12 votes
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What is the difference between Stochastic game and Bayesian game ?

In a Bayesian game, information is incomplete. To cope, players have beliefs about the state of the game. In a sense, each player strategizes as if the game was as he or she believes. So each player ...
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  • 1,216
9 votes

Bayesian Nash Equilibrium - Mixed Strategies

I believe that the answer given by @denesp is incorrect. The second method involves simply writing the game in strategic of "normal" form. I believe the first method is better (easier to use), but I ...
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7 votes
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Questions of the type "What do you think other people would think?"

Epistemic game theory would be the closest (sub-)field that deals with questions involving higher order beliefs among interacting agents. The introductory article by Dekel and Siniscalchi is a good ...
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6 votes
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Bayesian-Nash equilibrium in a first-price auction

It is actually assumed that $b_i(v_i)$ is of the form $\alpha_i+\beta_i \cdot v_i$. So it is an affine function. Linearity only works if the bottom of the uniform distribution is 0. A somewhat ...
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6 votes
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Take-it-or-leave-it PBE

After posting a bad solution yesterday I believe I got a better one: The strategy of the buyer consists of two functions, $(f_1(v,p_1),f_2(v,p_1,p_2))$ where both functions map to $\left\{A,R\right\}$...
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6 votes
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Adverse selection problem

Signaling is the informed side taking actions to reduce (or maintain, depending on the private types) the information asymmetry. For example, high skill workers getting certifications to signal their ...
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5 votes
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Social Welfare and Pareto Optimality in a Bayesian Game

While it is a bit unusual to describe a strategy profile as being Pareto optimal, especially in the context of Bayesian games, I guess you can still define Pareto optimality in different stages of ...
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5 votes
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Definition of Bayesian Nash equilibrium

Bayesian Nash equilibrium is a set of strategies $\{\sigma_i\}$ one for each player and some beliefs $\{\mu_i\}$ also one for each player such that $\sigma_i$ is a best response for player $i$ given ...
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  • 4,158
5 votes

Relationship among universal type space, Aumann's semantic knowledge model and Samet's syntactic knowledge model

There is no "universal Aumann model", as shown in Heifetz & Samet, GEB 1998, "Knowledge Spaces with Arbitrarily High Rank ", even though there is a universal type space. On a less technical level,...
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5 votes
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Models for online markets with reputation system

I would suggest you start by looking at C. Dellarocas. "The Digitization of Word-of-Mouth: Promise and Challenges of Online Reputation Systems". Management Science 49 (10), October 2003, 1407-...
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  • 16.6k
5 votes
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Bayes Correlated Equilibrium: obedience

To understand the "Obedience" inequality, notice that the player is "integrating out" everything that she is uncertain about, this is why the sum runs over the actions of other players, $a_{-i}$, the ...
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  • 4,158
5 votes
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Why is it called Bayesian Persuasion?

In a standard cheap-talk setting, a sender (S) has better information on a state of the world and wants to communicate this information to a receiver (R) who then takes an action. However, S and R ...
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5 votes
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Confusion about multiple information Sets

You could represent the game in extensive form like this: The dashed lines enclose player 2's information set. This encompasses all of player 2's nodes because player 2 observes neither nature's nor ...
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5 votes
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Finding Bayesian Nash Equilibrium

The Key to BNE is that players that know something (about the state of the world or their type) can condition their strategies with their information. That is, for example, in question 3, type A could ...
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  • 4,158
5 votes

I cannot understand Sobel & Crawford 1982 (cheap talk)

$U^S(\bar y,\bar m,b)=\max_{y\in Y}U^S(y,\bar m,b)$ is standard notation that says $\bar y$ is the action (taken by receiver) that would maximize sender's utility given message $\bar m$ and bias ...
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5 votes
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Difficulty in understanding the notation related from probability theory with game theory

We have that ${\cal I} = ((X^i)_i, \mu)$​ and ${\cal J} = ((Y^i)_i, \nu)$​ are two information structures. An Interpretation mapping for player $i$​​ is a mapping $\phi^i: X^i \to \Delta(Y^i)$​ so it ...
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  • 8,642
5 votes
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When should my beliefs be a martingale?

The expected posterior is the prior. But for this to hold, you have to take the expectation with respect to the prior. In your example, you describe a Bayesian but ev aluate their update with respect ...
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5 votes
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Correlation device that induces a specific transition probability

The common prior $p\in\Delta L$ and the transition probability $q:L\to\Delta A$ induce a joint distribution on $L\times A$ in which the pair $(l,a)$ is selected with probability $p_l\cdot q_l(a)$. You ...
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5 votes
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Why does the belief over information sets with probability zero matter in Perfect Bayesian Equilibrium?

The key is that "since both equilibria satisfy sequential rationality" is no longer true when you consider weak sequential equilibria. Both concepts satisfy sequential rationality on-path, ...
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5 votes
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Is the following claim written with the right way?

Would be strange to write it that way. If you had to define something like that, just do the following: Start with a type space $(T,\mu)$ with probability measure $\mu$. Let $\sigma: T \rightarrow \...
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5 votes

In games of Bayesian Persuasion, under what conditions is Receiver better off than under uninformative signals?

A receiver can always ignore any additional information and do what they would have done in the absence of informative signals. If they react optimally, they can therefore never be worse off. It is a ...
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4 votes
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Bayes-Nash equilibrium and correctness of beliefs

I think your definition is incorrect, or at least incomplete.** Usually, in a Bayesian game, there is assumed to be a prior distribution on $T$ (where $T = \times_i T_i$). This distribution is called ...
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  • 394
4 votes
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Game theory with rational and irrational players

Yes, a whole book has been written on Behavioral Game Theory. More specifically, standard solution concept such as Nash equilibrium requires that players best respond to a correct belief about other ...
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4 votes
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Bayesian game and the set of types

Sure. In the two type case there is an alternate solution that is frequently used. If $a \neq c$ and $b \neq d$, that is the types are really different in both attributes, then you can define a ...
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4 votes
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Are there multiple equilibria in the second price auction?

Sure. An example: if both valuations are drawn from the $[0,1]$ interval then the strategies $$ b_1(v_1) = v_1 $$ and $$ b_2(v_2) = \left\{\begin{array}{cc} v_2 & \text{ if } v_2 < 1 \\ 5 & ...
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4 votes
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Bayesian Nash Equilibria in Battle of Sexes

Yes, you are correct. All types $t_{1}$ choose O (B) and all types $t_{2}$ choose O (B) are both Bayesian equilibria. Note that there are other Bayesian equilibrium in this game, if you are interested ...
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  • 862
4 votes
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How to prove the WPBE is SE in signaling game?

Claim: If choice sets $T, M,$ and $A$ are finite, then an assessment $\{\beta^*_{r}, \beta^*_{s}, \mu^*\}$ is a WPBE (weak perfect Bayesian equilibrium) of the two-stage signalling game between ...
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  • 1,189
4 votes
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What does 'continuation' as in continuation games, strategies, plays, etc. exactly mean?

Consider a game with private information such as a privately known willingness-to-pay or any other type. We usually model this as a game in which at first "Nature" draws the type and then ...
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  • 5,090
4 votes
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Strong sequential equilibria and the existence of others

There are three classes of equilibria of this game. The first class is sequential: \begin{equation} (s_1,s_2)=(y,r) \end{equation} and the beliefs are \begin{equation} \mu_1(a)=\mu_1(b)=\mu_2(a\mid y)...
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