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Questions tagged [game-theory]

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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Nash equilibrium in second price sealed-bid auction

I'm trying to understand the basics of game theory and the topic of auctions has arisen. I understand the basic concepts of auctions but I'm struggling with second price sealed-bid auctions. I ...
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35 views

Duel question, why shoot when $P_1 (s)+P_2 (s)=1$

There are two duelists with one shot each, their probability to hit from a distance $x$ is given by two functions $P_1 (x)$, $P_2 (x)$ that are both continuously and strictly decreasing from $P(0)=1, ...
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26 views

Equilibrium in a Lottery

We have the following game with 1700 participants: Each participant can buy a lottery ticket for 1\$ (when a participant buys a ticket, he does not know how many other participants are buying tickets) ...
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18 views

Defining a cooperative game heavily dependent on edges

i'm trying to create a cooperative game based on graph $g = (N,E)$ where the value of the coalition $S$ must equal the weight ($w_i$) of the nodes in the set $\mathcal{V}$ which have at least 1 (one) ...
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Computing the Shapley value for a super-additive game

I have a specific problem I have been set, I'm asking here because I can't really find an answer anywhere else. Consider the scenario where a company offers some service to its users. The company has ...
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10 views

Is this simple demand-based prices game a submodular game?

I have this simple market game: $I=\{1,2...,n\}$ players $S_i$ strategy space of each player $i\in I$ $u_i(s_i,s_{-i})=R_i(s_i)-C(s_i,s_{-i})$ There's only one type of resource. The resource is ...
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1answer
20 views

Probability distributions in Bayesian games

In a Bayesian game, each player $i$ learns his own type, $\theta_i$, which is his private information, and then uses his prior $\phi_i$ to form posterior beliefs over the other types of players, using ...
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24 views

proving if player 1 has a pure optimal strategy, player 2 should as well

Problem: Prove that if in a matrix game 2x2 if the player 1 has a pure optimal strategy, so has player 2 Attempt: Given: We know that player 1 has pure optimal strategy, meaning: $$P(x, \overline{...
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40 views

relation between inquisitive logic and logic as games?

In the very intriguing thesis "Questions in Logic" Ivano A. Ciardelli shows how to build a semantics of questions that reduces to Truth Conditional logic for factual statements where ¬¬p = p, but has ...
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Finding a solution using backward induction for the game shown in tree

there is a exercise in Game Theory(Decisions, Interaction and Evolution) by James N Webb (Exercise 5.1), the question is "Finding a solution using backward induction for the game shown in Figure 5.2?" ...
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Proving that the Wardrop/Nash equilibirum flow on a graph is “stable”?

Let $[0, 1] \subset \mathbb{R}$ be denoted $\mathbb{R}_{[0, 1]}$. Let $[0, \infty) \subset \mathbb{R}$ be denoted $\mathbb{R}_{\geq 0}$. Let $G$ be a finite, connected graph. Associate with each edge ...
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How to find Nash equilibrium in pure and mixed strategies? Problem.

A two player game. Players choose a number from a segment $[0;1]$. A payoff function of the first player $f_1(x,y)=|x-y|,$ where $x$ and $y$ - the numbers chosen by the first and the second players ...
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Why does finding the partial derivative solve this payoff matrix?

So I'm taking this math course and I decided to do all the homework the first day of class, and as I was pulling an all-nighter doing it, I got to the topic of game theory and payoff matrices. I never ...
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Modeling multi-stage games as extensive form games

Consider a finitely repeated prisoner's dilemma game. In each of $T<\infty$ stages, two players play a prisoner's dilemma game. At the end of each stage, each incurs a payoff dictated by the played ...
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Expected Cost in the Second Price Auction

I am going through this paper https://arxiv.org/pdf/1803.02194.pdf on Bidding Machine. In the section 3.1 Problem definition, I got the part about finding the probability of winning at bidding price $...
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21 views

game theory-Are second priced bids always a nash equilibrium?

