In a Bayesian game, it is necessary to specify the strategy spaces, type spaces, payoff functions and beliefs for every player. A strategy for a player is a complete plan of actions that covers every contingency that might arise for every type that player might be.
A strategy must not only specify the actions of the player given the type that he is, but must specify the actions that he would take if he were of another type.
Strategy spaces are defined as above. A type space for a player is just the set of all possible types of that player. The beliefs of a player describe the uncertainty of that player about the types of the other players.
Each belief is the probability of the other players having particular types, given the type of the player with that belief i. A payoff function is a 2-place function of strategy profiles and types. Signalling games constitute an example of Bayesian games. In such a game, the informed party the agent knows their type, whereas the uninformed party the principal does not know the agent's type.
In some such games, it is possible for the principal to deduce the agent's type based on the actions the agent takes in the form of a signal sent to the principal in what is known as a separating equilibrium. A more specific example of a signalling game is a model of the job market. The players are the applicant agent and the employer principal. The employer will offer the applicant a contract based on how productive he thinks he will be.
Skilled workers are very productive generating a large payoff for the employer and unskilled workers are unproductive generating a low payoff for the employer. The payoff of the employer is determined thus by the skill of the applicant if the applicant accepts a contract and the wage paid. The applicant's action space comprises two actions, take a university education or do not.
It is less costly for the skilled worker to do so because he does not pay extra tuition fees, finds classes less taxing, etc. The employer's action space is the set of say natural numbers, which represents the wage of the applicant the applicant's action space might be extended to include acceptance of a wage, in which case it would be more appropriate to talk of his strategy space.
It might be possible for the employer to offer a wage that would compensate a skilled applicant sufficiently for acquiring a university education, but not an unskilled applicant, leading to a separating equilibrium where skilled applicants go to university and unskilled applicants do not, and skilled applicants workers command a high wage, whereas unskilled applicants workers receive a low wage.
The course will provide the basics: representing games and strategies, the extensive form which computer scientists call game trees , Bayesian games modeling things like auctions , repeated and stochastic games, and more. We'll include a variety of examples including classic games and a few applications.
It was such a helpful course that gave me the opportunity to learn few basic methods and terms about game theory through lots of interesting and to the point examples by three unique professors. I enjoyed learning about Game theory. The course syllabus was extremely interesting and pushed me to read and research more about Game theory.
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String Manipulation in R. Text Preprocessing In R. What sets apart US univs? Common Core standards in Math. Facebook recommends Math. England: GCSE. In Bayesian decision theory we are concerned with choosing between these different decision based on some information. Like our decision space, there is tremendous flexibility in what our information is univariate, multivariate, continuous, discrete, etc….
This requires that we first define some notion of what we want what are we trying to do? What does it mean to quantify? We may denote this loss function as. I find that the hard part of decision theory is often the choice of the loss function; this is really a subjective choice that should capture what matters to you. In the following examples I will focus on just a few very simple examples. What makes matters more complicated and the reason why this is related to Bayesian statistics is that we rarely know any information exactly, instead we often only have some beliefs about the information we want to use to make a decision.
I have written before on how crucial it can be to quantify our uncertainty in analyses or estimates. In short, it is one thing to estimate a quantity, but it can be far more powerful to not only estimate a quantity but actually quantify our uncertainty about an estimate as a distribution over possible values. However, any probability distribution over our information space can be used here. So why should we care about this?
In this form the Expected Loss does some pretty remarkable things in terms of decision making. Finally we come to the hard part. By best we will mean the decision that has the lowest expected loss. Because this part can get computationally intense quickly. Here, rather than continuing to go through the examples in the same manner I have done so far, I will instead focus just on the first example and go start-to-finish through a full worked Bayesian decision theory problem. Imagine I have an unscratched used phone that was made in that I want to sell.
Lets pretend I believe that there are 3 key variables that dictate whether a phone sold or not: whether it was scratched or not, the year it was made, and the price it was listed at.
Given this information I want to figure out what price I should list my phone at.
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