Questions tagged [game-theory]
The study of mathematical models of conflict and cooperation between intelligent rational decision-makers
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Variation on St. Petersburg Paradox, with total loss at the end
I’m not too sure how to answer these variations I thought of and was hoping someone could enlighten me.
A casino offers a game of chance for a single player in which a fair coin is tossed at each ...
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Probability of winning in the specific case [duplicate]
With 3 boxes, two of which are empty and one full, what is the probability of winning the full box by switching when there are two left? Considering that I initially choose one box, and then one empty ...
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One Card Poker Expected Value
At my local casino there is a one card poker game with the rules as follows:
The game is essentially WAR you get a card and the dealer gets a card. To beat the dealer you need a higher value card (in ...
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Analyzing Optimal Search Strategies for Rewards in a Game with Unbalanced Odds
I have received an invitation to participate in a game show where the amount I could potentially win depends on how well I perform. The game requires me to find diamonds hidden in 30 vases that are ...
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Given a set of random variables, how can I find a linear combination of these variables satisfying a constraint on the sum of their permuations?
Say I have n random variables, {X0...Xn}, n>9. I also have another set of random variables constructed from the first set, where each of these are the sum of 9 ...
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What does $\times_i \Sigma_i$ mean?
I cannot for the life of me figure this out. The context is from game theory (source: Game Theory by Fudenberg, Tirole):
... the space of mixed strategy profiles is denoted $\Sigma = \times_i \...
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Probability of having k+1 consecutive heads after k consecutive tails [duplicate]
Consider tossing a fair coin; you win only if you get k+1 consecutive heads after k successive tails. What is the probability you win in finite trials?
I encountered this problem in an interview. My ...
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Should $3\!-\!$pointers be worth $2.5{\it ?}$
I was seeking an alternative scoring rule sets instead of three points for a win (gained more engaging and balanced) to cancel the theory "Banking a draw meaning a new kind of loss".
It's ...
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Proof of $\left|c_{i}-c_{j}\right|\left\{\begin{matrix}\ge 2,{\rm of}\,3/4\,{\rm probability}\\<2,{\rm of}\,1/4\,{\rm probability}\end{matrix}\right.$
FA Premier League 2019/20. The season was affected by the COVID-19 Pandemic while each team had a so-called quarter of their schedule left. ("quarter" ? Since each team has 4/9 or 5/9 number ...
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How can we allocate when we have 150 open slots every day (5 days a week) for those 200 arrivals every day
My question is to solve a very basic problem related to the allocation of slots. Say there are 20 teams with 10 persons in each team.
I have 150 open slots every day (5 days a week) for those 20 teams ...
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(Open-ended?) Stat puzzle about expected value
A fair coin is flipped $200$ times and each time it lands on heads, $1$ dollar is added to a pot. After this process is over, an auction is held for the pot. There is exactly one other person at the ...
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Bernoulli distribution
Definition 3.3.1 (Bernoulli distribution). An r.v. X is said to have the Bernoulli distribution with parameter p if P(X = 1) = p and
P(X = 0) = 1 − p, where 0 < p < 1. We write this as X ~ Bern(...
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Difference between types of solutions in network matchings? [closed]
So there are so many different types of solutions: pareto, stable, rank-maximal, etc. etc.
How do anyone know which one to use for any given problem? Particularly when some of them may not exist?
And ...
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What are common statistical techniques applied to Game Theory?
I have been examining stochastic Bayesian games as a framework for modeling multi-agent games. As this looks like an active area of research, I was wondering if any particular techniques have emerged ...
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Kelly Criterion for game with payoff equal to normal distribution
I'm struggling to come up with an answer to a problem I've asked myself, if the answer is out there I must not be searching for the correct things. The problem formulation is this:
You're given the ...
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Interpreting SHAP Dependence Plot for Categorical Variables
I'm reading about the use of Shapley values for explaining complex machine learning models and I'm confused about how I should interpret the SHAP independence plot in the case of a categorical ...
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Do not vote, one vote will not reverse election results. What is wrong with this reasoning?
Do not vote, one vote will not reverse the election result. What's
more, the probability of injury in a traffic collision on the way to the
ballot box is much higher than your vote reversing the ...
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Queuing theory for elevators
It's been a while since I had my probability course based on Sheldon Ross' book "Probability Models", and while I never went into econometrics, I was very interested in the queuing theory section. I ...
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How to optimize a number of games with a limit set of second chances at guessing other options given probabilities?
Let's say I have a game with $M$ outcomes $\{O_i\}_{i=1}^{M}$ and I have a predictor $P$ that gives me probabilities for the outcomes of a game.
I now have $N$ independent games that I want to ...
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Why is Nash equilibrium stable?
As I understand it, a Nash equilibrium is an equilibrium because no player can single-handedly improve their situation.
But now consider a game where one of the Nash equilibria is worse for TWO of ...
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Strategy for game where larger number wins. Drawn from standard uniform distribution with one redraw allowed
Two players are playing a game where they each draw a secret random number uniformly between 0 and 1. If they are not satisfied with their draw they may redraw. The players do not know whether or not ...
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How to model game theory of poker-like card game
I'd like to get a general understanding of the optimal strategy from a game called Team Fight Tactics. To reduce this problem into a simpler form I'd like to analyze a simpler card game problem I've ...
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Two different dice games. What are the odds the game will be won?
Based on the images above, is it correct that the first game should be won 1 out of six times, the second game every time, and the 3rd game 1 out of six times?
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Problem of accepting a prize versus trying to get a better one
I don't know how to better formulate the general problem I am thinking about, let me try formulate an example.
Assume you are playing a game with N rounds, and at each round the following happens: ...
