# Tagged Questions

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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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $u_1,\ldots,u_n: \prod_{j=1}^n A_j \to \mathbb{R}$...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### Arrow's theorem, strategic thinking and utilitarianism

I have this problem as part of a course on Decision Theory, and was not sure about question a (4th condition of Arrow's theorem) and question dii (utilitarianism). I have provided the whole question ...
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### Sen's theorem on minimal liberalism

I want to solve the following question regarding Sen's theorem and preferences orderings. The bit I am having trouble with is 2b Question 1: Describe Sen's theorem ('paradox') There is no social ...
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### 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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### Move of Nature - Ultimatum game

Let $N=\left \{1,2 \right \}$, $A_1=\left \{(x,y): x+y= \omega \right \}$. Player 2, upon seeing a proposal $(x,y)$ can either accept or reject the offer. Player 2 (the responder) has an outside ...
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### Stochastic game matrix

I am trying to solve this exercise and I can't find all Nash Equilibrium. I posted as picture because I couldn't make this matrix!.
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### The variation of Nim game [duplicate]

Can you throw some tips to solve this one variation of Nim game? Two players. The first one throw some stones in heap, the second one throw another natural number in the second heap. Both players ...
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### The Nash Demand Game

I am having troubles to figure out how to find all pure strategies and two mixed strategies (one in which the shares received by players are always $0$, and one in which $\omega$ is often - but may be ...
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### Game theoretic generalization of decision graphs?

Decision trees can be generalized into game trees (to model interacting decisions amongst 2 or more players) and further into game trees with information sets (to recognize imperfect knowledge of game ...
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### Making sense of weak dominance exercise

I am working on the exercise $4.4$ of Game Theory: an Introduction by Steven Tadelis. The Chapter $4$ is about Beliefs, Best-Response Correspondences and Rationalizability in Game Theory. I am ...
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### How do you seed a “non-binary” tournament?

In a typical tournament, seeding is arranged to provide proportionate advantage to competitors based on their perceived relative ability. (Good teams get to play bad teams, bad teams have to play good ...
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### $N=\{1,2,3,4\}$ abd $v(124)=v(134)=v(234),v(1234)=1$ and $v(3)=0$

Let $N=\{1,2,3,4\}$ abd $v(124)=v(134)=v(234),v(1234)=1$ and $v(3)=0$ otherwise $(a)$ Is $C(v)\neq \phi$ ? $(b)$ Write down a stable set for the game Can anyone please tell how to solve above two ...
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### How to calculate the payoff in Battleship Game Theory

Consider a 3 by 3 board and suppose that Player I hides a destroyer(length 2 squares) vertically or horizontally on this board. Then Player II shoots by calling out squares of the board, one at a time....
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### Finding all mixed Nash equilibria in a $3\times 3$ game

I was looking at the exercise $2$ in this file http://isites.harvard.edu/fs/docs/icb.topic1531493.files/Practice%20Problem%20Solutions%20on%20Nash%20Equilibrium.pdf pages 4 to 7. I do not understand ...
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### Prisoner's Dilemma and Centipede Game - what's wrong with this analysis?

This is an analysis which to me seems trivial, but which I very rarely see brought up in any discussion of games like The Prisoner's Dilemma or The Centipede Game which are well known for having '...
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### Jumping on the Coordinate lattice grid

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show ...
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### What is the optimal strategy when playing head or tail per team

Introduction Once a week, we are playing head or tail in my favorite bar. There are $N$ people in the room and each person is guessing whether ...
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### Winning strategy of a game

The following is a game on monomials. Let $M(X,Y)$ denote the set of all monomials in $X$ and $Y$, i.e., $$M(X,Y)=\{X^aY^b\mid(a,b)\in\mathbb{N}^2\},$$ where $\mathbb{N}=\{0,1,\dots\}$. ...
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### Pursuers (a game)

From pure entertainment I am interested in the following questions: Let $N$ (particularly $N=1$ case is special) pursuers pursue running at max speed $u$ a runner running at max speed $v$. ...
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### Bayesian formulation of the Shapley Value

I wanted to know if there is a Bayesian formulation of the Shapley value in cooperative Games. I'm not sure if my problem really fits the definition of Bayesian games so here is the problem : For a ...
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### 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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### Choosing optimal strategy for real number assignment game

My opponent is dealt a real number $r\in [0, 1]$ uniformly at random and I know that with probability $1-r$ she chooses to discard that $r$ and be dealt a new number in $[0, 1]$ uniformly at random ...
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### How to simplify decision tree for sequential game

I am working on the exercise 2.8 of Game Theory: an Introduction by Steven Tadelis. I thought to solve this exercise by undertaking the steps sequentially, in order to minimize the loss in case of ...
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### Finding p positions with 2 subtraction sets in the take-away game [closed]

Find the set of P-positions for the takeaway game with the subtraction sets: $S = {1,3,5,7}$ $S = {1,2,4,8,16,32}$ Who wins each game when there are 100 tokens on the table to start, the first or the ...
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### Game, stealing edges in a graph.

I was inventing a problem for a math contest, I was really pleased with it, but then I found a mistake in my solution and have not been able to solve it. It is as follows: Alice and Bob play a game. ...
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### What's the difference between a Nash, Correlated, and Extreme equilibrium?

As the title states, what's the difference? As I understand it: The Nash Equilbirum (NE) is a solution concept in non-cooperative games where no player has incentive to unilaterally deviate from a ...
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### Find the Nash equilibria

A law is passed requiring a monopolistic soft-drink manufacturer to separate the production department and the marketing department. The marketing department chooses the price $P > 0$ to charge for ...
We consider the following public good provision game. There are 2 players, each choosing the amount of money $x_i$ ($i$ denotes 1 or 2) they will give to build a public good. We assume that each ...