I'm trying to understand the basics of game theory and the topic of auctions have come up. What I want to know is second priced bids always a nash equilibrium? Suppose we have this question. We ...
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When the set of correlated equilibria is a singleton?

Is there any case where I can say that the set of correlated equilibria is a singleton? I know the set of correlated equilibria includes the set of Nash equilibria, but in which cases they coincide? ...
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What's the chance of the next person leaving a joint venture when a person before leaves? [closed]

A quite interesting practical question came to my mind today. I don't know whether it's solvable and I'll try to put everything as clearly as I can. Question: Say you have 9 friends and you are ...
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20 views

How to check if mixed strategy leads to Nash equilibirum?

I have a simple payoff matrix defined here: matrix My question is: If both players play all 4 strategies with 1/4 probability, does that lead to Nash equilibrium? I can't quite figure this out. I ...
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12 views

How to start these questions from an epistemic game theory problem?

Upon reading an Epistmeic Game Theory book, I came upon a practical problem. However, as it is completely different than the examples in the book, I do not know how to approach it. Could somebody ...
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1answer
39 views

How do you solve the following differential equation (Proof of Lemma 3 of Hermalin, 1998)?

$(e(\theta) - s \theta)e'(\theta) = s(1-s)\theta$ The solution to the differential equation is given by: $e(\theta)=\frac{1}{2}(s+\sqrt{4s-3s^2})\theta$ Here, the dependent variable is $e(\theta)$, ...
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63 views

Game Theory - Mixed strategy ESS

I have a payoff matrix of a symmetric game like this: $ \begin{matrix} & A & B \\ A & (3, 3) & (1, 2) \\ B & (2, 1) & (5, 5) \end{matrix} $ I found that $(A, A)$ and $(B, ...
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1answer
33 views

Bounds and/or closed form for the “swerve numbers”?

You've probably experienced this: You're walking down a hallway when you notice you're on a collision course with somebody. Both of you swerve in one direction. Then you both swerve in the opposite ...
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45 views

Imbalance due to ELO rating

I want to use some rating system in my application. It will look like this: $TotalRating = rating_1*significanceCoefficient_1 + rating_2*significanceCoefficient_2 + ... + rating_n*...
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2answers
37 views

Does the von Neumann-Morgenstern utility theorem work for infinite outcomes?

The von Neumann-Morgenstern utility theorem is easy to prove for a finite number of outcomes. Is it still true for an infinite number of outcomes? With infinite outcomes, a lottery can now be any ...
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2answers
42 views

A game of nim determine value of X,Y [closed]

Alex and Bob plays a game A box contains numbers from 1 to 1024 Alex goes first Alex removes 512 numbers Bob removes 256 numbers Alex removes 128 numbers Bob removes 64 numbers Alex removes 32 ...
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41 views

final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...
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28 views

Nash Equilibrium and Mixed NE Problem

Does the game have a pure Nash equilibrium? Find all the mixed equilibria (note, there is at least one) I'm having some problems solving this exercise about the game between player 1 and 2 in the ...
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1answer
15 views

Nash Equilibrium for this sequential game

Player one (P1) picks a number between 100 and 0 inclusive. Let's call this number X. Whatever the X P1 has picked, it is subtracted from 100 and now is in front of P1. So P1 has 100 - X in front of ...
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Existence of Pure Nash Equilibrium in Global Connection Game?

I have a query in the proof of the following result: Result: A global connection game (GCG) always has a pure Nash equilibrium (PNE). This result is proved by showing that a global connection game ...
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calculation of probabilities in correlated equilibrium

I'm struggling to understand how to calculate the probability of each cell in a 3x3 matrics when trying to find the correlated equilibrium that maximizes the sum of players utilities?
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Primary source for definitions re partitions, coarsest common refinement, join/meet etc.?