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Neural Network and equally good predictions
There is a two-player game (discrete, deterministic, perfect information and so on) where - in some but not all states - a few moves may be equally good; i.e. they are symmetric and expert player will ...
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Best strategy to cheat on a lots game
Consider the game:
100 (n=100) people (inclusive you) are putting their names in a bowl for drawing a lot
There is 5 (p=5) prices, thus 95 wins nothing
Cheating
You can just throw instead of 1 ...
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Stochastic vs Adversarial Multi-Armed Bandit Problems
I know that the multi-armed bandit can be formalised in multiple ways - two of them being the stochastic and adversarial ways. I am familiar with the fact that adversarial way is a game theoretic ...
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Expected payoff from a weighted random sampling without replacement?
In an evolutionary game theoretic context, I am interested in calculating expected payoffs of different strategies in a 2x2 game, given a weighted random sampling without replacement from a population....
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Parameterizing Reinforcement Learning card game state space
I want to model a particular card game as a reinforcement learning problem. For simplicity let's say it is a single standard 52 card deck, and let's say it is just 2 players. The exact details are not ...
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What is the relation and/or difference between Game Theory and Markov Chain Model? [closed]
I am doing some work regarding my master's thesis in networks security.
I have decided to work with Game Theory, calculating the Nash Equilibrium for a two player zero sum game.
However, I have also ...
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Strategy to win a card game that follows Wallenius' noncentral hypergeometric distribution
Let's say that you have a two valued deck of N = 53 cards. Win and lose are the only values and we know that #win = 26 and #lose = 27. To play, we pay 1 unit of value. We draw one card at a time, but ...
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Multi-agent reinforcement learning: An introduction
I would like to make a Tic-Tac-Toe game with 10X10 board and I want to have 2 agents playing against each other and learning to play from interaction.
A naive approach could be taken - to simply let ...
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How to calculate Shapley value when order matters?
I am trying to figure out the Shapley value when order matters.
When using the R package GameTheoryAllocation you have to input the characteristic function.
For ...
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"Approved" switch criterion for the Secretary problem
The secretary problem ( 1, 2, 3, 4, 5 ) optimal stopping, says "stop and keep the best" in a randomized sequence of known length "k" where you can't select previously elements of the sequence.
One ...
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Why in fictitious self-play only average strategy converge to the Nash equilibrium?
What are the reasons that in fictitious self-play only average(not the best-response) strategy converge to the Nash equilibrium?
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Simulation of Secretary problem: optimal pool size given k=2?
Question:
Is it incorrect to think there is a "sweet spot" where more samples slightly decreases the likelihood of a "Best pick" in the Secretary Problem?
Details:
The "Secretary Problem" from "...
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Uniform random variables and optimal strategy
This comes from Fivethirtyeight's riddler weekly challenge...
Toddler poker is played by two players. Each is dealt a “card,” which
is actually a number randomly chosen uniformly from the ...
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Martingales: Why must expected posterior equal prior?
For a posterior distribution to be plausible in the Bayesian sense (Bayes' Plausible), it is said that:
$\mathbb{E}(\mu_{t+1} | \mu_t) = \mu_t$
where $\mu_t$ is the posterior distribution at time $...
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How to deal with unknown distribution?
I have a game with two players. Player 1's type is publicly known, while player 2's type is unknown (privately known to him) and distributed, say, from 0 to 1. Both players play their payoff ...
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In simple Russian Roulette how many times should you shoot before handing it off to the other person?
I'm assuming a game of Russian Roulette where you have a gun with 1 bullet and six chambers, and two people playing.
The rules are:
You must shoot at least once on your turn.
After shooting once, ...
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Shapley Value with incomplete information
I'm building an algorithm in R to calculate the Shapley Value for players in a collaborative game. However, I do not have an outcome value for all possible coalitions, partially because the number of ...
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How to know which method to use to work out hand probabilities in Texas Hold'Em Poker?
So I just asked earlier about how to calculate the probability of a starting pair making quads (four of a kind) on the flop and was informed that it's a hypergeometric distribution and essentially it'...
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How to make a confidence interval in a binomial model with fakes?
I'm writing a concept recognition self-test. (This is less weird than it sounds, but only slightly. Don't worry about it.) Partly to prevent people from clicking everything in sight, but mostly just ...
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Specifying a good goal in Multi-agent RL
I am reading a survey paper on Multiagent reinforcement learning (MARL) by Busoniu, Babuska, De Schutter (IEEE Transactions on Systems, Man and Cybernetics March 2008).
Under 'Challenges in MARL', ...
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What's the relation between game theory and reinforcement learning?
I'm interested in (Deep) Reinforcement Learning (RL). Before diving into this field should I take a course in Game Theory (GT)?
How are GT and RL related?
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Probability the next draw from a distribution is greater than some number given a previous draw
I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. I am looking to solve for two different probability functions, though I think ...
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Statistics guessing game
This is a game that I wanted to know the answer for.
The Game
There are many players in the game, who each get a guess.
There is a loss distribution, $L$, where:
$L$ ~ Gamma($\alpha$, $\beta$)
i....
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How to deal with negative Shapley coalitions/contributions
I have a moderate understanding around the Shapley value/theorem (I've been digging into many papers and research articles lately).
I'm trying to derive weights/attribution for each player when ...
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Expectations versus returns
I have calculated the probability of winning an event, say a horse race, and am comparing it with the odds on offer. I have two horses which look good prices.
My model says horse A has a 5% chance of ...
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How to get payoff function from probability?
Okay, I am studying for my game theory midterm tomorrow and I am stumped at the practice final and I don't want to just look at the answer key. I would like some knowledge to substantiate my work.
I ...
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