i = \frac{1 - \delta }{1 - \delta^T}\sum_{t=0}^{T-1} \delta^tg_i(a^t) $$ $\delta$ is the discount rate $...
0
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1answer
42 views

Does the first player have a winning strategy?

Two players play a game where they alternatively cross out a number from the numbers written on the board ($1-21$). They stop when two numbers are remaining. If thie sum of these two numbers is ...
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172 views

An envelope game

Each of two individuals receive a ticket on which there is an integer from one to five indicating an amount of money he may receive. The individuals' tickets are assigned randomly and independently. ...
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17 views

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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1answer
240 views

Coordination game

Consider the following game in normal form: $$\begin{bmatrix} & x_B=0& x_B=1 \\ x_A=0& 1-\theta_A,1-\theta_B & 0,0 \\ x_A=1 & 0,0 & 1+ \theta_A,1+\theta_B \end{bmatrix}$$ ...
1
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1answer
61 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 ...
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1answer
27 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
44 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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0answers
13 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 ...
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1answer
20 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 $|s|=1,...
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1answer
30 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 ...
2
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1answer
32 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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0answers
9 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 ...
0
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1answer
29 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 ...
0
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1answer
41 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
50 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 ...
5
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1answer
123 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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0answers
29 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
23 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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0answers
23 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, $B_1$) = 3$...
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0answers
16 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
30 views

Reference Book- Stochastic Games

Any suggestions for an introductory book on Stochastic Games.
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1answer
31 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
27 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 terminal ...
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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) we ...
0
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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 the ...
0
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1answer
17 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 3 ...
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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
46 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 do?...
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0answers
17 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 'game-...
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1answer
44 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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0answers
17 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
97 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 solution). ...
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18 views

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 $u_1,\ldots,u_n: \prod_{j=1}^n A_j \to \mathbb{R}$...
3
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0answers
72 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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15 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 $u_i$ ...
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20 views

Are these preferences rational according to utility theory?

I have this question about whether preferences satisfy the $6$ properties of utility, and am particularly stuck on the boundness, coherence and continuity conditions. Here is the problem: If one ...
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12 views

Does there exist a utility function for these gambles and preferences?

Suppose that you prefer a gamble which pays $£100$ with probability $0.5$ and $£10$ with probability $0.5$ to a gamble which pays $£200$ with probability $0.25$, $£50$ with probability $0.25$ and $£10$...
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32 views

Show, by example, that we can find social welfare functions which satisfy any three of the four Arrow's axioms.

I want to show, by example, that we can find social welfare functions which satisfy any three of the four Arrow's axioms. Given at least three rewards, and at least two individuals, there is no ...
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1answer
30 views

Pairwise majority voting and Arrow's axioms

The following is a question on Arrow's theorem with a pairwise majority decision. The bits I was unsure about was (bi) (is the 4th condition satisfied?) and also is (bii) correct? Thanks for your help ...