As much as I do appreciate the contributions on m.se, I need to be able to cite a primary source for definitions related to partitions, e.g., what a refinement/coarsening is, coarsest common ...
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23 views

Subgame Perfect Equilibria in a one-stage game

Consider the following one stage game, with two players A and B. There is a pie which is to be divided between the two players. A can offer B any fraction of the cake, which B can accept or reject. ...
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44 views

What is the winning strategy for this problem?

Before the game starts, there are a few "points" on the desktop, and then the two players take turns to do the following operations until the operation can not be completed: Starting from a "point" ...
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Finding Nash Equilibrium using Linear Program with strategy constraints

Finding the Nash Equilibrium p in mixed strategies of a 2-player, symmetric zero-sum game with 3 pure strategies can be done by solving LP: max $(0, 0, 0, 1)^{T}(p_1, \ p_2,\ p_3,\ \epsilon)$ s....
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Simple continuous game or not?

Two players simultaneously pick a number from [0, 1]. Payoff of the first player (equal to the loss of the second) is the distance between those numbers. Does there exist pure-strategy Nash ...
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Bayesian Games: Belief Functions

One way to formulate the Bayesian game in Game theory is to define the set of type combinations as $T:=\times_{j=1}^{n}T_j$ and say that the type combination $t=(t_1,\ldots,t_n)\in T$ is supposed to ...
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Pure Nash equililibrium

I’m really confused. I have the following game (zero sum). 1 2. 4 1. ...
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38 views

Pure Nash equilibrium in Zero sum [closed]

Do zero sum games have a pure Nash equilibrium and if so how do I find the pure Nash equilibrium
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47 views

Finding the winning strategy of a variation of the Nim game

Here is a variant of the Nim game which I could not find out the winning strategy, the game rule is like this: The games starts with 16 stones arranged as follow: o (first pile) ooo (second pile) ...
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26 views

Subgame Perfect Nash Equilibrium in Cournot oligopoly

I have some questions about how I'm supposed to find the SPNE in this particular Cournot oligopoly: Consider the following market game: An incumbent firm, called firm 3, is already in an industry. ...
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Probability of winning a board game by chance

What is the chance of winning a board game like Connect4 by chance? The player who makes the first turn could theoretically win in a perfect game all the time. How can I calculate the probability that ...
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How to create probability and scoring formulas?

I have seen questions regarding building scoring systems but the answers given apply only to the specific problem. I am trying to understand how people can come up with equations that express the ...
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1answer
36 views

Bet on the sum of two dices

There are two players and each one has a dice with six sides from 1 to 6. The probability of each side is equal. Now two players roll their dice, and they only know the number of their own dice. They ...
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Game Theory: Finding the nash Equilibirums of 2x3 bimatrix by graphically representing the best replies

I am having troubles understanding the graphical method for solving a 2 x 3 bimatrix and I turn to all of you for help: To give an overview of my doubt, consider a 2 x 3 bimatrix game, (A,B) ...
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King of the Centre - Is this an existing game?

Consider an $n$-player infinitely repeated game. First stage nature chooses for each player, $i$, a radius $r_{i}$. For each later stage $t$ each player $i$: The payer chooses a "target" $p_{i, t}$...
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Finding the Bayes-Nash Equilibrium for First-Price Auction with 2 bidders

Let $\sigma_i$ be the strategy profile for bidder $i$ that indicates how they should bid based on their value. As we know, if there are 2 bidders both with their values $v_1,v_2$ on $U[0,1]$ in a ...
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1answer
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number guessing game [duplicate]

Someone gets to choose a number between $1$ and $100$. I than have to try and guess by guessing a number and being told if the number is higher lower or equal. If I get the number on my first try I ...
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Game Theory Nash Equilibrium - Iterative Closed Bag Exchange Donation Game

I am making up a game for a Holiday party and became interested in checking if the game is somehow broken or if there is an optimal solution, which would make it less enjoyable. A Nash equilibrium may ...
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Cake cutting when some players don't like cake

I am beginning to acquaint myself with the fair division literature. So far, I've always encountered the assumption that the value functions $V_i$ are normalized, such that $V_i([0,1])=1$. But it ...